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

Theorem mamuval 21819
Description: Multiplication of two matrices. (Contributed by Stefan O'Rear, 2-Sep-2015.)
Hypotheses
Ref Expression
mamufval.f 𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑃⟩)
mamufval.b 𝐵 = (Base‘𝑅)
mamufval.t · = (.r𝑅)
mamufval.r (𝜑𝑅𝑉)
mamufval.m (𝜑𝑀 ∈ Fin)
mamufval.n (𝜑𝑁 ∈ Fin)
mamufval.p (𝜑𝑃 ∈ Fin)
mamuval.x (𝜑𝑋 ∈ (𝐵m (𝑀 × 𝑁)))
mamuval.y (𝜑𝑌 ∈ (𝐵m (𝑁 × 𝑃)))
Assertion
Ref Expression
mamuval (𝜑 → (𝑋𝐹𝑌) = (𝑖𝑀, 𝑘𝑃 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑋𝑗) · (𝑗𝑌𝑘))))))
Distinct variable groups:   𝑖,𝑗,𝑘,𝑀   𝑖,𝑁,𝑗,𝑘   𝑃,𝑖,𝑗,𝑘   𝑅,𝑖,𝑗,𝑘   𝑖,𝑋,𝑗,𝑘   𝑖,𝑌,𝑗,𝑘   𝜑,𝑖,𝑗,𝑘   · ,𝑖,𝑘
Allowed substitution hints:   𝐵(𝑖,𝑗,𝑘)   · (𝑗)   𝐹(𝑖,𝑗,𝑘)   𝑉(𝑖,𝑗,𝑘)

Proof of Theorem mamuval
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mamufval.f . . 3 𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑃⟩)
2 mamufval.b . . 3 𝐵 = (Base‘𝑅)
3 mamufval.t . . 3 · = (.r𝑅)
4 mamufval.r . . 3 (𝜑𝑅𝑉)
5 mamufval.m . . 3 (𝜑𝑀 ∈ Fin)
6 mamufval.n . . 3 (𝜑𝑁 ∈ Fin)
7 mamufval.p . . 3 (𝜑𝑃 ∈ Fin)
81, 2, 3, 4, 5, 6, 7mamufval 21818 . 2 (𝜑𝐹 = (𝑥 ∈ (𝐵m (𝑀 × 𝑁)), 𝑦 ∈ (𝐵m (𝑁 × 𝑃)) ↦ (𝑖𝑀, 𝑘𝑃 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑗𝑦𝑘)))))))
9 oveq 7400 . . . . . . 7 (𝑥 = 𝑋 → (𝑖𝑥𝑗) = (𝑖𝑋𝑗))
10 oveq 7400 . . . . . . 7 (𝑦 = 𝑌 → (𝑗𝑦𝑘) = (𝑗𝑌𝑘))
119, 10oveqan12d 7413 . . . . . 6 ((𝑥 = 𝑋𝑦 = 𝑌) → ((𝑖𝑥𝑗) · (𝑗𝑦𝑘)) = ((𝑖𝑋𝑗) · (𝑗𝑌𝑘)))
1211adantl 482 . . . . 5 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → ((𝑖𝑥𝑗) · (𝑗𝑦𝑘)) = ((𝑖𝑋𝑗) · (𝑗𝑌𝑘)))
1312mpteq2dv 5244 . . . 4 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑗𝑦𝑘))) = (𝑗𝑁 ↦ ((𝑖𝑋𝑗) · (𝑗𝑌𝑘))))
1413oveq2d 7410 . . 3 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑗𝑦𝑘)))) = (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑋𝑗) · (𝑗𝑌𝑘)))))
1514mpoeq3dv 7473 . 2 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑖𝑀, 𝑘𝑃 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑗𝑦𝑘))))) = (𝑖𝑀, 𝑘𝑃 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑋𝑗) · (𝑗𝑌𝑘))))))
16 mamuval.x . 2 (𝜑𝑋 ∈ (𝐵m (𝑀 × 𝑁)))
17 mamuval.y . 2 (𝜑𝑌 ∈ (𝐵m (𝑁 × 𝑃)))
18 mpoexga 8048 . . 3 ((𝑀 ∈ Fin ∧ 𝑃 ∈ Fin) → (𝑖𝑀, 𝑘𝑃 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑋𝑗) · (𝑗𝑌𝑘))))) ∈ V)
195, 7, 18syl2anc 584 . 2 (𝜑 → (𝑖𝑀, 𝑘𝑃 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑋𝑗) · (𝑗𝑌𝑘))))) ∈ V)
208, 15, 16, 17, 19ovmpod 7544 1 (𝜑 → (𝑋𝐹𝑌) = (𝑖𝑀, 𝑘𝑃 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑋𝑗) · (𝑗𝑌𝑘))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  Vcvv 3474  cotp 4631  cmpt 5225   × cxp 5668  cfv 6533  (class class class)co 7394  cmpo 7396  m cmap 8805  Fincfn 8924  Basecbs 17128  .rcmulr 17182   Σg cgsu 17370   maMul cmmul 21816
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5357  ax-pr 5421  ax-un 7709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-ot 4632  df-uni 4903  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5568  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7397  df-oprab 7398  df-mpo 7399  df-1st 7959  df-2nd 7960  df-mamu 21817
This theorem is referenced by:  mamufv  21820  mamures  21823  mamucl  21832  mpomatmul  21879  mamutpos  21891  mat1dimmul  21909  dmatmul  21930  madurid  22077  cramerimplem2  22117  mat2pmatmul  22164  decpmatmul  22205
  Copyright terms: Public domain W3C validator