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Theorem dmatval 21864
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 21862 . . . 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 7370 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat 𝑟) = (𝑁 Mat 𝑅))
54fveq2d 6850 . . . . . 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 6849 . . . . . . 7 (Base‘𝐴) = (Base‘(𝑁 Mat 𝑅))
96, 8eqtri 2761 . . . . . 6 𝐵 = (Base‘(𝑁 Mat 𝑅))
105, 9eqtr4di 2791 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = 𝐵)
11 simpl 484 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → 𝑛 = 𝑁)
12 fveq2 6846 . . . . . . . . . . 11 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
13 dmatval.0 . . . . . . . . . . 11 0 = (0g𝑅)
1412, 13eqtr4di 2791 . . . . . . . . . 10 (𝑟 = 𝑅 → (0g𝑟) = 0 )
1514adantl 483 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (0g𝑟) = 0 )
1615eqeq2d 2744 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → ((𝑖𝑚𝑗) = (0g𝑟) ↔ (𝑖𝑚𝑗) = 0 ))
1716imbi2d 341 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → ((𝑖𝑗 → (𝑖𝑚𝑗) = (0g𝑟)) ↔ (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )))
1811, 17raleqbidv 3318 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (∀𝑗𝑛 (𝑖𝑗 → (𝑖𝑚𝑗) = (0g𝑟)) ↔ ∀𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )))
1911, 18raleqbidv 3318 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → (∀𝑖𝑛𝑗𝑛 (𝑖𝑗 → (𝑖𝑚𝑗) = (0g𝑟)) ↔ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )))
2010, 19rabeqbidv 3423 . . . 4 ((𝑛 = 𝑁𝑟 = 𝑅) → {𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ∣ ∀𝑖𝑛𝑗𝑛 (𝑖𝑗 → (𝑖𝑚𝑗) = (0g𝑟))} = {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )})
2120adantl 483 . . 3 (((𝑁 ∈ Fin ∧ 𝑅𝑉) ∧ (𝑛 = 𝑁𝑟 = 𝑅)) → {𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ∣ ∀𝑖𝑛𝑗𝑛 (𝑖𝑗 → (𝑖𝑚𝑗) = (0g𝑟))} = {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )})
22 simpl 484 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑁 ∈ Fin)
23 elex 3465 . . . 4 (𝑅𝑉𝑅 ∈ V)
2423adantl 483 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑅 ∈ V)
256fvexi 6860 . . . 4 𝐵 ∈ V
26 rabexg 5292 . . . 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, 27ovmpod 7511 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑁 DMat 𝑅) = {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )})
291, 28eqtrid 2785 1 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝐷 = {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wne 2940  wral 3061  {crab 3406  Vcvv 3447  cfv 6500  (class class class)co 7361  cmpo 7363  Fincfn 8889  Basecbs 17091  0gc0g 17329   Mat cmat 21777   DMat cdmat 21860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-iota 6452  df-fun 6502  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-dmat 21862
This theorem is referenced by:  dmatel  21865  dmatmulcl  21872  scmatdmat  21887  dmatbas  46574
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