Theorem mvmulfv 21153

Description: A cell/element in the vector resulting from a multiplication of a vector with a matrix. (Contributed by AV, 23-Feb-2019.)

Hypotheses

Ref	Expression
mvmulfval.x	⊢ × = (𝑅 maVecMul ⟨𝑀, 𝑁⟩)
mvmulfval.b	⊢ 𝐵 = (Base‘𝑅)
mvmulfval.t	⊢ · = (.r‘𝑅)
mvmulfval.r	⊢ (𝜑 → 𝑅 ∈ 𝑉)
mvmulfval.m	⊢ (𝜑 → 𝑀 ∈ Fin)
mvmulfval.n	⊢ (𝜑 → 𝑁 ∈ Fin)
mvmulval.x	⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m (𝑀 × 𝑁)))
mvmulval.y	⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑m 𝑁))
mvmulfv.i	⊢ (𝜑 → 𝐼 ∈ 𝑀)

Assertion

Ref	Expression
mvmulfv	⊢ (𝜑 → ((𝑋 × 𝑌)‘𝐼) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗) · (𝑌‘𝑗)))))

Distinct variable groups:   𝜑,𝑗   𝑗,𝑀   𝑗,𝑁   𝑅,𝑗   𝑗,𝑋   𝑗,𝑌   𝑗,𝐼

Allowed substitution hints:   𝐵(𝑗)   · (𝑗)   × (𝑗)   𝑉(𝑗)

Proof of Theorem mvmulfv

Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mvmulfval.x	. . 3 ⊢ × = (𝑅 maVecMul ⟨𝑀, 𝑁⟩)
2		mvmulfval.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
3		mvmulfval.t	. . 3 ⊢ · = (.r‘𝑅)
4		mvmulfval.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑉)
5		mvmulfval.m	. . 3 ⊢ (𝜑 → 𝑀 ∈ Fin)
6		mvmulfval.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ Fin)
7		mvmulval.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m (𝑀 × 𝑁)))
8		mvmulval.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑m 𝑁))
9	1, 2, 3, 4, 5, 6, 7, 8	mvmulval 21152	. 2 ⊢ (𝜑 → (𝑋 × 𝑌) = (𝑖 ∈ 𝑀 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))))))
10		oveq1 7163	. . . . . 6 ⊢ (𝑖 = 𝐼 → (𝑖𝑋𝑗) = (𝐼𝑋𝑗))
11	10	adantl 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 = 𝐼) → (𝑖𝑋𝑗) = (𝐼𝑋𝑗))
12	11	oveq1d 7171	. . . 4 ⊢ ((𝜑 ∧ 𝑖 = 𝐼) → ((𝑖𝑋𝑗) · (𝑌‘𝑗)) = ((𝐼𝑋𝑗) · (𝑌‘𝑗)))
13	12	mpteq2dv 5162	. . 3 ⊢ ((𝜑 ∧ 𝑖 = 𝐼) → (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))) = (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗) · (𝑌‘𝑗))))
14	13	oveq2d 7172	. 2 ⊢ ((𝜑 ∧ 𝑖 = 𝐼) → (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗)))) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗) · (𝑌‘𝑗)))))
15		mvmulfv.i	. 2 ⊢ (𝜑 → 𝐼 ∈ 𝑀)
16		ovexd 7191	. 2 ⊢ (𝜑 → (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗) · (𝑌‘𝑗)))) ∈ V)
17	9, 14, 15, 16	fvmptd 6775	1 ⊢ (𝜑 → ((𝑋 × 𝑌)‘𝐼) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗) · (𝑌‘𝑗)))))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  Vcvv 3494  ⟨cop 4573   ↦ cmpt 5146   × cxp 5553  ‘cfv 6355  (class class class)co 7156   ↑_m cmap 8406  Fincfn 8509  Basecbs 16483  ._rcmulr 16566   Σ_g cgsu 16714   maVecMul cmvmul 21149

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-1st 7689  df-2nd 7690  df-mvmul 21150

This theorem is referenced by:  mvmumamul1  21163