Theorem matval 22305

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 3471	. . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
3		id 22	. . . . . . 7 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅)
4		id 22	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → 𝑛 = 𝑁)
5	4	sqxpeqd 5673	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (𝑛 × 𝑛) = (𝑁 × 𝑁))
6	3, 5	oveqan12rd 7410	. . . . . 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 2783	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑟 freeLMod (𝑛 × 𝑛)) = 𝐺)
9	4, 4, 4	oteq123d 4855	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → ⟨𝑛, 𝑛, 𝑛⟩ = ⟨𝑁, 𝑁, 𝑁⟩)
10	3, 9	oveqan12rd 7410	. . . . . . 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 2783	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩) = · )
13	12	opeq2d 4847	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → ⟨(.r‘ndx), (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)⟩ = ⟨(.r‘ndx), · ⟩)
14	8, 13	oveq12d 7408	. . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → ((𝑟 freeLMod (𝑛 × 𝑛)) sSet ⟨(.r‘ndx), (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)⟩) = (𝐺 sSet ⟨(.r‘ndx), · ⟩))
15		df-mat 22302	. . . 4 ⊢ Mat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ((𝑟 freeLMod (𝑛 × 𝑛)) sSet ⟨(.r‘ndx), (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)⟩))
16		ovex 7423	. . . 4 ⊢ (𝐺 sSet ⟨(.r‘ndx), · ⟩) ∈ V
17	14, 15, 16	ovmpoa 7547	. . 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 2777	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝐴 = (𝐺 sSet ⟨(.r‘ndx), · ⟩))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3450  ⟨cop 4598  ⟨cotp 4600   × cxp 5639  ‘cfv 6514  (class class class)co 7390  Fincfn 8921   sSet csts 17140  ndxcnx 17170  ._rcmulr 17228   freeLMod cfrlm 21662   maMul cmmul 22284   Mat cmat 22301

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-ot 4601  df-uni 4875  df-br 5111  df-opab 5173  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-iota 6467  df-fun 6516  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-mat 22302

This theorem is referenced by:  matbas  22307  matplusg  22308  matsca  22309  matvsca  22310  matmulr  22332