:: BCIALG_1 semantic presentation
:: deftheorem defines \ BCIALG_1:def 1 :
:: deftheorem defines ` BCIALG_1:def 2 :
:: deftheorem DefB defines being_B BCIALG_1:def 3 :
:: deftheorem DefC defines being_C BCIALG_1:def 4 :
:: deftheorem DefI defines being_I BCIALG_1:def 5 :
:: deftheorem DefK defines being_K BCIALG_1:def 6 :
:: deftheorem DefBCI4 defines being_BCI-4 BCIALG_1:def 7 :
:: deftheorem DefBCK5 defines being_BCK-5 BCIALG_1:def 8 :
:: deftheorem defines BCI-EXAMPLE BCIALG_1:def 9 :
Th1:
( BCI-EXAMPLE is_B & BCI-EXAMPLE is_C & BCI-EXAMPLE is_I & BCI-EXAMPLE is_BCI-4 )
:: deftheorem DSubAlg defines SubAlgebra BCIALG_1:def 10 :
theorem The1: :: BCIALG_1:1
:: deftheorem Def6 defines <= BCIALG_1:def 11 :
A00:
for X being BCI-algebra
for x being Element of X st x \ (0. X) = 0. X holds
x = 0. X
theorem Th00: :: BCIALG_1:2
theorem Th01: :: BCIALG_1:3
theorem Th02: :: BCIALG_1:4
theorem :: BCIALG_1:5
theorem :: BCIALG_1:6
theorem Th03: :: BCIALG_1:7
theorem Th05: :: BCIALG_1:8
theorem Th06: :: BCIALG_1:9
theorem T08: :: BCIALG_1:10
theorem :: BCIALG_1:11
theorem :: BCIALG_1:12
theorem :: BCIALG_1:13
theorem :: BCIALG_1:14
theorem :: BCIALG_1:15
theorem :: BCIALG_1:16
theorem :: BCIALG_1:17
theorem :: BCIALG_1:18
:: deftheorem defines BCK-part BCIALG_1:def 12 :
theorem BCKP1: :: BCIALG_1:19
theorem :: BCIALG_1:20
theorem :: BCIALG_1:21
theorem DEF114: :: BCIALG_1:22
:: deftheorem DTrivial defines proper BCIALG_1:def 13 :
:: deftheorem Defatom defines atom BCIALG_1:def 14 :
:: deftheorem defines AtomSet BCIALG_1:def 15 :
theorem Tm0: :: BCIALG_1:23
theorem Tm1: :: BCIALG_1:24
theorem :: BCIALG_1:25
theorem :: BCIALG_1:26
theorem :: BCIALG_1:27
theorem Tm5: :: BCIALG_1:28
theorem Tm6: :: BCIALG_1:29
theorem Tm7: :: BCIALG_1:30
theorem Tm8: :: BCIALG_1:31
theorem :: BCIALG_1:32
theorem TL133: :: BCIALG_1:33
theorem TL1: :: BCIALG_1:34
theorem TL134: :: BCIALG_1:35
:: deftheorem DefXX defines generated_by_atom BCIALG_1:def 16 :
:: deftheorem defines BranchV BCIALG_1:def 17 :
theorem :: BCIALG_1:36
theorem :: BCIALG_1:37
theorem TL13x: :: BCIALG_1:38
Lm136:
for X being BCI-algebra
for a being Element of AtomSet X
for x being Element of BranchV a holds a \ x = 0. X
theorem TL136: :: BCIALG_1:39
theorem :: BCIALG_1:40
theorem :: BCIALG_1:41
theorem :: BCIALG_1:42
:: deftheorem DIdeal defines Ideal BCIALG_1:def 18 :
:: deftheorem Close defines closed BCIALG_1:def 19 :
LM1610:
for X being BCI-algebra holds {(0. X)} is Ideal of X
LM161x:
for X being BCI-algebra
for X1 being Ideal of X st X1 = {(0. X)} holds
for x being Element of X1 holds x ` in {(0. X)}
theorem :: BCIALG_1:43
theorem :: BCIALG_1:44
theorem :: BCIALG_1:45
LM161:
for X being BCI-algebra
for IT being non empty Subset of X st IT is Ideal of X holds
for x, y being Element of IT holds { z where z is Element of X : z \ x <= y } c= IT
theorem :: BCIALG_1:46
:: deftheorem Assoc defines associative BCIALG_1:def 20 :
:: deftheorem Def72 defines quasi-associative BCIALG_1:def 21 :
:: deftheorem Def8 defines positive-implicative BCIALG_1:def 22 :
:: deftheorem Def9 defines weakly-positive-implicative BCIALG_1:def 23 :
:: deftheorem Defa defines implicative BCIALG_1:def 24 :
:: deftheorem defines weakly-implicative BCIALG_1:def 25 :
:: deftheorem Defb defines p-Semisimple BCIALG_1:def 26 :
:: deftheorem Defc defines alternative BCIALG_1:def 27 :
TT1:
for X being BCI-algebra st ( for x, y being Element of X holds y \ x = x \ y ) holds
X is associative
TT2:
for X being BCI-algebra st ( for x being Element of X holds x ` = x ) holds
for x, y being Element of X holds x \ y = y \ x
theorem TT3: :: BCIALG_1:47
theorem TT4: :: BCIALG_1:48
theorem Tq1: :: BCIALG_1:49
theorem Tq2: :: BCIALG_1:50
theorem :: BCIALG_1:51
theorem :: BCIALG_1:52
theorem :: BCIALG_1:53
T01:
for X being BCI-algebra holds
( ( for x being Element of X holds (x ` ) ` = x ) iff for x, y being Element of X holds y \ (y \ x) = x )
T02:
for X being BCI-algebra holds
( ( for x, y being Element of X holds y \ (y \ x) = x ) iff for x, y, z being Element of X holds (z \ y) \ (z \ x) = x \ y )
theorem TL2221: :: BCIALG_1:54
theorem TL2222: :: BCIALG_1:55
theorem TL2223: :: BCIALG_1:56
theorem TL2224: :: BCIALG_1:57
ThC4:
for X being BCI-algebra st X is p-Semisimple holds
for x, y, z, u being Element of X holds
( (x \ u) \ (z \ y) = (y \ u) \ (z \ x) & (x \ u) \ (z \ y) = (x \ z) \ (u \ y) )
ThC5:
for X being BCI-algebra st X is p-Semisimple holds
for x, y being Element of X holds (y ` ) \ ((0. X) \ x) = x \ y
ThC7:
for X being BCI-algebra st X is p-Semisimple holds
for x, y, z being Element of X holds (x \ y) \ (z \ y) = x \ z
ThC8:
for X being BCI-algebra st X is p-Semisimple holds
for x, y, z being Element of X st x \ y = x \ z holds
y = z
ThC9:
for X being BCI-algebra st X is p-Semisimple holds
for x, y, z being Element of X holds x \ (y \ z) = (z \ y) \ (x ` )
ThCa:
for X being BCI-algebra st X is p-Semisimple holds
for x, y, z being Element of X st y \ x = z \ x holds
y = z
theorem :: BCIALG_1:58
theorem TL2226: :: BCIALG_1:59
theorem :: BCIALG_1:60
theorem :: BCIALG_1:61
theorem ThC1: :: BCIALG_1:62
theorem TL2233: :: BCIALG_1:63
theorem :: BCIALG_1:64
theorem :: BCIALG_1:65
theorem :: BCIALG_1:66
theorem :: BCIALG_1:67
theorem :: BCIALG_1:68
theorem :: BCIALG_1:69
theorem :: BCIALG_1:70
LM2311:
for X being BCI-algebra st ( for x being Element of X holds x ` <= x ) holds
for x, y being Element of X holds (x \ y) ` = (y \ x) `
LM2312:
for X being BCI-algebra st ( for x, y being Element of X holds (x \ y) ` = (y \ x) ` ) holds
for x, y being Element of X holds (x ` ) \ y = (x \ y) `
LM2313:
for X being BCI-algebra st ( for x, y being Element of X holds (x ` ) \ y = (x \ y) ` ) holds
for x, y being Element of X holds (x \ y) \ (y \ x) in BCK-part X
LM2314:
for X being BCI-algebra st ( for x, y being Element of X holds (x \ y) \ (y \ x) in BCK-part X ) holds
X is quasi-associative
LM2315:
for X being BCI-algebra holds
( ( for x being Element of X holds x ` <= x ) iff for x, y, z being Element of X holds (x \ y) \ z <= x \ (y \ z) )
theorem TL2321: :: BCIALG_1:71
theorem TL2322: :: BCIALG_1:72
theorem TL2323: :: BCIALG_1:73
theorem :: BCIALG_1:74
theorem :: BCIALG_1:75
theorem ThA2: :: BCIALG_1:76
theorem :: BCIALG_1:77
theorem :: BCIALG_1:78
theorem :: BCIALG_1:79
theorem :: BCIALG_1:80
abc0:
for X being BCI-algebra holds
( X is alternative iff X is associative )
abc1:
for X being BCI-algebra st X is alternative holds
X is implicative
theorem :: BCIALG_1:81
theorem :: BCIALG_1:82
abc:
for X being BCI-algebra st X is associative holds
X is weakly-positive-implicative
bcd:
for X being BCI-algebra st X is p-Semisimple BCI-algebra holds
X is weakly-positive-implicative BCI-algebra
theorem :: BCIALG_1:83
theorem TL275: :: BCIALG_1:84
ThB0:
for X being BCI-algebra
for x, y being Element of X st X is weakly-positive-implicative holds
(x \ (x \ y)) \ (y \ x) = ((y \ (y \ x)) \ (y \ x)) \ (x \ y)
ThB1:
for X being BCI-algebra holds
( X is positive-implicative iff X is weakly-positive-implicative )
ThB2:
for X being BCI-algebra st X is alternative holds
X is weakly-positive-implicative
theorem :: BCIALG_1:85
theorem :: BCIALG_1:86