Theorem clm0

Description: The zero of the scalar ring of a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015)

	Ref	Expression
Hypothesis	clm0.f	⊢ 𝐹 = ( Scalar ‘ 𝑊 )
Assertion	clm0	⊢ ( 𝑊 ∈ ℂMod → 0 = ( 0g ‘ 𝐹 ) )

Proof

Step	Hyp	Ref	Expression
1		clm0.f	⊢ 𝐹 = ( Scalar ‘ 𝑊 )
2		eqid	⊢ ( Base ‘ 𝐹 ) = ( Base ‘ 𝐹 )
3	1 2	clmsubrg	⊢ ( 𝑊 ∈ ℂMod → ( Base ‘ 𝐹 ) ∈ ( SubRing ‘ ℂfld ) )
4		eqid	⊢ ( ℂfld ↾s ( Base ‘ 𝐹 ) ) = ( ℂfld ↾s ( Base ‘ 𝐹 ) )
5		cnfld0	⊢ 0 = ( 0g ‘ ℂfld )
6	4 5	subrg0	⊢ ( ( Base ‘ 𝐹 ) ∈ ( SubRing ‘ ℂfld ) → 0 = ( 0g ‘ ( ℂfld ↾s ( Base ‘ 𝐹 ) ) ) )
7	3 6	syl	⊢ ( 𝑊 ∈ ℂMod → 0 = ( 0g ‘ ( ℂfld ↾s ( Base ‘ 𝐹 ) ) ) )
8	1 2	clmsca	⊢ ( 𝑊 ∈ ℂMod → 𝐹 = ( ℂfld ↾s ( Base ‘ 𝐹 ) ) )
9	8	fveq2d	⊢ ( 𝑊 ∈ ℂMod → ( 0g ‘ 𝐹 ) = ( 0g ‘ ( ℂfld ↾s ( Base ‘ 𝐹 ) ) ) )
10	7 9	eqtr4d	⊢ ( 𝑊 ∈ ℂMod → 0 = ( 0g ‘ 𝐹 ) )