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 )