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Theorem mamufv 20926
Description: A cell in the 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)))
mamufv.i (𝜑𝐼𝑀)
mamufv.k (𝜑𝐾𝑃)
Assertion
Ref Expression
mamufv (𝜑 → (𝐼(𝑋𝐹𝑌)𝐾) = (𝑅 Σg (𝑗𝑁 ↦ ((𝐼𝑋𝑗) · (𝑗𝑌𝐾)))))
Distinct variable groups:   𝑗,𝑀   𝑗,𝑁   𝑃,𝑗   𝑅,𝑗   𝑗,𝑋   𝑗,𝑌   𝜑,𝑗   𝑗,𝐼   𝑗,𝐾
Allowed substitution hints:   𝐵(𝑗)   · (𝑗)   𝐹(𝑗)   𝑉(𝑗)

Proof of Theorem mamufv
Dummy variables 𝑖 𝑘 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)
8 mamuval.x . . 3 (𝜑𝑋 ∈ (𝐵m (𝑀 × 𝑁)))
9 mamuval.y . . 3 (𝜑𝑌 ∈ (𝐵m (𝑁 × 𝑃)))
101, 2, 3, 4, 5, 6, 7, 8, 9mamuval 20925 . 2 (𝜑 → (𝑋𝐹𝑌) = (𝑖𝑀, 𝑘𝑃 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑋𝑗) · (𝑗𝑌𝑘))))))
11 oveq1 7152 . . . . . 6 (𝑖 = 𝐼 → (𝑖𝑋𝑗) = (𝐼𝑋𝑗))
12 oveq2 7153 . . . . . 6 (𝑘 = 𝐾 → (𝑗𝑌𝑘) = (𝑗𝑌𝐾))
1311, 12oveqan12d 7164 . . . . 5 ((𝑖 = 𝐼𝑘 = 𝐾) → ((𝑖𝑋𝑗) · (𝑗𝑌𝑘)) = ((𝐼𝑋𝑗) · (𝑗𝑌𝐾)))
1413adantl 482 . . . 4 ((𝜑 ∧ (𝑖 = 𝐼𝑘 = 𝐾)) → ((𝑖𝑋𝑗) · (𝑗𝑌𝑘)) = ((𝐼𝑋𝑗) · (𝑗𝑌𝐾)))
1514mpteq2dv 5153 . . 3 ((𝜑 ∧ (𝑖 = 𝐼𝑘 = 𝐾)) → (𝑗𝑁 ↦ ((𝑖𝑋𝑗) · (𝑗𝑌𝑘))) = (𝑗𝑁 ↦ ((𝐼𝑋𝑗) · (𝑗𝑌𝐾))))
1615oveq2d 7161 . 2 ((𝜑 ∧ (𝑖 = 𝐼𝑘 = 𝐾)) → (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑋𝑗) · (𝑗𝑌𝑘)))) = (𝑅 Σg (𝑗𝑁 ↦ ((𝐼𝑋𝑗) · (𝑗𝑌𝐾)))))
17 mamufv.i . 2 (𝜑𝐼𝑀)
18 mamufv.k . 2 (𝜑𝐾𝑃)
19 ovexd 7180 . 2 (𝜑 → (𝑅 Σg (𝑗𝑁 ↦ ((𝐼𝑋𝑗) · (𝑗𝑌𝐾)))) ∈ V)
2010, 16, 17, 18, 19ovmpod 7291 1 (𝜑 → (𝐼(𝑋𝐹𝑌)𝐾) = (𝑅 Σg (𝑗𝑁 ↦ ((𝐼𝑋𝑗) · (𝑗𝑌𝐾)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  Vcvv 3492  cotp 4565  cmpt 5137   × cxp 5546  cfv 6348  (class class class)co 7145  m cmap 8395  Fincfn 8497  Basecbs 16471  .rcmulr 16554   Σg cgsu 16702   maMul cmmul 20922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-ot 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-mamu 20923
This theorem is referenced by:  mamuass  20939  mamudi  20940  mamudir  20941  mamuvs1  20942  mamuvs2  20943  mamulid  20978  mamurid  20979  matmulcell  20982  mavmulass  21086  mvmumamul1  21091  mdetmul  21160  decpmatmullem  21307  matunitlindflem2  34770
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