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Metamath Proof Explorer

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 ) )