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Metamath Proof Explorer

Theorem nmfval2

Description: The value of the norm function 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 = ( 0_g ‘ 𝑊 )
	nmfval2.d	⊢ 𝐷 = ( dist ‘ 𝑊 )
	nmfval2.e	⊢ 𝐸 = ( 𝐷 ↾ ( 𝑋 × 𝑋 ) )
Assertion	nmfval2	⊢ ( 𝑊 ∈ Grp → 𝑁 = ( 𝑥 ∈ 𝑋 ↦ ( 𝑥 𝐸 0 ) ) )

Proof

Step	Hyp	Ref	Expression
1		nmfval2.n	⊢ 𝑁 = ( norm ‘ 𝑊 )
2		nmfval2.x	⊢ 𝑋 = ( Base ‘ 𝑊 )
3		nmfval2.z	⊢ 0 = ( 0_g ‘ 𝑊 )
4		nmfval2.d	⊢ 𝐷 = ( dist ‘ 𝑊 )
5		nmfval2.e	⊢ 𝐸 = ( 𝐷 ↾ ( 𝑋 × 𝑋 ) )
6	2 3	grpidcl	⊢ ( 𝑊 ∈ Grp → 0 ∈ 𝑋 )
7	1 2 3 4 5	nmfval0	⊢ ( 0 ∈ 𝑋 → 𝑁 = ( 𝑥 ∈ 𝑋 ↦ ( 𝑥 𝐸 0 ) ) )
8	6 7	syl	⊢ ( 𝑊 ∈ Grp → 𝑁 = ( 𝑥 ∈ 𝑋 ↦ ( 𝑥 𝐸 0 ) ) )