Theorem tngip

Description: The inner product operation of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015)

	Ref	Expression
Hypotheses	tngbas.t	$$⊢T=G\mathrm{toNrmGrp}N$$
	tngip.2	$$⊢\underset{\dot{}}{,}={\cdot }_{𝑖}\left(G\right)$$
Assertion	tngip	$$⊢N\in V\to \underset{\dot{}}{,}={\cdot }_{𝑖}\left(T\right)$$

Proof

Step	Hyp	Ref	Expression
1		tngbas.t	$$⊢T=G\mathrm{toNrmGrp}N$$
2		tngip.2	$$⊢\underset{\dot{}}{,}={\cdot }_{𝑖}\left(G\right)$$
3		df-ip	$$⊢{\cdot }_{𝑖}=\mathrm{Slot}8$$
4		8nn	$$⊢8\in \mathbb{N}$$
5		8lt9	$$⊢8<9$$
6	1 3 4 5	tnglem	$$⊢N\in V\to {\cdot }_{𝑖}\left(G\right)={\cdot }_{𝑖}\left(T\right)$$
7	2 6	syl5eq	$$⊢N\in V\to \underset{\dot{}}{,}={\cdot }_{𝑖}\left(T\right)$$