Theorem dmatel 22515

Description: A 𝑁 x 𝑁 diagonal matrix over (a ring) 𝑅. (Contributed by AV, 18-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
dmatel	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑀 ∈ 𝐷 ↔ (𝑀 ∈ 𝐵 ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 ))))

Distinct variable groups:   𝑖,𝑁,𝑗   𝑅,𝑖,𝑗   𝑖,𝑀,𝑗

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

Proof of Theorem dmatel

Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmatval.a	. . . 4 ⊢ 𝐴 = (𝑁 Mat 𝑅)
2		dmatval.b	. . . 4 ⊢ 𝐵 = (Base‘𝐴)
3		dmatval.0	. . . 4 ⊢ 0 = (0g‘𝑅)
4		dmatval.d	. . . 4 ⊢ 𝐷 = (𝑁 DMat 𝑅)
5	1, 2, 3, 4	dmatval 22514	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝐷 = {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )})
6	5	eleq2d 2825	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑀 ∈ 𝐷 ↔ 𝑀 ∈ {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )}))
7		oveq 7437	. . . . . 6 ⊢ (𝑚 = 𝑀 → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
8	7	eqeq1d 2737	. . . . 5 ⊢ (𝑚 = 𝑀 → ((𝑖𝑚𝑗) = 0 ↔ (𝑖𝑀𝑗) = 0 ))
9	8	imbi2d 340	. . . 4 ⊢ (𝑚 = 𝑀 → ((𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 ) ↔ (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )))
10	9	2ralbidv 3219	. . 3 ⊢ (𝑚 = 𝑀 → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 ) ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )))
11	10	elrab 3695	. 2 ⊢ (𝑀 ∈ {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )} ↔ (𝑀 ∈ 𝐵 ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )))
12	6, 11	bitrdi 287	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑀 ∈ 𝐷 ↔ (𝑀 ∈ 𝐵 ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 ))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106   ≠ wne 2938  ∀wral 3059  {crab 3433  ‘cfv 6563  (class class class)co 7431  Fincfn 8984  Basecbs 17245  0_gc0g 17486   Mat cmat 22427   DMat cdmat 22510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-dmat 22512

This theorem is referenced by:  dmatmat  22516  dmatid  22517  dmatelnd  22518  dmatsubcl  22520  dmatscmcl  22525