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Metamath Proof Explorer

Theorem slmdvsass

Description: Associative law for scalar product. ( ax-hvmulass analog.) (Contributed by NM, 10-Jan-2014) (Revised by Mario Carneiro, 22-Sep-2015) (Revised by Thierry Arnoux, 1-Apr-2018)

Ref Expression
Hypotheses slmdvsass.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
slmdvsass.f ⊢ 𝐹 = ( Scalar ‘ 𝑊 )
slmdvsass.s ⊢ · = ( ·𝑠 ‘ 𝑊 )
slmdvsass.k ⊢ 𝐾 = ( Base ‘ 𝐹 )
slmdvsass.t ⊢ × = ( .r ‘ 𝐹 )
Assertion slmdvsass ( ( 𝑊 ∈ SLMod ∧ ( 𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ) ) → ( ( 𝑄 × 𝑅 ) · 𝑋 ) = ( 𝑄 · ( 𝑅 · 𝑋 ) ) )

Proof

Step Hyp Ref Expression
1 slmdvsass.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
2 slmdvsass.f ⊢ 𝐹 = ( Scalar ‘ 𝑊 )
3 slmdvsass.s ⊢ · = ( ·𝑠 ‘ 𝑊 )
4 slmdvsass.k ⊢ 𝐾 = ( Base ‘ 𝐹 )
5 slmdvsass.t ⊢ × = ( .r ‘ 𝐹 )
6 eqid ⊢ ( +g ‘ 𝑊 ) = ( +g ‘ 𝑊 )
7 eqid ⊢ ( 0g ‘ 𝑊 ) = ( 0g ‘ 𝑊 )
8 eqid ⊢ ( +g ‘ 𝐹 ) = ( +g ‘ 𝐹 )
9 eqid ⊢ ( 1r ‘ 𝐹 ) = ( 1r ‘ 𝐹 )
10 eqid ⊢ ( 0g ‘ 𝐹 ) = ( 0g ‘ 𝐹 )
11 1 6 3 7 2 4 8 5 9 10 slmdlema ⊢ ( ( 𝑊 ∈ SLMod ∧ ( 𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ) ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ) ) → ( ( ( 𝑅 · 𝑋 ) ∈ 𝑉 ∧ ( 𝑅 · ( 𝑋 ( +g ‘ 𝑊 ) 𝑋 ) ) = ( ( 𝑅 · 𝑋 ) ( +g ‘ 𝑊 ) ( 𝑅 · 𝑋 ) ) ∧ ( ( 𝑄 ( +g ‘ 𝐹 ) 𝑅 ) · 𝑋 ) = ( ( 𝑄 · 𝑋 ) ( +g ‘ 𝑊 ) ( 𝑅 · 𝑋 ) ) ) ∧ ( ( ( 𝑄 × 𝑅 ) · 𝑋 ) = ( 𝑄 · ( 𝑅 · 𝑋 ) ) ∧ ( ( 1r ‘ 𝐹 ) · 𝑋 ) = 𝑋 ∧ ( ( 0g ‘ 𝐹 ) · 𝑋 ) = ( 0g ‘ 𝑊 ) ) ) )
12 11 simprd ⊢ ( ( 𝑊 ∈ SLMod ∧ ( 𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ) ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ) ) → ( ( ( 𝑄 × 𝑅 ) · 𝑋 ) = ( 𝑄 · ( 𝑅 · 𝑋 ) ) ∧ ( ( 1r ‘ 𝐹 ) · 𝑋 ) = 𝑋 ∧ ( ( 0g ‘ 𝐹 ) · 𝑋 ) = ( 0g ‘ 𝑊 ) ) )
13 12 simp1d ⊢ ( ( 𝑊 ∈ SLMod ∧ ( 𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ) ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ) ) → ( ( 𝑄 × 𝑅 ) · 𝑋 ) = ( 𝑄 · ( 𝑅 · 𝑋 ) ) )
14 13 3expa ⊢ ( ( ( 𝑊 ∈ SLMod ∧ ( 𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ) ) ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ) ) → ( ( 𝑄 × 𝑅 ) · 𝑋 ) = ( 𝑄 · ( 𝑅 · 𝑋 ) ) )
15 14 anabsan2 ⊢ ( ( ( 𝑊 ∈ SLMod ∧ ( 𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ) ) ∧ 𝑋 ∈ 𝑉 ) → ( ( 𝑄 × 𝑅 ) · 𝑋 ) = ( 𝑄 · ( 𝑅 · 𝑋 ) ) )
16 15 exp42 ⊢ ( 𝑊 ∈ SLMod → ( 𝑄 ∈ 𝐾 → ( 𝑅 ∈ 𝐾 → ( 𝑋 ∈ 𝑉 → ( ( 𝑄 × 𝑅 ) · 𝑋 ) = ( 𝑄 · ( 𝑅 · 𝑋 ) ) ) ) ) )
17 16 3imp2 ⊢ ( ( 𝑊 ∈ SLMod ∧ ( 𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ) ) → ( ( 𝑄 × 𝑅 ) · 𝑋 ) = ( 𝑄 · ( 𝑅 · 𝑋 ) ) )