:: GR_CY_1 semantic presentation
:: deftheorem Def1 defines Segm GR_CY_1:def 1 :
Lemma2:
for b1 being set
for b2 being Nat holds
( b2 > 0 & b1 in Segm b2 implies b1 is Nat )
;
theorem Th1: :: GR_CY_1:1
canceled;
theorem Th2: :: GR_CY_1:2
canceled;
theorem Th3: :: GR_CY_1:3
canceled;
theorem Th4: :: GR_CY_1:4
canceled;
theorem Th5: :: GR_CY_1:5
canceled;
theorem Th6: :: GR_CY_1:6
canceled;
theorem Th7: :: GR_CY_1:7
canceled;
theorem Th8: :: GR_CY_1:8
canceled;
theorem Th9: :: GR_CY_1:9
canceled;
theorem Th10: :: GR_CY_1:10
theorem Th11: :: GR_CY_1:11
canceled;
theorem Th12: :: GR_CY_1:12
theorem Th13: :: GR_CY_1:13
definition
redefine func addint -> M6(
[:INT ,INT :],
INT )
means :: GR_CY_1:def 2
for b
1, b
2 being
Element of
INT holds a
1 . b
1,b
2 = addreal . b
1,b
2;
compatibility
for b1 being M6([:INT ,INT :], INT ) holds
( b1 = addint iff for b2, b3 being Element of INT holds b1 . b2,b3 = addreal . b2,b3 )
end;
:: deftheorem Def2 defines addint GR_CY_1:def 2 :
theorem Th14: :: GR_CY_1:14
theorem Th15: :: GR_CY_1:15
:: deftheorem Def3 defines Sum GR_CY_1:def 3 :
theorem Th16: :: GR_CY_1:16
canceled;
theorem Th17: :: GR_CY_1:17
canceled;
theorem Th18: :: GR_CY_1:18
canceled;
theorem Th19: :: GR_CY_1:19
canceled;
theorem Th20: :: GR_CY_1:20
theorem Th21: :: GR_CY_1:21
theorem Th22: :: GR_CY_1:22
Lemma8:
for b1 being Group
for b2 being Element of b1 holds Product (((len (<*> INT )) |-> b2) |^ (<*> INT )) = b2 |^ (Sum (<*> INT ))
Lemma9:
for b1 being Group
for b2 being Element of b1
for b3 being FinSequence of INT
for b4 being Element of INT holds
( Product (((len b3) |-> b2) |^ b3) = b2 |^ (Sum b3) implies Product (((len (b3 ^ <*b4*>)) |-> b2) |^ (b3 ^ <*b4*>)) = b2 |^ (Sum (b3 ^ <*b4*>)) )
theorem Th23: :: GR_CY_1:23
canceled;
theorem Th24: :: GR_CY_1:24
theorem Th25: :: GR_CY_1:25
theorem Th26: :: GR_CY_1:26
theorem Th27: :: GR_CY_1:27
theorem Th28: :: GR_CY_1:28
theorem Th29: :: GR_CY_1:29
theorem Th30: :: GR_CY_1:30
theorem Th31: :: GR_CY_1:31
theorem Th32: :: GR_CY_1:32
theorem Th33: :: GR_CY_1:33
:: deftheorem Def4 defines INT.Group GR_CY_1:def 4 :
:: deftheorem Def5 defines addint GR_CY_1:def 5 :
theorem Th34: :: GR_CY_1:34
:: deftheorem Def6 defines INT.Group GR_CY_1:def 6 :
theorem Th35: :: GR_CY_1:35
theorem Th36: :: GR_CY_1:36
:: deftheorem Def7 GR_CY_1:def 7 :
canceled;
:: deftheorem Def8 defines @' GR_CY_1:def 8 :
theorem Th37: :: GR_CY_1:37
Lemma25:
for b1, b2 being Element of INT.Group
for b3, b4 being Integer holds
( b1 = b3 & b2 = b4 implies b1 * b2 = b3 + b4 )
by Th14;
theorem Th38: :: GR_CY_1:38
for b
1 being
Nat holds
(@' 1) |^ b
1 = b
1
theorem Th39: :: GR_CY_1:39
Lemma28:
INT.Group = gr {(@' 1)}
:: deftheorem Def9 defines cyclic GR_CY_1:def 9 :
theorem Th40: :: GR_CY_1:40
theorem Th41: :: GR_CY_1:41
theorem Th42: :: GR_CY_1:42
theorem Th43: :: GR_CY_1:43
theorem Th44: :: GR_CY_1:44
theorem Th45: :: GR_CY_1:45
theorem Th46: :: GR_CY_1:46
theorem Th47: :: GR_CY_1:47
theorem Th48: :: GR_CY_1:48
theorem Th49: :: GR_CY_1:49
theorem Th50: :: GR_CY_1:50
theorem Th51: :: GR_CY_1:51