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

Theorem cxpsqrt 23648
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 10829 . . . . . 6  |-  ( 1  /  2 )  e.  CC
2 halfre 10828 . . . . . . 7  |-  ( 1  /  2 )  e.  RR
3 halfgt0 10830 . . . . . . 7  |-  0  <  ( 1  /  2
)
42, 3gt0ne0ii 10150 . . . . . 6  |-  ( 1  /  2 )  =/=  0
5 0cxp 23611 . . . . . 6  |-  ( ( ( 1  /  2
)  e.  CC  /\  ( 1  /  2
)  =/=  0 )  ->  ( 0  ^c  ( 1  / 
2 ) )  =  0 )
61, 4, 5mp2an 678 . . . . 5  |-  ( 0  ^c  ( 1  /  2 ) )  =  0
7 sqrt0 13305 . . . . 5  |-  ( sqr `  0 )  =  0
86, 7eqtr4i 2476 . . . 4  |-  ( 0  ^c  ( 1  /  2 ) )  =  ( sqr `  0
)
9 oveq1 6297 . . . 4  |-  ( A  =  0  ->  ( A  ^c  ( 1  /  2 ) )  =  ( 0  ^c  ( 1  / 
2 ) ) )
10 fveq2 5865 . . . 4  |-  ( A  =  0  ->  ( sqr `  A )  =  ( sqr `  0
) )
118, 9, 103eqtr4a 2511 . . 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 9598 . . . . . . . . . . . . . . . . 17  |-  _i  e.  CC
14 sqrtcl 13424 . . . . . . . . . . . . . . . . . 18  |-  ( A  e.  CC  ->  ( sqr `  A )  e.  CC )
1514ad2antrr 732 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( sqr `  A )  e.  CC )
16 sqmul 12338 . . . . . . . . . . . . . . . . 17  |-  ( ( _i  e.  CC  /\  ( sqr `  A )  e.  CC )  -> 
( ( _i  x.  ( sqr `  A ) ) ^ 2 )  =  ( ( _i
^ 2 )  x.  ( ( sqr `  A
) ^ 2 ) ) )
1713, 15, 16sylancr 669 . . . . . . . . . . . . . . . 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 12375 . . . . . . . . . . . . . . . . . 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 13427 . . . . . . . . . . . . . . . . . 18  |-  ( A  e.  CC  ->  (
( sqr `  A
) ^ 2 )  =  A )
2120ad2antrr 732 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
( sqr `  A
) ^ 2 )  =  A )
2219, 21oveq12d 6308 . . . . . . . . . . . . . . . 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 10060 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  CC  ->  ( -u 1  x.  A )  =  -u A )
2423ad2antrr 732 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( -u 1  x.  A )  =  -u A )
2517, 22, 243eqtrd 2489 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
( _i  x.  ( sqr `  A ) ) ^ 2 )  = 
-u A )
26 cxpsqrtlem 23647 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
_i  x.  ( sqr `  A ) )  e.  RR )
2726resqcld 12442 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
( _i  x.  ( sqr `  A ) ) ^ 2 )  e.  RR )
2825, 27eqeltrrd 2530 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  -u A  e.  RR )
29 negeq0 9928 . . . . . . . . . . . . . . . . . . . . 21  |-  ( A  e.  CC  ->  ( A  =  0  <->  -u A  =  0 ) )
3029necon3bid 2668 . . . . . . . . . . . . . . . . . . . 20  |-  ( A  e.  CC  ->  ( A  =/=  0  <->  -u A  =/=  0 ) )
3130biimpa 487 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  -u A  =/=  0 )
3231adantr 467 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  -u A  =/=  0 )
3325, 32eqnetrd 2691 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
( _i  x.  ( sqr `  A ) ) ^ 2 )  =/=  0 )
34 sq0i 12367 . . . . . . . . . . . . . . . . . 18  |-  ( ( _i  x.  ( sqr `  A ) )  =  0  ->  ( (
_i  x.  ( sqr `  A ) ) ^
2 )  =  0 )
3534necon3i 2656 . . . . . . . . . . . . . . . . 17  |-  ( ( ( _i  x.  ( sqr `  A ) ) ^ 2 )  =/=  0  ->  ( _i  x.  ( sqr `  A
) )  =/=  0
)
3633, 35syl 17 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
_i  x.  ( sqr `  A ) )  =/=  0 )
3726, 36sqgt0d 12444 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  0  <  ( ( _i  x.  ( sqr `  A ) ) ^ 2 ) )
3837, 25breqtrd 4427 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  0  <  -u A )
3928, 38elrpd 11338 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  -u A  e.  RR+ )
40 logneg 23537 . . . . . . . . . . . . 13  |-  ( -u A  e.  RR+  ->  ( log `  -u -u A )  =  ( ( log `  -u A
)  +  ( _i  x.  pi ) ) )
4139, 40syl 17 . . . . . . . . . . . 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 9924 . . . . . . . . . . . . . 14  |-  ( A  e.  CC  ->  -u -u A  =  A )
4342ad2antrr 732 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  -u -u A  =  A )
4443fveq2d 5869 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( log `  -u -u A )  =  ( log `  A
) )
4539relogcld 23572 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( log `  -u A )  e.  RR )
4645recnd 9669 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( log `  -u A )  e.  CC )
47 picn 23414 . . . . . . . . . . . . . 14  |-  pi  e.  CC
4813, 47mulcli 9648 . . . . . . . . . . . . 13  |-  ( _i  x.  pi )  e.  CC
49 addcom 9819 . . . . . . . . . . . . 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 668 . . . . . . . . . . . 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 2493 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( log `  A )  =  ( ( _i  x.  pi )  +  ( log `  -u A ) ) )
5251oveq2d 6306 . . . . . . . . . 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 9628 . . . . . . . . . . . 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 1354 . . . . . . . . . . 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 17 . . . . . . . . . 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 2485 . . . . . . . . 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 10680 . . . . . . . . . . . 12  |-  2  e.  CC
58 2ne0 10702 . . . . . . . . . . . 12  |-  2  =/=  0
59 divrec2 10287 . . . . . . . . . . . 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 1364 . . . . . . . . . . 11  |-  ( ( _i  x.  pi )  /  2 )  =  ( ( 1  / 
2 )  x.  (
_i  x.  pi )
)
6113, 47, 57, 58divassi 10363 . . . . . . . . . . 11  |-  ( ( _i  x.  pi )  /  2 )  =  ( _i  x.  (
pi  /  2 ) )
6260, 61eqtr3i 2475 . . . . . . . . . 10  |-  ( ( 1  /  2 )  x.  ( _i  x.  pi ) )  =  ( _i  x.  ( pi 
/  2 ) )
6362oveq1i 6300 . . . . . . . . 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 2501 . . . . . . . 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 5869 . . . . . . 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 10349 . . . . . . . . 9  |-  ( pi 
/  2 )  e.  CC
6713, 66mulcli 9648 . . . . . . . 8  |-  ( _i  x.  ( pi  / 
2 ) )  e.  CC
68 mulcl 9623 . . . . . . . . 9  |-  ( ( ( 1  /  2
)  e.  CC  /\  ( log `  -u A
)  e.  CC )  ->  ( ( 1  /  2 )  x.  ( log `  -u A
) )  e.  CC )
691, 46, 68sylancr 669 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
( 1  /  2
)  x.  ( log `  -u A ) )  e.  CC )
70 efadd 14148 . . . . . . . 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 669 . . . . . . 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 23426 . . . . . . . . 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 9875 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  -u A  e.  CC )
7574ad2antrr 732 . . . . . . . . . 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 23610 . . . . . . . . . 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 1268 . . . . . . . . 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 9597 . . . . . . . . . . . . . 14  |-  1  e.  CC
80 2halves 10841 . . . . . . . . . . . . . 14  |-  ( 1  e.  CC  ->  (
( 1  /  2
)  +  ( 1  /  2 ) )  =  1 )
8179, 80ax-mp 5 . . . . . . . . . . . . 13  |-  ( ( 1  /  2 )  +  ( 1  / 
2 ) )  =  1
8281oveq2i 6301 . . . . . . . . . . . 12  |-  ( -u A  ^c  ( ( 1  /  2 )  +  ( 1  / 
2 ) ) )  =  ( -u A  ^c  1 )
83 cxp1 23616 . . . . . . . . . . . . 13  |-  ( -u A  e.  CC  ->  (
-u A  ^c 
1 )  =  -u A )
8475, 83syl 17 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( -u A  ^c  1 )  =  -u A
)
8582, 84syl5eq 2497 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( -u A  ^c  ( ( 1  /  2
)  +  ( 1  /  2 ) ) )  =  -u A
)
86 rpcxpcl 23621 . . . . . . . . . . . . . . 15  |-  ( (
-u A  e.  RR+  /\  ( 1  /  2
)  e.  RR )  ->  ( -u A  ^c  ( 1  /  2 ) )  e.  RR+ )
8739, 2, 86sylancl 668 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( -u A  ^c  ( 1  /  2 ) )  e.  RR+ )
8887rpcnd 11343 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( -u A  ^c  ( 1  /  2 ) )  e.  CC )
8988sqvald 12413 . . . . . . . . . . . 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 23624 . . . . . . . . . . . . 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 1274 . . . . . . . . . . . 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 2488 . . . . . . . . . . 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 13501 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
( sqr `  -u A
) ^ 2 )  =  -u A )
9485, 92, 933eqtr4d 2495 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
( -u A  ^c 
( 1  /  2
) ) ^ 2 )  =  ( ( sqr `  -u A
) ^ 2 ) )
9587rprege0d 11348 . . . . . . . . . . 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 13473 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( sqr `  -u A )  e.  RR+ )
9796rprege0d 11348 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  (
( sqr `  -u A
)  e.  RR  /\  0  <_  ( sqr `  -u A
) ) )
98 sq11 12347 . . . . . . . . . . 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 667 . . . . . . . . . 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 214 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( -u A  ^c  ( 1  /  2 ) )  =  ( sqr `  -u A ) )
10178, 100eqtr3d 2487 . . . . . . . 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 6308 . . . . . . 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 2489 . . . . . 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 23610 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  (
1  /  2 )  e.  CC )  -> 
( A  ^c 
( 1  /  2
) )  =  ( exp `  ( ( 1  /  2 )  x.  ( log `  A
) ) ) )
1051, 104mp3an3 1353 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( A  ^c 
( 1  /  2
) )  =  ( exp `  ( ( 1  /  2 )  x.  ( log `  A
) ) ) )
106105adantr 467 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( A  ^c  ( 1  /  2 ) )  =  ( exp `  (
( 1  /  2
)  x.  ( log `  A ) ) ) )
10743fveq2d 5869 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( sqr `  -u -u A )  =  ( sqr `  A
) )
10839rpge0d 11345 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  0  <_ 
-u A )
10928, 108sqrtnegd 13483 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( sqr `  -u -u A )  =  ( _i  x.  ( sqr `  -u A ) ) )
110107, 109eqtr3d 2487 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( sqr `  A )  =  ( _i  x.  ( sqr `  -u A ) ) )
111103, 106, 1103eqtr4d 2495 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
) )  ->  ( A  ^c  ( 1  /  2 ) )  =  ( sqr `  A
) )
112111ex 436 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( A  ^c  ( 1  / 
2 ) )  = 
-u ( sqr `  A
)  ->  ( A  ^c  ( 1  /  2 ) )  =  ( sqr `  A
) ) )
11381oveq2i 6301 . . . . . . . . 9  |-  ( A  ^c  ( ( 1  /  2 )  +  ( 1  / 
2 ) ) )  =  ( A  ^c  1 )
114 cxpadd 23624 . . . . . . . . . 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 1356 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( A  ^c 
( ( 1  / 
2 )  +  ( 1  /  2 ) ) )  =  ( ( A  ^c 
( 1  /  2
) )  x.  ( A  ^c  ( 1  /  2 ) ) ) )
116 cxp1 23616 . . . . . . . . . 10  |-  ( A  e.  CC  ->  ( A  ^c  1 )  =  A )
117116adantr 467 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( A  ^c 
1 )  =  A )
118113, 115, 1173eqtr3a 2509 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( A  ^c  ( 1  / 
2 ) )  x.  ( A  ^c 
( 1  /  2
) ) )  =  A )
119 cxpcl 23619 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( 1  /  2
)  e.  CC )  ->  ( A  ^c  ( 1  / 
2 ) )  e.  CC )
1201, 119mpan2 677 . . . . . . . . . 10  |-  ( A  e.  CC  ->  ( A  ^c  ( 1  /  2 ) )  e.  CC )
121120sqvald 12413 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( A  ^c 
( 1  /  2
) ) ^ 2 )  =  ( ( A  ^c  ( 1  /  2 ) )  x.  ( A  ^c  ( 1  /  2 ) ) ) )
122121adantr 467 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( A  ^c  ( 1  / 
2 ) ) ^
2 )  =  ( ( A  ^c 
( 1  /  2
) )  x.  ( A  ^c  ( 1  /  2 ) ) ) )
12320adantr 467 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( sqr `  A
) ^ 2 )  =  A )
124118, 122, 1233eqtr4d 2495 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( A  ^c  ( 1  / 
2 ) ) ^
2 )  =  ( ( sqr `  A
) ^ 2 ) )
125 sqeqor 12388 . . . . . . . . 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 667 . . . . . . . 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 487 . . . . . . 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 473 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( A  ^c  ( 1  / 
2 ) )  =  ( sqr `  A
)  \/  ( A  ^c  ( 1  /  2 ) )  =  -u ( sqr `  A
) ) )
129128ord 379 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( -.  ( A  ^c  ( 1  /  2 ) )  =  ( sqr `  A
)  ->  ( A  ^c  ( 1  /  2 ) )  =  -u ( sqr `  A
) ) )
130129con1d 128 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( -.  ( A  ^c  ( 1  /  2 ) )  =  -u ( sqr `  A
)  ->  ( A  ^c  ( 1  /  2 ) )  =  ( sqr `  A
) ) )
131112, 130pm2.61d 162 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( A  ^c 
( 1  /  2
) )  =  ( sqr `  A ) )
132131ex 436 . 2  |-  ( A  e.  CC  ->  ( A  =/=  0  ->  ( A  ^c  ( 1  /  2 ) )  =  ( sqr `  A
) ) )
13312, 132pm2.61dne 2710 1  |-  ( A  e.  CC  ->  ( A  ^c  ( 1  /  2 ) )  =  ( sqr `  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    \/ wo 370    /\ wa 371    = wceq 1444    e. wcel 1887    =/= wne 2622   class class class wbr 4402   ` cfv 5582  (class class class)co 6290   CCcc 9537   RRcr 9538   0cc0 9539   1c1 9540   _ici 9541    + caddc 9542    x. cmul 9544    < clt 9675    <_ cle 9676   -ucneg 9861    / cdiv 10269   2c2 10659   RR+crp 11302   ^cexp 12272   sqrcsqrt 13296   expce 14114   picpi 14119   logclog 23504    ^c ccxp 23505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-ixp 7523  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7884  df-fi 7925  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ioc 11640  df-ico 11641  df-icc 11642  df-fz 11785  df-fzo 11916  df-fl 12028  df-mod 12097  df-seq 12214  df-exp 12273  df-fac 12460  df-bc 12488  df-hash 12516  df-shft 13130  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-limsup 13526  df-clim 13552  df-rlim 13553  df-sum 13753  df-ef 14121  df-sin 14123  df-cos 14124  df-pi 14126  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-starv 15205  df-sca 15206  df-vsca 15207  df-ip 15208  df-tset 15209  df-ple 15210  df-ds 15212  df-unif 15213  df-hom 15214  df-cco 15215  df-rest 15321  df-topn 15322  df-0g 15340  df-gsum 15341  df-topgen 15342  df-pt 15343  df-prds 15346  df-xrs 15400  df-qtop 15406  df-imas 15407  df-xps 15410  df-mre 15492  df-mrc 15493  df-acs 15495  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-submnd 16583  df-mulg 16676  df-cntz 16971  df-cmn 17432  df-psmet 18962  df-xmet 18963  df-met 18964  df-bl 18965  df-mopn 18966  df-fbas 18967  df-fg 18968  df-cnfld 18971  df-top 19921  df-bases 19922  df-topon 19923  df-topsp 19924  df-cld 20034  df-ntr 20035  df-cls 20036  df-nei 20114  df-lp 20152  df-perf 20153  df-cn 20243  df-cnp 20244  df-haus 20331  df-tx 20577  df-hmeo 20770  df-fil 20861  df-fm 20953  df-flim 20954  df-flf 20955  df-xms 21335  df-ms 21336  df-tms 21337  df-cncf 21910  df-limc 22821  df-dv 22822  df-log 23506  df-cxp 23507
This theorem is referenced by:  logsqrt  23649  dvsqrt  23682  dvcnsqrt  23684  resqrtcn  23689  sqrtcn  23690  efiatan  23838  efiatan2  23843  sqrtlim  23898  chpchtlim  24317
  Copyright terms: Public domain W3C validator