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Theorem sqrval 11722
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 2292 . . . 4  |-  ( y  =  A  ->  (
( x ^ 2 )  =  y  <->  ( x ^ 2 )  =  A ) )
213anbi1d 1256 . . 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 6306 . 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 11720 . 2  |-  sqr  =  ( y  e.  CC  |->  ( iota_ x  e.  CC ( ( x ^
2 )  =  y  /\  0  <_  (
Re `  x )  /\  ( _i  x.  x
)  e/  RR+ ) ) )
5 riotaex 6308 . 2  |-  ( iota_ x  e.  CC ( ( x ^ 2 )  =  A  /\  0  <_  ( Re `  x
)  /\  ( _i  x.  x )  e/  RR+ )
)  e.  _V
63, 4, 5fvmpt 5602 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 934    = wceq 1623    e. wcel 1684    e/ wnel 2447   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   iota_crio 6297   CCcc 8735   0cc0 8737   _ici 8739    x. cmul 8742    <_ cle 8868   2c2 9795   RR+crp 10354   ^cexp 11104   Recre 11582   sqrcsqr 11718
This theorem is referenced by:  sqr0  11727  resqrcl  11739  resqrthlem  11740  sqrneg  11753  sqrcl  11845  sqrthlem  11846
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-riota 6304  df-sqr 11720
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