MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  logdivlti Structured version   Unicode version

Theorem logdivlti 20507
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 12683 . . . . . . . . . . . 12  |-  _e  e.  RR
6 lelttr 9157 . . . . . . . . . . . 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 12798 . . . . . . . . . 10  |-  0  <  _e
11 0re 9083 . . . . . . . . . . . 12  |-  0  e.  RR
12 lttr 9144 . . . . . . . . . . . 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 10638 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  ->  B  e.  RR+ )
18 ltletr 9158 . . . . . . . . . . . 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 10638 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  ->  A  e.  RR+ )
2417, 23rpdivcld 10657 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( B  /  A
)  e.  RR+ )
25 relogcl 20465 . . . . . 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 10667 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( B  /  A
)  e.  RR )
28 1re 9082 . . . . . 6  |-  1  e.  RR
29 resubcl 9357 . . . . . 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 20465 . . . . . . 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 9108 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( ( B  /  A )  - 
1 )  x.  ( log `  A ) )  e.  RR )
34 reeflog 20467 . . . . . . . . 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 9040 . . . . . . . . 9  |-  1  e.  CC
3727recnd 9106 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( B  /  A
)  e.  CC )
38 pncan3 9305 . . . . . . . . 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 2470 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( exp `  ( log `  ( B  /  A ) ) )  =  ( 1  +  ( ( B  /  A )  -  1 ) ) )
414recnd 9106 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  ->  A  e.  CC )
4241mulid2d 9098 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( 1  x.  A
)  =  A )
4342, 3eqbrtrd 4224 . . . . . . . . . 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 9872 . . . . . . . . . . 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 10637 . . . . . . . . . 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 12708 . . . . . . . 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 4224 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( exp `  ( log `  ( B  /  A ) ) )  <  ( exp `  (
( B  /  A
)  -  1 ) ) )
54 eflt 12710 . . . . . . 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 9106 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( B  /  A )  -  1 )  e.  CC )
5857mulid1d 9097 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( ( B  /  A )  - 
1 )  x.  1 )  =  ( ( B  /  A )  -  1 ) )
59 df-e 12663 . . . . . . . . 9  |-  _e  =  ( exp `  1 )
60 reeflog 20467 . . . . . . . . . . 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 4230 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  ->  _e  <_  ( exp `  ( log `  A ) ) )
6359, 62syl5eqbrr 4238 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( exp `  1
)  <_  ( exp `  ( log `  A
) ) )
64 efle 12711 . . . . . . . . 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 9513 . . . . . . . . . 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 9855 . . . . . . . 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 4226 . . . . 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 9222 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( log `  ( B  /  A ) )  <  ( ( ( B  /  A )  -  1 )  x.  ( log `  A
) ) )
75 relogdiv 20479 . . . . 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 9106 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( log `  A
)  e.  CC )
7937, 77, 78subdird 9482 . . . . 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 9106 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  ->  B  e.  CC )
8123rpne0d 10645 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  ->  A  =/=  0 )
8280, 41, 78, 81div32d 9805 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( B  /  A )  x.  ( log `  A ) )  =  ( B  x.  ( ( log `  A
)  /  A ) ) )
8378mulid2d 9098 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( 1  x.  ( log `  A ) )  =  ( log `  A
) )
8482, 83oveq12d 6091 . . . . 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 2467 . . . 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 4233 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( log `  B
)  -  ( log `  A ) )  < 
( ( B  x.  ( ( log `  A
)  /  A ) )  -  ( log `  A ) ) )
87 relogcl 20465 . . . . 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 10667 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( log `  A
)  /  A )  e.  RR )
901, 89remulcld 9108 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( B  x.  (
( log `  A
)  /  A ) )  e.  RR )
9188, 90, 32ltsub1d 9627 . . 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 10687 . 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 1652    e. wcel 1725   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   CCcc 8980   RRcr 8981   0cc0 8982   1c1 8983    + caddc 8985    x. cmul 8987    < clt 9112    <_ cle 9113    - cmin 9283    / cdiv 9669   RR+crp 10604   expce 12656   _eceu 12657   logclog 20444
This theorem is referenced by:  logdivlt  20508
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061  ax-mulf 9062
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-fi 7408  df-sup 7438  df-oi 7471  df-card 7818  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-q 10567  df-rp 10605  df-xneg 10702  df-xadd 10703  df-xmul 10704  df-ioo 10912  df-ioc 10913  df-ico 10914  df-icc 10915  df-fz 11036  df-fzo 11128  df-fl 11194  df-mod 11243  df-seq 11316  df-exp 11375  df-fac 11559  df-bc 11586  df-hash 11611  df-shft 11874  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-limsup 12257  df-clim 12274  df-rlim 12275  df-sum 12472  df-ef 12662  df-e 12663  df-sin 12664  df-cos 12665  df-pi 12667  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-starv 13536  df-sca 13537  df-vsca 13538  df-tset 13540  df-ple 13541  df-ds 13543  df-unif 13544  df-hom 13545  df-cco 13546  df-rest 13642  df-topn 13643  df-topgen 13659  df-pt 13660  df-prds 13663  df-xrs 13718  df-0g 13719  df-gsum 13720  df-qtop 13725  df-imas 13726  df-xps 13728  df-mre 13803  df-mrc 13804  df-acs 13806  df-mnd 14682  df-submnd 14731  df-mulg 14807  df-cntz 15108  df-cmn 15406  df-psmet 16686  df-xmet 16687  df-met 16688  df-bl 16689  df-mopn 16690  df-fbas 16691  df-fg 16692  df-cnfld 16696  df-top 16955  df-bases 16957  df-topon 16958  df-topsp 16959  df-cld 17075  df-ntr 17076  df-cls 17077  df-nei 17154  df-lp 17192  df-perf 17193  df-cn 17283  df-cnp 17284  df-haus 17371  df-tx 17586  df-hmeo 17779  df-fil 17870  df-fm 17962  df-flim 17963  df-flf 17964  df-xms 18342  df-ms 18343  df-tms 18344  df-cncf 18900  df-limc 19745  df-dv 19746  df-log 20446
  Copyright terms: Public domain W3C validator