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Theorem matval 20948
Description: Value of the matrix algebra. (Contributed by Stefan O'Rear, 4-Sep-2015.)
Hypotheses
Ref Expression
matval.a 𝐴 = (𝑁 Mat 𝑅)
matval.g 𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))
matval.t · = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)
Assertion
Ref Expression
matval ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝐴 = (𝐺 sSet ⟨(.r‘ndx), · ⟩))

Proof of Theorem matval
Dummy variables 𝑛 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 matval.a . 2 𝐴 = (𝑁 Mat 𝑅)
2 elex 3510 . . 3 (𝑅𝑉𝑅 ∈ V)
3 id 22 . . . . . . 7 (𝑟 = 𝑅𝑟 = 𝑅)
4 id 22 . . . . . . . 8 (𝑛 = 𝑁𝑛 = 𝑁)
54sqxpeqd 5580 . . . . . . 7 (𝑛 = 𝑁 → (𝑛 × 𝑛) = (𝑁 × 𝑁))
63, 5oveqan12rd 7165 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑟 freeLMod (𝑛 × 𝑛)) = (𝑅 freeLMod (𝑁 × 𝑁)))
7 matval.g . . . . . 6 𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))
86, 7syl6eqr 2871 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑟 freeLMod (𝑛 × 𝑛)) = 𝐺)
94, 4, 4oteq123d 4810 . . . . . . . 8 (𝑛 = 𝑁 → ⟨𝑛, 𝑛, 𝑛⟩ = ⟨𝑁, 𝑁, 𝑁⟩)
103, 9oveqan12rd 7165 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩) = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩))
11 matval.t . . . . . . 7 · = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)
1210, 11syl6eqr 2871 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩) = · )
1312opeq2d 4802 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → ⟨(.r‘ndx), (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)⟩ = ⟨(.r‘ndx), · ⟩)
148, 13oveq12d 7163 . . . 4 ((𝑛 = 𝑁𝑟 = 𝑅) → ((𝑟 freeLMod (𝑛 × 𝑛)) sSet ⟨(.r‘ndx), (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)⟩) = (𝐺 sSet ⟨(.r‘ndx), · ⟩))
15 df-mat 20945 . . . 4 Mat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ((𝑟 freeLMod (𝑛 × 𝑛)) sSet ⟨(.r‘ndx), (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)⟩))
16 ovex 7178 . . . 4 (𝐺 sSet ⟨(.r‘ndx), · ⟩) ∈ V
1714, 15, 16ovmpoa 7294 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (𝑁 Mat 𝑅) = (𝐺 sSet ⟨(.r‘ndx), · ⟩))
182, 17sylan2 592 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑁 Mat 𝑅) = (𝐺 sSet ⟨(.r‘ndx), · ⟩))
191, 18syl5eq 2865 1 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝐴 = (𝐺 sSet ⟨(.r‘ndx), · ⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  Vcvv 3492  cop 4563  cotp 4565   × cxp 5546  cfv 6348  (class class class)co 7145  Fincfn 8497  ndxcnx 16468   sSet csts 16469  .rcmulr 16554   freeLMod cfrlm 20818   maMul cmmul 20922   Mat cmat 20944
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-sep 5194  ax-nul 5201  ax-pr 5320
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-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-ot 4566  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-mat 20945
This theorem is referenced by:  matbas  20950  matplusg  20951  matsca  20952  matvsca  20953  matmulr  20975
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