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Theorem cxpsqrt 22950
Description: The complex exponential function with exponent  1  /  2 exactly matches the complex square root function (the branch cut is in the same place for both functions), and thus serves as a suitable generalization to other  n-th roots and irrational roots. (Contributed by Mario Carneiro, 2-Aug-2014.)
Assertion
Ref Expression
cxpsqrt  |-  ( A  e.  CC  ->  ( A  ^c  ( 1  /  2 ) )  =  ( sqr `  A
) )

Proof of Theorem cxpsqrt
StepHypRef Expression
1 halfcn 10767 . . . . . 6  |-  ( 1  /  2 )  e.  CC
2 halfre 10766 . . . . . . 7  |-  ( 1  /  2 )  e.  RR
3 halfgt0 10768 . . . . . . 7  |-  0  <  ( 1  /  2
)
42, 3gt0ne0ii 10101 . . . . . 6  |-  ( 1  /  2 )  =/=  0
5 0cxp 22913 . . . . . 6  |-  ( ( ( 1  /  2
)  e.  CC  /\  ( 1  /  2
)  =/=  0 )  ->  ( 0  ^c  ( 1  / 
2 ) )  =  0 )
61, 4, 5mp2an 672 . . . . 5  |-  ( 0  ^c  ( 1  /  2 ) )  =  0
7 sqrt0 13055 . . . . 5  |-  ( sqr `  0 )  =  0
86, 7eqtr4i 2499 . . . 4  |-  ( 0  ^c  ( 1  /  2 ) )  =  ( sqr `  0
)
9 oveq1 6302 . . . 4  |-  ( A  =  0  ->  ( A  ^c  ( 1  /  2 ) )  =  ( 0  ^c  ( 1  / 
2 ) ) )
10 fveq2 5872 . . . 4  |-  ( A  =  0  ->  ( sqr `  A )  =  ( sqr `  0
) )
118, 9, 103eqtr4a 2534 . . 3  |-  ( A  =  0  ->  ( A  ^c  ( 1  /  2 ) )  =  ( sqr `  A
) )
1211a1i 11 . 2  |-  ( A  e.  CC  ->  ( A  =  0  ->  ( A  ^c  ( 1  /  2 ) )  =  ( sqr `  A ) ) )
13 ax-icn 9563 . . . . . . . . . . . . . . . . 17  |-  _i  e.  CC
14 sqrtcl 13174 . . . . . . . . . . . . . . . . . 18  |-  ( A  e.  CC  ->  ( sqr `  A )  e.  CC )
1514ad2antrr 725 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( sqr `  A )  e.  CC )
16 sqmul 12211 . . . . . . . . . . . . . . . . 17  |-  ( ( _i  e.  CC  /\  ( sqr `  A )  e.  CC )  -> 
( ( _i  x.  ( sqr `  A ) ) ^ 2 )  =  ( ( _i
^ 2 )  x.  ( ( sqr `  A
) ^ 2 ) ) )
1713, 15, 16sylancr 663 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
( _i  x.  ( sqr `  A ) ) ^ 2 )  =  ( ( _i ^
2 )  x.  (
( sqr `  A
) ^ 2 ) ) )
18 i2 12248 . . . . . . . . . . . . . . . . . 18  |-  ( _i
^ 2 )  = 
-u 1
1918a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
_i ^ 2 )  =  -u 1 )
20 sqrtth 13177 . . . . . . . . . . . . . . . . . 18  |-  ( A  e.  CC  ->  (
( sqr `  A
) ^ 2 )  =  A )
2120ad2antrr 725 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
( sqr `  A
) ^ 2 )  =  A )
2219, 21oveq12d 6313 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
( _i ^ 2 )  x.  ( ( sqr `  A ) ^ 2 ) )  =  ( -u 1  x.  A ) )
23 mulm1 10010 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  CC  ->  ( -u 1  x.  A )  =  -u A )
2423ad2antrr 725 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( -u 1  x.  A )  =  -u A )
2517, 22, 243eqtrd 2512 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
( _i  x.  ( sqr `  A ) ) ^ 2 )  = 
-u A )
26 cxpsqrtlem 22949 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
_i  x.  ( sqr `  A ) )  e.  RR )
2726resqcld 12316 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
( _i  x.  ( sqr `  A ) ) ^ 2 )  e.  RR )
2825, 27eqeltrrd 2556 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  -u A  e.  RR )
29 negeq0 9885 . . . . . . . . . . . . . . . . . . . . 21  |-  ( A  e.  CC  ->  ( A  =  0  <->  -u A  =  0 ) )
3029necon3bid 2725 . . . . . . . . . . . . . . . . . . . 20  |-  ( A  e.  CC  ->  ( A  =/=  0  <->  -u A  =/=  0 ) )
3130biimpa 484 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  -u A  =/=  0 )
3231adantr 465 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  -u A  =/=  0 )
3325, 32eqnetrd 2760 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
( _i  x.  ( sqr `  A ) ) ^ 2 )  =/=  0 )
34 sq0i 12240 . . . . . . . . . . . . . . . . . 18  |-  ( ( _i  x.  ( sqr `  A ) )  =  0  ->  ( (
_i  x.  ( sqr `  A ) ) ^
2 )  =  0 )
3534necon3i 2707 . . . . . . . . . . . . . . . . 17  |-  ( ( ( _i  x.  ( sqr `  A ) ) ^ 2 )  =/=  0  ->  ( _i  x.  ( sqr `  A
) )  =/=  0
)
3633, 35syl 16 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
_i  x.  ( sqr `  A ) )  =/=  0 )
3726, 36sqgt0d 12318 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  0  <  ( ( _i  x.  ( sqr `  A ) ) ^ 2 ) )
3837, 25breqtrd 4477 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  0  <  -u A )
3928, 38elrpd 11266 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  -u A  e.  RR+ )
40 logneg 22838 . . . . . . . . . . . . 13  |-  ( -u A  e.  RR+  ->  ( log `  -u -u A )  =  ( ( log `  -u A
)  +  ( _i  x.  pi ) ) )
4139, 40syl 16 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( log `  -u -u A )  =  ( ( log `  -u A
)  +  ( _i  x.  pi ) ) )
42 negneg 9881 . . . . . . . . . . . . . 14  |-  ( A  e.  CC  ->  -u -u A  =  A )
4342ad2antrr 725 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  -u -u A  =  A )
4443fveq2d 5876 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( log `  -u -u A )  =  ( log `  A
) )
4539relogcld 22874 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( log `  -u A )  e.  RR )
4645recnd 9634 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( log `  -u A )  e.  CC )
47 picn 22719 . . . . . . . . . . . . . 14  |-  pi  e.  CC
4813, 47mulcli 9613 . . . . . . . . . . . . 13  |-  ( _i  x.  pi )  e.  CC
49 addcom 9777 . . . . . . . . . . . . 13  |-  ( ( ( log `  -u A
)  e.  CC  /\  ( _i  x.  pi )  e.  CC )  ->  ( ( log `  -u A
)  +  ( _i  x.  pi ) )  =  ( ( _i  x.  pi )  +  ( log `  -u A
) ) )
5046, 48, 49sylancl 662 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
( log `  -u A
)  +  ( _i  x.  pi ) )  =  ( ( _i  x.  pi )  +  ( log `  -u A
) ) )
5141, 44, 503eqtr3d 2516 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( log `  A )  =  ( ( _i  x.  pi )  +  ( log `  -u A ) ) )
5251oveq2d 6311 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
( 1  /  2
)  x.  ( log `  A ) )  =  ( ( 1  / 
2 )  x.  (
( _i  x.  pi )  +  ( log `  -u A ) ) ) )
53 adddi 9593 . . . . . . . . . . . 12  |-  ( ( ( 1  /  2
)  e.  CC  /\  ( _i  x.  pi )  e.  CC  /\  ( log `  -u A )  e.  CC )  ->  (
( 1  /  2
)  x.  ( ( _i  x.  pi )  +  ( log `  -u A
) ) )  =  ( ( ( 1  /  2 )  x.  ( _i  x.  pi ) )  +  ( ( 1  /  2
)  x.  ( log `  -u A ) ) ) )
541, 48, 53mp3an12 1314 . . . . . . . . . . 11  |-  ( ( log `  -u A
)  e.  CC  ->  ( ( 1  /  2
)  x.  ( ( _i  x.  pi )  +  ( log `  -u A
) ) )  =  ( ( ( 1  /  2 )  x.  ( _i  x.  pi ) )  +  ( ( 1  /  2
)  x.  ( log `  -u A ) ) ) )
5546, 54syl 16 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
( 1  /  2
)  x.  ( ( _i  x.  pi )  +  ( log `  -u A
) ) )  =  ( ( ( 1  /  2 )  x.  ( _i  x.  pi ) )  +  ( ( 1  /  2
)  x.  ( log `  -u A ) ) ) )
5652, 55eqtrd 2508 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
( 1  /  2
)  x.  ( log `  A ) )  =  ( ( ( 1  /  2 )  x.  ( _i  x.  pi ) )  +  ( ( 1  /  2
)  x.  ( log `  -u A ) ) ) )
57 2cn 10618 . . . . . . . . . . . 12  |-  2  e.  CC
58 2ne0 10640 . . . . . . . . . . . 12  |-  2  =/=  0
59 divrec2 10236 . . . . . . . . . . . 12  |-  ( ( ( _i  x.  pi )  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  (
( _i  x.  pi )  /  2 )  =  ( ( 1  / 
2 )  x.  (
_i  x.  pi )
) )
6048, 57, 58, 59mp3an 1324 . . . . . . . . . . 11  |-  ( ( _i  x.  pi )  /  2 )  =  ( ( 1  / 
2 )  x.  (
_i  x.  pi )
)
6113, 47, 57, 58divassi 10312 . . . . . . . . . . 11  |-  ( ( _i  x.  pi )  /  2 )  =  ( _i  x.  (
pi  /  2 ) )
6260, 61eqtr3i 2498 . . . . . . . . . 10  |-  ( ( 1  /  2 )  x.  ( _i  x.  pi ) )  =  ( _i  x.  ( pi 
/  2 ) )
6362oveq1i 6305 . . . . . . . . 9  |-  ( ( ( 1  /  2
)  x.  ( _i  x.  pi ) )  +  ( ( 1  /  2 )  x.  ( log `  -u A
) ) )  =  ( ( _i  x.  ( pi  /  2
) )  +  ( ( 1  /  2
)  x.  ( log `  -u A ) ) )
6456, 63syl6eq 2524 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
( 1  /  2
)  x.  ( log `  A ) )  =  ( ( _i  x.  ( pi  /  2
) )  +  ( ( 1  /  2
)  x.  ( log `  -u A ) ) ) )
6564fveq2d 5876 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( exp `  ( ( 1  /  2 )  x.  ( log `  A
) ) )  =  ( exp `  (
( _i  x.  (
pi  /  2 ) )  +  ( ( 1  /  2 )  x.  ( log `  -u A
) ) ) ) )
6647, 57, 58divcli 10298 . . . . . . . . 9  |-  ( pi 
/  2 )  e.  CC
6713, 66mulcli 9613 . . . . . . . 8  |-  ( _i  x.  ( pi  / 
2 ) )  e.  CC
68 mulcl 9588 . . . . . . . . 9  |-  ( ( ( 1  /  2
)  e.  CC  /\  ( log `  -u A
)  e.  CC )  ->  ( ( 1  /  2 )  x.  ( log `  -u A
) )  e.  CC )
691, 46, 68sylancr 663 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
( 1  /  2
)  x.  ( log `  -u A ) )  e.  CC )
70 efadd 13708 . . . . . . . 8  |-  ( ( ( _i  x.  (
pi  /  2 ) )  e.  CC  /\  ( ( 1  / 
2 )  x.  ( log `  -u A ) )  e.  CC )  -> 
( exp `  (
( _i  x.  (
pi  /  2 ) )  +  ( ( 1  /  2 )  x.  ( log `  -u A
) ) ) )  =  ( ( exp `  ( _i  x.  (
pi  /  2 ) ) )  x.  ( exp `  ( ( 1  /  2 )  x.  ( log `  -u A
) ) ) ) )
7167, 69, 70sylancr 663 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( exp `  ( ( _i  x.  ( pi  / 
2 ) )  +  ( ( 1  / 
2 )  x.  ( log `  -u A ) ) ) )  =  ( ( exp `  (
_i  x.  ( pi  /  2 ) ) )  x.  ( exp `  (
( 1  /  2
)  x.  ( log `  -u A ) ) ) ) )
72 efhalfpi 22730 . . . . . . . . 9  |-  ( exp `  ( _i  x.  (
pi  /  2 ) ) )  =  _i
7372a1i 11 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( exp `  ( _i  x.  ( pi  /  2
) ) )  =  _i )
74 negcl 9832 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  -u A  e.  CC )
7574ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  -u A  e.  CC )
761a1i 11 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
1  /  2 )  e.  CC )
77 cxpef 22912 . . . . . . . . . 10  |-  ( (
-u A  e.  CC  /\  -u A  =/=  0  /\  ( 1  /  2
)  e.  CC )  ->  ( -u A  ^c  ( 1  /  2 ) )  =  ( exp `  (
( 1  /  2
)  x.  ( log `  -u A ) ) ) )
7875, 32, 76, 77syl3anc 1228 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( -u A  ^c  ( 1  /  2 ) )  =  ( exp `  ( ( 1  / 
2 )  x.  ( log `  -u A ) ) ) )
79 ax-1cn 9562 . . . . . . . . . . . . . 14  |-  1  e.  CC
80 2halves 10779 . . . . . . . . . . . . . 14  |-  ( 1  e.  CC  ->  (
( 1  /  2
)  +  ( 1  /  2 ) )  =  1 )
8179, 80ax-mp 5 . . . . . . . . . . . . 13  |-  ( ( 1  /  2 )  +  ( 1  / 
2 ) )  =  1
8281oveq2i 6306 . . . . . . . . . . . 12  |-  ( -u A  ^c  ( ( 1  /  2 )  +  ( 1  / 
2 ) ) )  =  ( -u A  ^c  1 )
83 cxp1 22918 . . . . . . . . . . . . 13  |-  ( -u A  e.  CC  ->  (
-u A  ^c 
1 )  =  -u A )
8475, 83syl 16 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( -u A  ^c  1 )  =  -u A
)
8582, 84syl5eq 2520 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( -u A  ^c  ( ( 1  /  2
)  +  ( 1  /  2 ) ) )  =  -u A
)
86 rpcxpcl 22923 . . . . . . . . . . . . . . 15  |-  ( (
-u A  e.  RR+  /\  ( 1  /  2
)  e.  RR )  ->  ( -u A  ^c  ( 1  /  2 ) )  e.  RR+ )
8739, 2, 86sylancl 662 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( -u A  ^c  ( 1  /  2 ) )  e.  RR+ )
8887rpcnd 11270 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( -u A  ^c  ( 1  /  2 ) )  e.  CC )
8988sqvald 12287 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
( -u A  ^c 
( 1  /  2
) ) ^ 2 )  =  ( (
-u A  ^c 
( 1  /  2
) )  x.  ( -u A  ^c  ( 1  /  2 ) ) ) )
90 cxpadd 22926 . . . . . . . . . . . . 13  |-  ( ( ( -u A  e.  CC  /\  -u A  =/=  0 )  /\  (
1  /  2 )  e.  CC  /\  (
1  /  2 )  e.  CC )  -> 
( -u A  ^c 
( ( 1  / 
2 )  +  ( 1  /  2 ) ) )  =  ( ( -u A  ^c  ( 1  / 
2 ) )  x.  ( -u A  ^c  ( 1  / 
2 ) ) ) )
9175, 32, 76, 76, 90syl211anc 1234 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( -u A  ^c  ( ( 1  /  2
)  +  ( 1  /  2 ) ) )  =  ( (
-u A  ^c 
( 1  /  2
) )  x.  ( -u A  ^c  ( 1  /  2 ) ) ) )
9289, 91eqtr4d 2511 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
( -u A  ^c 
( 1  /  2
) ) ^ 2 )  =  ( -u A  ^c  ( ( 1  /  2 )  +  ( 1  / 
2 ) ) ) )
9375sqsqrtd 13250 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
( sqr `  -u A
) ^ 2 )  =  -u A )
9485, 92, 933eqtr4d 2518 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
( -u A  ^c 
( 1  /  2
) ) ^ 2 )  =  ( ( sqr `  -u A
) ^ 2 ) )
9587rprege0d 11275 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
( -u A  ^c 
( 1  /  2
) )  e.  RR  /\  0  <_  ( -u A  ^c  ( 1  /  2 ) ) ) )
9639rpsqrtcld 13223 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( sqr `  -u A )  e.  RR+ )
9796rprege0d 11275 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
( sqr `  -u A
)  e.  RR  /\  0  <_  ( sqr `  -u A
) ) )
98 sq11 12220 . . . . . . . . . . 11  |-  ( ( ( ( -u A  ^c  ( 1  /  2 ) )  e.  RR  /\  0  <_  ( -u A  ^c  ( 1  / 
2 ) ) )  /\  ( ( sqr `  -u A )  e.  RR  /\  0  <_ 
( sqr `  -u A
) ) )  -> 
( ( ( -u A  ^c  ( 1  /  2 ) ) ^ 2 )  =  ( ( sqr `  -u A
) ^ 2 )  <-> 
( -u A  ^c 
( 1  /  2
) )  =  ( sqr `  -u A
) ) )
9995, 97, 98syl2anc 661 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
( ( -u A  ^c  ( 1  /  2 ) ) ^ 2 )  =  ( ( sqr `  -u A
) ^ 2 )  <-> 
( -u A  ^c 
( 1  /  2
) )  =  ( sqr `  -u A
) ) )
10094, 99mpbid 210 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( -u A  ^c  ( 1  /  2 ) )  =  ( sqr `  -u A ) )
10178, 100eqtr3d 2510 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( exp `  ( ( 1  /  2 )  x.  ( log `  -u A
) ) )  =  ( sqr `  -u A
) )
10273, 101oveq12d 6313 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
( exp `  (
_i  x.  ( pi  /  2 ) ) )  x.  ( exp `  (
( 1  /  2
)  x.  ( log `  -u A ) ) ) )  =  ( _i  x.  ( sqr `  -u A ) ) )
10365, 71, 1023eqtrd 2512 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( exp `  ( ( 1  /  2 )  x.  ( log `  A
) ) )  =  ( _i  x.  ( sqr `  -u A ) ) )
104 cxpef 22912 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  (
1  /  2 )  e.  CC )  -> 
( A  ^c 
( 1  /  2
) )  =  ( exp `  ( ( 1  /  2 )  x.  ( log `  A
) ) ) )
1051, 104mp3an3 1313 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( A  ^c 
( 1  /  2
) )  =  ( exp `  ( ( 1  /  2 )  x.  ( log `  A
) ) ) )
106105adantr 465 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( A  ^c  ( 1  /  2 ) )  =  ( exp `  (
( 1  /  2
)  x.  ( log `  A ) ) ) )
10743fveq2d 5876 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( sqr `  -u -u A )  =  ( sqr `  A
) )
10839rpge0d 11272 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  0  <_ 
-u A )
10928, 108sqrtnegd 13233 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( sqr `  -u -u A )  =  ( _i  x.  ( sqr `  -u A ) ) )
110107, 109eqtr3d 2510 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( sqr `  A )  =  ( _i  x.  ( sqr `  -u A ) ) )
111103, 106, 1103eqtr4d 2518 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( A  ^c  ( 1  /  2 ) )  =  ( sqr `  A
) )
112111ex 434 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
)  ->  ( A  ^c  ( 1  /  2 ) )  =  ( sqr `  A
) ) )
11381oveq2i 6306 . . . . . . . . 9  |-  ( A  ^c  ( ( 1  /  2 )  +  ( 1  / 
2 ) ) )  =  ( A  ^c  1 )
114 cxpadd 22926 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( 1  / 
2 )  e.  CC  /\  ( 1  /  2
)  e.  CC )  ->  ( A  ^c  ( ( 1  /  2 )  +  ( 1  /  2
) ) )  =  ( ( A  ^c  ( 1  / 
2 ) )  x.  ( A  ^c 
( 1  /  2
) ) ) )
1151, 1, 114mp3an23 1316 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( A  ^c 
( ( 1  / 
2 )  +  ( 1  /  2 ) ) )  =  ( ( A  ^c 
( 1  /  2
) )  x.  ( A  ^c  ( 1  /  2 ) ) ) )
116 cxp1 22918 . . . . . . . . . 10  |-  ( A  e.  CC  ->  ( A  ^c  1 )  =  A )
117116adantr 465 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( A  ^c 
1 )  =  A )
118113, 115, 1173eqtr3a 2532 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( A  ^c  ( 1  / 
2 ) )  x.  ( A  ^c 
( 1  /  2
) ) )  =  A )
119 cxpcl 22921 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( 1  /  2
)  e.  CC )  ->  ( A  ^c  ( 1  / 
2 ) )  e.  CC )
1201, 119mpan2 671 . . . . . . . . . 10  |-  ( A  e.  CC  ->  ( A  ^c  ( 1  /  2 ) )  e.  CC )
121120sqvald 12287 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( A  ^c 
( 1  /  2
) ) ^ 2 )  =  ( ( A  ^c  ( 1  /  2 ) )  x.  ( A  ^c  ( 1  /  2 ) ) ) )
122121adantr 465 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( A  ^c  ( 1  / 
2 ) ) ^
2 )  =  ( ( A  ^c 
( 1  /  2
) )  x.  ( A  ^c  ( 1  /  2 ) ) ) )
12320adantr 465 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( sqr `  A
) ^ 2 )  =  A )
124118, 122, 1233eqtr4d 2518 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( A  ^c  ( 1  / 
2 ) ) ^
2 )  =  ( ( sqr `  A
) ^ 2 ) )
125 sqeqor 12262 . . . . . . . . 9  |-  ( ( ( A  ^c 
( 1  /  2
) )  e.  CC  /\  ( sqr `  A
)  e.  CC )  ->  ( ( ( A  ^c  ( 1  /  2 ) ) ^ 2 )  =  ( ( sqr `  A ) ^ 2 )  <->  ( ( A  ^c  ( 1  /  2 ) )  =  ( sqr `  A
)  \/  ( A  ^c  ( 1  /  2 ) )  =  -u ( sqr `  A
) ) ) )
126120, 14, 125syl2anc 661 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( ( A  ^c  ( 1  / 
2 ) ) ^
2 )  =  ( ( sqr `  A
) ^ 2 )  <-> 
( ( A  ^c  ( 1  / 
2 ) )  =  ( sqr `  A
)  \/  ( A  ^c  ( 1  /  2 ) )  =  -u ( sqr `  A
) ) ) )
127126biimpa 484 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( ( A  ^c  ( 1  / 
2 ) ) ^
2 )  =  ( ( sqr `  A
) ^ 2 ) )  ->  ( ( A  ^c  ( 1  /  2 ) )  =  ( sqr `  A
)  \/  ( A  ^c  ( 1  /  2 ) )  =  -u ( sqr `  A
) ) )
128124, 127syldan 470 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( A  ^c  ( 1  / 
2 ) )  =  ( sqr `  A
)  \/  ( A  ^c  ( 1  /  2 ) )  =  -u ( sqr `  A
) ) )
129128ord 377 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( -.  ( A  ^c  ( 1  /  2 ) )  =  ( sqr `  A
)  ->  ( A  ^c  ( 1  /  2 ) )  =  -u ( sqr `  A
) ) )
130129con1d 124 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( -.  ( A  ^c  ( 1  /  2 ) )  =  -u ( sqr `  A
)  ->  ( A  ^c  ( 1  /  2 ) )  =  ( sqr `  A
) ) )
131112, 130pm2.61d 158 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( A  ^c 
( 1  /  2
) )  =  ( sqr `  A ) )
132131ex 434 . 2  |-  ( A  e.  CC  ->  ( A  =/=  0  ->  ( A  ^c  ( 1  /  2 ) )  =  ( sqr `  A
) ) )
13312, 132pm2.61dne 2784 1  |-  ( A  e.  CC  ->  ( A  ^c  ( 1  /  2 ) )  =  ( sqr `  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   CCcc 9502   RRcr 9503   0cc0 9504   1c1 9505   _ici 9506    + caddc 9507    x. cmul 9509    < clt 9640    <_ cle 9641   -ucneg 9818    / cdiv 10218   2c2 10597   RR+crp 11232   ^cexp 12146   sqrcsqrt 13046   expce 13676   picpi 13681   logclog 22808    ^c ccxp 22809
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582  ax-addf 9583  ax-mulf 9584
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-fi 7883  df-sup 7913  df-oi 7947  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-q 11195  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-ioo 11545  df-ioc 11546  df-ico 11547  df-icc 11548  df-fz 11685  df-fzo 11805  df-fl 11909  df-mod 11977  df-seq 12088  df-exp 12147  df-fac 12334  df-bc 12361  df-hash 12386  df-shft 12880  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-limsup 13274  df-clim 13291  df-rlim 13292  df-sum 13489  df-ef 13682  df-sin 13684  df-cos 13685  df-pi 13687  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-starv 14587  df-sca 14588  df-vsca 14589  df-ip 14590  df-tset 14591  df-ple 14592  df-ds 14594  df-unif 14595  df-hom 14596  df-cco 14597  df-rest 14695  df-topn 14696  df-0g 14714  df-gsum 14715  df-topgen 14716  df-pt 14717  df-prds 14720  df-xrs 14774  df-qtop 14779  df-imas 14780  df-xps 14782  df-mre 14858  df-mrc 14859  df-acs 14861  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15840  df-mulg 15932  df-cntz 16227  df-cmn 16673  df-psmet 18281  df-xmet 18282  df-met 18283  df-bl 18284  df-mopn 18285  df-fbas 18286  df-fg 18287  df-cnfld 18291  df-top 19268  df-bases 19270  df-topon 19271  df-topsp 19272  df-cld 19388  df-ntr 19389  df-cls 19390  df-nei 19467  df-lp 19505  df-perf 19506  df-cn 19596  df-cnp 19597  df-haus 19684  df-tx 19931  df-hmeo 20124  df-fil 20215  df-fm 20307  df-flim 20308  df-flf 20309  df-xms 20691  df-ms 20692  df-tms 20693  df-cncf 21250  df-limc 22138  df-dv 22139  df-log 22810  df-cxp 22811
This theorem is referenced by:  logsqrt  22951  dvsqrt  22984  resqrtcn  22989  sqrtcn  22990  efiatan  23109  efiatan2  23114  sqrtlim  23168  chpchtlim  23530  dvcnsqrt  30028
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