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Theorem mvmulfv 21693
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 21692 . 2 (𝜑 → (𝑋 × 𝑌) = (𝑖𝑀 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌𝑗))))))
10 oveq1 7282 . . . . . 6 (𝑖 = 𝐼 → (𝑖𝑋𝑗) = (𝐼𝑋𝑗))
1110adantl 482 . . . . 5 ((𝜑𝑖 = 𝐼) → (𝑖𝑋𝑗) = (𝐼𝑋𝑗))
1211oveq1d 7290 . . . 4 ((𝜑𝑖 = 𝐼) → ((𝑖𝑋𝑗) · (𝑌𝑗)) = ((𝐼𝑋𝑗) · (𝑌𝑗)))
1312mpteq2dv 5176 . . 3 ((𝜑𝑖 = 𝐼) → (𝑗𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌𝑗))) = (𝑗𝑁 ↦ ((𝐼𝑋𝑗) · (𝑌𝑗))))
1413oveq2d 7291 . 2 ((𝜑𝑖 = 𝐼) → (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌𝑗)))) = (𝑅 Σg (𝑗𝑁 ↦ ((𝐼𝑋𝑗) · (𝑌𝑗)))))
15 mvmulfv.i . 2 (𝜑𝐼𝑀)
16 ovexd 7310 . 2 (𝜑 → (𝑅 Σg (𝑗𝑁 ↦ ((𝐼𝑋𝑗) · (𝑌𝑗)))) ∈ V)
179, 14, 15, 16fvmptd 6882 1 (𝜑 → ((𝑋 × 𝑌)‘𝐼) = (𝑅 Σg (𝑗𝑁 ↦ ((𝐼𝑋𝑗) · (𝑌𝑗)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wcel 2106  Vcvv 3432  cop 4567  cmpt 5157   × cxp 5587  cfv 6433  (class class class)co 7275  m cmap 8615  Fincfn 8733  Basecbs 16912  .rcmulr 16963   Σg cgsu 17151   maVecMul cmvmul 21689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-mvmul 21690
This theorem is referenced by:  mvmumamul1  21703
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