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Theorem cxpsqr 22280
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 10651 . . . . . 6  |-  ( 1  /  2 )  e.  CC
2 halfre 10650 . . . . . . 7  |-  ( 1  /  2 )  e.  RR
3 halfgt0 10652 . . . . . . 7  |-  0  <  ( 1  /  2
)
42, 3gt0ne0ii 9986 . . . . . 6  |-  ( 1  /  2 )  =/=  0
5 0cxp 22243 . . . . . 6  |-  ( ( ( 1  /  2
)  e.  CC  /\  ( 1  /  2
)  =/=  0 )  ->  ( 0  ^c  ( 1  / 
2 ) )  =  0 )
61, 4, 5mp2an 672 . . . . 5  |-  ( 0  ^c  ( 1  /  2 ) )  =  0
7 sqr0 12848 . . . . 5  |-  ( sqr `  0 )  =  0
86, 7eqtr4i 2486 . . . 4  |-  ( 0  ^c  ( 1  /  2 ) )  =  ( sqr `  0
)
9 oveq1 6206 . . . 4  |-  ( A  =  0  ->  ( A  ^c  ( 1  /  2 ) )  =  ( 0  ^c  ( 1  / 
2 ) ) )
10 fveq2 5798 . . . 4  |-  ( A  =  0  ->  ( sqr `  A )  =  ( sqr `  0
) )
118, 9, 103eqtr4a 2521 . . 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 9451 . . . . . . . . . . . . . . . . 17  |-  _i  e.  CC
14 sqrcl 12966 . . . . . . . . . . . . . . . . . 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 12045 . . . . . . . . . . . . . . . . 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 12082 . . . . . . . . . . . . . . . . . 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 12969 . . . . . . . . . . . . . . . . . 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 6217 . . . . . . . . . . . . . . . 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 9896 . . . . . . . . . . . . . . . . 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 2499 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
( _i  x.  ( sqr `  A ) ) ^ 2 )  = 
-u A )
26 cxpsqrlem 22279 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
_i  x.  ( sqr `  A ) )  e.  RR )
2726resqcld 12150 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
( _i  x.  ( sqr `  A ) ) ^ 2 )  e.  RR )
2825, 27eqeltrrd 2543 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  -u A  e.  RR )
29 negeq0 9773 . . . . . . . . . . . . . . . . . . . . 21  |-  ( A  e.  CC  ->  ( A  =  0  <->  -u A  =  0 ) )
3029necon3bid 2709 . . . . . . . . . . . . . . . . . . . 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 2744 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
( _i  x.  ( sqr `  A ) ) ^ 2 )  =/=  0 )
34 sq0i 12074 . . . . . . . . . . . . . . . . . 18  |-  ( ( _i  x.  ( sqr `  A ) )  =  0  ->  ( (
_i  x.  ( sqr `  A ) ) ^
2 )  =  0 )
3534necon3i 2691 . . . . . . . . . . . . . . . . 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 12152 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  0  <  ( ( _i  x.  ( sqr `  A ) ) ^ 2 ) )
3837, 25breqtrd 4423 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  0  <  -u A )
3928, 38elrpd 11135 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  -u A  e.  RR+ )
40 logneg 22168 . . . . . . . . . . . . 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 9769 . . . . . . . . . . . . . 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 5802 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( log `  -u -u A )  =  ( log `  A
) )
4539relogcld 22204 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( log `  -u A )  e.  RR )
4645recnd 9522 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( log `  -u A )  e.  CC )
47 picn 22054 . . . . . . . . . . . . . 14  |-  pi  e.  CC
4813, 47mulcli 9501 . . . . . . . . . . . . 13  |-  ( _i  x.  pi )  e.  CC
49 addcom 9665 . . . . . . . . . . . . 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 2503 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( log `  A )  =  ( ( _i  x.  pi )  +  ( log `  -u A ) ) )
5251oveq2d 6215 . . . . . . . . . 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 9481 . . . . . . . . . . . 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 1305 . . . . . . . . . . 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 2495 . . . . . . . . 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 10502 . . . . . . . . . . . 12  |-  2  e.  CC
58 2ne0 10524 . . . . . . . . . . . 12  |-  2  =/=  0
59 divrec2 10121 . . . . . . . . . . . 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 1315 . . . . . . . . . . 11  |-  ( ( _i  x.  pi )  /  2 )  =  ( ( 1  / 
2 )  x.  (
_i  x.  pi )
)
6113, 47, 57, 58divassi 10197 . . . . . . . . . . 11  |-  ( ( _i  x.  pi )  /  2 )  =  ( _i  x.  (
pi  /  2 ) )
6260, 61eqtr3i 2485 . . . . . . . . . 10  |-  ( ( 1  /  2 )  x.  ( _i  x.  pi ) )  =  ( _i  x.  ( pi 
/  2 ) )
6362oveq1i 6209 . . . . . . . . 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 2511 . . . . . . . 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 5802 . . . . . . 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 10183 . . . . . . . . 9  |-  ( pi 
/  2 )  e.  CC
6713, 66mulcli 9501 . . . . . . . 8  |-  ( _i  x.  ( pi  / 
2 ) )  e.  CC
68 mulcl 9476 . . . . . . . . 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 13496 . . . . . . . 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 22065 . . . . . . . . 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 9720 . . . . . . . . . . 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 22242 . . . . . . . . . 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 1219 . . . . . . . . 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 9450 . . . . . . . . . . . . . 14  |-  1  e.  CC
80 2halves 10663 . . . . . . . . . . . . . 14  |-  ( 1  e.  CC  ->  (
( 1  /  2
)  +  ( 1  /  2 ) )  =  1 )
8179, 80ax-mp 5 . . . . . . . . . . . . 13  |-  ( ( 1  /  2 )  +  ( 1  / 
2 ) )  =  1
8281oveq2i 6210 . . . . . . . . . . . 12  |-  ( -u A  ^c  ( ( 1  /  2 )  +  ( 1  / 
2 ) ) )  =  ( -u A  ^c  1 )
83 cxp1 22248 . . . . . . . . . . . . 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 2507 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( -u A  ^c  ( ( 1  /  2
)  +  ( 1  /  2 ) ) )  =  -u A
)
86 rpcxpcl 22253 . . . . . . . . . . . . . . 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 11139 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( -u A  ^c  ( 1  /  2 ) )  e.  CC )
8988sqvald 12121 . . . . . . . . . . . 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 22256 . . . . . . . . . . . . 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 1225 . . . . . . . . . . . 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 2498 . . . . . . . . . . 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 13042 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
( sqr `  -u A
) ^ 2 )  =  -u A )
9485, 92, 933eqtr4d 2505 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
( -u A  ^c 
( 1  /  2
) ) ^ 2 )  =  ( ( sqr `  -u A
) ^ 2 ) )
9587rprege0d 11144 . . . . . . . . . . 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 13015 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( sqr `  -u A )  e.  RR+ )
9796rprege0d 11144 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
( sqr `  -u A
)  e.  RR  /\  0  <_  ( sqr `  -u A
) ) )
98 sq11 12054 . . . . . . . . . . 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 2497 . . . . . . . 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 6217 . . . . . . 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 2499 . . . . . 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 22242 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  (
1  /  2 )  e.  CC )  -> 
( A  ^c 
( 1  /  2
) )  =  ( exp `  ( ( 1  /  2 )  x.  ( log `  A
) ) ) )
1051, 104mp3an3 1304 . . . . . . 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 5802 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( sqr `  -u -u A )  =  ( sqr `  A
) )
10839rpge0d 11141 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  0  <_ 
-u A )
10928, 108sqrnegd 13025 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( sqr `  -u -u A )  =  ( _i  x.  ( sqr `  -u A ) ) )
110107, 109eqtr3d 2497 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( sqr `  A )  =  ( _i  x.  ( sqr `  -u A ) ) )
111103, 106, 1103eqtr4d 2505 . . . . 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 6210 . . . . . . . . 9  |-  ( A  ^c  ( ( 1  /  2 )  +  ( 1  / 
2 ) ) )  =  ( A  ^c  1 )
114 cxpadd 22256 . . . . . . . . . 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 1307 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( A  ^c 
( ( 1  / 
2 )  +  ( 1  /  2 ) ) )  =  ( ( A  ^c 
( 1  /  2
) )  x.  ( A  ^c  ( 1  /  2 ) ) ) )
116 cxp1 22248 . . . . . . . . . 10  |-  ( A  e.  CC  ->  ( A  ^c  1 )  =  A )
117116adantr 465 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( A  ^c 
1 )  =  A )
118113, 115, 1173eqtr3a 2519 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( A  ^c  ( 1  / 
2 ) )  x.  ( A  ^c 
( 1  /  2
) ) )  =  A )
119 cxpcl 22251 . . . . . . . . . . 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 12121 . . . . . . . . 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 2505 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( A  ^c  ( 1  / 
2 ) ) ^
2 )  =  ( ( sqr `  A
) ^ 2 ) )
125 sqeqor 12096 . . . . . . . . 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 2768 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 1370    e. wcel 1758    =/= wne 2647   class class class wbr 4399   ` cfv 5525  (class class class)co 6199   CCcc 9390   RRcr 9391   0cc0 9392   1c1 9393   _ici 9394    + caddc 9395    x. cmul 9397    < clt 9528    <_ cle 9529   -ucneg 9706    / cdiv 10103   2c2 10481   RR+crp 11101   ^cexp 11981   sqrcsqr 12839   expce 13464   picpi 13469   logclog 22138    ^c ccxp 22139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-inf2 7957  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469  ax-pre-sup 9470  ax-addf 9471  ax-mulf 9472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-iin 4281  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-se 4787  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-isom 5534  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-of 6429  df-om 6586  df-1st 6686  df-2nd 6687  df-supp 6800  df-recs 6941  df-rdg 6975  df-1o 7029  df-2o 7030  df-oadd 7033  df-er 7210  df-map 7325  df-pm 7326  df-ixp 7373  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-fsupp 7731  df-fi 7771  df-sup 7801  df-oi 7834  df-card 8219  df-cda 8447  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-div 10104  df-nn 10433  df-2 10490  df-3 10491  df-4 10492  df-5 10493  df-6 10494  df-7 10495  df-8 10496  df-9 10497  df-10 10498  df-n0 10690  df-z 10757  df-dec 10866  df-uz 10972  df-q 11064  df-rp 11102  df-xneg 11199  df-xadd 11200  df-xmul 11201  df-ioo 11414  df-ioc 11415  df-ico 11416  df-icc 11417  df-fz 11554  df-fzo 11665  df-fl 11758  df-mod 11825  df-seq 11923  df-exp 11982  df-fac 12168  df-bc 12195  df-hash 12220  df-shft 12673  df-cj 12705  df-re 12706  df-im 12707  df-sqr 12841  df-abs 12842  df-limsup 13066  df-clim 13083  df-rlim 13084  df-sum 13281  df-ef 13470  df-sin 13472  df-cos 13473  df-pi 13475  df-struct 14293  df-ndx 14294  df-slot 14295  df-base 14296  df-sets 14297  df-ress 14298  df-plusg 14369  df-mulr 14370  df-starv 14371  df-sca 14372  df-vsca 14373  df-ip 14374  df-tset 14375  df-ple 14376  df-ds 14378  df-unif 14379  df-hom 14380  df-cco 14381  df-rest 14479  df-topn 14480  df-0g 14498  df-gsum 14499  df-topgen 14500  df-pt 14501  df-prds 14504  df-xrs 14558  df-qtop 14563  df-imas 14564  df-xps 14566  df-mre 14642  df-mrc 14643  df-acs 14645  df-mnd 15533  df-submnd 15583  df-mulg 15666  df-cntz 15953  df-cmn 16399  df-psmet 17933  df-xmet 17934  df-met 17935  df-bl 17936  df-mopn 17937  df-fbas 17938  df-fg 17939  df-cnfld 17943  df-top 18634  df-bases 18636  df-topon 18637  df-topsp 18638  df-cld 18754  df-ntr 18755  df-cls 18756  df-nei 18833  df-lp 18871  df-perf 18872  df-cn 18962  df-cnp 18963  df-haus 19050  df-tx 19266  df-hmeo 19459  df-fil 19550  df-fm 19642  df-flim 19643  df-flf 19644  df-xms 20026  df-ms 20027  df-tms 20028  df-cncf 20585  df-limc 21473  df-dv 21474  df-log 22140  df-cxp 22141
This theorem is referenced by:  logsqr  22281  dvsqr  22314  resqrcn  22319  sqrcn  22320  efiatan  22439  efiatan2  22444  sqrlim  22498  chpchtlim  22860  dvcnsqr  28625
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