Theorem matval 22327

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 ∧ 𝑅 ∈ 𝑉) → 𝐴 = (𝐺 sSet ⟨(.r‘ndx), · ⟩))

Proof of Theorem matval

Dummy variables 𝑛 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		matval.a	. 2 ⊢ 𝐴 = (𝑁 Mat 𝑅)
2		elex 3458	. . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
3		id 22	. . . . . . 7 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅)
4		id 22	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → 𝑛 = 𝑁)
5	4	sqxpeqd 5651	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (𝑛 × 𝑛) = (𝑁 × 𝑁))
6	3, 5	oveqan12rd 7372	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑟 freeLMod (𝑛 × 𝑛)) = (𝑅 freeLMod (𝑁 × 𝑁)))
7		matval.g	. . . . . 6 ⊢ 𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))
8	6, 7	eqtr4di 2786	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑟 freeLMod (𝑛 × 𝑛)) = 𝐺)
9	4, 4, 4	oteq123d 4839	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → ⟨𝑛, 𝑛, 𝑛⟩ = ⟨𝑁, 𝑁, 𝑁⟩)
10	3, 9	oveqan12rd 7372	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩) = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩))
11		matval.t	. . . . . . 7 ⊢ · = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)
12	10, 11	eqtr4di 2786	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩) = · )
13	12	opeq2d 4831	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → ⟨(.r‘ndx), (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)⟩ = ⟨(.r‘ndx), · ⟩)
14	8, 13	oveq12d 7370	. . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → ((𝑟 freeLMod (𝑛 × 𝑛)) sSet ⟨(.r‘ndx), (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)⟩) = (𝐺 sSet ⟨(.r‘ndx), · ⟩))
15		df-mat 22324	. . . 4 ⊢ Mat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ((𝑟 freeLMod (𝑛 × 𝑛)) sSet ⟨(.r‘ndx), (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)⟩))
16		ovex 7385	. . . 4 ⊢ (𝐺 sSet ⟨(.r‘ndx), · ⟩) ∈ V
17	14, 15, 16	ovmpoa 7507	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (𝑁 Mat 𝑅) = (𝐺 sSet ⟨(.r‘ndx), · ⟩))
18	2, 17	sylan2 593	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑁 Mat 𝑅) = (𝐺 sSet ⟨(.r‘ndx), · ⟩))
19	1, 18	eqtrid 2780	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝐴 = (𝐺 sSet ⟨(.r‘ndx), · ⟩))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Vcvv 3437  ⟨cop 4581  ⟨cotp 4583   × cxp 5617  ‘cfv 6486  (class class class)co 7352  Fincfn 8875   sSet csts 17076  ndxcnx 17106  ._rcmulr 17164   freeLMod cfrlm 21685   maMul cmmul 22306   Mat cmat 22323

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-ot 4584  df-uni 4859  df-br 5094  df-opab 5156  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-iota 6442  df-fun 6488  df-fv 6494  df-ov 7355  df-oprab 7356  df-mpo 7357  df-mat 22324

This theorem is referenced by:  matbas  22329  matplusg  22330  matsca  22331  matvsca  22332  matmulr  22354