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Theorem mvmulfv 20169
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 · = (.r𝑅)
mvmulfval.r (𝜑𝑅𝑉)
mvmulfval.m (𝜑𝑀 ∈ Fin)
mvmulfval.n (𝜑𝑁 ∈ Fin)
mvmulval.x (𝜑𝑋 ∈ (𝐵𝑚 (𝑀 × 𝑁)))
mvmulval.y (𝜑𝑌 ∈ (𝐵𝑚 𝑁))
mvmulfv.i (𝜑𝐼𝑀)
Assertion
Ref Expression
mvmulfv (𝜑 → ((𝑋 × 𝑌)‘𝐼) = (𝑅 Σg (𝑗𝑁 ↦ ((𝐼𝑋𝑗) · (𝑌𝑗)))))
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.
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)
7 mvmulval.x . . 3 (𝜑𝑋 ∈ (𝐵𝑚 (𝑀 × 𝑁)))
8 mvmulval.y . . 3 (𝜑𝑌 ∈ (𝐵𝑚 𝑁))
91, 2, 3, 4, 5, 6, 7, 8mvmulval 20168 . 2 (𝜑 → (𝑋 × 𝑌) = (𝑖𝑀 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌𝑗))))))
10 oveq1 6556 . . . . . 6 (𝑖 = 𝐼 → (𝑖𝑋𝑗) = (𝐼𝑋𝑗))
1110adantl 481 . . . . 5 ((𝜑𝑖 = 𝐼) → (𝑖𝑋𝑗) = (𝐼𝑋𝑗))
1211oveq1d 6564 . . . 4 ((𝜑𝑖 = 𝐼) → ((𝑖𝑋𝑗) · (𝑌𝑗)) = ((𝐼𝑋𝑗) · (𝑌𝑗)))
1312mpteq2dv 4673 . . 3 ((𝜑𝑖 = 𝐼) → (𝑗𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌𝑗))) = (𝑗𝑁 ↦ ((𝐼𝑋𝑗) · (𝑌𝑗))))
1413oveq2d 6565 . 2 ((𝜑𝑖 = 𝐼) → (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌𝑗)))) = (𝑅 Σg (𝑗𝑁 ↦ ((𝐼𝑋𝑗) · (𝑌𝑗)))))
15 mvmulfv.i . 2 (𝜑𝐼𝑀)
16 ovex 6577 . . 3 (𝑅 Σg (𝑗𝑁 ↦ ((𝐼𝑋𝑗) · (𝑌𝑗)))) ∈ V
1716a1i 11 . 2 (𝜑 → (𝑅 Σg (𝑗𝑁 ↦ ((𝐼𝑋𝑗) · (𝑌𝑗)))) ∈ V)
189, 14, 15, 17fvmptd 6197 1 (𝜑 → ((𝑋 × 𝑌)‘𝐼) = (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:  mvmumamul1  20179
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