Theorem mvmulval 22533

Description: 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))

Assertion

Ref	Expression
mvmulval	⊢ (𝜑 → (𝑋 × 𝑌) = (𝑖 ∈ 𝑀 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))))))

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

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

Proof of Theorem mvmulval

Dummy variables 𝑥 𝑦 are mutually distinct and 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	1, 2, 3, 4, 5, 6	mvmulfval 22532	. 2 ⊢ (𝜑 → × = (𝑥 ∈ (𝐵 ↑m (𝑀 × 𝑁)), 𝑦 ∈ (𝐵 ↑m 𝑁) ↦ (𝑖 ∈ 𝑀 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦‘𝑗)))))))
8		oveq 7369	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑖𝑥𝑗) = (𝑖𝑋𝑗))
9		fveq1 6833	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑦‘𝑗) = (𝑌‘𝑗))
10	8, 9	oveqan12d 7382	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((𝑖𝑥𝑗) · (𝑦‘𝑗)) = ((𝑖𝑋𝑗) · (𝑌‘𝑗)))
11	10	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ((𝑖𝑥𝑗) · (𝑦‘𝑗)) = ((𝑖𝑋𝑗) · (𝑌‘𝑗)))
12	11	mpteq2dv 5173	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦‘𝑗))) = (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))))
13	12	oveq2d 7379	. . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦‘𝑗)))) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗)))))
14	13	mpteq2dv 5173	. 2 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑖 ∈ 𝑀 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦‘𝑗))))) = (𝑖 ∈ 𝑀 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))))))
15		mvmulval.x	. 2 ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m (𝑀 × 𝑁)))
16		mvmulval.y	. 2 ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑m 𝑁))
17	5	mptexd 7175	. 2 ⊢ (𝜑 → (𝑖 ∈ 𝑀 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))))) ∈ V)
18	7, 14, 15, 16, 17	ovmpod 7515	1 ⊢ (𝜑 → (𝑋 × 𝑌) = (𝑖 ∈ 𝑀 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))))))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  Vcvv 3432  ⟨cop 4568   ↦ cmpt 5160   × cxp 5623  ‘cfv 6492  (class class class)co 7363   ↑_m cmap 8770  Fincfn 8890  Basecbs 17177  ._rcmulr 17219   Σ_g cgsu 17401   maVecMul cmvmul 22530

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7366  df-oprab 7367  df-mpo 7368  df-1st 7938  df-2nd 7939  df-mvmul 22531

This theorem is referenced by:  mvmulfv  22534  mavmulval  22535