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Theorem sqrval 12005
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 2421 . . . 4  |-  ( y  =  A  ->  (
( x ^ 2 )  =  y  <->  ( x ^ 2 )  =  A ) )
213anbi1d 1258 . . 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 6518 . 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 12003 . 2  |-  sqr  =  ( y  e.  CC  |->  ( iota_ x  e.  CC ( ( x ^
2 )  =  y  /\  0  <_  (
Re `  x )  /\  ( _i  x.  x
)  e/  RR+ ) ) )
5 riotaex 6520 . 2  |-  ( iota_ x  e.  CC ( ( x ^ 2 )  =  A  /\  0  <_  ( Re `  x
)  /\  ( _i  x.  x )  e/  RR+ )
)  e.  _V
63, 4, 5fvmpt 5773 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 936    = wceq 1649    e. wcel 1721    e/ wnel 2576   class class class wbr 4180   ` cfv 5421  (class class class)co 6048   iota_crio 6509   CCcc 8952   0cc0 8954   _ici 8956    x. cmul 8959    <_ cle 9085   2c2 10013   RR+crp 10576   ^cexp 11345   Recre 11865   sqrcsqr 12001
This theorem is referenced by:  sqr0  12010  resqrcl  12022  resqrthlem  12023  sqrneg  12036  sqrcl  12128  sqrthlem  12129
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-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5385  df-fun 5423  df-fv 5429  df-riota 6516  df-sqr 12003
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