Theorem mvmulfv 22670

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 22669	. 2 ⊢ (𝜑 → (𝑋 × 𝑌) = (𝑖 ∈ 𝑀 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))))))
10		oveq1 7418	. . . . . 6 ⊢ (𝑖 = 𝐼 → (𝑖𝑋𝑗) = (𝐼𝑋𝑗))
11	10	adantl 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 = 𝐼) → (𝑖𝑋𝑗) = (𝐼𝑋𝑗))
12	11	oveq1d 7426	. . . 4 ⊢ ((𝜑 ∧ 𝑖 = 𝐼) → ((𝑖𝑋𝑗) · (𝑌‘𝑗)) = ((𝐼𝑋𝑗) · (𝑌‘𝑗)))
13	12	mpteq2dv 5209	. . 3 ⊢ ((𝜑 ∧ 𝑖 = 𝐼) → (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))) = (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗) · (𝑌‘𝑗))))
14	13	oveq2d 7427	. 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 7446	. 2 ⊢ (𝜑 → (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗) · (𝑌‘𝑗)))) ∈ V)
17	9, 14, 15, 16	fvmptd 6998	1 ⊢ (𝜑 → ((𝑋 × 𝑌)‘𝐼) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗) · (𝑌‘𝑗)))))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  Vcvv 3463  ⟨cop 4600   ↦ cmpt 5196   × cxp 5660  ‘cfv 6537  (class class class)co 7411   ↑_m cmap 8824  Fincfn 8943  Basecbs 17269  ._rcmulr 17311   Σ_g cgsu 17493   maVecMul cmvmul 22666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7986  df-2nd 7987  df-mvmul 22667

This theorem is referenced by:  mvmumamul1  22680