Theorem mvmulfv 21601

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 21600	. 2 ⊢ (𝜑 → (𝑋 × 𝑌) = (𝑖 ∈ 𝑀 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))))))
10		oveq1 7262	. . . . . 6 ⊢ (𝑖 = 𝐼 → (𝑖𝑋𝑗) = (𝐼𝑋𝑗))
11	10	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 = 𝐼) → (𝑖𝑋𝑗) = (𝐼𝑋𝑗))
12	11	oveq1d 7270	. . . 4 ⊢ ((𝜑 ∧ 𝑖 = 𝐼) → ((𝑖𝑋𝑗) · (𝑌‘𝑗)) = ((𝐼𝑋𝑗) · (𝑌‘𝑗)))
13	12	mpteq2dv 5172	. . 3 ⊢ ((𝜑 ∧ 𝑖 = 𝐼) → (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))) = (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗) · (𝑌‘𝑗))))
14	13	oveq2d 7271	. 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 7290	. 2 ⊢ (𝜑 → (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗) · (𝑌‘𝑗)))) ∈ V)
17	9, 14, 15, 16	fvmptd 6864	1 ⊢ (𝜑 → ((𝑋 × 𝑌)‘𝐼) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗) · (𝑌‘𝑗)))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  Vcvv 3422  ⟨cop 4564   ↦ cmpt 5153   × cxp 5578  ‘cfv 6418  (class class class)co 7255   ↑_m cmap 8573  Fincfn 8691  Basecbs 16840  ._rcmulr 16889   Σ_g cgsu 17068   maVecMul cmvmul 21597

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-mvmul 21598

This theorem is referenced by:  mvmumamul1  21611