Theorem clmvs1

Description: Scalar product with ring unity. ( lmodvs1 analog.) (Contributed by Mario Carneiro, 16-Oct-2015)

	Ref	Expression
Hypotheses	clmvs1.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
	clmvs1.s	⊢ · = ( ·𝑠 ‘ 𝑊 )
Assertion	clmvs1	⊢ ( ( 𝑊 ∈ ℂMod ∧ 𝑋 ∈ 𝑉 ) → ( 1 · 𝑋 ) = 𝑋 )

Proof

Step	Hyp	Ref	Expression
1		clmvs1.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
2		clmvs1.s	⊢ · = ( ·𝑠 ‘ 𝑊 )
3		eqid	⊢ ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑊 )
4	3	clm1	⊢ ( 𝑊 ∈ ℂMod → 1 = ( 1r ‘ ( Scalar ‘ 𝑊 ) ) )
5	4	adantr	⊢ ( ( 𝑊 ∈ ℂMod ∧ 𝑋 ∈ 𝑉 ) → 1 = ( 1r ‘ ( Scalar ‘ 𝑊 ) ) )
6	5	oveq1d	⊢ ( ( 𝑊 ∈ ℂMod ∧ 𝑋 ∈ 𝑉 ) → ( 1 · 𝑋 ) = ( ( 1r ‘ ( Scalar ‘ 𝑊 ) ) · 𝑋 ) )
7		clmlmod	⊢ ( 𝑊 ∈ ℂMod → 𝑊 ∈ LMod )
8		eqid	⊢ ( 1r ‘ ( Scalar ‘ 𝑊 ) ) = ( 1r ‘ ( Scalar ‘ 𝑊 ) )
9	1 3 2 8	lmodvs1	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( ( 1r ‘ ( Scalar ‘ 𝑊 ) ) · 𝑋 ) = 𝑋 )
10	7 9	sylan	⊢ ( ( 𝑊 ∈ ℂMod ∧ 𝑋 ∈ 𝑉 ) → ( ( 1r ‘ ( Scalar ‘ 𝑊 ) ) · 𝑋 ) = 𝑋 )
11	6 10	eqtrd	⊢ ( ( 𝑊 ∈ ℂMod ∧ 𝑋 ∈ 𝑉 ) → ( 1 · 𝑋 ) = 𝑋 )