Theorem mvmulval 22499

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 22498	. 2 ⊢ (𝜑 → × = (𝑥 ∈ (𝐵 ↑m (𝑀 × 𝑁)), 𝑦 ∈ (𝐵 ↑m 𝑁) ↦ (𝑖 ∈ 𝑀 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦‘𝑗)))))))
8		oveq 7374	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑖𝑥𝑗) = (𝑖𝑋𝑗))
9		fveq1 6841	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑦‘𝑗) = (𝑌‘𝑗))
10	8, 9	oveqan12d 7387	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((𝑖𝑥𝑗) · (𝑦‘𝑗)) = ((𝑖𝑋𝑗) · (𝑌‘𝑗)))
11	10	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ((𝑖𝑥𝑗) · (𝑦‘𝑗)) = ((𝑖𝑋𝑗) · (𝑌‘𝑗)))
12	11	mpteq2dv 5194	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦‘𝑗))) = (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))))
13	12	oveq2d 7384	. . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦‘𝑗)))) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗)))))
14	13	mpteq2dv 5194	. 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 7180	. 2 ⊢ (𝜑 → (𝑖 ∈ 𝑀 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))))) ∈ V)
18	7, 14, 15, 16, 17	ovmpod 7520	1 ⊢ (𝜑 → (𝑋 × 𝑌) = (𝑖 ∈ 𝑀 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3442  ⟨cop 4588   ↦ cmpt 5181   × cxp 5630  ‘cfv 6500  (class class class)co 7368   ↑_m cmap 8775  Fincfn 8895  Basecbs 17148  ._rcmulr 17190   Σ_g cgsu 17372   maVecMul cmvmul 22496

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-mvmul 22497

This theorem is referenced by:  mvmulfv  22500  mavmulval  22501