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Theorem dmatbas 44751
 Description: The set of all 𝑁 x 𝑁 diagonal matrices over (the ring) 𝑅 is the base set of the algebra of 𝑁 x 𝑁 diagonal matrices over (the ring) 𝑅. (Contributed by AV, 8-Dec-2019.)
Hypotheses
Ref Expression
dmatbas.a 𝐴 = (𝑁 Mat 𝑅)
dmatbas.b 𝐵 = (Base‘𝐴)
dmatbas.0 0 = (0g𝑅)
dmatbas.d 𝐷 = (𝑁 DMat 𝑅)
Assertion
Ref Expression
dmatbas ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝐷 = (Base‘(𝑁 DMatALT 𝑅)))

Proof of Theorem dmatbas
Dummy variables 𝑚 𝑖 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmatbas.a . . 3 𝐴 = (𝑁 Mat 𝑅)
2 dmatbas.b . . 3 𝐵 = (Base‘𝐴)
3 dmatbas.0 . . 3 0 = (0g𝑅)
4 dmatbas.d . . 3 𝐷 = (𝑁 DMat 𝑅)
51, 2, 3, 4dmatval 21095 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝐷 = {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )})
6 elex 3487 . . 3 (𝑅𝑉𝑅 ∈ V)
7 eqid 2822 . . . 4 (𝑁 DMatALT 𝑅) = (𝑁 DMatALT 𝑅)
81, 2, 3, 7dmatALTbas 44749 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (Base‘(𝑁 DMatALT 𝑅)) = {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )})
96, 8sylan2 595 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (Base‘(𝑁 DMatALT 𝑅)) = {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )})
105, 9eqtr4d 2860 1 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝐷 = (Base‘(𝑁 DMatALT 𝑅)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2114   ≠ wne 3011  ∀wral 3130  {crab 3134  Vcvv 3469  ‘cfv 6334  (class class class)co 7140  Fincfn 8496  Basecbs 16474  0gc0g 16704   Mat cmat 21010   DMat cdmat 21091   DMatALT cdmatalt 44744 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-cnex 10582  ax-1cn 10584  ax-addcl 10586 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-nn 11626  df-ndx 16477  df-slot 16478  df-base 16480  df-sets 16481  df-ress 16482  df-dmat 21093  df-dmatalt 44746 This theorem is referenced by: (None)
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