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Theorem cxple2 22147
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 1039 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  <  A )  ->  A  e.  RR )
2 simpr 461 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  <  A )  ->  0  <  A )
31, 2elrpd 11030 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  <  A )  ->  A  e.  RR+ )
43adantr 465 . . . 4  |-  ( ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  <  A )  /\  0  <  B )  ->  A  e.  RR+ )
5 simp2l 1014 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  ->  B  e.  RR )
65ad2antrr 725 . . . . 5  |-  ( ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  <  A )  /\  0  <  B )  ->  B  e.  RR )
7 simpr 461 . . . . 5  |-  ( ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  <  A )  /\  0  <  B )  ->  0  <  B )
86, 7elrpd 11030 . . . 4  |-  ( ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  <  A )  /\  0  <  B )  ->  B  e.  RR+ )
9 simp3 990 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  ->  C  e.  RR+ )
109ad2antrr 725 . . . 4  |-  ( ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  <  A )  /\  0  <  B )  ->  C  e.  RR+ )
11 simp3 990 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  C  e.  RR+ )
1211rpred 11032 . . . . . . 7  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  C  e.  RR )
13 relogcl 22032 . . . . . . . 8  |-  ( A  e.  RR+  ->  ( log `  A )  e.  RR )
14133ad2ant1 1009 . . . . . . 7  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  ( log `  A )  e.  RR )
1512, 14remulcld 9419 . . . . . 6  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  ( C  x.  ( log `  A
) )  e.  RR )
16 relogcl 22032 . . . . . . . 8  |-  ( B  e.  RR+  ->  ( log `  B )  e.  RR )
17163ad2ant2 1010 . . . . . . 7  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  ( log `  B )  e.  RR )
1812, 17remulcld 9419 . . . . . 6  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  ( C  x.  ( log `  B
) )  e.  RR )
19 efle 13407 . . . . . 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 661 . . . . 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 13407 . . . . . . 7  |-  ( ( ( log `  A
)  e.  RR  /\  ( log `  B )  e.  RR )  -> 
( ( log `  A
)  <_  ( log `  B )  <->  ( exp `  ( log `  A
) )  <_  ( exp `  ( log `  B
) ) ) )
2214, 17, 21syl2anc 661 . . . . . 6  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  ( ( log `  A )  <_  ( log `  B
)  <->  ( exp `  ( log `  A ) )  <_  ( exp `  ( log `  B ) ) ) )
2314, 17, 11lemul2d 11072 . . . . . 6  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  ( ( log `  A )  <_  ( log `  B
)  <->  ( C  x.  ( log `  A ) )  <_  ( C  x.  ( log `  B
) ) ) )
24 reeflog 22034 . . . . . . . 8  |-  ( A  e.  RR+  ->  ( exp `  ( log `  A
) )  =  A )
25243ad2ant1 1009 . . . . . . 7  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  ( exp `  ( log `  A
) )  =  A )
26 reeflog 22034 . . . . . . . 8  |-  ( B  e.  RR+  ->  ( exp `  ( log `  B
) )  =  B )
27263ad2ant2 1010 . . . . . . 7  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  ( exp `  ( log `  B
) )  =  B )
2825, 27breq12d 4310 . . . . . 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 11002 . . . . . . . . 9  |-  ( A  e.  RR+  ->  A  e.  RR )
31303ad2ant1 1009 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  A  e.  RR )
3231recnd 9417 . . . . . . 7  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  A  e.  CC )
33 rpne0 11011 . . . . . . . 8  |-  ( A  e.  RR+  ->  A  =/=  0 )
34333ad2ant1 1009 . . . . . . 7  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  A  =/=  0 )
3512recnd 9417 . . . . . . 7  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  C  e.  CC )
36 cxpef 22115 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  C  e.  CC )  ->  ( A  ^c  C )  =  ( exp `  ( C  x.  ( log `  A ) ) ) )
3732, 34, 35, 36syl3anc 1218 . . . . . 6  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  ( A  ^c  C )  =  ( exp `  ( C  x.  ( log `  A ) ) ) )
38 rpre 11002 . . . . . . . . 9  |-  ( B  e.  RR+  ->  B  e.  RR )
39383ad2ant2 1010 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  B  e.  RR )
4039recnd 9417 . . . . . . 7  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  B  e.  CC )
41 rpne0 11011 . . . . . . . 8  |-  ( B  e.  RR+  ->  B  =/=  0 )
42413ad2ant2 1010 . . . . . . 7  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  B  =/=  0 )
43 cxpef 22115 . . . . . . 7  |-  ( ( B  e.  CC  /\  B  =/=  0  /\  C  e.  CC )  ->  ( B  ^c  C )  =  ( exp `  ( C  x.  ( log `  B ) ) ) )
4440, 42, 35, 43syl3anc 1218 . . . . . 6  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  ( B  ^c  C )  =  ( exp `  ( C  x.  ( log `  B ) ) ) )
4537, 44breq12d 4310 . . . . 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 1218 . . 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 9391 . . . . . . . 8  |-  0  e.  RR
49 simp1l 1012 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  ->  A  e.  RR )
50 ltnle 9459 . . . . . . . 8  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <  A  <->  -.  A  <_  0 ) )
5148, 49, 50sylancr 663 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  ->  ( 0  <  A  <->  -.  A  <_  0 ) )
5251biimpa 484 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  <  A )  ->  -.  A  <_  0 )
539rpred 11032 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  ->  C  e.  RR )
5453adantr 465 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  <  A )  ->  C  e.  RR )
55 rpcxpcl 22126 . . . . . . . . 9  |-  ( ( A  e.  RR+  /\  C  e.  RR )  ->  ( A  ^c  C )  e.  RR+ )
563, 54, 55syl2anc 661 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  <  A )  ->  ( A  ^c  C )  e.  RR+ )
57 rpgt0 11007 . . . . . . . . 9  |-  ( ( A  ^c  C )  e.  RR+  ->  0  <  ( A  ^c  C ) )
58 rpre 11002 . . . . . . . . . 10  |-  ( ( A  ^c  C )  e.  RR+  ->  ( A  ^c  C )  e.  RR )
59 ltnle 9459 . . . . . . . . . 10  |-  ( ( 0  e.  RR  /\  ( A  ^c  C )  e.  RR )  ->  ( 0  < 
( A  ^c  C )  <->  -.  ( A  ^c  C )  <_  0 ) )
6048, 58, 59sylancr 663 . . . . . . . . 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 9417 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  ->  C  e.  CC )
649rpne0d 11037 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  ->  C  =/=  0 )
65 0cxp 22116 . . . . . . . . . 10  |-  ( ( C  e.  CC  /\  C  =/=  0 )  -> 
( 0  ^c  C )  =  0 )
6663, 64, 65syl2anc 661 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  ->  ( 0  ^c  C )  =  0 )
6766adantr 465 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  <  A )  ->  (
0  ^c  C )  =  0 )
6867breq2d 4309 . . . . . . 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 301 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  <  A )  ->  -.  ( A  ^c  C )  <_  (
0  ^c  C ) )
7052, 692falsed 351 . . . . 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 4301 . . . . . 6  |-  ( 0  =  B  ->  ( A  <_  0  <->  A  <_  B ) )
72 oveq1 6103 . . . . . . 7  |-  ( 0  =  B  ->  (
0  ^c  C )  =  ( B  ^c  C ) )
7372breq2d 4309 . . . . . 6  |-  ( 0  =  B  ->  (
( A  ^c  C )  <_  (
0  ^c  C )  <->  ( A  ^c  C )  <_  ( B  ^c  C ) ) )
7471, 73bibi12d 321 . . . . 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 429 . . 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 1015 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  ->  0  <_  B )
78 leloe 9466 . . . . . 6  |-  ( ( 0  e.  RR  /\  B  e.  RR )  ->  ( 0  <_  B  <->  ( 0  <  B  \/  0  =  B )
) )
7948, 5, 78sylancr 663 . . . . 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 465 . . 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 461 . . . 4  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  =  A )  ->  0  =  A )
84 simpl2r 1042 . . . 4  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  =  A )  ->  0  <_  B )
8583, 84eqbrtrrd 4319 . . 3  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  =  A )  ->  A  <_  B )
8666adantr 465 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  =  A )  ->  (
0  ^c  C )  =  0 )
8783oveq1d 6111 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  =  A )  ->  (
0  ^c  C )  =  ( A  ^c  C ) )
8886, 87eqtr3d 2477 . . . 4  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  =  A )  ->  0  =  ( A  ^c  C ) )
89 simpl2l 1041 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  =  A )  ->  B  e.  RR )
9053adantr 465 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  =  A )  ->  C  e.  RR )
91 cxpge0 22133 . . . . 5  |-  ( ( B  e.  RR  /\  0  <_  B  /\  C  e.  RR )  ->  0  <_  ( B  ^c  C ) )
9289, 84, 90, 91syl3anc 1218 . . . 4  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  =  A )  ->  0  <_  ( B  ^c  C ) )
9388, 92eqbrtrrd 4319 . . 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 1013 . . 3  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  ->  0  <_  A )
96 leloe 9466 . . . 4  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <_  A  <->  ( 0  <  A  \/  0  =  A )
) )
9748, 49, 96sylancr 663 . . 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 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2611   class class class wbr 4297   ` cfv 5423  (class class class)co 6096   CCcc 9285   RRcr 9286   0cc0 9287    x. cmul 9292    < clt 9423    <_ cle 9424   RR+crp 10996   expce 13352   logclog 22011    ^c ccxp 22012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365  ax-addf 9366  ax-mulf 9367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-iin 4179  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-of 6325  df-om 6482  df-1st 6582  df-2nd 6583  df-supp 6696  df-recs 6837  df-rdg 6871  df-1o 6925  df-2o 6926  df-oadd 6929  df-er 7106  df-map 7221  df-pm 7222  df-ixp 7269  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-fsupp 7626  df-fi 7666  df-sup 7696  df-oi 7729  df-card 8114  df-cda 8342  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-10 10393  df-n0 10585  df-z 10652  df-dec 10761  df-uz 10867  df-q 10959  df-rp 10997  df-xneg 11094  df-xadd 11095  df-xmul 11096  df-ioo 11309  df-ioc 11310  df-ico 11311  df-icc 11312  df-fz 11443  df-fzo 11554  df-fl 11647  df-mod 11714  df-seq 11812  df-exp 11871  df-fac 12057  df-bc 12084  df-hash 12109  df-shft 12561  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-limsup 12954  df-clim 12971  df-rlim 12972  df-sum 13169  df-ef 13358  df-sin 13360  df-cos 13361  df-pi 13363  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-mulr 14257  df-starv 14258  df-sca 14259  df-vsca 14260  df-ip 14261  df-tset 14262  df-ple 14263  df-ds 14265  df-unif 14266  df-hom 14267  df-cco 14268  df-rest 14366  df-topn 14367  df-0g 14385  df-gsum 14386  df-topgen 14387  df-pt 14388  df-prds 14391  df-xrs 14445  df-qtop 14450  df-imas 14451  df-xps 14453  df-mre 14529  df-mrc 14530  df-acs 14532  df-mnd 15420  df-submnd 15470  df-mulg 15553  df-cntz 15840  df-cmn 16284  df-psmet 17814  df-xmet 17815  df-met 17816  df-bl 17817  df-mopn 17818  df-fbas 17819  df-fg 17820  df-cnfld 17824  df-top 18508  df-bases 18510  df-topon 18511  df-topsp 18512  df-cld 18628  df-ntr 18629  df-cls 18630  df-nei 18707  df-lp 18745  df-perf 18746  df-cn 18836  df-cnp 18837  df-haus 18924  df-tx 19140  df-hmeo 19333  df-fil 19424  df-fm 19516  df-flim 19517  df-flf 19518  df-xms 19900  df-ms 19901  df-tms 19902  df-cncf 20459  df-limc 21346  df-dv 21347  df-log 22013  df-cxp 22014
This theorem is referenced by:  cxplt2  22148  cxple2a  22149  cxple2d  22177
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