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Theorem sqrtrval 10740
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 2127 . . . 4  |-  ( y  =  A  ->  (
( x ^ 2 )  =  y  <->  ( x ^ 2 )  =  A ) )
21anbi1d 460 . . 3  |-  ( y  =  A  ->  (
( ( x ^
2 )  =  y  /\  0  <_  x
)  <->  ( ( x ^ 2 )  =  A  /\  0  <_  x ) ) )
32riotabidv 5700 . 2  |-  ( y  =  A  ->  ( iota_ x  e.  RR  (
( x ^ 2 )  =  y  /\  0  <_  x ) )  =  ( iota_ x  e.  RR  ( ( x ^ 2 )  =  A  /\  0  <_  x ) ) )
4 df-rsqrt 10738 . 2  |-  sqr  =  ( y  e.  RR  |->  ( iota_ x  e.  RR  ( ( x ^
2 )  =  y  /\  0  <_  x
) ) )
5 reex 7722 . . 3  |-  RR  e.  _V
6 riotaexg 5702 . . 3  |-  ( RR  e.  _V  ->  ( iota_ x  e.  RR  (
( x ^ 2 )  =  A  /\  0  <_  x ) )  e.  _V )
75, 6ax-mp 5 . 2  |-  ( iota_ x  e.  RR  ( ( x ^ 2 )  =  A  /\  0  <_  x ) )  e. 
_V
83, 4, 7fvmpt 5466 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 103    = wceq 1316    e. wcel 1465   _Vcvv 2660   class class class wbr 3899   ` cfv 5093   iota_crio 5697  (class class class)co 5742   RRcr 7587   0cc0 7588    <_ cle 7769   2c2 8739   ^cexp 10260   sqrcsqrt 10736
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-cnex 7679  ax-resscn 7680
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fv 5101  df-riota 5698  df-rsqrt 10738
This theorem is referenced by:  sqrt0  10744  resqrtcl  10769  rersqrtthlem  10770  sqrtsq  10784
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