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Theorem logdivlti 19971
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 959 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  ->  B  e.  RR )
2 simpl3 960 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  ->  _e  <_  A )
3 simpr 447 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  ->  A  <  B )
4 simpl1 958 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  ->  A  e.  RR )
5 ere 12370 . . . . . . . . . . . 12  |-  _e  e.  RR
6 lelttr 8912 . . . . . . . . . . . 12  |-  ( ( _e  e.  RR  /\  A  e.  RR  /\  B  e.  RR )  ->  (
( _e  <_  A  /\  A  <  B )  ->  _e  <  B
) )
75, 6mp3an1 1264 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( _e  <_  A  /\  A  <  B
)  ->  _e  <  B ) )
84, 1, 7syl2anc 642 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( _e  <_  A  /\  A  <  B
)  ->  _e  <  B ) )
92, 3, 8mp2and 660 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  ->  _e  <  B )
10 epos 12485 . . . . . . . . . 10  |-  0  <  _e
11 0re 8838 . . . . . . . . . . . 12  |-  0  e.  RR
12 lttr 8899 . . . . . . . . . . . 12  |-  ( ( 0  e.  RR  /\  _e  e.  RR  /\  B  e.  RR )  ->  (
( 0  <  _e  /\  _e  <  B )  ->  0  <  B
) )
1311, 5, 12mp3an12 1267 . . . . . . . . . . 11  |-  ( B  e.  RR  ->  (
( 0  <  _e  /\  _e  <  B )  ->  0  <  B
) )
141, 13syl 15 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( 0  < 
_e  /\  _e  <  B )  ->  0  <  B ) )
1510, 14mpani 657 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( _e  <  B  ->  0  <  B ) )
169, 15mpd 14 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
0  <  B )
171, 16elrpd 10388 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  ->  B  e.  RR+ )
18 ltletr 8913 . . . . . . . . . . . 12  |-  ( ( 0  e.  RR  /\  _e  e.  RR  /\  A  e.  RR )  ->  (
( 0  <  _e  /\  _e  <_  A )  ->  0  <  A ) )
1911, 5, 18mp3an12 1267 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  (
( 0  <  _e  /\  _e  <_  A )  ->  0  <  A ) )
204, 19syl 15 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( 0  < 
_e  /\  _e  <_  A )  ->  0  <  A ) )
2110, 20mpani 657 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( _e  <_  A  ->  0  <  A ) )
222, 21mpd 14 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
0  <  A )
234, 22elrpd 10388 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  ->  A  e.  RR+ )
2417, 23rpdivcld 10407 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( B  /  A
)  e.  RR+ )
25 relogcl 19932 . . . . . 6  |-  ( ( B  /  A )  e.  RR+  ->  ( log `  ( B  /  A
) )  e.  RR )
2624, 25syl 15 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( log `  ( B  /  A ) )  e.  RR )
271, 23rerpdivcld 10417 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( B  /  A
)  e.  RR )
28 1re 8837 . . . . . 6  |-  1  e.  RR
29 resubcl 9111 . . . . . 6  |-  ( ( ( B  /  A
)  e.  RR  /\  1  e.  RR )  ->  ( ( B  /  A )  -  1 )  e.  RR )
3027, 28, 29sylancl 643 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( B  /  A )  -  1 )  e.  RR )
31 relogcl 19932 . . . . . . 7  |-  ( A  e.  RR+  ->  ( log `  A )  e.  RR )
3223, 31syl 15 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( log `  A
)  e.  RR )
3330, 32remulcld 8863 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( ( B  /  A )  - 
1 )  x.  ( log `  A ) )  e.  RR )
34 reeflog 19934 . . . . . . . . 9  |-  ( ( B  /  A )  e.  RR+  ->  ( exp `  ( log `  ( B  /  A ) ) )  =  ( B  /  A ) )
3524, 34syl 15 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( exp `  ( log `  ( B  /  A ) ) )  =  ( B  /  A ) )
36 ax-1cn 8795 . . . . . . . . 9  |-  1  e.  CC
3727recnd 8861 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( B  /  A
)  e.  CC )
38 pncan3 9059 . . . . . . . . 9  |-  ( ( 1  e.  CC  /\  ( B  /  A
)  e.  CC )  ->  ( 1  +  ( ( B  /  A )  -  1 ) )  =  ( B  /  A ) )
3936, 37, 38sylancr 644 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( 1  +  ( ( B  /  A
)  -  1 ) )  =  ( B  /  A ) )
4035, 39eqtr4d 2318 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( exp `  ( log `  ( B  /  A ) ) )  =  ( 1  +  ( ( B  /  A )  -  1 ) ) )
414recnd 8861 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  ->  A  e.  CC )
4241mulid2d 8853 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( 1  x.  A
)  =  A )
4342, 3eqbrtrd 4043 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( 1  x.  A
)  <  B )
4428a1i 10 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
1  e.  RR )
45 ltmuldiv 9626 . . . . . . . . . . 11  |-  ( ( 1  e.  RR  /\  B  e.  RR  /\  ( A  e.  RR  /\  0  <  A ) )  -> 
( ( 1  x.  A )  <  B  <->  1  <  ( B  /  A ) ) )
4644, 1, 4, 22, 45syl112anc 1186 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( 1  x.  A )  <  B  <->  1  <  ( B  /  A ) ) )
4743, 46mpbid 201 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
1  <  ( B  /  A ) )
48 difrp 10387 . . . . . . . . . 10  |-  ( ( 1  e.  RR  /\  ( B  /  A
)  e.  RR )  ->  ( 1  < 
( B  /  A
)  <->  ( ( B  /  A )  - 
1 )  e.  RR+ ) )
4928, 27, 48sylancr 644 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( 1  <  ( B  /  A )  <->  ( ( B  /  A )  - 
1 )  e.  RR+ ) )
5047, 49mpbid 201 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( B  /  A )  -  1 )  e.  RR+ )
51 efgt1p 12395 . . . . . . . 8  |-  ( ( ( B  /  A
)  -  1 )  e.  RR+  ->  ( 1  +  ( ( B  /  A )  - 
1 ) )  < 
( exp `  (
( B  /  A
)  -  1 ) ) )
5250, 51syl 15 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( 1  +  ( ( B  /  A
)  -  1 ) )  <  ( exp `  ( ( B  /  A )  -  1 ) ) )
5340, 52eqbrtrd 4043 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( exp `  ( log `  ( B  /  A ) ) )  <  ( exp `  (
( B  /  A
)  -  1 ) ) )
54 eflt 12397 . . . . . . 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 642 . . . . . 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 223 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( log `  ( B  /  A ) )  <  ( ( B  /  A )  - 
1 ) )
5730recnd 8861 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( B  /  A )  -  1 )  e.  CC )
5857mulid1d 8852 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( ( B  /  A )  - 
1 )  x.  1 )  =  ( ( B  /  A )  -  1 ) )
59 df-e 12350 . . . . . . . . 9  |-  _e  =  ( exp `  1 )
60 reeflog 19934 . . . . . . . . . . 11  |-  ( A  e.  RR+  ->  ( exp `  ( log `  A
) )  =  A )
6123, 60syl 15 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( exp `  ( log `  A ) )  =  A )
622, 61breqtrrd 4049 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  ->  _e  <_  ( exp `  ( log `  A ) ) )
6359, 62syl5eqbrr 4057 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( exp `  1
)  <_  ( exp `  ( log `  A
) ) )
64 efle 12398 . . . . . . . . 9  |-  ( ( 1  e.  RR  /\  ( log `  A )  e.  RR )  -> 
( 1  <_  ( log `  A )  <->  ( exp `  1 )  <_  ( exp `  ( log `  A
) ) ) )
6528, 32, 64sylancr 644 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( 1  <_  ( log `  A )  <->  ( exp `  1 )  <_  ( exp `  ( log `  A
) ) ) )
6663, 65mpbird 223 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
1  <_  ( log `  A ) )
67 posdif 9267 . . . . . . . . . 10  |-  ( ( 1  e.  RR  /\  ( B  /  A
)  e.  RR )  ->  ( 1  < 
( B  /  A
)  <->  0  <  (
( B  /  A
)  -  1 ) ) )
6828, 27, 67sylancr 644 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( 1  <  ( B  /  A )  <->  0  <  ( ( B  /  A
)  -  1 ) ) )
6947, 68mpbid 201 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
0  <  ( ( B  /  A )  - 
1 ) )
70 lemul2 9609 . . . . . . . 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 1186 . . . . . . 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 201 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( ( B  /  A )  - 
1 )  x.  1 )  <_  ( (
( B  /  A
)  -  1 )  x.  ( log `  A
) ) )
7358, 72eqbrtrrd 4045 . . . . 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 8976 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( log `  ( B  /  A ) )  <  ( ( ( B  /  A )  -  1 )  x.  ( log `  A
) ) )
75 relogdiv 19946 . . . . 5  |-  ( ( B  e.  RR+  /\  A  e.  RR+ )  ->  ( log `  ( B  /  A ) )  =  ( ( log `  B
)  -  ( log `  A ) ) )
7617, 23, 75syl2anc 642 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( log `  ( B  /  A ) )  =  ( ( log `  B )  -  ( log `  A ) ) )
7736a1i 10 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
1  e.  CC )
7832recnd 8861 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( log `  A
)  e.  CC )
7937, 77, 78subdird 9236 . . . . 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 8861 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  ->  B  e.  CC )
8123rpne0d 10395 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  ->  A  =/=  0 )
8280, 41, 78, 81div32d 9559 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( B  /  A )  x.  ( log `  A ) )  =  ( B  x.  ( ( log `  A
)  /  A ) ) )
8378mulid2d 8853 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( 1  x.  ( log `  A ) )  =  ( log `  A
) )
8482, 83oveq12d 5876 . . . . 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 2315 . . . 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 4052 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( log `  B
)  -  ( log `  A ) )  < 
( ( B  x.  ( ( log `  A
)  /  A ) )  -  ( log `  A ) ) )
87 relogcl 19932 . . . . 5  |-  ( B  e.  RR+  ->  ( log `  B )  e.  RR )
8817, 87syl 15 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( log `  B
)  e.  RR )
8932, 23rerpdivcld 10417 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( log `  A
)  /  A )  e.  RR )
901, 89remulcld 8863 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( B  x.  (
( log `  A
)  /  A ) )  e.  RR )
9188, 90, 32ltsub1d 9381 . . 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 223 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( log `  B
)  <  ( B  x.  ( ( log `  A
)  /  A ) ) )
9388, 89, 17ltdivmuld 10437 . 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 223 1  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( log `  B
)  /  B )  <  ( ( log `  A )  /  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    < clt 8867    <_ cle 8868    - cmin 9037    / cdiv 9423   RR+crp 10354   expce 12343   _eceu 12344   logclog 19912
This theorem is referenced by:  logdivlt  19972
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-e 12350  df-sin 12351  df-cos 12352  df-pi 12354  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217  df-log 19914
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