Theorem sqrtrval 11147

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 2203	. . . 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 5876	. 2 |- ( y = A -> ( iota_ x e. RR ( ( x ^ 2 ) = y /\ 0 <_ x ) ) = ( iota_ x e. RR ( ( x ^ 2 ) = A /\ 0 <_ x ) ) )
4		df-rsqrt 11145	. 2 |- sqr = ( y e. RR |-> ( iota_ x e. RR ( ( x ^ 2 ) = y /\ 0 <_ x ) ) )
5		reex 8008	. . 3 |- RR e. _V
6		riotaexg 5878	. . 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 5635	1 |- ( A e. RR -> ( sqr ` A ) = ( iota_ x e. RR ( ( x ^ 2 ) = A /\ 0 <_ x ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   _Vcvv 2760   class class class wbr 4030   ` cfv 5255   iota_crio 5873  (class class class)co 5919   RRcr 7873   0cc0 7874   <_ cle 8057   2c2 9035   ^cexp 10612   sqrcsqrt 11143

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-cnex 7965  ax-resscn 7966

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-iota 5216  df-fun 5257  df-fv 5263  df-riota 5874  df-rsqrt 11145

This theorem is referenced by:  sqrt0  11151  resqrtcl  11176  rersqrtthlem  11177  sqrtsq  11191