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Theorem cxpsqr 22033
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
cxpsqr  |-  ( A  e.  CC  ->  ( A  ^c  ( 1  /  2 ) )  =  ( sqr `  A
) )

Proof of Theorem cxpsqr
StepHypRef Expression
1 halfcn 10529 . . . . . 6  |-  ( 1  /  2 )  e.  CC
2 halfre 10528 . . . . . . 7  |-  ( 1  /  2 )  e.  RR
3 halfgt0 10530 . . . . . . 7  |-  0  <  ( 1  /  2
)
42, 3gt0ne0ii 9864 . . . . . 6  |-  ( 1  /  2 )  =/=  0
5 0cxp 21996 . . . . . 6  |-  ( ( ( 1  /  2
)  e.  CC  /\  ( 1  /  2
)  =/=  0 )  ->  ( 0  ^c  ( 1  / 
2 ) )  =  0 )
61, 4, 5mp2an 665 . . . . 5  |-  ( 0  ^c  ( 1  /  2 ) )  =  0
7 sqr0 12715 . . . . 5  |-  ( sqr `  0 )  =  0
86, 7eqtr4i 2456 . . . 4  |-  ( 0  ^c  ( 1  /  2 ) )  =  ( sqr `  0
)
9 oveq1 6087 . . . 4  |-  ( A  =  0  ->  ( A  ^c  ( 1  /  2 ) )  =  ( 0  ^c  ( 1  / 
2 ) ) )
10 fveq2 5679 . . . 4  |-  ( A  =  0  ->  ( sqr `  A )  =  ( sqr `  0
) )
118, 9, 103eqtr4a 2491 . . 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 9329 . . . . . . . . . . . . . . . . 17  |-  _i  e.  CC
14 sqrcl 12833 . . . . . . . . . . . . . . . . . 18  |-  ( A  e.  CC  ->  ( sqr `  A )  e.  CC )
1514ad2antrr 718 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( sqr `  A )  e.  CC )
16 sqmul 11913 . . . . . . . . . . . . . . . . 17  |-  ( ( _i  e.  CC  /\  ( sqr `  A )  e.  CC )  -> 
( ( _i  x.  ( sqr `  A ) ) ^ 2 )  =  ( ( _i
^ 2 )  x.  ( ( sqr `  A
) ^ 2 ) ) )
1713, 15, 16sylancr 656 . . . . . . . . . . . . . . . 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 11950 . . . . . . . . . . . . . . . . . 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 sqrth 12836 . . . . . . . . . . . . . . . . . 18  |-  ( A  e.  CC  ->  (
( sqr `  A
) ^ 2 )  =  A )
2120ad2antrr 718 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
( sqr `  A
) ^ 2 )  =  A )
2219, 21oveq12d 6098 . . . . . . . . . . . . . . . 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 9774 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  CC  ->  ( -u 1  x.  A )  =  -u A )
2423ad2antrr 718 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( -u 1  x.  A )  =  -u A )
2517, 22, 243eqtrd 2469 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
( _i  x.  ( sqr `  A ) ) ^ 2 )  = 
-u A )
26 cxpsqrlem 22032 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
_i  x.  ( sqr `  A ) )  e.  RR )
2726resqcld 12018 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
( _i  x.  ( sqr `  A ) ) ^ 2 )  e.  RR )
2825, 27eqeltrrd 2508 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  -u A  e.  RR )
29 negeq0 9651 . . . . . . . . . . . . . . . . . . . . 21  |-  ( A  e.  CC  ->  ( A  =  0  <->  -u A  =  0 ) )
3029necon3bid 2633 . . . . . . . . . . . . . . . . . . . 20  |-  ( A  e.  CC  ->  ( A  =/=  0  <->  -u A  =/=  0 ) )
3130biimpa 481 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  -u A  =/=  0 )
3231adantr 462 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  -u A  =/=  0 )
3325, 32eqnetrd 2616 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
( _i  x.  ( sqr `  A ) ) ^ 2 )  =/=  0 )
34 sq0i 11942 . . . . . . . . . . . . . . . . . 18  |-  ( ( _i  x.  ( sqr `  A ) )  =  0  ->  ( (
_i  x.  ( sqr `  A ) ) ^
2 )  =  0 )
3534necon3i 2640 . . . . . . . . . . . . . . . . 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 12020 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  0  <  ( ( _i  x.  ( sqr `  A ) ) ^ 2 ) )
3837, 25breqtrd 4304 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  0  <  -u A )
3928, 38elrpd 11013 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  -u A  e.  RR+ )
40 logneg 21921 . . . . . . . . . . . . 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 9647 . . . . . . . . . . . . . 14  |-  ( A  e.  CC  ->  -u -u A  =  A )
4342ad2antrr 718 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  -u -u A  =  A )
4443fveq2d 5683 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( log `  -u -u A )  =  ( log `  A
) )
4539relogcld 21957 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( log `  -u A )  e.  RR )
4645recnd 9400 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( log `  -u A )  e.  CC )
47 picn 21807 . . . . . . . . . . . . . 14  |-  pi  e.  CC
4813, 47mulcli 9379 . . . . . . . . . . . . 13  |-  ( _i  x.  pi )  e.  CC
49 addcom 9543 . . . . . . . . . . . . 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 655 . . . . . . . . . . . 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 2473 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( log `  A )  =  ( ( _i  x.  pi )  +  ( log `  -u A ) ) )
5251oveq2d 6096 . . . . . . . . . 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 9359 . . . . . . . . . . . 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 1297 . . . . . . . . . . 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 2465 . . . . . . . . 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 10380 . . . . . . . . . . . 12  |-  2  e.  CC
58 2ne0 10402 . . . . . . . . . . . 12  |-  2  =/=  0
59 divrec2 9999 . . . . . . . . . . . 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 1307 . . . . . . . . . . 11  |-  ( ( _i  x.  pi )  /  2 )  =  ( ( 1  / 
2 )  x.  (
_i  x.  pi )
)
6113, 47, 57, 58divassi 10075 . . . . . . . . . . 11  |-  ( ( _i  x.  pi )  /  2 )  =  ( _i  x.  (
pi  /  2 ) )
6260, 61eqtr3i 2455 . . . . . . . . . 10  |-  ( ( 1  /  2 )  x.  ( _i  x.  pi ) )  =  ( _i  x.  ( pi 
/  2 ) )
6362oveq1i 6090 . . . . . . . . 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 2481 . . . . . . . 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 5683 . . . . . . 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 10061 . . . . . . . . 9  |-  ( pi 
/  2 )  e.  CC
6713, 66mulcli 9379 . . . . . . . 8  |-  ( _i  x.  ( pi  / 
2 ) )  e.  CC
68 mulcl 9354 . . . . . . . . 9  |-  ( ( ( 1  /  2
)  e.  CC  /\  ( log `  -u A
)  e.  CC )  ->  ( ( 1  /  2 )  x.  ( log `  -u A
) )  e.  CC )
691, 46, 68sylancr 656 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
( 1  /  2
)  x.  ( log `  -u A ) )  e.  CC )
70 efadd 13362 . . . . . . . 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 656 . . . . . . 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 21818 . . . . . . . . 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 9598 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  -u A  e.  CC )
7574ad2antrr 718 . . . . . . . . . 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 21995 . . . . . . . . . 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 1211 . . . . . . . . 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 9328 . . . . . . . . . . . . . 14  |-  1  e.  CC
80 2halves 10541 . . . . . . . . . . . . . 14  |-  ( 1  e.  CC  ->  (
( 1  /  2
)  +  ( 1  /  2 ) )  =  1 )
8179, 80ax-mp 5 . . . . . . . . . . . . 13  |-  ( ( 1  /  2 )  +  ( 1  / 
2 ) )  =  1
8281oveq2i 6091 . . . . . . . . . . . 12  |-  ( -u A  ^c  ( ( 1  /  2 )  +  ( 1  / 
2 ) ) )  =  ( -u A  ^c  1 )
83 cxp1 22001 . . . . . . . . . . . . 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 2477 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( -u A  ^c  ( ( 1  /  2
)  +  ( 1  /  2 ) ) )  =  -u A
)
86 rpcxpcl 22006 . . . . . . . . . . . . . . 15  |-  ( (
-u A  e.  RR+  /\  ( 1  /  2
)  e.  RR )  ->  ( -u A  ^c  ( 1  /  2 ) )  e.  RR+ )
8739, 2, 86sylancl 655 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( -u A  ^c  ( 1  /  2 ) )  e.  RR+ )
8887rpcnd 11017 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( -u A  ^c  ( 1  /  2 ) )  e.  CC )
8988sqvald 11989 . . . . . . . . . . . 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 22009 . . . . . . . . . . . . 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 1217 . . . . . . . . . . . 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 2468 . . . . . . . . . . 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 ) ) ) )
9375sqsqrd 12909 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
( sqr `  -u A
) ^ 2 )  =  -u A )
9485, 92, 933eqtr4d 2475 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
( -u A  ^c 
( 1  /  2
) ) ^ 2 )  =  ( ( sqr `  -u A
) ^ 2 ) )
9587rprege0d 11022 . . . . . . . . . . 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 ) ) ) )
9639rpsqrcld 12882 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( sqr `  -u A )  e.  RR+ )
9796rprege0d 11022 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
( sqr `  -u A
)  e.  RR  /\  0  <_  ( sqr `  -u A
) ) )
98 sq11 11922 . . . . . . . . . . 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 654 . . . . . . . . . 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 2467 . . . . . . . 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 6098 . . . . . . 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 2469 . . . . . 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 21995 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  (
1  /  2 )  e.  CC )  -> 
( A  ^c 
( 1  /  2
) )  =  ( exp `  ( ( 1  /  2 )  x.  ( log `  A
) ) ) )
1051, 104mp3an3 1296 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( A  ^c 
( 1  /  2
) )  =  ( exp `  ( ( 1  /  2 )  x.  ( log `  A
) ) ) )
106105adantr 462 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( A  ^c  ( 1  /  2 ) )  =  ( exp `  (
( 1  /  2
)  x.  ( log `  A ) ) ) )
10743fveq2d 5683 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( sqr `  -u -u A )  =  ( sqr `  A
) )
10839rpge0d 11019 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  0  <_ 
-u A )
10928, 108sqrnegd 12892 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( sqr `  -u -u A )  =  ( _i  x.  ( sqr `  -u A ) ) )
110107, 109eqtr3d 2467 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( sqr `  A )  =  ( _i  x.  ( sqr `  -u A ) ) )
111103, 106, 1103eqtr4d 2475 . . . . 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 6091 . . . . . . . . 9  |-  ( A  ^c  ( ( 1  /  2 )  +  ( 1  / 
2 ) ) )  =  ( A  ^c  1 )
114 cxpadd 22009 . . . . . . . . . 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 1299 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( A  ^c 
( ( 1  / 
2 )  +  ( 1  /  2 ) ) )  =  ( ( A  ^c 
( 1  /  2
) )  x.  ( A  ^c  ( 1  /  2 ) ) ) )
116 cxp1 22001 . . . . . . . . . 10  |-  ( A  e.  CC  ->  ( A  ^c  1 )  =  A )
117116adantr 462 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( A  ^c 
1 )  =  A )
118113, 115, 1173eqtr3a 2489 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( A  ^c  ( 1  / 
2 ) )  x.  ( A  ^c 
( 1  /  2
) ) )  =  A )
119 cxpcl 22004 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( 1  /  2
)  e.  CC )  ->  ( A  ^c  ( 1  / 
2 ) )  e.  CC )
1201, 119mpan2 664 . . . . . . . . . 10  |-  ( A  e.  CC  ->  ( A  ^c  ( 1  /  2 ) )  e.  CC )
121120sqvald 11989 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( A  ^c 
( 1  /  2
) ) ^ 2 )  =  ( ( A  ^c  ( 1  /  2 ) )  x.  ( A  ^c  ( 1  /  2 ) ) ) )
122121adantr 462 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( A  ^c  ( 1  / 
2 ) ) ^
2 )  =  ( ( A  ^c 
( 1  /  2
) )  x.  ( A  ^c  ( 1  /  2 ) ) ) )
12320adantr 462 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( sqr `  A
) ^ 2 )  =  A )
124118, 122, 1233eqtr4d 2475 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( A  ^c  ( 1  / 
2 ) ) ^
2 )  =  ( ( sqr `  A
) ^ 2 ) )
125 sqeqor 11964 . . . . . . . . 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 654 . . . . . . . 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 481 . . . . . . 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 467 . . . . . 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 2678 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 1362    e. wcel 1755    =/= wne 2596   class class class wbr 4280   ` cfv 5406  (class class class)co 6080   CCcc 9268   RRcr 9269   0cc0 9270   1c1 9271   _ici 9272    + caddc 9273    x. cmul 9275    < clt 9406    <_ cle 9407   -ucneg 9584    / cdiv 9981   2c2 10359   RR+crp 10979   ^cexp 11849   sqrcsqr 12706   expce 13330   picpi 13335   logclog 21891    ^c ccxp 21892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348  ax-addf 9349  ax-mulf 9350
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-iin 4162  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309  df-om 6466  df-1st 6566  df-2nd 6567  df-supp 6680  df-recs 6818  df-rdg 6852  df-1o 6908  df-2o 6909  df-oadd 6912  df-er 7089  df-map 7204  df-pm 7205  df-ixp 7252  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-fsupp 7609  df-fi 7649  df-sup 7679  df-oi 7712  df-card 8097  df-cda 8325  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-2 10368  df-3 10369  df-4 10370  df-5 10371  df-6 10372  df-7 10373  df-8 10374  df-9 10375  df-10 10376  df-n0 10568  df-z 10635  df-dec 10744  df-uz 10850  df-q 10942  df-rp 10980  df-xneg 11077  df-xadd 11078  df-xmul 11079  df-ioo 11292  df-ioc 11293  df-ico 11294  df-icc 11295  df-fz 11425  df-fzo 11533  df-fl 11626  df-mod 11693  df-seq 11791  df-exp 11850  df-fac 12036  df-bc 12063  df-hash 12088  df-shft 12540  df-cj 12572  df-re 12573  df-im 12574  df-sqr 12708  df-abs 12709  df-limsup 12933  df-clim 12950  df-rlim 12951  df-sum 13148  df-ef 13336  df-sin 13338  df-cos 13339  df-pi 13341  df-struct 14159  df-ndx 14160  df-slot 14161  df-base 14162  df-sets 14163  df-ress 14164  df-plusg 14234  df-mulr 14235  df-starv 14236  df-sca 14237  df-vsca 14238  df-ip 14239  df-tset 14240  df-ple 14241  df-ds 14243  df-unif 14244  df-hom 14245  df-cco 14246  df-rest 14344  df-topn 14345  df-0g 14363  df-gsum 14364  df-topgen 14365  df-pt 14366  df-prds 14369  df-xrs 14423  df-qtop 14428  df-imas 14429  df-xps 14431  df-mre 14507  df-mrc 14508  df-acs 14510  df-mnd 15398  df-submnd 15448  df-mulg 15528  df-cntz 15815  df-cmn 16259  df-psmet 17653  df-xmet 17654  df-met 17655  df-bl 17656  df-mopn 17657  df-fbas 17658  df-fg 17659  df-cnfld 17663  df-top 18345  df-bases 18347  df-topon 18348  df-topsp 18349  df-cld 18465  df-ntr 18466  df-cls 18467  df-nei 18544  df-lp 18582  df-perf 18583  df-cn 18673  df-cnp 18674  df-haus 18761  df-tx 18977  df-hmeo 19170  df-fil 19261  df-fm 19353  df-flim 19354  df-flf 19355  df-xms 19737  df-ms 19738  df-tms 19739  df-cncf 20296  df-limc 21183  df-dv 21184  df-log 21893  df-cxp 21894
This theorem is referenced by:  logsqr  22034  dvsqr  22067  resqrcn  22072  sqrcn  22073  efiatan  22192  efiatan2  22197  sqrlim  22251  chpchtlim  22613  dvcnsqr  28322
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