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Theorem logdivlti 20382
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 961 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  ->  B  e.  RR )
2 simpl3 962 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  ->  _e  <_  A )
3 simpr 448 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  ->  A  <  B )
4 simpl1 960 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  ->  A  e.  RR )
5 ere 12618 . . . . . . . . . . . 12  |-  _e  e.  RR
6 lelttr 9098 . . . . . . . . . . . 12  |-  ( ( _e  e.  RR  /\  A  e.  RR  /\  B  e.  RR )  ->  (
( _e  <_  A  /\  A  <  B )  ->  _e  <  B
) )
75, 6mp3an1 1266 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( _e  <_  A  /\  A  <  B
)  ->  _e  <  B ) )
84, 1, 7syl2anc 643 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( _e  <_  A  /\  A  <  B
)  ->  _e  <  B ) )
92, 3, 8mp2and 661 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  ->  _e  <  B )
10 epos 12733 . . . . . . . . . 10  |-  0  <  _e
11 0re 9024 . . . . . . . . . . . 12  |-  0  e.  RR
12 lttr 9085 . . . . . . . . . . . 12  |-  ( ( 0  e.  RR  /\  _e  e.  RR  /\  B  e.  RR )  ->  (
( 0  <  _e  /\  _e  <  B )  ->  0  <  B
) )
1311, 5, 12mp3an12 1269 . . . . . . . . . . 11  |-  ( B  e.  RR  ->  (
( 0  <  _e  /\  _e  <  B )  ->  0  <  B
) )
141, 13syl 16 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( 0  < 
_e  /\  _e  <  B )  ->  0  <  B ) )
1510, 14mpani 658 . . . . . . . . 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 10578 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  ->  B  e.  RR+ )
18 ltletr 9099 . . . . . . . . . . . 12  |-  ( ( 0  e.  RR  /\  _e  e.  RR  /\  A  e.  RR )  ->  (
( 0  <  _e  /\  _e  <_  A )  ->  0  <  A ) )
1911, 5, 18mp3an12 1269 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  (
( 0  <  _e  /\  _e  <_  A )  ->  0  <  A ) )
204, 19syl 16 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( 0  < 
_e  /\  _e  <_  A )  ->  0  <  A ) )
2110, 20mpani 658 . . . . . . . . 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 10578 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  ->  A  e.  RR+ )
2417, 23rpdivcld 10597 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( B  /  A
)  e.  RR+ )
25 relogcl 20340 . . . . . 6  |-  ( ( B  /  A )  e.  RR+  ->  ( log `  ( B  /  A
) )  e.  RR )
2624, 25syl 16 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( log `  ( B  /  A ) )  e.  RR )
271, 23rerpdivcld 10607 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( B  /  A
)  e.  RR )
28 1re 9023 . . . . . 6  |-  1  e.  RR
29 resubcl 9297 . . . . . 6  |-  ( ( ( B  /  A
)  e.  RR  /\  1  e.  RR )  ->  ( ( B  /  A )  -  1 )  e.  RR )
3027, 28, 29sylancl 644 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( B  /  A )  -  1 )  e.  RR )
31 relogcl 20340 . . . . . . 7  |-  ( A  e.  RR+  ->  ( log `  A )  e.  RR )
3223, 31syl 16 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( log `  A
)  e.  RR )
3330, 32remulcld 9049 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( ( B  /  A )  - 
1 )  x.  ( log `  A ) )  e.  RR )
34 reeflog 20342 . . . . . . . . 9  |-  ( ( B  /  A )  e.  RR+  ->  ( exp `  ( log `  ( B  /  A ) ) )  =  ( B  /  A ) )
3524, 34syl 16 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( exp `  ( log `  ( B  /  A ) ) )  =  ( B  /  A ) )
36 ax-1cn 8981 . . . . . . . . 9  |-  1  e.  CC
3727recnd 9047 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( B  /  A
)  e.  CC )
38 pncan3 9245 . . . . . . . . 9  |-  ( ( 1  e.  CC  /\  ( B  /  A
)  e.  CC )  ->  ( 1  +  ( ( B  /  A )  -  1 ) )  =  ( B  /  A ) )
3936, 37, 38sylancr 645 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( 1  +  ( ( B  /  A
)  -  1 ) )  =  ( B  /  A ) )
4035, 39eqtr4d 2422 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( exp `  ( log `  ( B  /  A ) ) )  =  ( 1  +  ( ( B  /  A )  -  1 ) ) )
414recnd 9047 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  ->  A  e.  CC )
4241mulid2d 9039 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( 1  x.  A
)  =  A )
4342, 3eqbrtrd 4173 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( 1  x.  A
)  <  B )
4428a1i 11 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
1  e.  RR )
45 ltmuldiv 9812 . . . . . . . . . . 11  |-  ( ( 1  e.  RR  /\  B  e.  RR  /\  ( A  e.  RR  /\  0  <  A ) )  -> 
( ( 1  x.  A )  <  B  <->  1  <  ( B  /  A ) ) )
4644, 1, 4, 22, 45syl112anc 1188 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( 1  x.  A )  <  B  <->  1  <  ( B  /  A ) ) )
4743, 46mpbid 202 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
1  <  ( B  /  A ) )
48 difrp 10577 . . . . . . . . . 10  |-  ( ( 1  e.  RR  /\  ( B  /  A
)  e.  RR )  ->  ( 1  < 
( B  /  A
)  <->  ( ( B  /  A )  - 
1 )  e.  RR+ ) )
4928, 27, 48sylancr 645 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( 1  <  ( B  /  A )  <->  ( ( B  /  A )  - 
1 )  e.  RR+ ) )
5047, 49mpbid 202 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( B  /  A )  -  1 )  e.  RR+ )
51 efgt1p 12643 . . . . . . . 8  |-  ( ( ( B  /  A
)  -  1 )  e.  RR+  ->  ( 1  +  ( ( B  /  A )  - 
1 ) )  < 
( exp `  (
( B  /  A
)  -  1 ) ) )
5250, 51syl 16 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( 1  +  ( ( B  /  A
)  -  1 ) )  <  ( exp `  ( ( B  /  A )  -  1 ) ) )
5340, 52eqbrtrd 4173 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( exp `  ( log `  ( B  /  A ) ) )  <  ( exp `  (
( B  /  A
)  -  1 ) ) )
54 eflt 12645 . . . . . . 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 643 . . . . . 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 224 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( log `  ( B  /  A ) )  <  ( ( B  /  A )  - 
1 ) )
5730recnd 9047 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( B  /  A )  -  1 )  e.  CC )
5857mulid1d 9038 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( ( B  /  A )  - 
1 )  x.  1 )  =  ( ( B  /  A )  -  1 ) )
59 df-e 12598 . . . . . . . . 9  |-  _e  =  ( exp `  1 )
60 reeflog 20342 . . . . . . . . . . 11  |-  ( A  e.  RR+  ->  ( exp `  ( log `  A
) )  =  A )
6123, 60syl 16 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( exp `  ( log `  A ) )  =  A )
622, 61breqtrrd 4179 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  ->  _e  <_  ( exp `  ( log `  A ) ) )
6359, 62syl5eqbrr 4187 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( exp `  1
)  <_  ( exp `  ( log `  A
) ) )
64 efle 12646 . . . . . . . . 9  |-  ( ( 1  e.  RR  /\  ( log `  A )  e.  RR )  -> 
( 1  <_  ( log `  A )  <->  ( exp `  1 )  <_  ( exp `  ( log `  A
) ) ) )
6528, 32, 64sylancr 645 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( 1  <_  ( log `  A )  <->  ( exp `  1 )  <_  ( exp `  ( log `  A
) ) ) )
6663, 65mpbird 224 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
1  <_  ( log `  A ) )
67 posdif 9453 . . . . . . . . . 10  |-  ( ( 1  e.  RR  /\  ( B  /  A
)  e.  RR )  ->  ( 1  < 
( B  /  A
)  <->  0  <  (
( B  /  A
)  -  1 ) ) )
6828, 27, 67sylancr 645 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( 1  <  ( B  /  A )  <->  0  <  ( ( B  /  A
)  -  1 ) ) )
6947, 68mpbid 202 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
0  <  ( ( B  /  A )  - 
1 ) )
70 lemul2 9795 . . . . . . . 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 1188 . . . . . . 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 202 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( ( B  /  A )  - 
1 )  x.  1 )  <_  ( (
( B  /  A
)  -  1 )  x.  ( log `  A
) ) )
7358, 72eqbrtrrd 4175 . . . . 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 9162 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( log `  ( B  /  A ) )  <  ( ( ( B  /  A )  -  1 )  x.  ( log `  A
) ) )
75 relogdiv 20354 . . . . 5  |-  ( ( B  e.  RR+  /\  A  e.  RR+ )  ->  ( log `  ( B  /  A ) )  =  ( ( log `  B
)  -  ( log `  A ) ) )
7617, 23, 75syl2anc 643 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( log `  ( B  /  A ) )  =  ( ( log `  B )  -  ( log `  A ) ) )
7736a1i 11 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
1  e.  CC )
7832recnd 9047 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( log `  A
)  e.  CC )
7937, 77, 78subdird 9422 . . . . 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 9047 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  ->  B  e.  CC )
8123rpne0d 10585 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  ->  A  =/=  0 )
8280, 41, 78, 81div32d 9745 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( B  /  A )  x.  ( log `  A ) )  =  ( B  x.  ( ( log `  A
)  /  A ) ) )
8378mulid2d 9039 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( 1  x.  ( log `  A ) )  =  ( log `  A
) )
8482, 83oveq12d 6038 . . . . 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 2419 . . . 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 4182 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( log `  B
)  -  ( log `  A ) )  < 
( ( B  x.  ( ( log `  A
)  /  A ) )  -  ( log `  A ) ) )
87 relogcl 20340 . . . . 5  |-  ( B  e.  RR+  ->  ( log `  B )  e.  RR )
8817, 87syl 16 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( log `  B
)  e.  RR )
8932, 23rerpdivcld 10607 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( log `  A
)  /  A )  e.  RR )
901, 89remulcld 9049 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( B  x.  (
( log `  A
)  /  A ) )  e.  RR )
9188, 90, 32ltsub1d 9567 . . 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 224 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( log `  B
)  <  ( B  x.  ( ( log `  A
)  /  A ) ) )
9388, 89, 17ltdivmuld 10627 . 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 224 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   class class class wbr 4153   ` cfv 5394  (class class class)co 6020   CCcc 8921   RRcr 8922   0cc0 8923   1c1 8924    + caddc 8926    x. cmul 8928    < clt 9053    <_ cle 9054    - cmin 9223    / cdiv 9609   RR+crp 10544   expce 12591   _eceu 12592   logclog 20319
This theorem is referenced by:  logdivlt  20383
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001  ax-addf 9002  ax-mulf 9003
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-iin 4038  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-of 6244  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-2o 6661  df-oadd 6664  df-er 6841  df-map 6956  df-pm 6957  df-ixp 7000  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-fi 7351  df-sup 7381  df-oi 7412  df-card 7759  df-cda 7981  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-4 9992  df-5 9993  df-6 9994  df-7 9995  df-8 9996  df-9 9997  df-10 9998  df-n0 10154  df-z 10215  df-dec 10315  df-uz 10421  df-q 10507  df-rp 10545  df-xneg 10642  df-xadd 10643  df-xmul 10644  df-ioo 10852  df-ioc 10853  df-ico 10854  df-icc 10855  df-fz 10976  df-fzo 11066  df-fl 11129  df-mod 11178  df-seq 11251  df-exp 11310  df-fac 11494  df-bc 11521  df-hash 11546  df-shft 11809  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968  df-limsup 12192  df-clim 12209  df-rlim 12210  df-sum 12407  df-ef 12597  df-e 12598  df-sin 12599  df-cos 12600  df-pi 12602  df-struct 13398  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-ress 13403  df-plusg 13469  df-mulr 13470  df-starv 13471  df-sca 13472  df-vsca 13473  df-tset 13475  df-ple 13476  df-ds 13478  df-unif 13479  df-hom 13480  df-cco 13481  df-rest 13577  df-topn 13578  df-topgen 13594  df-pt 13595  df-prds 13598  df-xrs 13653  df-0g 13654  df-gsum 13655  df-qtop 13660  df-imas 13661  df-xps 13663  df-mre 13738  df-mrc 13739  df-acs 13741  df-mnd 14617  df-submnd 14666  df-mulg 14742  df-cntz 15043  df-cmn 15341  df-xmet 16619  df-met 16620  df-bl 16621  df-mopn 16622  df-fbas 16623  df-fg 16624  df-cnfld 16627  df-top 16886  df-bases 16888  df-topon 16889  df-topsp 16890  df-cld 17006  df-ntr 17007  df-cls 17008  df-nei 17085  df-lp 17123  df-perf 17124  df-cn 17213  df-cnp 17214  df-haus 17301  df-tx 17515  df-hmeo 17708  df-fil 17799  df-fm 17891  df-flim 17892  df-flf 17893  df-xms 18259  df-ms 18260  df-tms 18261  df-cncf 18779  df-limc 19620  df-dv 19621  df-log 20321
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