MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mvmulfv Structured version   Visualization version   GIF version

Theorem mvmulfv 21307
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 21306 . 2 (𝜑 → (𝑋 × 𝑌) = (𝑖𝑀 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌𝑗))))))
10 oveq1 7189 . . . . . 6 (𝑖 = 𝐼 → (𝑖𝑋𝑗) = (𝐼𝑋𝑗))
1110adantl 485 . . . . 5 ((𝜑𝑖 = 𝐼) → (𝑖𝑋𝑗) = (𝐼𝑋𝑗))
1211oveq1d 7197 . . . 4 ((𝜑𝑖 = 𝐼) → ((𝑖𝑋𝑗) · (𝑌𝑗)) = ((𝐼𝑋𝑗) · (𝑌𝑗)))
1312mpteq2dv 5136 . . 3 ((𝜑𝑖 = 𝐼) → (𝑗𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌𝑗))) = (𝑗𝑁 ↦ ((𝐼𝑋𝑗) · (𝑌𝑗))))
1413oveq2d 7198 . 2 ((𝜑𝑖 = 𝐼) → (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌𝑗)))) = (𝑅 Σg (𝑗𝑁 ↦ ((𝐼𝑋𝑗) · (𝑌𝑗)))))
15 mvmulfv.i . 2 (𝜑𝐼𝑀)
16 ovexd 7217 . 2 (𝜑 → (𝑅 Σg (𝑗𝑁 ↦ ((𝐼𝑋𝑗) · (𝑌𝑗)))) ∈ V)
179, 14, 15, 16fvmptd 6794 1 (𝜑 → ((𝑋 × 𝑌)‘𝐼) = (𝑅 Σg (𝑗𝑁 ↦ ((𝐼𝑋𝑗) · (𝑌𝑗)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  wcel 2114  Vcvv 3400  cop 4532  cmpt 5120   × cxp 5533  cfv 6349  (class class class)co 7182  m cmap 8449  Fincfn 8567  Basecbs 16598  .rcmulr 16681   Σg cgsu 16829   maVecMul cmvmul 21303
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5242  ax-pr 5306  ax-un 7491
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3402  df-sbc 3686  df-csb 3801  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-iun 4893  df-br 5041  df-opab 5103  df-mpt 5121  df-id 5439  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-iota 6307  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7185  df-oprab 7186  df-mpo 7187  df-1st 7726  df-2nd 7727  df-mvmul 21304
This theorem is referenced by:  mvmumamul1  21317
  Copyright terms: Public domain W3C validator