:: POWER semantic presentation
theorem Th1: :: POWER:1
theorem Th2: :: POWER:2
theorem Th3: :: POWER:3
for b
1 being
real number for b
2 being
Nat holds
( not ( not b
1 >= 0 & ( for b
3 being
Nat holds
not b
2 = 2
* b
3 ) ) implies b
1 |^ b
2 >= 0 )
:: deftheorem Def1 defines -root POWER:def 1 :
theorem Th4: :: POWER:4
canceled;
theorem Th5: :: POWER:5
for b
1 being
real number for b
2 being
Nat holds
( not ( not ( b
2 >= 1 & b
1 >= 0 ) & ( for b
3 being
Nat holds
not b
2 = (2 * b3) + 1 ) ) implies (
(b2 -root b1) |^ b
2 = b
1 & b
2 -root (b1 |^ b2) = b
1 ) )
theorem Th6: :: POWER:6
for b
1 being
Nat holds
( b
1 >= 1 implies b
1 -root 0
= 0 )
theorem Th7: :: POWER:7
for b
1 being
Nat holds
( b
1 >= 1 implies b
1 -root 1
= 1 )
theorem Th8: :: POWER:8
theorem Th9: :: POWER:9
for b
1 being
Nat holds
( ex b
2 being
Nat st b
1 = (2 * b2) + 1 implies b
1 -root (- 1) = - 1 )
theorem Th10: :: POWER:10
theorem Th11: :: POWER:11
theorem Th12: :: POWER:12
theorem Th13: :: POWER:13
theorem Th14: :: POWER:14
theorem Th15: :: POWER:15
for b
1 being
real number for b
2, b
3 being
Nat holds
( not ( not ( b
1 >= 0 & b
2 >= 1 & b
3 >= 1 ) & ( for b
4, b
5 being
Nat holds
not ( b
2 = (2 * b4) + 1 & b
3 = (2 * b5) + 1 ) ) ) implies b
2 -root (b3 -root b1) = (b2 * b3) -root b
1 )
theorem Th16: :: POWER:16
for b
1 being
real number for b
2, b
3 being
Nat holds
( not ( not ( b
1 >= 0 & b
2 >= 1 & b
3 >= 1 ) & ( for b
4, b
5 being
Nat holds
not ( b
2 = (2 * b4) + 1 & b
3 = (2 * b5) + 1 ) ) ) implies
(b2 -root b1) * (b3 -root b1) = (b2 * b3) -root (b1 |^ (b2 + b3)) )
theorem Th17: :: POWER:17
for b
1, b
2 being
real number for b
3 being
Nat holds
( b
1 <= b
2 & not ( not ( 0
<= b
1 & b
3 >= 1 ) & ( for b
4 being
Nat holds
not b
3 = (2 * b4) + 1 ) ) implies b
3 -root b
1 <= b
3 -root b
2 )
theorem Th18: :: POWER:18
for b
1, b
2 being
real number for b
3 being
Nat holds
not ( b
1 < b
2 & not ( not ( b
1 >= 0 & b
3 >= 1 ) & ( for b
4 being
Nat holds
not b
3 = (2 * b4) + 1 ) ) & not b
3 -root b
1 < b
3 -root b
2 )
theorem Th19: :: POWER:19
theorem Th20: :: POWER:20
theorem Th21: :: POWER:21
theorem Th22: :: POWER:22
theorem Th23: :: POWER:23
theorem Th24: :: POWER:24
:: deftheorem Def2 defines to_power POWER:def 2 :
theorem Th25: :: POWER:25
canceled;
theorem Th26: :: POWER:26
canceled;
theorem Th27: :: POWER:27
canceled;
theorem Th28: :: POWER:28
canceled;
theorem Th29: :: POWER:29
theorem Th30: :: POWER:30
theorem Th31: :: POWER:31
theorem Th32: :: POWER:32
theorem Th33: :: POWER:33
theorem Th34: :: POWER:34
theorem Th35: :: POWER:35
theorem Th36: :: POWER:36
theorem Th37: :: POWER:37
theorem Th38: :: POWER:38
theorem Th39: :: POWER:39
theorem Th40: :: POWER:40
theorem Th41: :: POWER:41
theorem Th42: :: POWER:42
theorem Th43: :: POWER:43
theorem Th44: :: POWER:44
theorem Th45: :: POWER:45
theorem Th46: :: POWER:46
theorem Th47: :: POWER:47
theorem Th48: :: POWER:48
theorem Th49: :: POWER:49
theorem Th50: :: POWER:50
theorem Th51: :: POWER:51
theorem Th52: :: POWER:52
theorem Th53: :: POWER:53
theorem Th54: :: POWER:54
theorem Th55: :: POWER:55
theorem Th56: :: POWER:56
theorem Th57: :: POWER:57
:: deftheorem Def3 defines log POWER:def 3 :
theorem Th58: :: POWER:58
canceled;
theorem Th59: :: POWER:59
theorem Th60: :: POWER:60
theorem Th61: :: POWER:61
theorem Th62: :: POWER:62
theorem Th63: :: POWER:63
theorem Th64: :: POWER:64
theorem Th65: :: POWER:65
for b
1, b
2, b
3 being
real number holds
not ( b
1 > 1 & b
2 > 0 & b
3 > b
2 & not
log b
1,b
3 > log b
1,b
2 )
theorem Th66: :: POWER:66
for b
1, b
2, b
3 being
real number holds
not ( b
1 > 0 & b
1 < 1 & b
2 > 0 & b
3 > b
2 & not
log b
1,b
3 < log b
1,b
2 )
theorem Th67: :: POWER:67
:: deftheorem Def4 defines number_e POWER:def 4 :