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Theorem mvmulval 20168
 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 · = (.r𝑅)
mvmulfval.r (𝜑𝑅𝑉)
mvmulfval.m (𝜑𝑀 ∈ Fin)
mvmulfval.n (𝜑𝑁 ∈ Fin)
mvmulval.x (𝜑𝑋 ∈ (𝐵𝑚 (𝑀 × 𝑁)))
mvmulval.y (𝜑𝑌 ∈ (𝐵𝑚 𝑁))
Assertion
Ref Expression
mvmulval (𝜑 → (𝑋 × 𝑌) = (𝑖𝑀 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌𝑗))))))
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.
StepHypRef Expression
1 mvmulfval.x . . 3 × = (𝑅 maVecMul ⟨𝑀, 𝑁⟩)
2 mvmulfval.b . . 3 𝐵 = (Base‘𝑅)
3 mvmulfval.t . . 3 · = (.r𝑅)
4 mvmulfval.r . . 3 (𝜑𝑅𝑉)
5 mvmulfval.m . . 3 (𝜑𝑀 ∈ Fin)
6 mvmulfval.n . . 3 (𝜑𝑁 ∈ Fin)
71, 2, 3, 4, 5, 6mvmulfval 20167 . 2 (𝜑× = (𝑥 ∈ (𝐵𝑚 (𝑀 × 𝑁)), 𝑦 ∈ (𝐵𝑚 𝑁) ↦ (𝑖𝑀 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦𝑗)))))))
8 oveq 6555 . . . . . . 7 (𝑥 = 𝑋 → (𝑖𝑥𝑗) = (𝑖𝑋𝑗))
9 fveq1 6102 . . . . . . 7 (𝑦 = 𝑌 → (𝑦𝑗) = (𝑌𝑗))
108, 9oveqan12d 6568 . . . . . 6 ((𝑥 = 𝑋𝑦 = 𝑌) → ((𝑖𝑥𝑗) · (𝑦𝑗)) = ((𝑖𝑋𝑗) · (𝑌𝑗)))
1110adantl 481 . . . . 5 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → ((𝑖𝑥𝑗) · (𝑦𝑗)) = ((𝑖𝑋𝑗) · (𝑌𝑗)))
1211mpteq2dv 4673 . . . 4 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦𝑗))) = (𝑗𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌𝑗))))
1312oveq2d 6565 . . 3 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦𝑗)))) = (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌𝑗)))))
1413mpteq2dv 4673 . 2 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑖𝑀 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦𝑗))))) = (𝑖𝑀 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌𝑗))))))
15 mvmulval.x . 2 (𝜑𝑋 ∈ (𝐵𝑚 (𝑀 × 𝑁)))
16 mvmulval.y . 2 (𝜑𝑌 ∈ (𝐵𝑚 𝑁))
17 mptexg 6389 . . 3 (𝑀 ∈ Fin → (𝑖𝑀 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌𝑗))))) ∈ V)
185, 17syl 17 . 2 (𝜑 → (𝑖𝑀 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌𝑗))))) ∈ V)
197, 14, 15, 16, 18ovmpt2d 6686 1 (𝜑 → (𝑋 × 𝑌) = (𝑖𝑀 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌𝑗))))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ⟨cop 4131   ↦ cmpt 4643   × cxp 5036  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744  Fincfn 7841  Basecbs 15695  .rcmulr 15769   Σg cgsu 15924   maVecMul cmvmul 20165 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-mvmul 20166 This theorem is referenced by:  mvmulfv  20169  mavmulval  20170
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