Theorem clmzlmvsca

Description: The scalar product of a subcomplex module matches the scalar product of the derived ZZ -module, which implies, together with zlmbas and zlmplusg , that any module over ZZ is structure-equivalent to the canonical ZZ -module ZModG . (Contributed by Mario Carneiro, 30-Oct-2015)

	Ref	Expression
Hypotheses	zlmclm.w	⊢ 𝑊 = ( ℤMod ‘ 𝐺 )
	clmzlmvsca.x	⊢ 𝑋 = ( Base ‘ 𝐺 )
Assertion	clmzlmvsca	⊢ ( ( 𝐺 ∈ ℂMod ∧ ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ 𝑋 ) ) → ( 𝐴 ( ·𝑠 ‘ 𝐺 ) 𝐵 ) = ( 𝐴 ( ·𝑠 ‘ 𝑊 ) 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		zlmclm.w	⊢ 𝑊 = ( ℤMod ‘ 𝐺 )
2		clmzlmvsca.x	⊢ 𝑋 = ( Base ‘ 𝐺 )
3		eqid	⊢ ( .g ‘ 𝐺 ) = ( .g ‘ 𝐺 )
4	1 3	zlmvsca	⊢ ( .g ‘ 𝐺 ) = ( ·𝑠 ‘ 𝑊 )
5	4	eqcomi	⊢ ( ·𝑠 ‘ 𝑊 ) = ( .g ‘ 𝐺 )
6		eqid	⊢ ( ·𝑠 ‘ 𝐺 ) = ( ·𝑠 ‘ 𝐺 )
7	2 5 6	clmmulg	⊢ ( ( 𝐺 ∈ ℂMod ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 ( ·𝑠 ‘ 𝑊 ) 𝐵 ) = ( 𝐴 ( ·𝑠 ‘ 𝐺 ) 𝐵 ) )
8	7	eqcomd	⊢ ( ( 𝐺 ∈ ℂMod ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 ( ·𝑠 ‘ 𝐺 ) 𝐵 ) = ( 𝐴 ( ·𝑠 ‘ 𝑊 ) 𝐵 ) )
9	8	3expb	⊢ ( ( 𝐺 ∈ ℂMod ∧ ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ 𝑋 ) ) → ( 𝐴 ( ·𝑠 ‘ 𝐺 ) 𝐵 ) = ( 𝐴 ( ·𝑠 ‘ 𝑊 ) 𝐵 ) )