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Theorem mvmulfv 20847
 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 20846 . 2 (𝜑 → (𝑋 × 𝑌) = (𝑖𝑀 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌𝑗))))))
10 oveq1 6977 . . . . . 6 (𝑖 = 𝐼 → (𝑖𝑋𝑗) = (𝐼𝑋𝑗))
1110adantl 474 . . . . 5 ((𝜑𝑖 = 𝐼) → (𝑖𝑋𝑗) = (𝐼𝑋𝑗))
1211oveq1d 6985 . . . 4 ((𝜑𝑖 = 𝐼) → ((𝑖𝑋𝑗) · (𝑌𝑗)) = ((𝐼𝑋𝑗) · (𝑌𝑗)))
1312mpteq2dv 5017 . . 3 ((𝜑𝑖 = 𝐼) → (𝑗𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌𝑗))) = (𝑗𝑁 ↦ ((𝐼𝑋𝑗) · (𝑌𝑗))))
1413oveq2d 6986 . 2 ((𝜑𝑖 = 𝐼) → (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌𝑗)))) = (𝑅 Σg (𝑗𝑁 ↦ ((𝐼𝑋𝑗) · (𝑌𝑗)))))
15 mvmulfv.i . 2 (𝜑𝐼𝑀)
16 ovexd 7004 . 2 (𝜑 → (𝑅 Σg (𝑗𝑁 ↦ ((𝐼𝑋𝑗) · (𝑌𝑗)))) ∈ V)
179, 14, 15, 16fvmptd 6595 1 (𝜑 → ((𝑋 × 𝑌)‘𝐼) = (𝑅 Σg (𝑗𝑁 ↦ ((𝐼𝑋𝑗) · (𝑌𝑗)))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 387   = wceq 1507   ∈ wcel 2048  Vcvv 3409  ⟨cop 4441   ↦ cmpt 5002   × cxp 5398  ‘cfv 6182  (class class class)co 6970   ↑𝑚 cmap 8198  Fincfn 8298  Basecbs 16329  .rcmulr 16412   Σg cgsu 16560   maVecMul cmvmul 20843 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5305  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-ov 6973  df-oprab 6974  df-mpo 6975  df-1st 7494  df-2nd 7495  df-mvmul 20844 This theorem is referenced by:  mvmumamul1  20857
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