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Theorem cxple2 23249
Description: Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 8-Sep-2014.)
Assertion
Ref Expression
cxple2  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  ->  ( A  <_  B  <->  ( A  ^c  C )  <_  ( B  ^c  C )
) )

Proof of Theorem cxple2
StepHypRef Expression
1 simpl1l 1045 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  <  A )  ->  A  e.  RR )
2 simpr 459 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  <  A )  ->  0  <  A )
31, 2elrpd 11256 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  <  A )  ->  A  e.  RR+ )
43adantr 463 . . . 4  |-  ( ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  <  A )  /\  0  <  B )  ->  A  e.  RR+ )
5 simp2l 1020 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  ->  B  e.  RR )
65ad2antrr 723 . . . . 5  |-  ( ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  <  A )  /\  0  <  B )  ->  B  e.  RR )
7 simpr 459 . . . . 5  |-  ( ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  <  A )  /\  0  <  B )  ->  0  <  B )
86, 7elrpd 11256 . . . 4  |-  ( ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  <  A )  /\  0  <  B )  ->  B  e.  RR+ )
9 simp3 996 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  ->  C  e.  RR+ )
109ad2antrr 723 . . . 4  |-  ( ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  <  A )  /\  0  <  B )  ->  C  e.  RR+ )
11 simp3 996 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  C  e.  RR+ )
1211rpred 11259 . . . . . . 7  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  C  e.  RR )
13 relogcl 23132 . . . . . . . 8  |-  ( A  e.  RR+  ->  ( log `  A )  e.  RR )
14133ad2ant1 1015 . . . . . . 7  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  ( log `  A )  e.  RR )
1512, 14remulcld 9613 . . . . . 6  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  ( C  x.  ( log `  A
) )  e.  RR )
16 relogcl 23132 . . . . . . . 8  |-  ( B  e.  RR+  ->  ( log `  B )  e.  RR )
17163ad2ant2 1016 . . . . . . 7  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  ( log `  B )  e.  RR )
1812, 17remulcld 9613 . . . . . 6  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  ( C  x.  ( log `  B
) )  e.  RR )
19 efle 13938 . . . . . 6  |-  ( ( ( C  x.  ( log `  A ) )  e.  RR  /\  ( C  x.  ( log `  B ) )  e.  RR )  ->  (
( C  x.  ( log `  A ) )  <_  ( C  x.  ( log `  B ) )  <->  ( exp `  ( C  x.  ( log `  A ) ) )  <_  ( exp `  ( C  x.  ( log `  B ) ) ) ) )
2015, 18, 19syl2anc 659 . . . . 5  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  ( ( C  x.  ( log `  A ) )  <_ 
( C  x.  ( log `  B ) )  <-> 
( exp `  ( C  x.  ( log `  A ) ) )  <_  ( exp `  ( C  x.  ( log `  B ) ) ) ) )
21 efle 13938 . . . . . . 7  |-  ( ( ( log `  A
)  e.  RR  /\  ( log `  B )  e.  RR )  -> 
( ( log `  A
)  <_  ( log `  B )  <->  ( exp `  ( log `  A
) )  <_  ( exp `  ( log `  B
) ) ) )
2214, 17, 21syl2anc 659 . . . . . 6  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  ( ( log `  A )  <_  ( log `  B
)  <->  ( exp `  ( log `  A ) )  <_  ( exp `  ( log `  B ) ) ) )
2314, 17, 11lemul2d 11299 . . . . . 6  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  ( ( log `  A )  <_  ( log `  B
)  <->  ( C  x.  ( log `  A ) )  <_  ( C  x.  ( log `  B
) ) ) )
24 reeflog 23137 . . . . . . . 8  |-  ( A  e.  RR+  ->  ( exp `  ( log `  A
) )  =  A )
25243ad2ant1 1015 . . . . . . 7  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  ( exp `  ( log `  A
) )  =  A )
26 reeflog 23137 . . . . . . . 8  |-  ( B  e.  RR+  ->  ( exp `  ( log `  B
) )  =  B )
27263ad2ant2 1016 . . . . . . 7  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  ( exp `  ( log `  B
) )  =  B )
2825, 27breq12d 4452 . . . . . 6  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  ( ( exp `  ( log `  A ) )  <_ 
( exp `  ( log `  B ) )  <-> 
A  <_  B )
)
2922, 23, 283bitr3rd 284 . . . . 5  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  ( A  <_  B  <->  ( C  x.  ( log `  A
) )  <_  ( C  x.  ( log `  B ) ) ) )
30 rpre 11227 . . . . . . . . 9  |-  ( A  e.  RR+  ->  A  e.  RR )
31303ad2ant1 1015 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  A  e.  RR )
3231recnd 9611 . . . . . . 7  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  A  e.  CC )
33 rpne0 11236 . . . . . . . 8  |-  ( A  e.  RR+  ->  A  =/=  0 )
34333ad2ant1 1015 . . . . . . 7  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  A  =/=  0 )
3512recnd 9611 . . . . . . 7  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  C  e.  CC )
36 cxpef 23217 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  C  e.  CC )  ->  ( A  ^c  C )  =  ( exp `  ( C  x.  ( log `  A ) ) ) )
3732, 34, 35, 36syl3anc 1226 . . . . . 6  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  ( A  ^c  C )  =  ( exp `  ( C  x.  ( log `  A ) ) ) )
38 rpre 11227 . . . . . . . . 9  |-  ( B  e.  RR+  ->  B  e.  RR )
39383ad2ant2 1016 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  B  e.  RR )
4039recnd 9611 . . . . . . 7  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  B  e.  CC )
41 rpne0 11236 . . . . . . . 8  |-  ( B  e.  RR+  ->  B  =/=  0 )
42413ad2ant2 1016 . . . . . . 7  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  B  =/=  0 )
43 cxpef 23217 . . . . . . 7  |-  ( ( B  e.  CC  /\  B  =/=  0  /\  C  e.  CC )  ->  ( B  ^c  C )  =  ( exp `  ( C  x.  ( log `  B ) ) ) )
4440, 42, 35, 43syl3anc 1226 . . . . . 6  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  ( B  ^c  C )  =  ( exp `  ( C  x.  ( log `  B ) ) ) )
4537, 44breq12d 4452 . . . . 5  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  ( ( A  ^c  C )  <_  ( B  ^c  C )  <->  ( exp `  ( C  x.  ( log `  A
) ) )  <_ 
( exp `  ( C  x.  ( log `  B ) ) ) ) )
4620, 29, 453bitr4d 285 . . . 4  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  ( A  <_  B  <->  ( A  ^c  C )  <_  ( B  ^c  C ) ) )
474, 8, 10, 46syl3anc 1226 . . 3  |-  ( ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  <  A )  /\  0  <  B )  ->  ( A  <_  B  <->  ( A  ^c  C )  <_  ( B  ^c  C ) ) )
48 0re 9585 . . . . . . . 8  |-  0  e.  RR
49 simp1l 1018 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  ->  A  e.  RR )
50 ltnle 9653 . . . . . . . 8  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <  A  <->  -.  A  <_  0 ) )
5148, 49, 50sylancr 661 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  ->  ( 0  <  A  <->  -.  A  <_  0 ) )
5251biimpa 482 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  <  A )  ->  -.  A  <_  0 )
539rpred 11259 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  ->  C  e.  RR )
5453adantr 463 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  <  A )  ->  C  e.  RR )
55 rpcxpcl 23228 . . . . . . . . 9  |-  ( ( A  e.  RR+  /\  C  e.  RR )  ->  ( A  ^c  C )  e.  RR+ )
563, 54, 55syl2anc 659 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  <  A )  ->  ( A  ^c  C )  e.  RR+ )
57 rpgt0 11232 . . . . . . . . 9  |-  ( ( A  ^c  C )  e.  RR+  ->  0  <  ( A  ^c  C ) )
58 rpre 11227 . . . . . . . . . 10  |-  ( ( A  ^c  C )  e.  RR+  ->  ( A  ^c  C )  e.  RR )
59 ltnle 9653 . . . . . . . . . 10  |-  ( ( 0  e.  RR  /\  ( A  ^c  C )  e.  RR )  ->  ( 0  < 
( A  ^c  C )  <->  -.  ( A  ^c  C )  <_  0 ) )
6048, 58, 59sylancr 661 . . . . . . . . 9  |-  ( ( A  ^c  C )  e.  RR+  ->  ( 0  <  ( A  ^c  C )  <->  -.  ( A  ^c  C )  <_  0
) )
6157, 60mpbid 210 . . . . . . . 8  |-  ( ( A  ^c  C )  e.  RR+  ->  -.  ( A  ^c  C )  <_  0
)
6256, 61syl 16 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  <  A )  ->  -.  ( A  ^c  C )  <_  0
)
6353recnd 9611 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  ->  C  e.  CC )
649rpne0d 11264 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  ->  C  =/=  0 )
65 0cxp 23218 . . . . . . . . . 10  |-  ( ( C  e.  CC  /\  C  =/=  0 )  -> 
( 0  ^c  C )  =  0 )
6663, 64, 65syl2anc 659 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  ->  ( 0  ^c  C )  =  0 )
6766adantr 463 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  <  A )  ->  (
0  ^c  C )  =  0 )
6867breq2d 4451 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  <  A )  ->  (
( A  ^c  C )  <_  (
0  ^c  C )  <->  ( A  ^c  C )  <_  0
) )
6962, 68mtbird 299 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  <  A )  ->  -.  ( A  ^c  C )  <_  (
0  ^c  C ) )
7052, 692falsed 349 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  <  A )  ->  ( A  <_  0  <->  ( A  ^c  C )  <_  ( 0  ^c  C ) ) )
71 breq2 4443 . . . . . 6  |-  ( 0  =  B  ->  ( A  <_  0  <->  A  <_  B ) )
72 oveq1 6277 . . . . . . 7  |-  ( 0  =  B  ->  (
0  ^c  C )  =  ( B  ^c  C ) )
7372breq2d 4451 . . . . . 6  |-  ( 0  =  B  ->  (
( A  ^c  C )  <_  (
0  ^c  C )  <->  ( A  ^c  C )  <_  ( B  ^c  C ) ) )
7471, 73bibi12d 319 . . . . 5  |-  ( 0  =  B  ->  (
( A  <_  0  <->  ( A  ^c  C )  <_  ( 0  ^c  C ) )  <->  ( A  <_  B 
<->  ( A  ^c  C )  <_  ( B  ^c  C ) ) ) )
7570, 74syl5ibcom 220 . . . 4  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  <  A )  ->  (
0  =  B  -> 
( A  <_  B  <->  ( A  ^c  C )  <_  ( B  ^c  C )
) ) )
7675imp 427 . . 3  |-  ( ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  <  A )  /\  0  =  B )  ->  ( A  <_  B  <->  ( A  ^c  C )  <_  ( B  ^c  C ) ) )
77 simp2r 1021 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  ->  0  <_  B )
78 leloe 9660 . . . . . 6  |-  ( ( 0  e.  RR  /\  B  e.  RR )  ->  ( 0  <_  B  <->  ( 0  <  B  \/  0  =  B )
) )
7948, 5, 78sylancr 661 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  ->  ( 0  <_  B  <->  ( 0  <  B  \/  0  =  B )
) )
8077, 79mpbid 210 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  ->  ( 0  <  B  \/  0  =  B
) )
8180adantr 463 . . 3  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  <  A )  ->  (
0  <  B  \/  0  =  B )
)
8247, 76, 81mpjaodan 784 . 2  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  <  A )  ->  ( A  <_  B  <->  ( A  ^c  C )  <_  ( B  ^c  C ) ) )
83 simpr 459 . . . 4  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  =  A )  ->  0  =  A )
84 simpl2r 1048 . . . 4  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  =  A )  ->  0  <_  B )
8583, 84eqbrtrrd 4461 . . 3  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  =  A )  ->  A  <_  B )
8666adantr 463 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  =  A )  ->  (
0  ^c  C )  =  0 )
8783oveq1d 6285 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  =  A )  ->  (
0  ^c  C )  =  ( A  ^c  C ) )
8886, 87eqtr3d 2497 . . . 4  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  =  A )  ->  0  =  ( A  ^c  C ) )
89 simpl2l 1047 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  =  A )  ->  B  e.  RR )
9053adantr 463 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  =  A )  ->  C  e.  RR )
91 cxpge0 23235 . . . . 5  |-  ( ( B  e.  RR  /\  0  <_  B  /\  C  e.  RR )  ->  0  <_  ( B  ^c  C ) )
9289, 84, 90, 91syl3anc 1226 . . . 4  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  =  A )  ->  0  <_  ( B  ^c  C ) )
9388, 92eqbrtrrd 4461 . . 3  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  =  A )  ->  ( A  ^c  C )  <_  ( B  ^c  C ) )
9485, 932thd 240 . 2  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  =  A )  ->  ( A  <_  B  <->  ( A  ^c  C )  <_  ( B  ^c  C ) ) )
95 simp1r 1019 . . 3  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  ->  0  <_  A )
96 leloe 9660 . . . 4  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <_  A  <->  ( 0  <  A  \/  0  =  A )
) )
9748, 49, 96sylancr 661 . . 3  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  ->  ( 0  <_  A  <->  ( 0  <  A  \/  0  =  A )
) )
9895, 97mpbid 210 . 2  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  ->  ( 0  <  A  \/  0  =  A
) )
9982, 94, 98mpjaodan 784 1  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  ->  ( A  <_  B  <->  ( A  ^c  C )  <_  ( B  ^c  C )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   CCcc 9479   RRcr 9480   0cc0 9481    x. cmul 9486    < clt 9617    <_ cle 9618   RR+crp 11221   expce 13882   logclog 23111    ^c ccxp 23112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-fi 7863  df-sup 7893  df-oi 7927  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-ioo 11536  df-ioc 11537  df-ico 11538  df-icc 11539  df-fz 11676  df-fzo 11800  df-fl 11910  df-mod 11979  df-seq 12093  df-exp 12152  df-fac 12339  df-bc 12366  df-hash 12391  df-shft 12985  df-cj 13017  df-re 13018  df-im 13019  df-sqrt 13153  df-abs 13154  df-limsup 13379  df-clim 13396  df-rlim 13397  df-sum 13594  df-ef 13888  df-sin 13890  df-cos 13891  df-pi 13893  df-struct 14721  df-ndx 14722  df-slot 14723  df-base 14724  df-sets 14725  df-ress 14726  df-plusg 14800  df-mulr 14801  df-starv 14802  df-sca 14803  df-vsca 14804  df-ip 14805  df-tset 14806  df-ple 14807  df-ds 14809  df-unif 14810  df-hom 14811  df-cco 14812  df-rest 14915  df-topn 14916  df-0g 14934  df-gsum 14935  df-topgen 14936  df-pt 14937  df-prds 14940  df-xrs 14994  df-qtop 14999  df-imas 15000  df-xps 15002  df-mre 15078  df-mrc 15079  df-acs 15081  df-mgm 16074  df-sgrp 16113  df-mnd 16123  df-submnd 16169  df-mulg 16262  df-cntz 16557  df-cmn 17002  df-psmet 18609  df-xmet 18610  df-met 18611  df-bl 18612  df-mopn 18613  df-fbas 18614  df-fg 18615  df-cnfld 18619  df-top 19569  df-bases 19571  df-topon 19572  df-topsp 19573  df-cld 19690  df-ntr 19691  df-cls 19692  df-nei 19769  df-lp 19807  df-perf 19808  df-cn 19898  df-cnp 19899  df-haus 19986  df-tx 20232  df-hmeo 20425  df-fil 20516  df-fm 20608  df-flim 20609  df-flf 20610  df-xms 20992  df-ms 20993  df-tms 20994  df-cncf 21551  df-limc 22439  df-dv 22440  df-log 23113  df-cxp 23114
This theorem is referenced by:  cxplt2  23250  cxple2a  23251  cxple2d  23279
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