Theorem dmatval 22405

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

Hypotheses

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

Assertion

Ref	Expression
dmatval	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝐷 = {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)   𝑉(𝑖,𝑗,𝑚)   0 (𝑖,𝑗,𝑚)

Proof of Theorem dmatval

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

Step	Hyp	Ref	Expression
1		dmatval.d	. 2 ⊢ 𝐷 = (𝑁 DMat 𝑅)
2		df-dmat 22403	. . . 4 ⊢ DMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ {𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ∣ ∀𝑖 ∈ 𝑛 ∀𝑗 ∈ 𝑛 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = (0g‘𝑟))})
3	2	a1i 11	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → DMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ {𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ∣ ∀𝑖 ∈ 𝑛 ∀𝑗 ∈ 𝑛 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = (0g‘𝑟))}))
4		oveq12 7355	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑛 Mat 𝑟) = (𝑁 Mat 𝑅))
5	4	fveq2d 6826	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = (Base‘(𝑁 Mat 𝑅)))
6		dmatval.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐴)
7		dmatval.a	. . . . . . . 8 ⊢ 𝐴 = (𝑁 Mat 𝑅)
8	7	fveq2i 6825	. . . . . . 7 ⊢ (Base‘𝐴) = (Base‘(𝑁 Mat 𝑅))
9	6, 8	eqtri 2754	. . . . . 6 ⊢ 𝐵 = (Base‘(𝑁 Mat 𝑅))
10	5, 9	eqtr4di 2784	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = 𝐵)
11		simpl 482	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → 𝑛 = 𝑁)
12		fveq2 6822	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅))
13		dmatval.0	. . . . . . . . . . 11 ⊢ 0 = (0g‘𝑅)
14	12, 13	eqtr4di 2784	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 )
15	14	adantl 481	. . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (0g‘𝑟) = 0 )
16	15	eqeq2d 2742	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → ((𝑖𝑚𝑗) = (0g‘𝑟) ↔ (𝑖𝑚𝑗) = 0 ))
17	16	imbi2d 340	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → ((𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = (0g‘𝑟)) ↔ (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )))
18	11, 17	raleqbidv 3312	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (∀𝑗 ∈ 𝑛 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = (0g‘𝑟)) ↔ ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )))
19	11, 18	raleqbidv 3312	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (∀𝑖 ∈ 𝑛 ∀𝑗 ∈ 𝑛 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = (0g‘𝑟)) ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )))
20	10, 19	rabeqbidv 3413	. . . 4 ⊢ ((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 )})
21	20	adantl 481	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) ∧ (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 )})
22		simpl 482	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑁 ∈ Fin)
23		elex 3457	. . . 4 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
24	23	adantl 481	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑅 ∈ V)
25	6	fvexi 6836	. . . 4 ⊢ 𝐵 ∈ V
26		rabexg 5275	. . . 4 ⊢ (𝐵 ∈ V → {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )} ∈ V)
27	25, 26	mp1i 13	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )} ∈ V)
28	3, 21, 22, 24, 27	ovmpod 7498	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑁 DMat 𝑅) = {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )})
29	1, 28	eqtrid 2778	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝐷 = {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  ∀wral 3047  {crab 3395  Vcvv 3436  ‘cfv 6481  (class class class)co 7346   ∈ cmpo 7348  Fincfn 8869  Basecbs 17117  0_gc0g 17340   Mat cmat 22320   DMat cdmat 22401

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3742  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-iota 6437  df-fun 6483  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-dmat 22403

This theorem is referenced by:  dmatel  22406  dmatmulcl  22413  scmatdmat  22428  dmatbas  48434