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Metamath Proof Explorer

Theorem mavmulval

Description: Multiplication of a vector with a square matrix. (Contributed by AV, 23-Feb-2019)

Ref Expression
Hypotheses mavmulval.a ⊢ 𝐴 = ( 𝑁 Mat 𝑅 )
mavmulval.m ⊢ × = ( 𝑅 maVecMul ⟨ 𝑁 , 𝑁 ⟩ )
mavmulval.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
mavmulval.t ⊢ · = ( .r ‘ 𝑅 )
mavmulval.r ⊢ ( 𝜑 → 𝑅 ∈ 𝑉 )
mavmulval.n ⊢ ( 𝜑 → 𝑁 ∈ Fin )
mavmulval.x ⊢ ( 𝜑 → 𝑋 ∈ ( Base ‘ 𝐴 ) )
mavmulval.y ⊢ ( 𝜑 → 𝑌 ∈ ( 𝐵 ↑m 𝑁 ) )
Assertion mavmulval ( 𝜑 → ( 𝑋 × 𝑌 ) = ( 𝑖 ∈ 𝑁 ↦ ( 𝑅 Σg ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑋 𝑗 ) · ( 𝑌 ‘ 𝑗 ) ) ) ) ) )

Proof

Step Hyp Ref Expression
1 mavmulval.a ⊢ 𝐴 = ( 𝑁 Mat 𝑅 )
2 mavmulval.m ⊢ × = ( 𝑅 maVecMul ⟨ 𝑁 , 𝑁 ⟩ )
3 mavmulval.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
4 mavmulval.t ⊢ · = ( .r ‘ 𝑅 )
5 mavmulval.r ⊢ ( 𝜑 → 𝑅 ∈ 𝑉 )
6 mavmulval.n ⊢ ( 𝜑 → 𝑁 ∈ Fin )
7 mavmulval.x ⊢ ( 𝜑 → 𝑋 ∈ ( Base ‘ 𝐴 ) )
8 mavmulval.y ⊢ ( 𝜑 → 𝑌 ∈ ( 𝐵 ↑m 𝑁 ) )
9 1 3 matbas2 ⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ) → ( 𝐵 ↑m ( 𝑁 × 𝑁 ) ) = ( Base ‘ 𝐴 ) )
10 6 5 9 syl2anc ⊢ ( 𝜑 → ( 𝐵 ↑m ( 𝑁 × 𝑁 ) ) = ( Base ‘ 𝐴 ) )
11 7 10 eleqtrrd ⊢ ( 𝜑 → 𝑋 ∈ ( 𝐵 ↑m ( 𝑁 × 𝑁 ) ) )
12 2 3 4 5 6 6 11 8 mvmulval ⊢ ( 𝜑 → ( 𝑋 × 𝑌 ) = ( 𝑖 ∈ 𝑁 ↦ ( 𝑅 Σg ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑋 𝑗 ) · ( 𝑌 ‘ 𝑗 ) ) ) ) ) )