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Theorem sqrtrval 10765
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 2147 . . . 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 5725 . 2  |-  ( y  =  A  ->  ( iota_ x  e.  RR  (
( x ^ 2 )  =  y  /\  0  <_  x ) )  =  ( iota_ x  e.  RR  ( ( x ^ 2 )  =  A  /\  0  <_  x ) ) )
4 df-rsqrt 10763 . 2  |-  sqr  =  ( y  e.  RR  |->  ( iota_ x  e.  RR  ( ( x ^
2 )  =  y  /\  0  <_  x
) ) )
5 reex 7747 . . 3  |-  RR  e.  _V
6 riotaexg 5727 . . 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 5491 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 1331    e. wcel 1480   _Vcvv 2681   class class class wbr 3924   ` cfv 5118   iota_crio 5722  (class class class)co 5767   RRcr 7612   0cc0 7613    <_ cle 7794   2c2 8764   ^cexp 10285   sqrcsqrt 10761
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-cnex 7704  ax-resscn 7705
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-iota 5083  df-fun 5120  df-fv 5126  df-riota 5723  df-rsqrt 10763
This theorem is referenced by:  sqrt0  10769  resqrtcl  10794  rersqrtthlem  10795  sqrtsq  10809
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