Theorem tchval 23017

Description: Define a function to augment a subcomplex pre-Hilbert space with norm. (Contributed by Mario Carneiro, 7-Oct-2015.)

Hypotheses

Ref	Expression
tchval.n	|- G = (toCHil ` W )
tchval.v	|- V = ( Base ` W )
tchval.h	|- ., = ( .i ` W )

Assertion

Ref	Expression
tchval	|- G = ( W toNrmGrp ( x e. V |-> ( sqr ` ( x ., x ) ) ) )

Distinct variable groups:   x, .,   x, G   x, V   x, W

Proof of Theorem tchval

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tchval.n	. 2 |- G = (toCHil ` W )
2		id 22	. . . . 5 |- ( w = W -> w = W )
3		fveq2 6191	. . . . . . 7 |- ( w = W -> ( Base ` w ) = ( Base ` W ) )
4		tchval.v	. . . . . . 7 |- V = ( Base ` W )
5	3, 4	syl6eqr 2674	. . . . . 6 |- ( w = W -> ( Base ` w ) = V )
6		fveq2 6191	. . . . . . . . 9 |- ( w = W -> ( .i ` w ) = ( .i ` W ) )
7		tchval.h	. . . . . . . . 9 |- ., = ( .i ` W )
8	6, 7	syl6eqr 2674	. . . . . . . 8 |- ( w = W -> ( .i ` w ) = ., )
9	8	oveqd 6667	. . . . . . 7 |- ( w = W -> ( x ( .i ` w ) x ) = ( x ., x ) )
10	9	fveq2d 6195	. . . . . 6 |- ( w = W -> ( sqr ` ( x ( .i ` w ) x ) ) = ( sqr ` ( x ., x ) ) )
11	5, 10	mpteq12dv 4733	. . . . 5 |- ( w = W -> ( x e. ( Base ` w ) |-> ( sqr ` ( x ( .i ` w ) x ) ) ) = ( x e. V |-> ( sqr ` ( x ., x ) ) ) )
12	2, 11	oveq12d 6668	. . . 4 |- ( w = W -> ( w toNrmGrp ( x e. ( Base ` w ) |-> ( sqr ` ( x ( .i ` w ) x ) ) ) ) = ( W toNrmGrp ( x e. V |-> ( sqr ` ( x ., x ) ) ) ) )
13		df-tch 22969	. . . 4 |- toCHil = ( w e. _V |-> ( w toNrmGrp ( x e. ( Base ` w ) |-> ( sqr ` ( x ( .i ` w ) x ) ) ) ) )
14		ovex 6678	. . . 4 |- ( W toNrmGrp ( x e. V |-> ( sqr ` ( x ., x ) ) ) ) e. _V
15	12, 13, 14	fvmpt 6282	. . 3 |- ( W e. _V -> (toCHil ` W ) = ( W toNrmGrp ( x e. V |-> ( sqr ` ( x ., x ) ) ) ) )
16		fvprc 6185	. . . 4 |- ( -. W e. _V -> (toCHil ` W ) = (/) )
17		reldmtng 22442	. . . . 5 |- Rel dom toNrmGrp
18	17	ovprc1 6684	. . . 4 |- ( -. W e. _V -> ( W toNrmGrp ( x e. V |-> ( sqr ` ( x ., x ) ) ) ) = (/) )
19	16, 18	eqtr4d 2659	. . 3 |- ( -. W e. _V -> (toCHil ` W ) = ( W toNrmGrp ( x e. V |-> ( sqr ` ( x ., x ) ) ) ) )
20	15, 19	pm2.61i 176	. 2 |- (toCHil ` W ) = ( W toNrmGrp ( x e. V |-> ( sqr ` ( x ., x ) ) ) )
21	1, 20	eqtri 2644	1 |- G = ( W toNrmGrp ( x e. V |-> ( sqr ` ( x ., x ) ) ) )

Syntax hints:   -. wn 3   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   sqrcsqrt 13973   Basecbs 15857   .icip 15946   toNrmGrp ctng 22383  toCHilctch 22967

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-8 1992  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-pow 4843  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-ov 6653  df-oprab 6654  df-mpt2 6655  df-tng 22389  df-tch 22969

This theorem is referenced by:  tchbas  23018  tchplusg  23019  tchmulr  23021  tchsca  23022  tchvsca  23023  tchip  23024  tchtopn  23025  tchnmfval  23027  tchds  23030  tchcph  23036