Theorem sqrtrval 11766

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 2248	. . . 4 |- ( y = A -> ( ( x ^ 2 ) = y <-> ( x ^ 2 ) = A ) )
2	1	anbi1d 469	. . 3 |- ( y = A -> ( ( ( x ^ 2 ) = y /\ 0 <_ x ) <-> ( ( x ^ 2 ) = A /\ 0 <_ x ) ) )
3	2	riotabidv 6040	. 2 |- ( y = A -> ( iota_ x e. RR ( ( x ^ 2 ) = y /\ 0 <_ x ) ) = ( iota_ x e. RR ( ( x ^ 2 ) = A /\ 0 <_ x ) ) )
4		df-rsqrt 11764	. 2 |- sqr = ( y e. RR |-> ( iota_ x e. RR ( ( x ^ 2 ) = y /\ 0 <_ x ) ) )
5		reex 8313	. . 3 |- RR e. _V
6		riotaexg 6042	. . 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 5782	1 |- ( A e. RR -> ( sqr ` A ) = ( iota_ x e. RR ( ( x ^ 2 ) = A /\ 0 <_ x ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   _Vcvv 2821   class class class wbr 4130   ` cfv 5377   iota_crio 6037  (class class class)co 6085   RRcr 8178   0cc0 8179   <_ cle 8361   2c2 9355   ^cexp 10975   sqrcsqrt 11762

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-cnex 8270  ax-resscn 8271

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-iota 5337  df-fun 5379  df-fv 5385  df-riota 6038  df-rsqrt 11764

This theorem is used by:  sqrt0  11770  resqrtcl  11795  rersqrtthlem  11796  sqrtsq  11810