Theorem mamufv

Description: A cell in the 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 ) ) )
	mamufv.i	⊢ ( 𝜑 → 𝐼 ∈ 𝑀 )
	mamufv.k	⊢ ( 𝜑 → 𝐾 ∈ 𝑃 )
Assertion	mamufv	⊢ ( 𝜑 → ( 𝐼 ( 𝑋 𝐹 𝑌 ) 𝐾 ) = ( 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		mamufv.i	⊢ ( 𝜑 → 𝐼 ∈ 𝑀 )
11		mamufv.k	⊢ ( 𝜑 → 𝐾 ∈ 𝑃 )
12	1 2 3 4 5 6 7 8 9	mamuval	⊢ ( 𝜑 → ( 𝑋 𝐹 𝑌 ) = ( 𝑖 ∈ 𝑀 , 𝑘 ∈ 𝑃 ↦ ( 𝑅 Σg ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑋 𝑗 ) · ( 𝑗 𝑌 𝑘 ) ) ) ) ) )
13		oveq1	⊢ ( 𝑖 = 𝐼 → ( 𝑖 𝑋 𝑗 ) = ( 𝐼 𝑋 𝑗 ) )
14		oveq2	⊢ ( 𝑘 = 𝐾 → ( 𝑗 𝑌 𝑘 ) = ( 𝑗 𝑌 𝐾 ) )
15	13 14	oveqan12d	⊢ ( ( 𝑖 = 𝐼 ∧ 𝑘 = 𝐾 ) → ( ( 𝑖 𝑋 𝑗 ) · ( 𝑗 𝑌 𝑘 ) ) = ( ( 𝐼 𝑋 𝑗 ) · ( 𝑗 𝑌 𝐾 ) ) )
16	15	adantl	⊢ ( ( 𝜑 ∧ ( 𝑖 = 𝐼 ∧ 𝑘 = 𝐾 ) ) → ( ( 𝑖 𝑋 𝑗 ) · ( 𝑗 𝑌 𝑘 ) ) = ( ( 𝐼 𝑋 𝑗 ) · ( 𝑗 𝑌 𝐾 ) ) )
17	16	mpteq2dv	⊢ ( ( 𝜑 ∧ ( 𝑖 = 𝐼 ∧ 𝑘 = 𝐾 ) ) → ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑋 𝑗 ) · ( 𝑗 𝑌 𝑘 ) ) ) = ( 𝑗 ∈ 𝑁 ↦ ( ( 𝐼 𝑋 𝑗 ) · ( 𝑗 𝑌 𝐾 ) ) ) )
18	17	oveq2d	⊢ ( ( 𝜑 ∧ ( 𝑖 = 𝐼 ∧ 𝑘 = 𝐾 ) ) → ( 𝑅 Σg ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑋 𝑗 ) · ( 𝑗 𝑌 𝑘 ) ) ) ) = ( 𝑅 Σg ( 𝑗 ∈ 𝑁 ↦ ( ( 𝐼 𝑋 𝑗 ) · ( 𝑗 𝑌 𝐾 ) ) ) ) )
19		ovexd	⊢ ( 𝜑 → ( 𝑅 Σg ( 𝑗 ∈ 𝑁 ↦ ( ( 𝐼 𝑋 𝑗 ) · ( 𝑗 𝑌 𝐾 ) ) ) ) ∈ V )
20	12 18 10 11 19	ovmpod	⊢ ( 𝜑 → ( 𝐼 ( 𝑋 𝐹 𝑌 ) 𝐾 ) = ( 𝑅 Σg ( 𝑗 ∈ 𝑁 ↦ ( ( 𝐼 𝑋 𝑗 ) · ( 𝑗 𝑌 𝐾 ) ) ) ) )