Theorem tcphip

Description: The inner product of a subcomplex pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015)

	Ref	Expression
Hypotheses	tcphval.n	|- G = ( toCPreHil ` W )
	tcphip.s	|- .x. = ( .i ` W )
Assertion	tcphip	|- .x. = ( .i ` G )

Proof

Step	Hyp	Ref	Expression
1		tcphval.n	|- G = ( toCPreHil ` W )
2		tcphip.s	|- .x. = ( .i ` W )
3		eqid	|- ( Base ` W ) = ( Base ` W )
4	3	tcphex	|- ( x e. ( Base ` W ) |-> ( sqrt ` ( x .x. x ) ) ) e. _V
5	1 3 2	tcphval	|- G = ( W toNrmGrp ( x e. ( Base ` W ) |-> ( sqrt ` ( x .x. x ) ) ) )
6	5 2	tngip	|- ( ( x e. ( Base ` W ) |-> ( sqrt ` ( x .x. x ) ) ) e. _V -> .x. = ( .i ` G ) )
7	4 6	ax-mp	|- .x. = ( .i ` G )