:: IRRAT_1 semantic presentation
theorem Th1: :: IRRAT_1:1
theorem Th2: :: IRRAT_1:2
:: deftheorem Def1 defines aseq IRRAT_1:def 1 :
:: deftheorem Def2 defines bseq IRRAT_1:def 2 :
:: deftheorem Def3 defines cseq IRRAT_1:def 3 :
theorem Th3: :: IRRAT_1:3
:: deftheorem Def4 defines dseq IRRAT_1:def 4 :
:: deftheorem Def5 defines eseq IRRAT_1:def 5 :
theorem Th4: :: IRRAT_1:4
for b
1, b
2 being
Nat holds
( b
1 > 0 implies b
1 .^. (- (b2 + 1)) = (b1 .^. (- b2)) / b
1 )
Lemma9:
for b1, b2, b3, b4, b5 being real number holds b1 / ((b2 * b3) * (b4 / b5)) = (b5 / b3) * (b1 / (b2 * b4))
by XCMPLX_1:235;
theorem Th5: :: IRRAT_1:5
canceled;
theorem Th6: :: IRRAT_1:6
theorem Th7: :: IRRAT_1:7
theorem Th8: :: IRRAT_1:8
for b
1, b
2 being
Nat holds
( b
1 > 0 implies
(aseq b2) . b
1 = 1
- (b2 / b1) )
theorem Th9: :: IRRAT_1:9
theorem Th10: :: IRRAT_1:10
theorem Th11: :: IRRAT_1:11
theorem Th12: :: IRRAT_1:12
for b
1 being
Nat holds
(1 / (b1 + 1)) * (1 / (b1 ! )) = 1
/ ((b1 + 1) ! )
theorem Th13: :: IRRAT_1:13
theorem Th14: :: IRRAT_1:14
for b
1, b
2 being
Nat holds
( b
1 < b
2 implies ( 0
< (aseq b1) . b
2 &
(aseq b1) . b
2 <= 1 ) )
theorem Th15: :: IRRAT_1:15
theorem Th16: :: IRRAT_1:16
theorem Th17: :: IRRAT_1:17
theorem Th18: :: IRRAT_1:18
theorem Th19: :: IRRAT_1:19
theorem Th20: :: IRRAT_1:20
theorem Th21: :: IRRAT_1:21
theorem Th22: :: IRRAT_1:22
theorem Th23: :: IRRAT_1:23
theorem Th24: :: IRRAT_1:24
theorem Th25: :: IRRAT_1:25
theorem Th26: :: IRRAT_1:26
theorem Th27: :: IRRAT_1:27
theorem Th28: :: IRRAT_1:28
theorem Th29: :: IRRAT_1:29
theorem Th30: :: IRRAT_1:30
theorem Th31: :: IRRAT_1:31
:: deftheorem Def6 defines number_e IRRAT_1:def 6 :
:: deftheorem Def7 defines number_e IRRAT_1:def 7 :
theorem Th32: :: IRRAT_1:32
theorem Th33: :: IRRAT_1:33
theorem Th34: :: IRRAT_1:34
for b
1, b
2 being
Nat holds
(b1 ! ) / (b2 ! ) > 0
theorem Th35: :: IRRAT_1:35
theorem Th36: :: IRRAT_1:36
theorem Th37: :: IRRAT_1:37
for b
1, b
2 being
Nat holds
( b
1 <= b
2 implies
(b2 ! ) / (b1 ! ) is
integer )
theorem Th38: :: IRRAT_1:38
theorem Th39: :: IRRAT_1:39
theorem Th40: :: IRRAT_1:40
theorem Th41: :: IRRAT_1:41
for b
1, b
2 being
real number holds
not ( b
2 >= 2 & b
1 = 1
/ (b2 + 1) & not b
1 / (1 - b1) < 1 )
theorem Th42: :: IRRAT_1:42