Theorem mavmulfv

Description: A cell/element in the vector resulting from a multiplication of a vector with a square matrix. (Contributed by AV, 6-Dec-2018) (Revised by AV, 18-Feb-2019) (Revised 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 ) )
	mavmulfv.i	⊢ ( 𝜑 → 𝐼 ∈ 𝑁 )
Assertion	mavmulfv	⊢ ( 𝜑 → ( ( 𝑋 × 𝑌 ) ‘ 𝐼 ) = ( 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		mavmulfv.i	⊢ ( 𝜑 → 𝐼 ∈ 𝑁 )
10	1 2 3 4 5 6 7 8	mavmulval	⊢ ( 𝜑 → ( 𝑋 × 𝑌 ) = ( 𝑖 ∈ 𝑁 ↦ ( 𝑅 Σg ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑋 𝑗 ) · ( 𝑌 ‘ 𝑗 ) ) ) ) ) )
11		oveq1	⊢ ( 𝑖 = 𝐼 → ( 𝑖 𝑋 𝑗 ) = ( 𝐼 𝑋 𝑗 ) )
12	11	adantl	⊢ ( ( 𝜑 ∧ 𝑖 = 𝐼 ) → ( 𝑖 𝑋 𝑗 ) = ( 𝐼 𝑋 𝑗 ) )
13	12	oveq1d	⊢ ( ( 𝜑 ∧ 𝑖 = 𝐼 ) → ( ( 𝑖 𝑋 𝑗 ) · ( 𝑌 ‘ 𝑗 ) ) = ( ( 𝐼 𝑋 𝑗 ) · ( 𝑌 ‘ 𝑗 ) ) )
14	13	mpteq2dv	⊢ ( ( 𝜑 ∧ 𝑖 = 𝐼 ) → ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑋 𝑗 ) · ( 𝑌 ‘ 𝑗 ) ) ) = ( 𝑗 ∈ 𝑁 ↦ ( ( 𝐼 𝑋 𝑗 ) · ( 𝑌 ‘ 𝑗 ) ) ) )
15	14	oveq2d	⊢ ( ( 𝜑 ∧ 𝑖 = 𝐼 ) → ( 𝑅 Σg ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑋 𝑗 ) · ( 𝑌 ‘ 𝑗 ) ) ) ) = ( 𝑅 Σg ( 𝑗 ∈ 𝑁 ↦ ( ( 𝐼 𝑋 𝑗 ) · ( 𝑌 ‘ 𝑗 ) ) ) ) )
16		ovexd	⊢ ( 𝜑 → ( 𝑅 Σg ( 𝑗 ∈ 𝑁 ↦ ( ( 𝐼 𝑋 𝑗 ) · ( 𝑌 ‘ 𝑗 ) ) ) ) ∈ V )
17	10 15 9 16	fvmptd	⊢ ( 𝜑 → ( ( 𝑋 × 𝑌 ) ‘ 𝐼 ) = ( 𝑅 Σg ( 𝑗 ∈ 𝑁 ↦ ( ( 𝐼 𝑋 𝑗 ) · ( 𝑌 ‘ 𝑗 ) ) ) ) )