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Theorem dmatALTbas 47338
Description: The base set of the algebra of 𝑁 x 𝑁 diagonal matrices over a ring 𝑅, i.e. the set of all 𝑁 x 𝑁 diagonal matrices over the ring 𝑅. (Contributed by AV, 8-Dec-2019.)
Hypotheses
Ref Expression
dmatALTval.a 𝐴 = (𝑁 Mat 𝑅)
dmatALTval.b 𝐵 = (Base‘𝐴)
dmatALTval.0 0 = (0g𝑅)
dmatALTval.d 𝐷 = (𝑁 DMatALT 𝑅)
Assertion
Ref Expression
dmatALTbas ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (Base‘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)   0 (𝑖,𝑗,𝑚)

Proof of Theorem dmatALTbas
StepHypRef Expression
1 dmatALTval.a . . . 4 𝐴 = (𝑁 Mat 𝑅)
2 dmatALTval.b . . . 4 𝐵 = (Base‘𝐴)
3 dmatALTval.0 . . . 4 0 = (0g𝑅)
4 dmatALTval.d . . . 4 𝐷 = (𝑁 DMatALT 𝑅)
51, 2, 3, 4dmatALTval 47337 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → 𝐷 = (𝐴s {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )}))
65fveq2d 6888 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (Base‘𝐷) = (Base‘(𝐴s {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )})))
72fvexi 6898 . . . 4 𝐵 ∈ V
8 rabexg 5324 . . . 4 (𝐵 ∈ V → {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )} ∈ V)
97, 8mp1i 13 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )} ∈ V)
10 eqid 2726 . . . 4 (𝐴s {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )}) = (𝐴s {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )})
1110, 2ressbas 17186 . . 3 ({𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )} ∈ V → ({𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )} ∩ 𝐵) = (Base‘(𝐴s {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )})))
129, 11syl 17 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → ({𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )} ∩ 𝐵) = (Base‘(𝐴s {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )})))
13 inrab2 4302 . . 3 ({𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )} ∩ 𝐵) = {𝑚 ∈ (𝐵𝐵) ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )}
14 inidm 4213 . . . 4 (𝐵𝐵) = 𝐵
15 rabeq 3440 . . . 4 ((𝐵𝐵) = 𝐵 → {𝑚 ∈ (𝐵𝐵) ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )} = {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )})
1614, 15mp1i 13 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → {𝑚 ∈ (𝐵𝐵) ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )} = {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )})
1713, 16eqtrid 2778 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → ({𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )} ∩ 𝐵) = {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )})
186, 12, 173eqtr2d 2772 1 ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (Base‘𝐷) = {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  wne 2934  wral 3055  {crab 3426  Vcvv 3468  cin 3942  cfv 6536  (class class class)co 7404  Fincfn 8938  Basecbs 17151  s cress 17180  0gc0g 17392   Mat cmat 22258   DMatALT cdmatalt 47333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-cnex 11165  ax-1cn 11167  ax-addcl 11169
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7852  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408  df-nn 12214  df-sets 17104  df-slot 17122  df-ndx 17134  df-base 17152  df-ress 17181  df-dmatalt 47335
This theorem is referenced by:  dmatALTbasel  47339  dmatbas  47340
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