Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  matval Structured version   Visualization version   GIF version

Theorem matval 21111
 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 3428 . . 3 (𝑅𝑉𝑅 ∈ V)
3 id 22 . . . . . . 7 (𝑟 = 𝑅𝑟 = 𝑅)
4 id 22 . . . . . . . 8 (𝑛 = 𝑁𝑛 = 𝑁)
54sqxpeqd 5556 . . . . . . 7 (𝑛 = 𝑁 → (𝑛 × 𝑛) = (𝑁 × 𝑁))
63, 5oveqan12rd 7170 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑟 freeLMod (𝑛 × 𝑛)) = (𝑅 freeLMod (𝑁 × 𝑁)))
7 matval.g . . . . . 6 𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))
86, 7eqtr4di 2811 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑟 freeLMod (𝑛 × 𝑛)) = 𝐺)
94, 4, 4oteq123d 4778 . . . . . . . 8 (𝑛 = 𝑁 → ⟨𝑛, 𝑛, 𝑛⟩ = ⟨𝑁, 𝑁, 𝑁⟩)
103, 9oveqan12rd 7170 . . . . . . 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 2811 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩) = · )
1312opeq2d 4770 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → ⟨(.r‘ndx), (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)⟩ = ⟨(.r‘ndx), · ⟩)
148, 13oveq12d 7168 . . . 4 ((𝑛 = 𝑁𝑟 = 𝑅) → ((𝑟 freeLMod (𝑛 × 𝑛)) sSet ⟨(.r‘ndx), (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)⟩) = (𝐺 sSet ⟨(.r‘ndx), · ⟩))
15 df-mat 21108 . . . 4 Mat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ((𝑟 freeLMod (𝑛 × 𝑛)) sSet ⟨(.r‘ndx), (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)⟩))
16 ovex 7183 . . . 4 (𝐺 sSet ⟨(.r‘ndx), · ⟩) ∈ V
1714, 15, 16ovmpoa 7300 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (𝑁 Mat 𝑅) = (𝐺 sSet ⟨(.r‘ndx), · ⟩))
182, 17sylan2 595 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑁 Mat 𝑅) = (𝐺 sSet ⟨(.r‘ndx), · ⟩))
191, 18syl5eq 2805 1 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝐴 = (𝐺 sSet ⟨(.r‘ndx), · ⟩))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3409  ⟨cop 4528  ⟨cotp 4530   × cxp 5522  ‘cfv 6335  (class class class)co 7150  Fincfn 8527  ndxcnx 16538   sSet csts 16539  .rcmulr 16624   freeLMod cfrlm 20511   maMul cmmul 21085   Mat cmat 21107 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-ot 4531  df-uni 4799  df-br 5033  df-opab 5095  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-iota 6294  df-fun 6337  df-fv 6343  df-ov 7153  df-oprab 7154  df-mpo 7155  df-mat 21108 This theorem is referenced by:  matbas  21113  matplusg  21114  matsca  21115  matvsca  21116  matmulr  21138
 Copyright terms: Public domain W3C validator