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Theorem sqrval 11929
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 2375 . . . 4  |-  ( y  =  A  ->  (
( x ^ 2 )  =  y  <->  ( x ^ 2 )  =  A ) )
213anbi1d 1257 . . 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 6448 . 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 11927 . 2  |-  sqr  =  ( y  e.  CC  |->  ( iota_ x  e.  CC ( ( x ^
2 )  =  y  /\  0  <_  (
Re `  x )  /\  ( _i  x.  x
)  e/  RR+ ) ) )
5 riotaex 6450 . 2  |-  ( iota_ x  e.  CC ( ( x ^ 2 )  =  A  /\  0  <_  ( Re `  x
)  /\  ( _i  x.  x )  e/  RR+ )
)  e.  _V
63, 4, 5fvmpt 5709 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 935    = wceq 1647    e. wcel 1715    e/ wnel 2530   class class class wbr 4125   ` cfv 5358  (class class class)co 5981   iota_crio 6439   CCcc 8882   0cc0 8884   _ici 8886    x. cmul 8889    <_ cle 9015   2c2 9942   RR+crp 10505   ^cexp 11269   Recre 11789   sqrcsqr 11925
This theorem is referenced by:  sqr0  11934  resqrcl  11946  resqrthlem  11947  sqrneg  11960  sqrcl  12052  sqrthlem  12053
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-iota 5322  df-fun 5360  df-fv 5366  df-riota 6446  df-sqr 11927
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