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

Metamath Proof Explorer

Theorem slmdvs1

Description: Scalar product with ring unity. ( ax-hvmulid analog.) (Contributed by NM, 10-Jan-2014) (Revised by Mario Carneiro, 19-Jun-2014) (Revised by Thierry Arnoux, 1-Apr-2018)

Ref Expression
Hypotheses slmdvs1.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
slmdvs1.f ⊢ 𝐹 = ( Scalar ‘ 𝑊 )
slmdvs1.s ⊢ · = ( ·𝑠 ‘ 𝑊 )
slmdvs1.u ⊢ 1 = ( 1r ‘ 𝐹 )
Assertion slmdvs1 ( ( 𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉 ) → ( 1 · 𝑋 ) = 𝑋 )

Proof

Step Hyp Ref Expression
1 slmdvs1.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
2 slmdvs1.f ⊢ 𝐹 = ( Scalar ‘ 𝑊 )
3 slmdvs1.s ⊢ · = ( ·𝑠 ‘ 𝑊 )
4 slmdvs1.u ⊢ 1 = ( 1r ‘ 𝐹 )
5 simpl ⊢ ( ( 𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉 ) → 𝑊 ∈ SLMod )
6 eqid ⊢ ( Base ‘ 𝐹 ) = ( Base ‘ 𝐹 )
7 2 6 4 slmd1cl ⊢ ( 𝑊 ∈ SLMod → 1 ∈ ( Base ‘ 𝐹 ) )
8 7 adantr ⊢ ( ( 𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉 ) → 1 ∈ ( Base ‘ 𝐹 ) )
9 simpr ⊢ ( ( 𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉 ) → 𝑋 ∈ 𝑉 )
10 eqid ⊢ ( +g ‘ 𝑊 ) = ( +g ‘ 𝑊 )
11 eqid ⊢ ( 0g ‘ 𝑊 ) = ( 0g ‘ 𝑊 )
12 eqid ⊢ ( +g ‘ 𝐹 ) = ( +g ‘ 𝐹 )
13 eqid ⊢ ( .r ‘ 𝐹 ) = ( .r ‘ 𝐹 )
14 eqid ⊢ ( 0g ‘ 𝐹 ) = ( 0g ‘ 𝐹 )
15 1 10 3 11 2 6 12 13 4 14 slmdlema ⊢ ( ( 𝑊 ∈ SLMod ∧ ( 1 ∈ ( Base ‘ 𝐹 ) ∧ 1 ∈ ( Base ‘ 𝐹 ) ) ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ) ) → ( ( ( 1 · 𝑋 ) ∈ 𝑉 ∧ ( 1 · ( 𝑋 ( +g ‘ 𝑊 ) 𝑋 ) ) = ( ( 1 · 𝑋 ) ( +g ‘ 𝑊 ) ( 1 · 𝑋 ) ) ∧ ( ( 1 ( +g ‘ 𝐹 ) 1 ) · 𝑋 ) = ( ( 1 · 𝑋 ) ( +g ‘ 𝑊 ) ( 1 · 𝑋 ) ) ) ∧ ( ( ( 1 ( .r ‘ 𝐹 ) 1 ) · 𝑋 ) = ( 1 · ( 1 · 𝑋 ) ) ∧ ( 1 · 𝑋 ) = 𝑋 ∧ ( ( 0g ‘ 𝐹 ) · 𝑋 ) = ( 0g ‘ 𝑊 ) ) ) )
16 15 simprd ⊢ ( ( 𝑊 ∈ SLMod ∧ ( 1 ∈ ( Base ‘ 𝐹 ) ∧ 1 ∈ ( Base ‘ 𝐹 ) ) ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ) ) → ( ( ( 1 ( .r ‘ 𝐹 ) 1 ) · 𝑋 ) = ( 1 · ( 1 · 𝑋 ) ) ∧ ( 1 · 𝑋 ) = 𝑋 ∧ ( ( 0g ‘ 𝐹 ) · 𝑋 ) = ( 0g ‘ 𝑊 ) ) )
17 16 simp2d ⊢ ( ( 𝑊 ∈ SLMod ∧ ( 1 ∈ ( Base ‘ 𝐹 ) ∧ 1 ∈ ( Base ‘ 𝐹 ) ) ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ) ) → ( 1 · 𝑋 ) = 𝑋 )
18 5 8 8 9 9 17 syl122anc ⊢ ( ( 𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉 ) → ( 1 · 𝑋 ) = 𝑋 )