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Theorem matval 22431
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 3499 . . 3 (𝑅𝑉𝑅 ∈ V)
3 id 22 . . . . . . 7 (𝑟 = 𝑅𝑟 = 𝑅)
4 id 22 . . . . . . . 8 (𝑛 = 𝑁𝑛 = 𝑁)
54sqxpeqd 5721 . . . . . . 7 (𝑛 = 𝑁 → (𝑛 × 𝑛) = (𝑁 × 𝑁))
63, 5oveqan12rd 7451 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑟 freeLMod (𝑛 × 𝑛)) = (𝑅 freeLMod (𝑁 × 𝑁)))
7 matval.g . . . . . 6 𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))
86, 7eqtr4di 2793 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑟 freeLMod (𝑛 × 𝑛)) = 𝐺)
94, 4, 4oteq123d 4893 . . . . . . . 8 (𝑛 = 𝑁 → ⟨𝑛, 𝑛, 𝑛⟩ = ⟨𝑁, 𝑁, 𝑁⟩)
103, 9oveqan12rd 7451 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩) = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩))
11 matval.t . . . . . . 7 · = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)
1210, 11eqtr4di 2793 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩) = · )
1312opeq2d 4885 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → ⟨(.r‘ndx), (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)⟩ = ⟨(.r‘ndx), · ⟩)
148, 13oveq12d 7449 . . . 4 ((𝑛 = 𝑁𝑟 = 𝑅) → ((𝑟 freeLMod (𝑛 × 𝑛)) sSet ⟨(.r‘ndx), (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)⟩) = (𝐺 sSet ⟨(.r‘ndx), · ⟩))
15 df-mat 22428 . . . 4 Mat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ((𝑟 freeLMod (𝑛 × 𝑛)) sSet ⟨(.r‘ndx), (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)⟩))
16 ovex 7464 . . . 4 (𝐺 sSet ⟨(.r‘ndx), · ⟩) ∈ V
1714, 15, 16ovmpoa 7588 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (𝑁 Mat 𝑅) = (𝐺 sSet ⟨(.r‘ndx), · ⟩))
182, 17sylan2 593 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑁 Mat 𝑅) = (𝐺 sSet ⟨(.r‘ndx), · ⟩))
191, 18eqtrid 2787 1 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝐴 = (𝐺 sSet ⟨(.r‘ndx), · ⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  cop 4637  cotp 4639   × cxp 5687  cfv 6563  (class class class)co 7431  Fincfn 8984   sSet csts 17197  ndxcnx 17227  .rcmulr 17299   freeLMod cfrlm 21784   maMul cmmul 22410   Mat cmat 22427
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-ot 4640  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-mat 22428
This theorem is referenced by:  matbas  22433  matplusg  22434  matsca  22435  matscaOLD  22436  matvsca  22437  matvscaOLD  22438  matmulr  22460
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