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Metamath Proof Explorer

Theorem lmodvsass

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

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

Proof

Step Hyp Ref Expression
1 lmodvsass.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
2 lmodvsass.f ⊢ 𝐹 = ( Scalar ‘ 𝑊 )
3 lmodvsass.s ⊢ · = ( ·𝑠 ‘ 𝑊 )
4 lmodvsass.k ⊢ 𝐾 = ( Base ‘ 𝐹 )
5 lmodvsass.t ⊢ × = ( .r ‘ 𝐹 )
6 eqid ⊢ ( +g ‘ 𝑊 ) = ( +g ‘ 𝑊 )
7 eqid ⊢ ( +g ‘ 𝐹 ) = ( +g ‘ 𝐹 )
8 eqid ⊢ ( 1r ‘ 𝐹 ) = ( 1r ‘ 𝐹 )
9 1 6 3 2 4 7 5 8 lmodlema ⊢ ( ( 𝑊 ∈ LMod ∧ ( 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 ) ) )
10 9 simprld ⊢ ( ( 𝑊 ∈ LMod ∧ ( 𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ) ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ) ) → ( ( 𝑄 × 𝑅 ) · 𝑋 ) = ( 𝑄 · ( 𝑅 · 𝑋 ) ) )
11 10 3expa ⊢ ( ( ( 𝑊 ∈ LMod ∧ ( 𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ) ) ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ) ) → ( ( 𝑄 × 𝑅 ) · 𝑋 ) = ( 𝑄 · ( 𝑅 · 𝑋 ) ) )
12 11 anabsan2 ⊢ ( ( ( 𝑊 ∈ LMod ∧ ( 𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ) ) ∧ 𝑋 ∈ 𝑉 ) → ( ( 𝑄 × 𝑅 ) · 𝑋 ) = ( 𝑄 · ( 𝑅 · 𝑋 ) ) )
13 12 exp42 ⊢ ( 𝑊 ∈ LMod → ( 𝑄 ∈ 𝐾 → ( 𝑅 ∈ 𝐾 → ( 𝑋 ∈ 𝑉 → ( ( 𝑄 × 𝑅 ) · 𝑋 ) = ( 𝑄 · ( 𝑅 · 𝑋 ) ) ) ) ) )
14 13 3imp2 ⊢ ( ( 𝑊 ∈ LMod ∧ ( 𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ) ) → ( ( 𝑄 × 𝑅 ) · 𝑋 ) = ( 𝑄 · ( 𝑅 · 𝑋 ) ) )