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

Theorem sqrval 12839
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 2467 . . . 4  |-  ( y  =  A  ->  (
( x ^ 2 )  =  y  <->  ( x ^ 2 )  =  A ) )
213anbi1d 1294 . . 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 6158 . 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 12837 . 2  |-  sqr  =  ( y  e.  CC  |->  ( iota_ x  e.  CC  ( ( x ^
2 )  =  y  /\  0  <_  (
Re `  x )  /\  ( _i  x.  x
)  e/  RR+ ) ) )
5 riotaex 6160 . 2  |-  ( iota_ x  e.  CC  ( ( x ^ 2 )  =  A  /\  0  <_  ( Re `  x
)  /\  ( _i  x.  x )  e/  RR+ )
)  e.  _V
63, 4, 5fvmpt 5878 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 setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1370    e. wcel 1758    e/ wnel 2646   class class class wbr 4395   ` cfv 5521   iota_crio 6155  (class class class)co 6195   CCcc 9386   0cc0 9388   _ici 9390    x. cmul 9393    <_ cle 9525   2c2 10477   RR+crp 11097   ^cexp 11977   Recre 12699   sqrcsqr 12835
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pr 4634
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-iota 5484  df-fun 5523  df-fv 5529  df-riota 6156  df-sqr 12837
This theorem is referenced by:  sqr0  12844  resqrcl  12856  resqrthlem  12857  sqrneg  12870  sqrcl  12962  sqrthlem  12963
  Copyright terms: Public domain W3C validator