This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 )