Theorem tngip

Description: The inner product operation of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015) (Revised by AV, 31-Oct-2024)

	Ref	Expression
Hypotheses	tngbas.t	⊢ 𝑇 = ( 𝐺 toNrmGrp 𝑁 )
	tngip.2	⊢ , = ( ·𝑖 ‘ 𝐺 )
Assertion	tngip	⊢ ( 𝑁 ∈ 𝑉 → , = ( ·𝑖 ‘ 𝑇 ) )

Proof

Step	Hyp	Ref	Expression
1		tngbas.t	⊢ 𝑇 = ( 𝐺 toNrmGrp 𝑁 )
2		tngip.2	⊢ , = ( ·𝑖 ‘ 𝐺 )
3		ipid	⊢ ·𝑖 = Slot ( ·𝑖 ‘ ndx )
4		slotstnscsi	⊢ ( ( TopSet ‘ ndx ) ≠ ( Scalar ‘ ndx ) ∧ ( TopSet ‘ ndx ) ≠ ( ·𝑠 ‘ ndx ) ∧ ( TopSet ‘ ndx ) ≠ ( ·𝑖 ‘ ndx ) )
5	4	simp3i	⊢ ( TopSet ‘ ndx ) ≠ ( ·𝑖 ‘ ndx )
6	5	necomi	⊢ ( ·𝑖 ‘ ndx ) ≠ ( TopSet ‘ ndx )
7		slotsdnscsi	⊢ ( ( dist ‘ ndx ) ≠ ( Scalar ‘ ndx ) ∧ ( dist ‘ ndx ) ≠ ( ·𝑠 ‘ ndx ) ∧ ( dist ‘ ndx ) ≠ ( ·𝑖 ‘ ndx ) )
8	7	simp3i	⊢ ( dist ‘ ndx ) ≠ ( ·𝑖 ‘ ndx )
9	8	necomi	⊢ ( ·𝑖 ‘ ndx ) ≠ ( dist ‘ ndx )
10	1 3 6 9	tnglem	⊢ ( 𝑁 ∈ 𝑉 → ( ·𝑖 ‘ 𝐺 ) = ( ·𝑖 ‘ 𝑇 ) )
11	2 10	eqtrid	⊢ ( 𝑁 ∈ 𝑉 → , = ( ·𝑖 ‘ 𝑇 ) )