Theorem dmatALTval 48876

Description: The algebra of 𝑁 x 𝑁 diagonal matrices over a ring 𝑅. (Contributed by AV, 8-Dec-2019.)

Hypotheses

Ref	Expression
dmatALTval.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
dmatALTval.b	⊢ 𝐵 = (Base‘𝐴)
dmatALTval.0	⊢ 0 = (0g‘𝑅)
dmatALTval.d	⊢ 𝐷 = (𝑁 DMatALT 𝑅)

Assertion

Ref	Expression
dmatALTval	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → 𝐷 = (𝐴 ↾s {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )}))

Distinct variable groups:   𝐵,𝑚   𝑖,𝑁,𝑗,𝑚   𝑅,𝑖,𝑗,𝑚

Allowed substitution hints:   𝐴(𝑖,𝑗,𝑚)   𝐵(𝑖,𝑗)   𝐷(𝑖,𝑗,𝑚)   0 (𝑖,𝑗,𝑚)

Proof of Theorem dmatALTval

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

Step	Hyp	Ref	Expression
1		dmatALTval.d	. 2 ⊢ 𝐷 = (𝑁 DMatALT 𝑅)
2		ovexd 7402	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑛 Mat 𝑟) ∈ V)
3		id 22	. . . . . . 7 ⊢ (𝑎 = (𝑛 Mat 𝑟) → 𝑎 = (𝑛 Mat 𝑟))
4		fveq2 6840	. . . . . . . 8 ⊢ (𝑎 = (𝑛 Mat 𝑟) → (Base‘𝑎) = (Base‘(𝑛 Mat 𝑟)))
5	4	rabeqdv 3404	. . . . . . 7 ⊢ (𝑎 = (𝑛 Mat 𝑟) → {𝑚 ∈ (Base‘𝑎) ∣ ∀𝑖 ∈ 𝑛 ∀𝑗 ∈ 𝑛 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = (0g‘𝑟))} = {𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ∣ ∀𝑖 ∈ 𝑛 ∀𝑗 ∈ 𝑛 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = (0g‘𝑟))})
6	3, 5	oveq12d 7385	. . . . . 6 ⊢ (𝑎 = (𝑛 Mat 𝑟) → (𝑎 ↾s {𝑚 ∈ (Base‘𝑎) ∣ ∀𝑖 ∈ 𝑛 ∀𝑗 ∈ 𝑛 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = (0g‘𝑟))}) = ((𝑛 Mat 𝑟) ↾s {𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ∣ ∀𝑖 ∈ 𝑛 ∀𝑗 ∈ 𝑛 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = (0g‘𝑟))}))
7	6	adantl 481	. . . . 5 ⊢ (((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) ∧ 𝑎 = (𝑛 Mat 𝑟)) → (𝑎 ↾s {𝑚 ∈ (Base‘𝑎) ∣ ∀𝑖 ∈ 𝑛 ∀𝑗 ∈ 𝑛 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = (0g‘𝑟))}) = ((𝑛 Mat 𝑟) ↾s {𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ∣ ∀𝑖 ∈ 𝑛 ∀𝑗 ∈ 𝑛 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = (0g‘𝑟))}))
8	2, 7	csbied 3873	. . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → ⦋(𝑛 Mat 𝑟) / 𝑎⦌(𝑎 ↾s {𝑚 ∈ (Base‘𝑎) ∣ ∀𝑖 ∈ 𝑛 ∀𝑗 ∈ 𝑛 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = (0g‘𝑟))}) = ((𝑛 Mat 𝑟) ↾s {𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ∣ ∀𝑖 ∈ 𝑛 ∀𝑗 ∈ 𝑛 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = (0g‘𝑟))}))
9		oveq12 7376	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑛 Mat 𝑟) = (𝑁 Mat 𝑅))
10		dmatALTval.a	. . . . . 6 ⊢ 𝐴 = (𝑁 Mat 𝑅)
11	9, 10	eqtr4di 2789	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑛 Mat 𝑟) = 𝐴)
12	11	fveq2d 6844	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = (Base‘𝐴))
13		dmatALTval.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐴)
14	12, 13	eqtr4di 2789	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = 𝐵)
15		simpl 482	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → 𝑛 = 𝑁)
16		fveq2 6840	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅))
17		dmatALTval.0	. . . . . . . . . . . 12 ⊢ 0 = (0g‘𝑅)
18	16, 17	eqtr4di 2789	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 )
19	18	adantl 481	. . . . . . . . . 10 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (0g‘𝑟) = 0 )
20	19	eqeq2d 2747	. . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → ((𝑖𝑚𝑗) = (0g‘𝑟) ↔ (𝑖𝑚𝑗) = 0 ))
21	20	imbi2d 340	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → ((𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = (0g‘𝑟)) ↔ (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )))
22	15, 21	raleqbidv 3311	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (∀𝑗 ∈ 𝑛 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = (0g‘𝑟)) ↔ ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )))
23	15, 22	raleqbidv 3311	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (∀𝑖 ∈ 𝑛 ∀𝑗 ∈ 𝑛 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = (0g‘𝑟)) ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )))
24	14, 23	rabeqbidv 3407	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → {𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ∣ ∀𝑖 ∈ 𝑛 ∀𝑗 ∈ 𝑛 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = (0g‘𝑟))} = {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )})
25	11, 24	oveq12d 7385	. . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → ((𝑛 Mat 𝑟) ↾s {𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ∣ ∀𝑖 ∈ 𝑛 ∀𝑗 ∈ 𝑛 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = (0g‘𝑟))}) = (𝐴 ↾s {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )}))
26	8, 25	eqtrd 2771	. . 3 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → ⦋(𝑛 Mat 𝑟) / 𝑎⦌(𝑎 ↾s {𝑚 ∈ (Base‘𝑎) ∣ ∀𝑖 ∈ 𝑛 ∀𝑗 ∈ 𝑛 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = (0g‘𝑟))}) = (𝐴 ↾s {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )}))
27		df-dmatalt 48874	. . 3 ⊢ DMatALT = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ⦋(𝑛 Mat 𝑟) / 𝑎⦌(𝑎 ↾s {𝑚 ∈ (Base‘𝑎) ∣ ∀𝑖 ∈ 𝑛 ∀𝑗 ∈ 𝑛 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = (0g‘𝑟))}))
28		ovex 7400	. . 3 ⊢ (𝐴 ↾s {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )}) ∈ V
29	26, 27, 28	ovmpoa 7522	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (𝑁 DMatALT 𝑅) = (𝐴 ↾s {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )}))
30	1, 29	eqtrid 2783	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → 𝐷 = (𝐴 ↾s {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  ∀wral 3051  {crab 3389  Vcvv 3429  ⦋csb 3837  ‘cfv 6498  (class class class)co 7367  Fincfn 8893  Basecbs 17179   ↾_s cress 17200  0_gc0g 17402   Mat cmat 22372   DMatALT cdmatalt 48872

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6454  df-fun 6500  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-dmatalt 48874

This theorem is referenced by:  dmatALTbas  48877