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

Theorem logdivlti 19987
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 12386 . . . . . . . . . . . 12  |-  _e  e.  RR
6 lelttr 8928 . . . . . . . . . . . 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 12501 . . . . . . . . . 10  |-  0  <  _e
11 0re 8854 . . . . . . . . . . . 12  |-  0  e.  RR
12 lttr 8915 . . . . . . . . . . . 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 10404 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  ->  B  e.  RR+ )
18 ltletr 8929 . . . . . . . . . . . 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 10404 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  ->  A  e.  RR+ )
2417, 23rpdivcld 10423 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( B  /  A
)  e.  RR+ )
25 relogcl 19948 . . . . . 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 10433 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( B  /  A
)  e.  RR )
28 1re 8853 . . . . . 6  |-  1  e.  RR
29 resubcl 9127 . . . . . 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 19948 . . . . . . 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 8879 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( ( B  /  A )  - 
1 )  x.  ( log `  A ) )  e.  RR )
34 reeflog 19950 . . . . . . . . 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 8811 . . . . . . . . 9  |-  1  e.  CC
3727recnd 8877 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( B  /  A
)  e.  CC )
38 pncan3 9075 . . . . . . . . 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 2331 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( exp `  ( log `  ( B  /  A ) ) )  =  ( 1  +  ( ( B  /  A )  -  1 ) ) )
414recnd 8877 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  ->  A  e.  CC )
4241mulid2d 8869 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( 1  x.  A
)  =  A )
4342, 3eqbrtrd 4059 . . . . . . . . . 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 9642 . . . . . . . . . . 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 10403 . . . . . . . . . 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 12411 . . . . . . . 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 4059 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( exp `  ( log `  ( B  /  A ) ) )  <  ( exp `  (
( B  /  A
)  -  1 ) ) )
54 eflt 12413 . . . . . . 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 8877 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( B  /  A )  -  1 )  e.  CC )
5857mulid1d 8868 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( ( B  /  A )  - 
1 )  x.  1 )  =  ( ( B  /  A )  -  1 ) )
59 df-e 12366 . . . . . . . . 9  |-  _e  =  ( exp `  1 )
60 reeflog 19950 . . . . . . . . . . 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 4065 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  ->  _e  <_  ( exp `  ( log `  A ) ) )
6359, 62syl5eqbrr 4073 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( exp `  1
)  <_  ( exp `  ( log `  A
) ) )
64 efle 12414 . . . . . . . . 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 9283 . . . . . . . . . 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 9625 . . . . . . . 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 4061 . . . . 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 8992 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( log `  ( B  /  A ) )  <  ( ( ( B  /  A )  -  1 )  x.  ( log `  A
) ) )
75 relogdiv 19962 . . . . 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 8877 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( log `  A
)  e.  CC )
7937, 77, 78subdird 9252 . . . . 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 8877 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  ->  B  e.  CC )
8123rpne0d 10411 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  ->  A  =/=  0 )
8280, 41, 78, 81div32d 9575 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( B  /  A )  x.  ( log `  A ) )  =  ( B  x.  ( ( log `  A
)  /  A ) ) )
8378mulid2d 8869 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( 1  x.  ( log `  A ) )  =  ( log `  A
) )
8482, 83oveq12d 5892 . . . . 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 2328 . . . 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 4068 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( log `  B
)  -  ( log `  A ) )  < 
( ( B  x.  ( ( log `  A
)  /  A ) )  -  ( log `  A ) ) )
87 relogcl 19948 . . . . 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 10433 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( ( log `  A
)  /  A )  e.  RR )
901, 89remulcld 8879 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  -> 
( B  x.  (
( log `  A
)  /  A ) )  e.  RR )
9188, 90, 32ltsub1d 9397 . . 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 10453 . 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 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    < clt 8883    <_ cle 8884    - cmin 9053    / cdiv 9439   RR+crp 10370   expce 12359   _eceu 12360   logclog 19928
This theorem is referenced by:  logdivlt  19988
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ioc 10677  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-fac 11305  df-bc 11332  df-hash 11354  df-shft 11578  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-limsup 11961  df-clim 11978  df-rlim 11979  df-sum 12175  df-ef 12365  df-e 12366  df-sin 12367  df-cos 12368  df-pi 12370  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884  df-perf 16885  df-cn 16973  df-cnp 16974  df-haus 17059  df-tx 17273  df-hmeo 17462  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-xms 17901  df-ms 17902  df-tms 17903  df-cncf 18398  df-limc 19232  df-dv 19233  df-log 19930
  Copyright terms: Public domain W3C validator