Theorem sqrtrval 11230

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.

Step	Hyp	Ref	Expression
1		eqeq2 2214	. . . 4 |- ( y = A -> ( ( x ^ 2 ) = y <-> ( x ^ 2 ) = A ) )
2	1	anbi1d 465	. . 3 |- ( y = A -> ( ( ( x ^ 2 ) = y /\ 0 <_ x ) <-> ( ( x ^ 2 ) = A /\ 0 <_ x ) ) )
3	2	riotabidv 5891	. 2 |- ( y = A -> ( iota_ x e. RR ( ( x ^ 2 ) = y /\ 0 <_ x ) ) = ( iota_ x e. RR ( ( x ^ 2 ) = A /\ 0 <_ x ) ) )
4		df-rsqrt 11228	. 2 |- sqr = ( y e. RR |-> ( iota_ x e. RR ( ( x ^ 2 ) = y /\ 0 <_ x ) ) )
5		reex 8041	. . 3 |- RR e. _V
6		riotaexg 5893	. . 3 |- ( RR e. _V -> ( iota_ x e. RR ( ( x ^ 2 ) = A /\ 0 <_ x ) ) e. _V )
7	5, 6	ax-mp 5	. 2 |- ( iota_ x e. RR ( ( x ^ 2 ) = A /\ 0 <_ x ) ) e. _V
8	3, 4, 7	fvmpt 5650	1 |- ( A e. RR -> ( sqr ` A ) = ( iota_ x e. RR ( ( x ^ 2 ) = A /\ 0 <_ x ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1372   e. wcel 2175   _Vcvv 2771   class class class wbr 4043   ` cfv 5268   iota_crio 5888  (class class class)co 5934   RRcr 7906   0cc0 7907   <_ cle 8090   2c2 9069   ^cexp 10664   sqrcsqrt 11226

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4478  ax-cnex 7998  ax-resscn 7999

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-sbc 2998  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4338  df-xp 4679  df-rel 4680  df-cnv 4681  df-co 4682  df-dm 4683  df-iota 5229  df-fun 5270  df-fv 5276  df-riota 5889  df-rsqrt 11228

This theorem is referenced by:  sqrt0  11234  resqrtcl  11259  rersqrtthlem  11260  sqrtsq  11274