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Theorem dmatval 21077
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 21075 . . . 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 7142 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat 𝑟) = (𝑁 Mat 𝑅))
54fveq2d 6650 . . . . . 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 6649 . . . . . . 7 (Base‘𝐴) = (Base‘(𝑁 Mat 𝑅))
96, 8eqtri 2843 . . . . . 6 𝐵 = (Base‘(𝑁 Mat 𝑅))
105, 9syl6eqr 2873 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = 𝐵)
11 simpl 485 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → 𝑛 = 𝑁)
12 fveq2 6646 . . . . . . . . . . 11 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
13 dmatval.0 . . . . . . . . . . 11 0 = (0g𝑅)
1412, 13syl6eqr 2873 . . . . . . . . . 10 (𝑟 = 𝑅 → (0g𝑟) = 0 )
1514adantl 484 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (0g𝑟) = 0 )
1615eqeq2d 2831 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → ((𝑖𝑚𝑗) = (0g𝑟) ↔ (𝑖𝑚𝑗) = 0 ))
1716imbi2d 343 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → ((𝑖𝑗 → (𝑖𝑚𝑗) = (0g𝑟)) ↔ (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )))
1811, 17raleqbidv 3388 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (∀𝑗𝑛 (𝑖𝑗 → (𝑖𝑚𝑗) = (0g𝑟)) ↔ ∀𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )))
1911, 18raleqbidv 3388 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → (∀𝑖𝑛𝑗𝑛 (𝑖𝑗 → (𝑖𝑚𝑗) = (0g𝑟)) ↔ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )))
2010, 19rabeqbidv 3464 . . . 4 ((𝑛 = 𝑁𝑟 = 𝑅) → {𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ∣ ∀𝑖𝑛𝑗𝑛 (𝑖𝑗 → (𝑖𝑚𝑗) = (0g𝑟))} = {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )})
2120adantl 484 . . 3 (((𝑁 ∈ Fin ∧ 𝑅𝑉) ∧ (𝑛 = 𝑁𝑟 = 𝑅)) → {𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ∣ ∀𝑖𝑛𝑗𝑛 (𝑖𝑗 → (𝑖𝑚𝑗) = (0g𝑟))} = {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )})
22 simpl 485 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑁 ∈ Fin)
23 elex 3491 . . . 4 (𝑅𝑉𝑅 ∈ V)
2423adantl 484 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑅 ∈ V)
256fvexi 6660 . . . 4 𝐵 ∈ V
26 rabexg 5210 . . . 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 7279 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑁 DMat 𝑅) = {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )})
291, 28syl5eq 2867 1 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝐷 = {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1537  wcel 2114  wne 3006  wral 3125  {crab 3129  Vcvv 3473  cfv 6331  (class class class)co 7133  cmpo 7135  Fincfn 8487  Basecbs 16462  0gc0g 16692   Mat cmat 20992   DMat cdmat 21073
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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pr 5306
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3753  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-br 5043  df-opab 5105  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-iota 6290  df-fun 6333  df-fv 6339  df-ov 7136  df-oprab 7137  df-mpo 7138  df-dmat 21075
This theorem is referenced by:  dmatel  21078  dmatmulcl  21085  scmatdmat  21100  dmatbas  44603
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