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Metamath Proof Explorer

Theorem nmf

Description: The norm on a normed group is a function into the reals. (Contributed by Mario Carneiro, 4-Oct-2015)

Ref Expression
Hypotheses nmf.x ⊢ X = Base G
nmf.n ⊢ N = norm ⁡ G
Assertion nmf ⊢ G ∈ NrmGrp → N : X ⟶ ℝ

Proof

Step Hyp Ref Expression
1 nmf.x ⊢ X = Base G
2 nmf.n ⊢ N = norm ⁡ G
3 ngpgrp ⊢ G ∈ NrmGrp → G ∈ Grp
4 eqid ⊢ dist ⁡ G ↾ X × X = dist ⁡ G ↾ X × X
5 1 4 ngpmet ⊢ G ∈ NrmGrp → dist ⁡ G ↾ X × X ∈ Met ⁡ X
6 eqid ⊢ dist ⁡ G = dist ⁡ G
7 2 1 6 4 nmf2 ⊢ G ∈ Grp ∧ dist ⁡ G ↾ X × X ∈ Met ⁡ X → N : X ⟶ ℝ
8 3 5 7 syl2anc ⊢ G ∈ NrmGrp → N : X ⟶ ℝ