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Theorem dmatbas 48248
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 22513 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝐷 = {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )})
6 elex 3498 . . 3 (𝑅𝑉𝑅 ∈ V)
7 eqid 2734 . . . 4 (𝑁 DMatALT 𝑅) = (𝑁 DMatALT 𝑅)
81, 2, 3, 7dmatALTbas 48246 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (Base‘(𝑁 DMatALT 𝑅)) = {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )})
96, 8sylan2 593 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (Base‘(𝑁 DMatALT 𝑅)) = {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )})
105, 9eqtr4d 2777 1 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝐷 = (Base‘(𝑁 DMatALT 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wcel 2105  wne 2937  wral 3058  {crab 3432  Vcvv 3477  cfv 6562  (class class class)co 7430  Fincfn 8983  Basecbs 17244  0gc0g 17485   Mat cmat 22426   DMat cdmat 22509   DMatALT cdmatalt 48241
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-1cn 11210  ax-addcl 11212
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-nn 12264  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  df-dmat 22511  df-dmatalt 48243
This theorem is referenced by: (None)
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