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Theorem dmatval 20516
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 = (0g𝑅)
dmatval.d 𝐷 = (𝑁 DMat 𝑅)
Assertion
Ref Expression
dmatval ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝐷 = {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (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.
StepHypRef Expression
1 dmatval.d . 2 𝐷 = (𝑁 DMat 𝑅)
2 df-dmat 20514 . . . 4 DMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ {𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ∣ ∀𝑖𝑛𝑗𝑛 (𝑖𝑗 → (𝑖𝑚𝑗) = (0g𝑟))})
32a1i 11 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → DMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ {𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ∣ ∀𝑖𝑛𝑗𝑛 (𝑖𝑗 → (𝑖𝑚𝑗) = (0g𝑟))}))
4 oveq12 6805 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat 𝑟) = (𝑁 Mat 𝑅))
54fveq2d 6337 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = (Base‘(𝑁 Mat 𝑅)))
6 dmatval.b . . . . . . 7 𝐵 = (Base‘𝐴)
7 dmatval.a . . . . . . . 8 𝐴 = (𝑁 Mat 𝑅)
87fveq2i 6336 . . . . . . 7 (Base‘𝐴) = (Base‘(𝑁 Mat 𝑅))
96, 8eqtri 2793 . . . . . 6 𝐵 = (Base‘(𝑁 Mat 𝑅))
105, 9syl6eqr 2823 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = 𝐵)
11 simpl 468 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → 𝑛 = 𝑁)
12 fveq2 6333 . . . . . . . . . . 11 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
13 dmatval.0 . . . . . . . . . . 11 0 = (0g𝑅)
1412, 13syl6eqr 2823 . . . . . . . . . 10 (𝑟 = 𝑅 → (0g𝑟) = 0 )
1514adantl 467 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (0g𝑟) = 0 )
1615eqeq2d 2781 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → ((𝑖𝑚𝑗) = (0g𝑟) ↔ (𝑖𝑚𝑗) = 0 ))
1716imbi2d 329 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → ((𝑖𝑗 → (𝑖𝑚𝑗) = (0g𝑟)) ↔ (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )))
1811, 17raleqbidv 3301 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (∀𝑗𝑛 (𝑖𝑗 → (𝑖𝑚𝑗) = (0g𝑟)) ↔ ∀𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )))
1911, 18raleqbidv 3301 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → (∀𝑖𝑛𝑗𝑛 (𝑖𝑗 → (𝑖𝑚𝑗) = (0g𝑟)) ↔ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )))
2010, 19rabeqbidv 3345 . . . 4 ((𝑛 = 𝑁𝑟 = 𝑅) → {𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ∣ ∀𝑖𝑛𝑗𝑛 (𝑖𝑗 → (𝑖𝑚𝑗) = (0g𝑟))} = {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )})
2120adantl 467 . . 3 (((𝑁 ∈ Fin ∧ 𝑅𝑉) ∧ (𝑛 = 𝑁𝑟 = 𝑅)) → {𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ∣ ∀𝑖𝑛𝑗𝑛 (𝑖𝑗 → (𝑖𝑚𝑗) = (0g𝑟))} = {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )})
22 simpl 468 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑁 ∈ Fin)
23 elex 3364 . . . 4 (𝑅𝑉𝑅 ∈ V)
2423adantl 467 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑅 ∈ V)
256fvexi 6345 . . . 4 𝐵 ∈ V
26 rabexg 4946 . . . 4 (𝐵 ∈ V → {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )} ∈ V)
2725, 26mp1i 13 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )} ∈ V)
283, 21, 22, 24, 27ovmpt2d 6939 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑁 DMat 𝑅) = {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )})
291, 28syl5eq 2817 1 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝐷 = {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wne 2943  wral 3061  {crab 3065  Vcvv 3351  cfv 6030  (class class class)co 6796  cmpt2 6798  Fincfn 8113  Basecbs 16064  0gc0g 16308   Mat cmat 20430   DMat cdmat 20512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-iota 5993  df-fun 6032  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-dmat 20514
This theorem is referenced by:  dmatel  20517  dmatmulcl  20524  scmatdmat  20539  dmatbas  42717
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