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Theorem matval 22329
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 Γ— 𝑁))
matval.t Β· = (𝑅 maMul βŸ¨π‘, 𝑁, π‘βŸ©)
Assertion
Ref Expression
matval ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) β†’ 𝐴 = (𝐺 sSet ⟨(.rβ€˜ndx), Β· ⟩))

Proof of Theorem matval
Dummy variables 𝑛 π‘Ÿ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 matval.a . 2 𝐴 = (𝑁 Mat 𝑅)
2 elex 3490 . . 3 (𝑅 ∈ 𝑉 β†’ 𝑅 ∈ V)
3 id 22 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ π‘Ÿ = 𝑅)
4 id 22 . . . . . . . 8 (𝑛 = 𝑁 β†’ 𝑛 = 𝑁)
54sqxpeqd 5712 . . . . . . 7 (𝑛 = 𝑁 β†’ (𝑛 Γ— 𝑛) = (𝑁 Γ— 𝑁))
63, 5oveqan12rd 7444 . . . . . 6 ((𝑛 = 𝑁 ∧ π‘Ÿ = 𝑅) β†’ (π‘Ÿ freeLMod (𝑛 Γ— 𝑛)) = (𝑅 freeLMod (𝑁 Γ— 𝑁)))
7 matval.g . . . . . 6 𝐺 = (𝑅 freeLMod (𝑁 Γ— 𝑁))
86, 7eqtr4di 2785 . . . . 5 ((𝑛 = 𝑁 ∧ π‘Ÿ = 𝑅) β†’ (π‘Ÿ freeLMod (𝑛 Γ— 𝑛)) = 𝐺)
94, 4, 4oteq123d 4891 . . . . . . . 8 (𝑛 = 𝑁 β†’ βŸ¨π‘›, 𝑛, π‘›βŸ© = βŸ¨π‘, 𝑁, π‘βŸ©)
103, 9oveqan12rd 7444 . . . . . . 7 ((𝑛 = 𝑁 ∧ π‘Ÿ = 𝑅) β†’ (π‘Ÿ maMul βŸ¨π‘›, 𝑛, π‘›βŸ©) = (𝑅 maMul βŸ¨π‘, 𝑁, π‘βŸ©))
11 matval.t . . . . . . 7 Β· = (𝑅 maMul βŸ¨π‘, 𝑁, π‘βŸ©)
1210, 11eqtr4di 2785 . . . . . 6 ((𝑛 = 𝑁 ∧ π‘Ÿ = 𝑅) β†’ (π‘Ÿ maMul βŸ¨π‘›, 𝑛, π‘›βŸ©) = Β· )
1312opeq2d 4883 . . . . 5 ((𝑛 = 𝑁 ∧ π‘Ÿ = 𝑅) β†’ ⟨(.rβ€˜ndx), (π‘Ÿ maMul βŸ¨π‘›, 𝑛, π‘›βŸ©)⟩ = ⟨(.rβ€˜ndx), Β· ⟩)
148, 13oveq12d 7442 . . . 4 ((𝑛 = 𝑁 ∧ π‘Ÿ = 𝑅) β†’ ((π‘Ÿ freeLMod (𝑛 Γ— 𝑛)) sSet ⟨(.rβ€˜ndx), (π‘Ÿ maMul βŸ¨π‘›, 𝑛, π‘›βŸ©)⟩) = (𝐺 sSet ⟨(.rβ€˜ndx), Β· ⟩))
15 df-mat 22326 . . . 4 Mat = (𝑛 ∈ Fin, π‘Ÿ ∈ V ↦ ((π‘Ÿ freeLMod (𝑛 Γ— 𝑛)) sSet ⟨(.rβ€˜ndx), (π‘Ÿ maMul βŸ¨π‘›, 𝑛, π‘›βŸ©)⟩))
16 ovex 7457 . . . 4 (𝐺 sSet ⟨(.rβ€˜ndx), Β· ⟩) ∈ V
1714, 15, 16ovmpoa 7580 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) β†’ (𝑁 Mat 𝑅) = (𝐺 sSet ⟨(.rβ€˜ndx), Β· ⟩))
182, 17sylan2 591 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) β†’ (𝑁 Mat 𝑅) = (𝐺 sSet ⟨(.rβ€˜ndx), Β· ⟩))
191, 18eqtrid 2779 1 ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) β†’ 𝐴 = (𝐺 sSet ⟨(.rβ€˜ndx), Β· ⟩))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  Vcvv 3471  βŸ¨cop 4636  βŸ¨cotp 4638   Γ— cxp 5678  β€˜cfv 6551  (class class class)co 7424  Fincfn 8968   sSet csts 17137  ndxcnx 17167  .rcmulr 17239   freeLMod cfrlm 21685   maMul cmmul 22303   Mat cmat 22325
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-ot 4639  df-uni 4911  df-br 5151  df-opab 5213  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-iota 6503  df-fun 6553  df-fv 6559  df-ov 7427  df-oprab 7428  df-mpo 7429  df-mat 22326
This theorem is referenced by:  matbas  22331  matplusg  22332  matsca  22333  matscaOLD  22334  matvsca  22335  matvscaOLD  22336  matmulr  22358
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