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Theorem sqrval 12047
Description: Value of square root function. (Contributed by Mario Carneiro, 8-Jul-2013.)
Assertion
Ref Expression
sqrval  |-  ( A  e.  CC  ->  ( sqr `  A )  =  ( iota_ x  e.  CC ( ( x ^
2 )  =  A  /\  0  <_  (
Re `  x )  /\  ( _i  x.  x
)  e/  RR+ ) ) )
Distinct variable group:    x, A

Proof of Theorem sqrval
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eqeq2 2447 . . . 4  |-  ( y  =  A  ->  (
( x ^ 2 )  =  y  <->  ( x ^ 2 )  =  A ) )
213anbi1d 1259 . . 3  |-  ( y  =  A  ->  (
( ( x ^
2 )  =  y  /\  0  <_  (
Re `  x )  /\  ( _i  x.  x
)  e/  RR+ )  <->  ( (
x ^ 2 )  =  A  /\  0  <_  ( Re `  x
)  /\  ( _i  x.  x )  e/  RR+ )
) )
32riotabidv 6554 . 2  |-  ( y  =  A  ->  ( iota_ x  e.  CC ( ( x ^ 2 )  =  y  /\  0  <_  ( Re `  x )  /\  (
_i  x.  x )  e/  RR+ ) )  =  ( iota_ x  e.  CC ( ( x ^
2 )  =  A  /\  0  <_  (
Re `  x )  /\  ( _i  x.  x
)  e/  RR+ ) ) )
4 df-sqr 12045 . 2  |-  sqr  =  ( y  e.  CC  |->  ( iota_ x  e.  CC ( ( x ^
2 )  =  y  /\  0  <_  (
Re `  x )  /\  ( _i  x.  x
)  e/  RR+ ) ) )
5 riotaex 6556 . 2  |-  ( iota_ x  e.  CC ( ( x ^ 2 )  =  A  /\  0  <_  ( Re `  x
)  /\  ( _i  x.  x )  e/  RR+ )
)  e.  _V
63, 4, 5fvmpt 5809 1  |-  ( A  e.  CC  ->  ( sqr `  A )  =  ( iota_ x  e.  CC ( ( x ^
2 )  =  A  /\  0  <_  (
Re `  x )  /\  ( _i  x.  x
)  e/  RR+ ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 937    = wceq 1653    e. wcel 1726    e/ wnel 2602   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   iota_crio 6545   CCcc 8993   0cc0 8995   _ici 8997    x. cmul 9000    <_ cle 9126   2c2 10054   RR+crp 10617   ^cexp 11387   Recre 11907   sqrcsqr 12043
This theorem is referenced by:  sqr0  12052  resqrcl  12064  resqrthlem  12065  sqrneg  12078  sqrcl  12170  sqrthlem  12171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-riota 6552  df-sqr 12045
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