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Theorem cxpsqr 20547
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 1re 9046 . . . . . . . 8  |-  1  e.  RR
21rehalfcli 10172 . . . . . . 7  |-  ( 1  /  2 )  e.  RR
32recni 9058 . . . . . 6  |-  ( 1  /  2 )  e.  CC
4 halfgt0 10144 . . . . . . 7  |-  0  <  ( 1  /  2
)
52, 4gt0ne0ii 9519 . . . . . 6  |-  ( 1  /  2 )  =/=  0
6 0cxp 20510 . . . . . 6  |-  ( ( ( 1  /  2
)  e.  CC  /\  ( 1  /  2
)  =/=  0 )  ->  ( 0  ^ c  ( 1  / 
2 ) )  =  0 )
73, 5, 6mp2an 654 . . . . 5  |-  ( 0  ^ c  ( 1  /  2 ) )  =  0
8 sqr0 12002 . . . . 5  |-  ( sqr `  0 )  =  0
97, 8eqtr4i 2427 . . . 4  |-  ( 0  ^ c  ( 1  /  2 ) )  =  ( sqr `  0
)
10 oveq1 6047 . . . 4  |-  ( A  =  0  ->  ( A  ^ c  ( 1  /  2 ) )  =  ( 0  ^ c  ( 1  / 
2 ) ) )
11 fveq2 5687 . . . 4  |-  ( A  =  0  ->  ( sqr `  A )  =  ( sqr `  0
) )
129, 10, 113eqtr4a 2462 . . 3  |-  ( A  =  0  ->  ( A  ^ c  ( 1  /  2 ) )  =  ( sqr `  A
) )
1312a1i 11 . 2  |-  ( A  e.  CC  ->  ( A  =  0  ->  ( A  ^ c  ( 1  /  2 ) )  =  ( sqr `  A ) ) )
14 ax-icn 9005 . . . . . . . . . . . . . . . . 17  |-  _i  e.  CC
15 sqrcl 12120 . . . . . . . . . . . . . . . . . 18  |-  ( A  e.  CC  ->  ( sqr `  A )  e.  CC )
1615ad2antrr 707 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^ c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( sqr `  A )  e.  CC )
17 sqmul 11400 . . . . . . . . . . . . . . . . 17  |-  ( ( _i  e.  CC  /\  ( sqr `  A )  e.  CC )  -> 
( ( _i  x.  ( sqr `  A ) ) ^ 2 )  =  ( ( _i
^ 2 )  x.  ( ( sqr `  A
) ^ 2 ) ) )
1814, 16, 17sylancr 645 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^ c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
( _i  x.  ( sqr `  A ) ) ^ 2 )  =  ( ( _i ^
2 )  x.  (
( sqr `  A
) ^ 2 ) ) )
19 i2 11436 . . . . . . . . . . . . . . . . . 18  |-  ( _i
^ 2 )  = 
-u 1
2019a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^ c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
_i ^ 2 )  =  -u 1 )
21 sqrth 12123 . . . . . . . . . . . . . . . . . 18  |-  ( A  e.  CC  ->  (
( sqr `  A
) ^ 2 )  =  A )
2221ad2antrr 707 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^ c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
( sqr `  A
) ^ 2 )  =  A )
2320, 22oveq12d 6058 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^ c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
( _i ^ 2 )  x.  ( ( sqr `  A ) ^ 2 ) )  =  ( -u 1  x.  A ) )
24 mulm1 9431 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  CC  ->  ( -u 1  x.  A )  =  -u A )
2524ad2antrr 707 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^ c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( -u 1  x.  A )  =  -u A )
2618, 23, 253eqtrd 2440 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^ c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
( _i  x.  ( sqr `  A ) ) ^ 2 )  = 
-u A )
27 cxpsqrlem 20546 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^ c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
_i  x.  ( sqr `  A ) )  e.  RR )
2827resqcld 11504 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^ c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
( _i  x.  ( sqr `  A ) ) ^ 2 )  e.  RR )
2926, 28eqeltrrd 2479 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^ c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  -u A  e.  RR )
30 negeq0 9311 . . . . . . . . . . . . . . . . . . . . 21  |-  ( A  e.  CC  ->  ( A  =  0  <->  -u A  =  0 ) )
3130necon3bid 2602 . . . . . . . . . . . . . . . . . . . 20  |-  ( A  e.  CC  ->  ( A  =/=  0  <->  -u A  =/=  0 ) )
3231biimpa 471 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  -u A  =/=  0 )
3332adantr 452 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^ c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  -u A  =/=  0 )
3426, 33eqnetrd 2585 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^ c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
( _i  x.  ( sqr `  A ) ) ^ 2 )  =/=  0 )
35 sq0i 11429 . . . . . . . . . . . . . . . . . 18  |-  ( ( _i  x.  ( sqr `  A ) )  =  0  ->  ( (
_i  x.  ( sqr `  A ) ) ^
2 )  =  0 )
3635necon3i 2606 . . . . . . . . . . . . . . . . 17  |-  ( ( ( _i  x.  ( sqr `  A ) ) ^ 2 )  =/=  0  ->  ( _i  x.  ( sqr `  A
) )  =/=  0
)
3734, 36syl 16 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^ c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
_i  x.  ( sqr `  A ) )  =/=  0 )
3827, 37sqgt0d 11506 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^ c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  0  <  ( ( _i  x.  ( sqr `  A ) ) ^ 2 ) )
3938, 26breqtrd 4196 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^ c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  0  <  -u A )
4029, 39elrpd 10602 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^ c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  -u A  e.  RR+ )
41 logneg 20435 . . . . . . . . . . . . 13  |-  ( -u A  e.  RR+  ->  ( log `  -u -u A )  =  ( ( log `  -u A
)  +  ( _i  x.  pi ) ) )
4240, 41syl 16 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^ c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( log `  -u -u A )  =  ( ( log `  -u A
)  +  ( _i  x.  pi ) ) )
43 negneg 9307 . . . . . . . . . . . . . 14  |-  ( A  e.  CC  ->  -u -u A  =  A )
4443ad2antrr 707 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^ c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  -u -u A  =  A )
4544fveq2d 5691 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^ c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( log `  -u -u A )  =  ( log `  A
) )
4640relogcld 20471 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^ c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( log `  -u A )  e.  RR )
4746recnd 9070 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^ c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( log `  -u A )  e.  CC )
48 pire 20325 . . . . . . . . . . . . . . 15  |-  pi  e.  RR
4948recni 9058 . . . . . . . . . . . . . 14  |-  pi  e.  CC
5014, 49mulcli 9051 . . . . . . . . . . . . 13  |-  ( _i  x.  pi )  e.  CC
51 addcom 9208 . . . . . . . . . . . . 13  |-  ( ( ( log `  -u A
)  e.  CC  /\  ( _i  x.  pi )  e.  CC )  ->  ( ( log `  -u A
)  +  ( _i  x.  pi ) )  =  ( ( _i  x.  pi )  +  ( log `  -u A
) ) )
5247, 50, 51sylancl 644 . . . . . . . . . . . 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
) ) )
5342, 45, 523eqtr3d 2444 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^ c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( log `  A )  =  ( ( _i  x.  pi )  +  ( log `  -u A ) ) )
5453oveq2d 6056 . . . . . . . . . 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 ) ) ) )
55 adddi 9035 . . . . . . . . . . . 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 ) ) ) )
563, 50, 55mp3an12 1269 . . . . . . . . . . 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 ) ) ) )
5747, 56syl 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 ) ) ) )
5854, 57eqtrd 2436 . . . . . . . . 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 ) ) ) )
59 2cn 10026 . . . . . . . . . . . 12  |-  2  e.  CC
60 2ne0 10039 . . . . . . . . . . . 12  |-  2  =/=  0
61 divrec2 9651 . . . . . . . . . . . 12  |-  ( ( ( _i  x.  pi )  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  (
( _i  x.  pi )  /  2 )  =  ( ( 1  / 
2 )  x.  (
_i  x.  pi )
) )
6250, 59, 60, 61mp3an 1279 . . . . . . . . . . 11  |-  ( ( _i  x.  pi )  /  2 )  =  ( ( 1  / 
2 )  x.  (
_i  x.  pi )
)
6314, 49, 59, 60divassi 9726 . . . . . . . . . . 11  |-  ( ( _i  x.  pi )  /  2 )  =  ( _i  x.  (
pi  /  2 ) )
6462, 63eqtr3i 2426 . . . . . . . . . 10  |-  ( ( 1  /  2 )  x.  ( _i  x.  pi ) )  =  ( _i  x.  ( pi 
/  2 ) )
6564oveq1i 6050 . . . . . . . . 9  |-  ( ( ( 1  /  2
)  x.  ( _i  x.  pi ) )  +  ( ( 1  /  2 )  x.  ( log `  -u A
) ) )  =  ( ( _i  x.  ( pi  /  2
) )  +  ( ( 1  /  2
)  x.  ( log `  -u A ) ) )
6658, 65syl6eq 2452 . . . . . . . 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 ) ) ) )
6766fveq2d 5691 . . . . . . 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
) ) ) ) )
6849, 59, 60divcli 9712 . . . . . . . . 9  |-  ( pi 
/  2 )  e.  CC
6914, 68mulcli 9051 . . . . . . . 8  |-  ( _i  x.  ( pi  / 
2 ) )  e.  CC
70 mulcl 9030 . . . . . . . . 9  |-  ( ( ( 1  /  2
)  e.  CC  /\  ( log `  -u A
)  e.  CC )  ->  ( ( 1  /  2 )  x.  ( log `  -u A
) )  e.  CC )
713, 47, 70sylancr 645 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^ c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
( 1  /  2
)  x.  ( log `  -u A ) )  e.  CC )
72 efadd 12651 . . . . . . . 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
) ) ) ) )
7369, 71, 72sylancr 645 . . . . . . 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 ) ) ) ) )
74 efhalfpi 20332 . . . . . . . . 9  |-  ( exp `  ( _i  x.  (
pi  /  2 ) ) )  =  _i
7574a1i 11 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^ c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( exp `  ( _i  x.  ( pi  /  2
) ) )  =  _i )
76 negcl 9262 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  -u A  e.  CC )
7776ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^ c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  -u A  e.  CC )
783a1i 11 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^ c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
1  /  2 )  e.  CC )
79 cxpef 20509 . . . . . . . . . 10  |-  ( (
-u A  e.  CC  /\  -u A  =/=  0  /\  ( 1  /  2
)  e.  CC )  ->  ( -u A  ^ c  ( 1  /  2 ) )  =  ( exp `  (
( 1  /  2
)  x.  ( log `  -u A ) ) ) )
8077, 33, 78, 79syl3anc 1184 . . . . . . . . 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 ) ) ) )
81 ax-1cn 9004 . . . . . . . . . . . . . 14  |-  1  e.  CC
82 2halves 10152 . . . . . . . . . . . . . 14  |-  ( 1  e.  CC  ->  (
( 1  /  2
)  +  ( 1  /  2 ) )  =  1 )
8381, 82ax-mp 8 . . . . . . . . . . . . 13  |-  ( ( 1  /  2 )  +  ( 1  / 
2 ) )  =  1
8483oveq2i 6051 . . . . . . . . . . . 12  |-  ( -u A  ^ c  ( ( 1  /  2 )  +  ( 1  / 
2 ) ) )  =  ( -u A  ^ c  1 )
85 cxp1 20515 . . . . . . . . . . . . 13  |-  ( -u A  e.  CC  ->  (
-u A  ^ c 
1 )  =  -u A )
8677, 85syl 16 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^ c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( -u A  ^ c  1 )  =  -u A
)
8784, 86syl5eq 2448 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^ c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( -u A  ^ c  ( ( 1  /  2
)  +  ( 1  /  2 ) ) )  =  -u A
)
88 rpcxpcl 20520 . . . . . . . . . . . . . . 15  |-  ( (
-u A  e.  RR+  /\  ( 1  /  2
)  e.  RR )  ->  ( -u A  ^ c  ( 1  /  2 ) )  e.  RR+ )
8940, 2, 88sylancl 644 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^ c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( -u A  ^ c  ( 1  /  2 ) )  e.  RR+ )
9089rpcnd 10606 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^ c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( -u A  ^ c  ( 1  /  2 ) )  e.  CC )
9190sqvald 11475 . . . . . . . . . . . 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 ) ) ) )
92 cxpadd 20523 . . . . . . . . . . . . 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 ) ) ) )
9377, 33, 78, 78, 92syl211anc 1190 . . . . . . . . . . . 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 ) ) ) )
9491, 93eqtr4d 2439 . . . . . . . . . . 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 ) ) ) )
9577sqsqrd 12196 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^ c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
( sqr `  -u A
) ^ 2 )  =  -u A )
9687, 94, 953eqtr4d 2446 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^ c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
( -u A  ^ c 
( 1  /  2
) ) ^ 2 )  =  ( ( sqr `  -u A
) ^ 2 ) )
9789rprege0d 10611 . . . . . . . . . . 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 ) ) ) )
9840rpsqrcld 12169 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^ c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( sqr `  -u A )  e.  RR+ )
9998rprege0d 10611 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^ c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
( sqr `  -u A
)  e.  RR  /\  0  <_  ( sqr `  -u A
) ) )
100 sq11 11409 . . . . . . . . . . 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
) ) )
10197, 99, 100syl2anc 643 . . . . . . . . . 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
) ) )
10296, 101mpbid 202 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^ c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( -u A  ^ c  ( 1  /  2 ) )  =  ( sqr `  -u A ) )
10380, 102eqtr3d 2438 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^ c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( exp `  ( ( 1  /  2 )  x.  ( log `  -u A
) ) )  =  ( sqr `  -u A
) )
10475, 103oveq12d 6058 . . . . . . 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 ) ) )
10567, 73, 1043eqtrd 2440 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^ c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( exp `  ( ( 1  /  2 )  x.  ( log `  A
) ) )  =  ( _i  x.  ( sqr `  -u A ) ) )
106 cxpef 20509 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  (
1  /  2 )  e.  CC )  -> 
( A  ^ c 
( 1  /  2
) )  =  ( exp `  ( ( 1  /  2 )  x.  ( log `  A
) ) ) )
1073, 106mp3an3 1268 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( A  ^ c 
( 1  /  2
) )  =  ( exp `  ( ( 1  /  2 )  x.  ( log `  A
) ) ) )
108107adantr 452 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^ c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( A  ^ c  ( 1  /  2 ) )  =  ( exp `  (
( 1  /  2
)  x.  ( log `  A ) ) ) )
10944fveq2d 5691 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^ c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( sqr `  -u -u A )  =  ( sqr `  A
) )
11040rpge0d 10608 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^ c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  0  <_ 
-u A )
11129, 110sqrnegd 12179 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^ c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( sqr `  -u -u A )  =  ( _i  x.  ( sqr `  -u A ) ) )
112109, 111eqtr3d 2438 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^ c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( sqr `  A )  =  ( _i  x.  ( sqr `  -u A ) ) )
113105, 108, 1123eqtr4d 2446 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^ c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( A  ^ c  ( 1  /  2 ) )  =  ( sqr `  A
) )
114113ex 424 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( A  ^ c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
)  ->  ( A  ^ c  ( 1  /  2 ) )  =  ( sqr `  A
) ) )
11583oveq2i 6051 . . . . . . . . 9  |-  ( A  ^ c  ( ( 1  /  2 )  +  ( 1  / 
2 ) ) )  =  ( A  ^ c  1 )
116 cxpadd 20523 . . . . . . . . . 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
) ) ) )
1173, 3, 116mp3an23 1271 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( A  ^ c 
( ( 1  / 
2 )  +  ( 1  /  2 ) ) )  =  ( ( A  ^ c 
( 1  /  2
) )  x.  ( A  ^ c  ( 1  /  2 ) ) ) )
118 cxp1 20515 . . . . . . . . . 10  |-  ( A  e.  CC  ->  ( A  ^ c  1 )  =  A )
119118adantr 452 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( A  ^ c 
1 )  =  A )
120115, 117, 1193eqtr3a 2460 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( A  ^ c  ( 1  / 
2 ) )  x.  ( A  ^ c 
( 1  /  2
) ) )  =  A )
121 cxpcl 20518 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( 1  /  2
)  e.  CC )  ->  ( A  ^ c  ( 1  / 
2 ) )  e.  CC )
1223, 121mpan2 653 . . . . . . . . . 10  |-  ( A  e.  CC  ->  ( A  ^ c  ( 1  /  2 ) )  e.  CC )
123122sqvald 11475 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( A  ^ c 
( 1  /  2
) ) ^ 2 )  =  ( ( A  ^ c  ( 1  /  2 ) )  x.  ( A  ^ c  ( 1  /  2 ) ) ) )
124123adantr 452 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( A  ^ c  ( 1  / 
2 ) ) ^
2 )  =  ( ( A  ^ c 
( 1  /  2
) )  x.  ( A  ^ c  ( 1  /  2 ) ) ) )
12521adantr 452 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( sqr `  A
) ^ 2 )  =  A )
126120, 124, 1253eqtr4d 2446 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( A  ^ c  ( 1  / 
2 ) ) ^
2 )  =  ( ( sqr `  A
) ^ 2 ) )
127 sqeqor 11450 . . . . . . . . 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
) ) ) )
128122, 15, 127syl2anc 643 . . . . . . . 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
) ) ) )
129128biimpa 471 . . . . . . 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
) ) )
130126, 129syldan 457 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( A  ^ c  ( 1  / 
2 ) )  =  ( sqr `  A
)  \/  ( A  ^ c  ( 1  /  2 ) )  =  -u ( sqr `  A
) ) )
131130ord 367 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( -.  ( A  ^ c  ( 1  /  2 ) )  =  ( sqr `  A
)  ->  ( A  ^ c  ( 1  /  2 ) )  =  -u ( sqr `  A
) ) )
132131con1d 118 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( -.  ( A  ^ c  ( 1  /  2 ) )  =  -u ( sqr `  A
)  ->  ( A  ^ c  ( 1  /  2 ) )  =  ( sqr `  A
) ) )
133114, 132pm2.61d 152 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( A  ^ c 
( 1  /  2
) )  =  ( sqr `  A ) )
134133ex 424 . 2  |-  ( A  e.  CC  ->  ( A  =/=  0  ->  ( A  ^ c  ( 1  /  2 ) )  =  ( sqr `  A
) ) )
13513, 134pm2.61dne 2644 1  |-  ( A  e.  CC  ->  ( A  ^ c  ( 1  /  2 ) )  =  ( sqr `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947   _ici 8948    + caddc 8949    x. cmul 8951    < clt 9076    <_ cle 9077   -ucneg 9248    / cdiv 9633   2c2 10005   RR+crp 10568   ^cexp 11337   sqrcsqr 11993   expce 12619   picpi 12624   logclog 20405    ^ c ccxp 20406
This theorem is referenced by:  logsqr  20548  dvsqr  20581  resqrcn  20586  sqrcn  20587  efiatan  20705  efiatan2  20710  sqrlim  20764  chpchtlim  21126
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ioc 10877  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-shft 11837  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-sum 12435  df-ef 12625  df-sin 12627  df-cos 12628  df-pi 12630  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-limc 19706  df-dv 19707  df-log 20407  df-cxp 20408
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