Theorem sqrtval 13977

Description: Value of square root function. (Contributed by Mario Carneiro, 8-Jul-2013.)

Assertion

Ref	Expression
sqrtval	|- ( A e. CC -> ( sqr ` A ) = ( iota_ x e. CC ( ( x ^ 2 ) = A /\ 0 <_ ( Re ` x ) /\ ( _i x. x ) e/ RR+ ) ) )

Distinct variable group:   x, A

Proof of Theorem sqrtval

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq2 2633	. . . 4 |- ( y = A -> ( ( x ^ 2 ) = y <-> ( x ^ 2 ) = A ) )
2	1	3anbi1d 1403	. . 3 |- ( y = A -> ( ( ( x ^ 2 ) = y /\ 0 <_ ( Re ` x ) /\ ( _i x. x ) e/ RR+ ) <-> ( ( x ^ 2 ) = A /\ 0 <_ ( Re ` x ) /\ ( _i x. x ) e/ RR+ ) ) )
3	2	riotabidv 6613	. 2 |- ( y = A -> ( iota_ x e. CC ( ( x ^ 2 ) = y /\ 0 <_ ( Re ` x ) /\ ( _i x. x ) e/ RR+ ) ) = ( iota_ x e. CC ( ( x ^ 2 ) = A /\ 0 <_ ( Re ` x ) /\ ( _i x. x ) e/ RR+ ) ) )
4		df-sqrt 13975	. 2 |- sqr = ( y e. CC |-> ( iota_ x e. CC ( ( x ^ 2 ) = y /\ 0 <_ ( Re ` x ) /\ ( _i x. x ) e/ RR+ ) ) )
5		riotaex 6615	. 2 |- ( iota_ x e. CC ( ( x ^ 2 ) = A /\ 0 <_ ( Re ` x ) /\ ( _i x. x ) e/ RR+ ) ) e. _V
6	3, 4, 5	fvmpt 6282	1 |- ( A e. CC -> ( sqr ` A ) = ( iota_ x e. CC ( ( x ^ 2 ) = A /\ 0 <_ ( Re ` x ) /\ ( _i x. x ) e/ RR+ ) ) )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   e/ wnel 2897   class class class wbr 4653   ` cfv 5888   iota_crio 6610  (class class class)co 6650   CCcc 9934   0cc0 9936   _ici 9938   x. cmul 9941   <_ cle 10075   2c2 11070   RR+crp 11832   ^cexp 12860   Recre 13837   sqrcsqrt 13973

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-riota 6611  df-sqrt 13975

This theorem is referenced by:  sqrt0  13982  resqrtcl  13994  resqrtthlem  13995  sqrtneg  14008  sqrtcl  14101  sqrtthlem  14102