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Theorem sqrtrval 10013
Description: Value of square root function. (Contributed by Jim Kingdon, 23-Aug-2020.)
Assertion
Ref Expression
sqrtrval  |-  ( A  e.  RR  ->  ( sqr `  A )  =  ( iota_ x  e.  RR  ( ( x ^
2 )  =  A  /\  0  <_  x
) ) )
Distinct variable group:    x, A

Proof of Theorem sqrtrval
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eqeq2 2091 . . . 4  |-  ( y  =  A  ->  (
( x ^ 2 )  =  y  <->  ( x ^ 2 )  =  A ) )
21anbi1d 453 . . 3  |-  ( y  =  A  ->  (
( ( x ^
2 )  =  y  /\  0  <_  x
)  <->  ( ( x ^ 2 )  =  A  /\  0  <_  x ) ) )
32riotabidv 5495 . 2  |-  ( y  =  A  ->  ( iota_ x  e.  RR  (
( x ^ 2 )  =  y  /\  0  <_  x ) )  =  ( iota_ x  e.  RR  ( ( x ^ 2 )  =  A  /\  0  <_  x ) ) )
4 df-rsqrt 10011 . 2  |-  sqr  =  ( y  e.  RR  |->  ( iota_ x  e.  RR  ( ( x ^
2 )  =  y  /\  0  <_  x
) ) )
5 reex 7158 . . 3  |-  RR  e.  _V
6 riotaexg 5497 . . 3  |-  ( RR  e.  _V  ->  ( iota_ x  e.  RR  (
( x ^ 2 )  =  A  /\  0  <_  x ) )  e.  _V )
75, 6ax-mp 7 . 2  |-  ( iota_ x  e.  RR  ( ( x ^ 2 )  =  A  /\  0  <_  x ) )  e. 
_V
83, 4, 7fvmpt 5275 1  |-  ( A  e.  RR  ->  ( sqr `  A )  =  ( iota_ x  e.  RR  ( ( x ^
2 )  =  A  /\  0  <_  x
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434   _Vcvv 2602   class class class wbr 3787   ` cfv 4926   iota_crio 5492  (class class class)co 5537   RRcr 7031   0cc0 7032    <_ cle 7205   2c2 8145   ^cexp 9561   sqrcsqrt 10009
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966  ax-un 4190  ax-cnex 7118  ax-resscn 7119
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-br 3788  df-opab 3842  df-mpt 3843  df-id 4050  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-iota 4891  df-fun 4928  df-fv 4934  df-riota 5493  df-rsqrt 10011
This theorem is referenced by:  sqrt0  10017  resqrtcl  10042  rersqrtthlem  10043  sqrtsq  10057
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