Theorem tngsca

Description: The scalar ring 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 )
	tngsca.2	⊢ 𝐹 = ( Scalar ‘ 𝐺 )
Assertion	tngsca	⊢ ( 𝑁 ∈ 𝑉 → 𝐹 = ( Scalar ‘ 𝑇 ) )

Proof

Step	Hyp	Ref	Expression
1		tngbas.t	⊢ 𝑇 = ( 𝐺 toNrmGrp 𝑁 )
2		tngsca.2	⊢ 𝐹 = ( Scalar ‘ 𝐺 )
3		scaid	⊢ Scalar = Slot ( Scalar ‘ ndx )
4		slotstnscsi	⊢ ( ( TopSet ‘ ndx ) ≠ ( Scalar ‘ ndx ) ∧ ( TopSet ‘ ndx ) ≠ ( ·𝑠 ‘ ndx ) ∧ ( TopSet ‘ ndx ) ≠ ( ·𝑖 ‘ ndx ) )
5	4	simp1i	⊢ ( TopSet ‘ ndx ) ≠ ( Scalar ‘ ndx )
6	5	necomi	⊢ ( Scalar ‘ ndx ) ≠ ( TopSet ‘ ndx )
7		slotsdnscsi	⊢ ( ( dist ‘ ndx ) ≠ ( Scalar ‘ ndx ) ∧ ( dist ‘ ndx ) ≠ ( ·𝑠 ‘ ndx ) ∧ ( dist ‘ ndx ) ≠ ( ·𝑖 ‘ ndx ) )
8	7	simp1i	⊢ ( dist ‘ ndx ) ≠ ( Scalar ‘ ndx )
9	8	necomi	⊢ ( Scalar ‘ ndx ) ≠ ( dist ‘ ndx )
10	1 3 6 9	tnglem	⊢ ( 𝑁 ∈ 𝑉 → ( Scalar ‘ 𝐺 ) = ( Scalar ‘ 𝑇 ) )
11	2 10	eqtrid	⊢ ( 𝑁 ∈ 𝑉 → 𝐹 = ( Scalar ‘ 𝑇 ) )