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	$$⊢G=\mathrm{toCPreHil}\left(W\right)$$
	tcphip.s	$$⊢\underset{\dot{}}{·}={\cdot }_{𝑖}\left(W\right)$$
Assertion	tcphip	$$⊢\underset{\dot{}}{·}={\cdot }_{𝑖}\left(G\right)$$

Proof

Step	Hyp	Ref	Expression
1		tcphval.n	$$⊢G=\mathrm{toCPreHil}\left(W\right)$$
2		tcphip.s	$$⊢\underset{\dot{}}{·}={\cdot }_{𝑖}\left(W\right)$$
3		eqid	$$⊢{\mathrm{Base}}_W={\mathrm{Base}}_W$$
4	3	tcphex	$$⊢\left(x\in {\mathrm{Base}}_W⟼\sqrt{x\underset{\dot{}}{·}x}\right)\in \mathrm{V}$$
5	1 3 2	tcphval	$$⊢G=W\mathrm{toNrmGrp}\left(x\in {\mathrm{Base}}_W⟼\sqrt{x\underset{\dot{}}{·}x}\right)$$
6	5 2	tngip	$$⊢\left(x\in {\mathrm{Base}}_W⟼\sqrt{x\underset{\dot{}}{·}x}\right)\in \mathrm{V}\to \underset{\dot{}}{·}={\cdot }_{𝑖}\left(G\right)$$
7	4 6	ax-mp	$$⊢\underset{\dot{}}{·}={\cdot }_{𝑖}\left(G\right)$$