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Theorem logdivlti 23189
Description: The  log x  /  x function is strictly decreasing on the reals greater than  _e. (Contributed by Mario Carneiro, 14-Mar-2014.)
Assertion
Ref Expression
logdivlti  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( log `  B
)  /  B )  <  ( ( log `  A )  /  A
) )

Proof of Theorem logdivlti
StepHypRef Expression
1 simpl2 999 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  ->  B  e.  RR )
2 simpl3 1000 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  ->  _e  <_  A )
3 simpr 459 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  ->  A  <  B )
4 simpl1 998 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  ->  A  e.  RR )
5 ere 13923 . . . . . . . . . . . 12  |-  _e  e.  RR
6 lelttr 9624 . . . . . . . . . . . 12  |-  ( ( _e  e.  RR  /\  A  e.  RR  /\  B  e.  RR )  ->  (
( _e  <_  A  /\  A  <  B )  ->  _e  <  B
) )
75, 6mp3an1 1311 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( _e  <_  A  /\  A  <  B
)  ->  _e  <  B ) )
84, 1, 7syl2anc 659 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( _e  <_  A  /\  A  <  B
)  ->  _e  <  B ) )
92, 3, 8mp2and 677 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  ->  _e  <  B )
10 epos 14039 . . . . . . . . . 10  |-  0  <  _e
11 0re 9544 . . . . . . . . . . . 12  |-  0  e.  RR
12 lttr 9610 . . . . . . . . . . . 12  |-  ( ( 0  e.  RR  /\  _e  e.  RR  /\  B  e.  RR )  ->  (
( 0  <  _e  /\  _e  <  B )  ->  0  <  B
) )
1311, 5, 12mp3an12 1314 . . . . . . . . . . 11  |-  ( B  e.  RR  ->  (
( 0  <  _e  /\  _e  <  B )  ->  0  <  B
) )
141, 13syl 17 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( 0  < 
_e  /\  _e  <  B )  ->  0  <  B ) )
1510, 14mpani 674 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( _e  <  B  ->  0  <  B ) )
169, 15mpd 15 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
0  <  B )
171, 16elrpd 11217 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  ->  B  e.  RR+ )
18 ltletr 9625 . . . . . . . . . . . 12  |-  ( ( 0  e.  RR  /\  _e  e.  RR  /\  A  e.  RR )  ->  (
( 0  <  _e  /\  _e  <_  A )  ->  0  <  A ) )
1911, 5, 18mp3an12 1314 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  (
( 0  <  _e  /\  _e  <_  A )  ->  0  <  A ) )
204, 19syl 17 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( 0  < 
_e  /\  _e  <_  A )  ->  0  <  A ) )
2110, 20mpani 674 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( _e  <_  A  ->  0  <  A ) )
222, 21mpd 15 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
0  <  A )
234, 22elrpd 11217 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  ->  A  e.  RR+ )
2417, 23rpdivcld 11237 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( B  /  A
)  e.  RR+ )
25 relogcl 23145 . . . . . 6  |-  ( ( B  /  A )  e.  RR+  ->  ( log `  ( B  /  A
) )  e.  RR )
2624, 25syl 17 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( log `  ( B  /  A ) )  e.  RR )
271, 23rerpdivcld 11247 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( B  /  A
)  e.  RR )
28 1re 9543 . . . . . 6  |-  1  e.  RR
29 resubcl 9837 . . . . . 6  |-  ( ( ( B  /  A
)  e.  RR  /\  1  e.  RR )  ->  ( ( B  /  A )  -  1 )  e.  RR )
3027, 28, 29sylancl 660 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( B  /  A )  -  1 )  e.  RR )
31 relogcl 23145 . . . . . . 7  |-  ( A  e.  RR+  ->  ( log `  A )  e.  RR )
3223, 31syl 17 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( log `  A
)  e.  RR )
3330, 32remulcld 9572 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( ( B  /  A )  - 
1 )  x.  ( log `  A ) )  e.  RR )
34 reeflog 23150 . . . . . . . . 9  |-  ( ( B  /  A )  e.  RR+  ->  ( exp `  ( log `  ( B  /  A ) ) )  =  ( B  /  A ) )
3524, 34syl 17 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( exp `  ( log `  ( B  /  A ) ) )  =  ( B  /  A ) )
36 ax-1cn 9498 . . . . . . . . 9  |-  1  e.  CC
3727recnd 9570 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( B  /  A
)  e.  CC )
38 pncan3 9782 . . . . . . . . 9  |-  ( ( 1  e.  CC  /\  ( B  /  A
)  e.  CC )  ->  ( 1  +  ( ( B  /  A )  -  1 ) )  =  ( B  /  A ) )
3936, 37, 38sylancr 661 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( 1  +  ( ( B  /  A
)  -  1 ) )  =  ( B  /  A ) )
4035, 39eqtr4d 2444 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( exp `  ( log `  ( B  /  A ) ) )  =  ( 1  +  ( ( B  /  A )  -  1 ) ) )
414recnd 9570 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  ->  A  e.  CC )
4241mulid2d 9562 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( 1  x.  A
)  =  A )
4342, 3eqbrtrd 4412 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( 1  x.  A
)  <  B )
44 1red 9559 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
1  e.  RR )
45 ltmuldiv 10374 . . . . . . . . . . 11  |-  ( ( 1  e.  RR  /\  B  e.  RR  /\  ( A  e.  RR  /\  0  <  A ) )  -> 
( ( 1  x.  A )  <  B  <->  1  <  ( B  /  A ) ) )
4644, 1, 4, 22, 45syl112anc 1232 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( 1  x.  A )  <  B  <->  1  <  ( B  /  A ) ) )
4743, 46mpbid 210 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
1  <  ( B  /  A ) )
48 difrp 11216 . . . . . . . . . 10  |-  ( ( 1  e.  RR  /\  ( B  /  A
)  e.  RR )  ->  ( 1  < 
( B  /  A
)  <->  ( ( B  /  A )  - 
1 )  e.  RR+ ) )
4928, 27, 48sylancr 661 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( 1  <  ( B  /  A )  <->  ( ( B  /  A )  - 
1 )  e.  RR+ ) )
5047, 49mpbid 210 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( B  /  A )  -  1 )  e.  RR+ )
51 efgt1p 13949 . . . . . . . 8  |-  ( ( ( B  /  A
)  -  1 )  e.  RR+  ->  ( 1  +  ( ( B  /  A )  - 
1 ) )  < 
( exp `  (
( B  /  A
)  -  1 ) ) )
5250, 51syl 17 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( 1  +  ( ( B  /  A
)  -  1 ) )  <  ( exp `  ( ( B  /  A )  -  1 ) ) )
5340, 52eqbrtrd 4412 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( exp `  ( log `  ( B  /  A ) ) )  <  ( exp `  (
( B  /  A
)  -  1 ) ) )
54 eflt 13951 . . . . . . 7  |-  ( ( ( log `  ( B  /  A ) )  e.  RR  /\  (
( B  /  A
)  -  1 )  e.  RR )  -> 
( ( log `  ( B  /  A ) )  <  ( ( B  /  A )  - 
1 )  <->  ( exp `  ( log `  ( B  /  A ) ) )  <  ( exp `  ( ( B  /  A )  -  1 ) ) ) )
5526, 30, 54syl2anc 659 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( log `  ( B  /  A ) )  <  ( ( B  /  A )  - 
1 )  <->  ( exp `  ( log `  ( B  /  A ) ) )  <  ( exp `  ( ( B  /  A )  -  1 ) ) ) )
5653, 55mpbird 232 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( log `  ( B  /  A ) )  <  ( ( B  /  A )  - 
1 ) )
5730recnd 9570 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( B  /  A )  -  1 )  e.  CC )
5857mulid1d 9561 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( ( B  /  A )  - 
1 )  x.  1 )  =  ( ( B  /  A )  -  1 ) )
59 df-e 13903 . . . . . . . . 9  |-  _e  =  ( exp `  1 )
60 reeflog 23150 . . . . . . . . . . 11  |-  ( A  e.  RR+  ->  ( exp `  ( log `  A
) )  =  A )
6123, 60syl 17 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( exp `  ( log `  A ) )  =  A )
622, 61breqtrrd 4418 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  ->  _e  <_  ( exp `  ( log `  A ) ) )
6359, 62syl5eqbrr 4426 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( exp `  1
)  <_  ( exp `  ( log `  A
) ) )
64 efle 13952 . . . . . . . . 9  |-  ( ( 1  e.  RR  /\  ( log `  A )  e.  RR )  -> 
( 1  <_  ( log `  A )  <->  ( exp `  1 )  <_  ( exp `  ( log `  A
) ) ) )
6528, 32, 64sylancr 661 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( 1  <_  ( log `  A )  <->  ( exp `  1 )  <_  ( exp `  ( log `  A
) ) ) )
6663, 65mpbird 232 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
1  <_  ( log `  A ) )
67 posdif 10004 . . . . . . . . . 10  |-  ( ( 1  e.  RR  /\  ( B  /  A
)  e.  RR )  ->  ( 1  < 
( B  /  A
)  <->  0  <  (
( B  /  A
)  -  1 ) ) )
6828, 27, 67sylancr 661 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( 1  <  ( B  /  A )  <->  0  <  ( ( B  /  A
)  -  1 ) ) )
6947, 68mpbid 210 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
0  <  ( ( B  /  A )  - 
1 ) )
70 lemul2 10354 . . . . . . . 8  |-  ( ( 1  e.  RR  /\  ( log `  A )  e.  RR  /\  (
( ( B  /  A )  -  1 )  e.  RR  /\  0  <  ( ( B  /  A )  - 
1 ) ) )  ->  ( 1  <_ 
( log `  A
)  <->  ( ( ( B  /  A )  -  1 )  x.  1 )  <_  (
( ( B  /  A )  -  1 )  x.  ( log `  A ) ) ) )
7144, 32, 30, 69, 70syl112anc 1232 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( 1  <_  ( log `  A )  <->  ( (
( B  /  A
)  -  1 )  x.  1 )  <_ 
( ( ( B  /  A )  - 
1 )  x.  ( log `  A ) ) ) )
7266, 71mpbid 210 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( ( B  /  A )  - 
1 )  x.  1 )  <_  ( (
( B  /  A
)  -  1 )  x.  ( log `  A
) ) )
7358, 72eqbrtrrd 4414 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( B  /  A )  -  1 )  <_  ( (
( B  /  A
)  -  1 )  x.  ( log `  A
) ) )
7426, 30, 33, 56, 73ltletrd 9694 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( log `  ( B  /  A ) )  <  ( ( ( B  /  A )  -  1 )  x.  ( log `  A
) ) )
75 relogdiv 23162 . . . . 5  |-  ( ( B  e.  RR+  /\  A  e.  RR+ )  ->  ( log `  ( B  /  A ) )  =  ( ( log `  B
)  -  ( log `  A ) ) )
7617, 23, 75syl2anc 659 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( log `  ( B  /  A ) )  =  ( ( log `  B )  -  ( log `  A ) ) )
77 1cnd 9560 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
1  e.  CC )
7832recnd 9570 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( log `  A
)  e.  CC )
7937, 77, 78subdird 9972 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( ( B  /  A )  - 
1 )  x.  ( log `  A ) )  =  ( ( ( B  /  A )  x.  ( log `  A
) )  -  (
1  x.  ( log `  A ) ) ) )
801recnd 9570 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  ->  B  e.  CC )
8123rpne0d 11225 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  ->  A  =/=  0 )
8280, 41, 78, 81div32d 10302 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( B  /  A )  x.  ( log `  A ) )  =  ( B  x.  ( ( log `  A
)  /  A ) ) )
8378mulid2d 9562 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( 1  x.  ( log `  A ) )  =  ( log `  A
) )
8482, 83oveq12d 6250 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( ( B  /  A )  x.  ( log `  A
) )  -  (
1  x.  ( log `  A ) ) )  =  ( ( B  x.  ( ( log `  A )  /  A
) )  -  ( log `  A ) ) )
8579, 84eqtrd 2441 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( ( B  /  A )  - 
1 )  x.  ( log `  A ) )  =  ( ( B  x.  ( ( log `  A )  /  A
) )  -  ( log `  A ) ) )
8674, 76, 853brtr3d 4421 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( log `  B
)  -  ( log `  A ) )  < 
( ( B  x.  ( ( log `  A
)  /  A ) )  -  ( log `  A ) ) )
87 relogcl 23145 . . . . 5  |-  ( B  e.  RR+  ->  ( log `  B )  e.  RR )
8817, 87syl 17 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( log `  B
)  e.  RR )
8932, 23rerpdivcld 11247 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( log `  A
)  /  A )  e.  RR )
901, 89remulcld 9572 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( B  x.  (
( log `  A
)  /  A ) )  e.  RR )
9188, 90, 32ltsub1d 10119 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( log `  B
)  <  ( B  x.  ( ( log `  A
)  /  A ) )  <->  ( ( log `  B )  -  ( log `  A ) )  <  ( ( B  x.  ( ( log `  A )  /  A
) )  -  ( log `  A ) ) ) )
9286, 91mpbird 232 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( log `  B
)  <  ( B  x.  ( ( log `  A
)  /  A ) ) )
9388, 89, 17ltdivmuld 11267 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( ( log `  B )  /  B
)  <  ( ( log `  A )  /  A )  <->  ( log `  B )  <  ( B  x.  ( ( log `  A )  /  A ) ) ) )
9492, 93mpbird 232 1  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( log `  B
)  /  B )  <  ( ( log `  A )  /  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 972    = wceq 1403    e. wcel 1840   class class class wbr 4392   ` cfv 5523  (class class class)co 6232   CCcc 9438   RRcr 9439   0cc0 9440   1c1 9441    + caddc 9443    x. cmul 9445    < clt 9576    <_ cle 9577    - cmin 9759    / cdiv 10165   RR+crp 11181   expce 13896   _eceu 13897   logclog 23124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-inf2 8009  ax-cnex 9496  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516  ax-pre-mulgt0 9517  ax-pre-sup 9518  ax-addf 9519  ax-mulf 9520
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-fal 1409  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-int 4225  df-iun 4270  df-iin 4271  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-se 4780  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-isom 5532  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-of 6475  df-om 6637  df-1st 6736  df-2nd 6737  df-supp 6855  df-recs 6997  df-rdg 7031  df-1o 7085  df-2o 7086  df-oadd 7089  df-er 7266  df-map 7377  df-pm 7378  df-ixp 7426  df-en 7473  df-dom 7474  df-sdom 7475  df-fin 7476  df-fsupp 7782  df-fi 7823  df-sup 7853  df-oi 7887  df-card 8270  df-cda 8498  df-pnf 9578  df-mnf 9579  df-xr 9580  df-ltxr 9581  df-le 9582  df-sub 9761  df-neg 9762  df-div 10166  df-nn 10495  df-2 10553  df-3 10554  df-4 10555  df-5 10556  df-6 10557  df-7 10558  df-8 10559  df-9 10560  df-10 10561  df-n0 10755  df-z 10824  df-dec 10938  df-uz 11044  df-q 11144  df-rp 11182  df-xneg 11287  df-xadd 11288  df-xmul 11289  df-ioo 11502  df-ioc 11503  df-ico 11504  df-icc 11505  df-fz 11642  df-fzo 11766  df-fl 11877  df-mod 11946  df-seq 12060  df-exp 12119  df-fac 12306  df-bc 12333  df-hash 12358  df-shft 12954  df-cj 12986  df-re 12987  df-im 12988  df-sqrt 13122  df-abs 13123  df-limsup 13348  df-clim 13365  df-rlim 13366  df-sum 13563  df-ef 13902  df-e 13903  df-sin 13904  df-cos 13905  df-pi 13907  df-struct 14733  df-ndx 14734  df-slot 14735  df-base 14736  df-sets 14737  df-ress 14738  df-plusg 14812  df-mulr 14813  df-starv 14814  df-sca 14815  df-vsca 14816  df-ip 14817  df-tset 14818  df-ple 14819  df-ds 14821  df-unif 14822  df-hom 14823  df-cco 14824  df-rest 14927  df-topn 14928  df-0g 14946  df-gsum 14947  df-topgen 14948  df-pt 14949  df-prds 14952  df-xrs 15006  df-qtop 15011  df-imas 15012  df-xps 15014  df-mre 15090  df-mrc 15091  df-acs 15093  df-mgm 16086  df-sgrp 16125  df-mnd 16135  df-submnd 16181  df-mulg 16274  df-cntz 16569  df-cmn 17014  df-psmet 18621  df-xmet 18622  df-met 18623  df-bl 18624  df-mopn 18625  df-fbas 18626  df-fg 18627  df-cnfld 18631  df-top 19581  df-bases 19583  df-topon 19584  df-topsp 19585  df-cld 19702  df-ntr 19703  df-cls 19704  df-nei 19782  df-lp 19820  df-perf 19821  df-cn 19911  df-cnp 19912  df-haus 19999  df-tx 20245  df-hmeo 20438  df-fil 20529  df-fm 20621  df-flim 20622  df-flf 20623  df-xms 21005  df-ms 21006  df-tms 21007  df-cncf 21564  df-limc 22452  df-dv 22453  df-log 23126
This theorem is referenced by:  logdivlt  23190
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