Theorem matval 20217

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 3212	. . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
3		id 22	. . . . . . 7 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅)
4		id 22	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → 𝑛 = 𝑁)
5	4	sqxpeqd 5141	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (𝑛 × 𝑛) = (𝑁 × 𝑁))
6	3, 5	oveqan12rd 6670	. . . . . 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	syl6eqr 2674	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑟 freeLMod (𝑛 × 𝑛)) = 𝐺)
9	4, 4, 4	oteq123d 4417	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → ⟨𝑛, 𝑛, 𝑛⟩ = ⟨𝑁, 𝑁, 𝑁⟩)
10	3, 9	oveqan12rd 6670	. . . . . . 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	syl6eqr 2674	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩) = · )
13	12	opeq2d 4409	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → ⟨(.r‘ndx), (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)⟩ = ⟨(.r‘ndx), · ⟩)
14	8, 13	oveq12d 6668	. . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → ((𝑟 freeLMod (𝑛 × 𝑛)) sSet ⟨(.r‘ndx), (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)⟩) = (𝐺 sSet ⟨(.r‘ndx), · ⟩))
15		df-mat 20214	. . . 4 ⊢ Mat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ((𝑟 freeLMod (𝑛 × 𝑛)) sSet ⟨(.r‘ndx), (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)⟩))
16		ovex 6678	. . . 4 ⊢ (𝐺 sSet ⟨(.r‘ndx), · ⟩) ∈ V
17	14, 15, 16	ovmpt2a 6791	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (𝑁 Mat 𝑅) = (𝐺 sSet ⟨(.r‘ndx), · ⟩))
18	2, 17	sylan2 491	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑁 Mat 𝑅) = (𝐺 sSet ⟨(.r‘ndx), · ⟩))
19	1, 18	syl5eq 2668	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝐴 = (𝐺 sSet ⟨(.r‘ndx), · ⟩))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200  ⟨cop 4183  ⟨cotp 4185   × cxp 5112  ‘cfv 5888  (class class class)co 6650  Fincfn 7955  ndxcnx 15854   sSet csts 15855  ._rcmulr 15942   freeLMod cfrlm 20090   maMul cmmul 20189   Mat cmat 20213

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-ot 4186  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-mat 20214

This theorem is referenced by:  matbas  20219  matplusg  20220  matsca  20221  matvsca  20222  matmulr  20244