Theorem nmval2

Description: The value of the norm on a group as the distance restricted to the elements of the base set to zero. (Contributed by Mario Carneiro, 2-Oct-2015)

	Ref	Expression
Hypotheses	nmfval2.n	⊢ 𝑁 = ( norm ‘ 𝑊 )
	nmfval2.x	⊢ 𝑋 = ( Base ‘ 𝑊 )
	nmfval2.z	⊢ 0 = ( 0g ‘ 𝑊 )
	nmfval2.d	⊢ 𝐷 = ( dist ‘ 𝑊 )
	nmfval2.e	⊢ 𝐸 = ( 𝐷 ↾ ( 𝑋 × 𝑋 ) )
Assertion	nmval2	⊢ ( ( 𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑋 ) → ( 𝑁 ‘ 𝐴 ) = ( 𝐴 𝐸 0 ) )

Proof

Step	Hyp	Ref	Expression
1		nmfval2.n	⊢ 𝑁 = ( norm ‘ 𝑊 )
2		nmfval2.x	⊢ 𝑋 = ( Base ‘ 𝑊 )
3		nmfval2.z	⊢ 0 = ( 0g ‘ 𝑊 )
4		nmfval2.d	⊢ 𝐷 = ( dist ‘ 𝑊 )
5		nmfval2.e	⊢ 𝐸 = ( 𝐷 ↾ ( 𝑋 × 𝑋 ) )
6	1 2 3 4	nmval	⊢ ( 𝐴 ∈ 𝑋 → ( 𝑁 ‘ 𝐴 ) = ( 𝐴 𝐷 0 ) )
7	6	adantl	⊢ ( ( 𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑋 ) → ( 𝑁 ‘ 𝐴 ) = ( 𝐴 𝐷 0 ) )
8	5	oveqi	⊢ ( 𝐴 𝐸 0 ) = ( 𝐴 ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) 0 )
9		id	⊢ ( 𝐴 ∈ 𝑋 → 𝐴 ∈ 𝑋 )
10	2 3	grpidcl	⊢ ( 𝑊 ∈ Grp → 0 ∈ 𝑋 )
11		ovres	⊢ ( ( 𝐴 ∈ 𝑋 ∧ 0 ∈ 𝑋 ) → ( 𝐴 ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) 0 ) = ( 𝐴 𝐷 0 ) )
12	9 10 11	syl2anr	⊢ ( ( 𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑋 ) → ( 𝐴 ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) 0 ) = ( 𝐴 𝐷 0 ) )
13	8 12	eqtr2id	⊢ ( ( 𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑋 ) → ( 𝐴 𝐷 0 ) = ( 𝐴 𝐸 0 ) )
14	7 13	eqtrd	⊢ ( ( 𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑋 ) → ( 𝑁 ‘ 𝐴 ) = ( 𝐴 𝐸 0 ) )