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Theorem matval 21758
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 3463 . . 3 (𝑅𝑉𝑅 ∈ V)
3 id 22 . . . . . . 7 (𝑟 = 𝑅𝑟 = 𝑅)
4 id 22 . . . . . . . 8 (𝑛 = 𝑁𝑛 = 𝑁)
54sqxpeqd 5665 . . . . . . 7 (𝑛 = 𝑁 → (𝑛 × 𝑛) = (𝑁 × 𝑁))
63, 5oveqan12rd 7377 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑟 freeLMod (𝑛 × 𝑛)) = (𝑅 freeLMod (𝑁 × 𝑁)))
7 matval.g . . . . . 6 𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))
86, 7eqtr4di 2794 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑟 freeLMod (𝑛 × 𝑛)) = 𝐺)
94, 4, 4oteq123d 4845 . . . . . . . 8 (𝑛 = 𝑁 → ⟨𝑛, 𝑛, 𝑛⟩ = ⟨𝑁, 𝑁, 𝑁⟩)
103, 9oveqan12rd 7377 . . . . . . 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 2794 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩) = · )
1312opeq2d 4837 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → ⟨(.r‘ndx), (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)⟩ = ⟨(.r‘ndx), · ⟩)
148, 13oveq12d 7375 . . . 4 ((𝑛 = 𝑁𝑟 = 𝑅) → ((𝑟 freeLMod (𝑛 × 𝑛)) sSet ⟨(.r‘ndx), (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)⟩) = (𝐺 sSet ⟨(.r‘ndx), · ⟩))
15 df-mat 21755 . . . 4 Mat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ((𝑟 freeLMod (𝑛 × 𝑛)) sSet ⟨(.r‘ndx), (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)⟩))
16 ovex 7390 . . . 4 (𝐺 sSet ⟨(.r‘ndx), · ⟩) ∈ V
1714, 15, 16ovmpoa 7510 . . 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 2788 1 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝐴 = (𝐺 sSet ⟨(.r‘ndx), · ⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  Vcvv 3445  cop 4592  cotp 4594   × cxp 5631  cfv 6496  (class class class)co 7357  Fincfn 8883   sSet csts 17035  ndxcnx 17065  .rcmulr 17134   freeLMod cfrlm 21152   maMul cmmul 21732   Mat cmat 21754
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 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-ot 4595  df-uni 4866  df-br 5106  df-opab 5168  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-iota 6448  df-fun 6498  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-mat 21755
This theorem is referenced by:  matbas  21760  matplusg  21761  matsca  21762  matscaOLD  21763  matvsca  21764  matvscaOLD  21765  matmulr  21787
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