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Theorem matval 22415
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 3501 . . 3 (𝑅𝑉𝑅 ∈ V)
3 id 22 . . . . . . 7 (𝑟 = 𝑅𝑟 = 𝑅)
4 id 22 . . . . . . . 8 (𝑛 = 𝑁𝑛 = 𝑁)
54sqxpeqd 5717 . . . . . . 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 2795 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑟 freeLMod (𝑛 × 𝑛)) = 𝐺)
94, 4, 4oteq123d 4888 . . . . . . . 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 2795 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩) = · )
1312opeq2d 4880 . . . . 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 22412 . . . 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 2789 1 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝐴 = (𝐺 sSet ⟨(.r‘ndx), · ⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  cop 4632  cotp 4634   × cxp 5683  cfv 6561  (class class class)co 7431  Fincfn 8985   sSet csts 17200  ndxcnx 17230  .rcmulr 17298   freeLMod cfrlm 21766   maMul cmmul 22394   Mat cmat 22411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-ot 4635  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-mat 22412
This theorem is referenced by:  matbas  22417  matplusg  22418  matsca  22419  matscaOLD  22420  matvsca  22421  matvscaOLD  22422  matmulr  22444
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