MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cxple2 Unicode version

Theorem cxple2 20060
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 1006 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  <  A )  ->  A  e.  RR )
2 simpr 447 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  <  A )  ->  0  <  A )
31, 2elrpd 10404 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  <  A )  ->  A  e.  RR+ )
43adantr 451 . . . 4  |-  ( ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  <  A )  /\  0  <  B )  ->  A  e.  RR+ )
5 simp2l 981 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  ->  B  e.  RR )
65ad2antrr 706 . . . . 5  |-  ( ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  <  A )  /\  0  <  B )  ->  B  e.  RR )
7 simpr 447 . . . . 5  |-  ( ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  <  A )  /\  0  <  B )  ->  0  <  B )
86, 7elrpd 10404 . . . 4  |-  ( ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  <  A )  /\  0  <  B )  ->  B  e.  RR+ )
9 simp3 957 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  ->  C  e.  RR+ )
109ad2antrr 706 . . . 4  |-  ( ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  <  A )  /\  0  <  B )  ->  C  e.  RR+ )
11 simp3 957 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  C  e.  RR+ )
1211rpred 10406 . . . . . . 7  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  C  e.  RR )
13 relogcl 19948 . . . . . . . 8  |-  ( A  e.  RR+  ->  ( log `  A )  e.  RR )
14133ad2ant1 976 . . . . . . 7  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  ( log `  A )  e.  RR )
1512, 14remulcld 8879 . . . . . 6  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  ( C  x.  ( log `  A
) )  e.  RR )
16 relogcl 19948 . . . . . . . 8  |-  ( B  e.  RR+  ->  ( log `  B )  e.  RR )
17163ad2ant2 977 . . . . . . 7  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  ( log `  B )  e.  RR )
1812, 17remulcld 8879 . . . . . 6  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  ( C  x.  ( log `  B
) )  e.  RR )
19 efle 12414 . . . . . 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 642 . . . . 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 12414 . . . . . . 7  |-  ( ( ( log `  A
)  e.  RR  /\  ( log `  B )  e.  RR )  -> 
( ( log `  A
)  <_  ( log `  B )  <->  ( exp `  ( log `  A
) )  <_  ( exp `  ( log `  B
) ) ) )
2214, 17, 21syl2anc 642 . . . . . 6  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  ( ( log `  A )  <_  ( log `  B
)  <->  ( exp `  ( log `  A ) )  <_  ( exp `  ( log `  B ) ) ) )
2314, 17, 11lemul2d 10446 . . . . . 6  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  ( ( log `  A )  <_  ( log `  B
)  <->  ( C  x.  ( log `  A ) )  <_  ( C  x.  ( log `  B
) ) ) )
24 reeflog 19950 . . . . . . . 8  |-  ( A  e.  RR+  ->  ( exp `  ( log `  A
) )  =  A )
25243ad2ant1 976 . . . . . . 7  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  ( exp `  ( log `  A
) )  =  A )
26 reeflog 19950 . . . . . . . 8  |-  ( B  e.  RR+  ->  ( exp `  ( log `  B
) )  =  B )
27263ad2ant2 977 . . . . . . 7  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  ( exp `  ( log `  B
) )  =  B )
2825, 27breq12d 4052 . . . . . 6  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  ( ( exp `  ( log `  A ) )  <_ 
( exp `  ( log `  B ) )  <-> 
A  <_  B )
)
2922, 23, 283bitr3rd 275 . . . . 5  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  ( A  <_  B  <->  ( C  x.  ( log `  A
) )  <_  ( C  x.  ( log `  B ) ) ) )
30 rpre 10376 . . . . . . . . 9  |-  ( A  e.  RR+  ->  A  e.  RR )
31303ad2ant1 976 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  A  e.  RR )
3231recnd 8877 . . . . . . 7  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  A  e.  CC )
33 rpne0 10385 . . . . . . . 8  |-  ( A  e.  RR+  ->  A  =/=  0 )
34333ad2ant1 976 . . . . . . 7  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  A  =/=  0 )
3512recnd 8877 . . . . . . 7  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  C  e.  CC )
36 cxpef 20028 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  C  e.  CC )  ->  ( A  ^ c  C )  =  ( exp `  ( C  x.  ( log `  A ) ) ) )
3732, 34, 35, 36syl3anc 1182 . . . . . 6  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  ( A  ^ c  C )  =  ( exp `  ( C  x.  ( log `  A ) ) ) )
38 rpre 10376 . . . . . . . . 9  |-  ( B  e.  RR+  ->  B  e.  RR )
39383ad2ant2 977 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  B  e.  RR )
4039recnd 8877 . . . . . . 7  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  B  e.  CC )
41 rpne0 10385 . . . . . . . 8  |-  ( B  e.  RR+  ->  B  =/=  0 )
42413ad2ant2 977 . . . . . . 7  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  B  =/=  0 )
43 cxpef 20028 . . . . . . 7  |-  ( ( B  e.  CC  /\  B  =/=  0  /\  C  e.  CC )  ->  ( B  ^ c  C )  =  ( exp `  ( C  x.  ( log `  B ) ) ) )
4440, 42, 35, 43syl3anc 1182 . . . . . 6  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  ( B  ^ c  C )  =  ( exp `  ( C  x.  ( log `  B ) ) ) )
4537, 44breq12d 4052 . . . . 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 276 . . . 4  |-  ( ( A  e.  RR+  /\  B  e.  RR+  /\  C  e.  RR+ )  ->  ( A  <_  B  <->  ( A  ^ c  C )  <_  ( B  ^ c  C ) ) )
474, 8, 10, 46syl3anc 1182 . . 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 8854 . . . . . . . 8  |-  0  e.  RR
49 simp1l 979 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  ->  A  e.  RR )
50 ltnle 8918 . . . . . . . 8  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <  A  <->  -.  A  <_  0 ) )
5148, 49, 50sylancr 644 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  ->  ( 0  <  A  <->  -.  A  <_  0 ) )
5251biimpa 470 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  <  A )  ->  -.  A  <_  0 )
539rpred 10406 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  ->  C  e.  RR )
5453adantr 451 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  <  A )  ->  C  e.  RR )
55 rpcxpcl 20039 . . . . . . . . 9  |-  ( ( A  e.  RR+  /\  C  e.  RR )  ->  ( A  ^ c  C )  e.  RR+ )
563, 54, 55syl2anc 642 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  <  A )  ->  ( A  ^ c  C )  e.  RR+ )
57 rpgt0 10381 . . . . . . . . 9  |-  ( ( A  ^ c  C
)  e.  RR+  ->  0  <  ( A  ^ c  C ) )
58 rpre 10376 . . . . . . . . . 10  |-  ( ( A  ^ c  C
)  e.  RR+  ->  ( A  ^ c  C
)  e.  RR )
59 ltnle 8918 . . . . . . . . . 10  |-  ( ( 0  e.  RR  /\  ( A  ^ c  C )  e.  RR )  ->  ( 0  < 
( A  ^ c  C )  <->  -.  ( A  ^ c  C )  <_  0 ) )
6048, 58, 59sylancr 644 . . . . . . . . 9  |-  ( ( A  ^ c  C
)  e.  RR+  ->  ( 0  <  ( A  ^ c  C )  <->  -.  ( A  ^ c  C )  <_  0
) )
6157, 60mpbid 201 . . . . . . . 8  |-  ( ( A  ^ c  C
)  e.  RR+  ->  -.  ( A  ^ c  C )  <_  0
)
6256, 61syl 15 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  <  A )  ->  -.  ( A  ^ c  C )  <_  0
)
6353recnd 8877 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  ->  C  e.  CC )
649rpne0d 10411 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  ->  C  =/=  0 )
65 0cxp 20029 . . . . . . . . . 10  |-  ( ( C  e.  CC  /\  C  =/=  0 )  -> 
( 0  ^ c  C )  =  0 )
6663, 64, 65syl2anc 642 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  ->  ( 0  ^ c  C )  =  0 )
6766adantr 451 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  <  A )  ->  (
0  ^ c  C
)  =  0 )
6867breq2d 4051 . . . . . . 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 292 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  <  A )  ->  -.  ( A  ^ c  C )  <_  (
0  ^ c  C
) )
7052, 692falsed 340 . . . . 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 4043 . . . . . 6  |-  ( 0  =  B  ->  ( A  <_  0  <->  A  <_  B ) )
72 oveq1 5881 . . . . . . 7  |-  ( 0  =  B  ->  (
0  ^ c  C
)  =  ( B  ^ c  C ) )
7372breq2d 4051 . . . . . 6  |-  ( 0  =  B  ->  (
( A  ^ c  C )  <_  (
0  ^ c  C
)  <->  ( A  ^ c  C )  <_  ( B  ^ c  C ) ) )
7471, 73bibi12d 312 . . . . 5  |-  ( 0  =  B  ->  (
( A  <_  0  <->  ( A  ^ c  C
)  <_  ( 0  ^ c  C ) )  <->  ( A  <_  B 
<->  ( A  ^ c  C )  <_  ( B  ^ c  C ) ) ) )
7570, 74syl5ibcom 211 . . . 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 418 . . 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 982 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  ->  0  <_  B )
78 leloe 8924 . . . . . 6  |-  ( ( 0  e.  RR  /\  B  e.  RR )  ->  ( 0  <_  B  <->  ( 0  <  B  \/  0  =  B )
) )
7948, 5, 78sylancr 644 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  ->  ( 0  <_  B  <->  ( 0  <  B  \/  0  =  B )
) )
8077, 79mpbid 201 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  ->  ( 0  <  B  \/  0  =  B
) )
8180adantr 451 . . 3  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  <  A )  ->  (
0  <  B  \/  0  =  B )
)
8247, 76, 81mpjaodan 761 . 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 447 . . . 4  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  =  A )  ->  0  =  A )
84 simpl2r 1009 . . . 4  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  =  A )  ->  0  <_  B )
8583, 84eqbrtrrd 4061 . . 3  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  =  A )  ->  A  <_  B )
8666adantr 451 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  =  A )  ->  (
0  ^ c  C
)  =  0 )
8783oveq1d 5889 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  =  A )  ->  (
0  ^ c  C
)  =  ( A  ^ c  C ) )
8886, 87eqtr3d 2330 . . . 4  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  =  A )  ->  0  =  ( A  ^ c  C ) )
89 simpl2l 1008 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  =  A )  ->  B  e.  RR )
9053adantr 451 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  =  A )  ->  C  e.  RR )
91 cxpge0 20046 . . . . 5  |-  ( ( B  e.  RR  /\  0  <_  B  /\  C  e.  RR )  ->  0  <_  ( B  ^ c  C ) )
9289, 84, 90, 91syl3anc 1182 . . . 4  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  =  A )  ->  0  <_  ( B  ^ c  C ) )
9388, 92eqbrtrrd 4061 . . 3  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  /\  0  =  A )  ->  ( A  ^ c  C )  <_  ( B  ^ c  C ) )
9485, 932thd 231 . 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 980 . . 3  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  ->  0  <_  A )
96 leloe 8924 . . . 4  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <_  A  <->  ( 0  <  A  \/  0  =  A )
) )
9748, 49, 96sylancr 644 . . 3  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  ->  ( 0  <_  A  <->  ( 0  <  A  \/  0  =  A )
) )
9895, 97mpbid 201 . 2  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  ->  ( 0  <  A  \/  0  =  A
) )
9982, 94, 98mpjaodan 761 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753    x. cmul 8758    < clt 8883    <_ cle 8884   RR+crp 10370   expce 12359   logclog 19928    ^ c ccxp 19929
This theorem is referenced by:  cxplt2  20061  cxple2a  20062  cxple2d  20090
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-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  df-cxp 19931
  Copyright terms: Public domain W3C validator