Theorem zlmnm

Description: Norm of a ZZ -module (if present). (Contributed by Thierry Arnoux, 8-Nov-2017)

	Ref	Expression
Hypotheses	zlmlem2.1	⊢ 𝑊 = ( ℤMod ‘ 𝐺 )
	zlmnm.1	⊢ 𝑁 = ( norm ‘ 𝐺 )
Assertion	zlmnm	⊢ ( 𝐺 ∈ 𝑉 → 𝑁 = ( norm ‘ 𝑊 ) )

Proof

Step	Hyp	Ref	Expression
1		zlmlem2.1	⊢ 𝑊 = ( ℤMod ‘ 𝐺 )
2		zlmnm.1	⊢ 𝑁 = ( norm ‘ 𝐺 )
3		eqid	⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 )
4	1 3	zlmbas	⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝑊 )
5	4	a1i	⊢ ( 𝐺 ∈ 𝑉 → ( Base ‘ 𝐺 ) = ( Base ‘ 𝑊 ) )
6		eqid	⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 )
7	1 6	zlmplusg	⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝑊 )
8	7	a1i	⊢ ( 𝐺 ∈ 𝑉 → ( +g ‘ 𝐺 ) = ( +g ‘ 𝑊 ) )
9		eqid	⊢ ( dist ‘ 𝐺 ) = ( dist ‘ 𝐺 )
10	1 9	zlmds	⊢ ( 𝐺 ∈ 𝑉 → ( dist ‘ 𝐺 ) = ( dist ‘ 𝑊 ) )
11	5 8 10	nmpropd	⊢ ( 𝐺 ∈ 𝑉 → ( norm ‘ 𝐺 ) = ( norm ‘ 𝑊 ) )
12	2 11	eqtrid	⊢ ( 𝐺 ∈ 𝑉 → 𝑁 = ( norm ‘ 𝑊 ) )