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Metamath Proof Explorer

Theorem mamuval

Description: Multiplication of two matrices. (Contributed by Stefan O'Rear, 2-Sep-2015)

Ref Expression
Hypotheses mamufval.f ⊢ 𝐹 = ( 𝑅 maMul ⟨ 𝑀 , 𝑁 , 𝑃 ⟩ )
mamufval.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
mamufval.t ⊢ · = ( .r ‘ 𝑅 )
mamufval.r ⊢ ( 𝜑 → 𝑅 ∈ 𝑉 )
mamufval.m ⊢ ( 𝜑 → 𝑀 ∈ Fin )
mamufval.n ⊢ ( 𝜑 → 𝑁 ∈ Fin )
mamufval.p ⊢ ( 𝜑 → 𝑃 ∈ Fin )
mamuval.x ⊢ ( 𝜑 → 𝑋 ∈ ( 𝐵 ↑m ( 𝑀 × 𝑁 ) ) )
mamuval.y ⊢ ( 𝜑 → 𝑌 ∈ ( 𝐵 ↑m ( 𝑁 × 𝑃 ) ) )
Assertion mamuval ( 𝜑 → ( 𝑋 𝐹 𝑌 ) = ( 𝑖 ∈ 𝑀 , 𝑘 ∈ 𝑃 ↦ ( 𝑅 Σg ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑋 𝑗 ) · ( 𝑗 𝑌 𝑘 ) ) ) ) ) )

Proof

Step Hyp Ref Expression
1 mamufval.f ⊢ 𝐹 = ( 𝑅 maMul ⟨ 𝑀 , 𝑁 , 𝑃 ⟩ )
2 mamufval.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
3 mamufval.t ⊢ · = ( .r ‘ 𝑅 )
4 mamufval.r ⊢ ( 𝜑 → 𝑅 ∈ 𝑉 )
5 mamufval.m ⊢ ( 𝜑 → 𝑀 ∈ Fin )
6 mamufval.n ⊢ ( 𝜑 → 𝑁 ∈ Fin )
7 mamufval.p ⊢ ( 𝜑 → 𝑃 ∈ Fin )
8 mamuval.x ⊢ ( 𝜑 → 𝑋 ∈ ( 𝐵 ↑m ( 𝑀 × 𝑁 ) ) )
9 mamuval.y ⊢ ( 𝜑 → 𝑌 ∈ ( 𝐵 ↑m ( 𝑁 × 𝑃 ) ) )
10 1 2 3 4 5 6 7 mamufval ⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ ( 𝐵 ↑m ( 𝑀 × 𝑁 ) ) , 𝑦 ∈ ( 𝐵 ↑m ( 𝑁 × 𝑃 ) ) ↦ ( 𝑖 ∈ 𝑀 , 𝑘 ∈ 𝑃 ↦ ( 𝑅 Σg ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑥 𝑗 ) · ( 𝑗 𝑦 𝑘 ) ) ) ) ) ) )
11 oveq ⊢ ( 𝑥 = 𝑋 → ( 𝑖 𝑥 𝑗 ) = ( 𝑖 𝑋 𝑗 ) )
12 oveq ⊢ ( 𝑦 = 𝑌 → ( 𝑗 𝑦 𝑘 ) = ( 𝑗 𝑌 𝑘 ) )
13 11 12 oveqan12d ⊢ ( ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) → ( ( 𝑖 𝑥 𝑗 ) · ( 𝑗 𝑦 𝑘 ) ) = ( ( 𝑖 𝑋 𝑗 ) · ( 𝑗 𝑌 𝑘 ) ) )
14 13 adantl ⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → ( ( 𝑖 𝑥 𝑗 ) · ( 𝑗 𝑦 𝑘 ) ) = ( ( 𝑖 𝑋 𝑗 ) · ( 𝑗 𝑌 𝑘 ) ) )
15 14 mpteq2dv ⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑥 𝑗 ) · ( 𝑗 𝑦 𝑘 ) ) ) = ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑋 𝑗 ) · ( 𝑗 𝑌 𝑘 ) ) ) )
16 15 oveq2d ⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → ( 𝑅 Σg ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑥 𝑗 ) · ( 𝑗 𝑦 𝑘 ) ) ) ) = ( 𝑅 Σg ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑋 𝑗 ) · ( 𝑗 𝑌 𝑘 ) ) ) ) )
17 16 mpoeq3dv ⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → ( 𝑖 ∈ 𝑀 , 𝑘 ∈ 𝑃 ↦ ( 𝑅 Σg ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑥 𝑗 ) · ( 𝑗 𝑦 𝑘 ) ) ) ) ) = ( 𝑖 ∈ 𝑀 , 𝑘 ∈ 𝑃 ↦ ( 𝑅 Σg ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑋 𝑗 ) · ( 𝑗 𝑌 𝑘 ) ) ) ) ) )
18 mpoexga ⊢ ( ( 𝑀 ∈ Fin ∧ 𝑃 ∈ Fin ) → ( 𝑖 ∈ 𝑀 , 𝑘 ∈ 𝑃 ↦ ( 𝑅 Σg ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑋 𝑗 ) · ( 𝑗 𝑌 𝑘 ) ) ) ) ) ∈ V )
19 5 7 18 syl2anc ⊢ ( 𝜑 → ( 𝑖 ∈ 𝑀 , 𝑘 ∈ 𝑃 ↦ ( 𝑅 Σg ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑋 𝑗 ) · ( 𝑗 𝑌 𝑘 ) ) ) ) ) ∈ V )
20 10 17 8 9 19 ovmpod ⊢ ( 𝜑 → ( 𝑋 𝐹 𝑌 ) = ( 𝑖 ∈ 𝑀 , 𝑘 ∈ 𝑃 ↦ ( 𝑅 Σg ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑋 𝑗 ) · ( 𝑗 𝑌 𝑘 ) ) ) ) ) )