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Theorem sqrtrval 10964
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 2180 . . . 4  |-  ( y  =  A  ->  (
( x ^ 2 )  =  y  <->  ( x ^ 2 )  =  A ) )
21anbi1d 462 . . 3  |-  ( y  =  A  ->  (
( ( x ^
2 )  =  y  /\  0  <_  x
)  <->  ( ( x ^ 2 )  =  A  /\  0  <_  x ) ) )
32riotabidv 5811 . 2  |-  ( y  =  A  ->  ( iota_ x  e.  RR  (
( x ^ 2 )  =  y  /\  0  <_  x ) )  =  ( iota_ x  e.  RR  ( ( x ^ 2 )  =  A  /\  0  <_  x ) ) )
4 df-rsqrt 10962 . 2  |-  sqr  =  ( y  e.  RR  |->  ( iota_ x  e.  RR  ( ( x ^
2 )  =  y  /\  0  <_  x
) ) )
5 reex 7908 . . 3  |-  RR  e.  _V
6 riotaexg 5813 . . 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 5573 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 1348    e. wcel 2141   _Vcvv 2730   class class class wbr 3989   ` cfv 5198   iota_crio 5808  (class class class)co 5853   RRcr 7773   0cc0 7774    <_ cle 7955   2c2 8929   ^cexp 10475   sqrcsqrt 10960
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-cnex 7865  ax-resscn 7866
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-rsqrt 10962
This theorem is referenced by:  sqrt0  10968  resqrtcl  10993  rersqrtthlem  10994  sqrtsq  11008
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