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Theorem dmatALTbas 44628
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 44627 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → 𝐷 = (𝐴s {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )}))
65fveq2d 6647 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (Base‘𝐷) = (Base‘(𝐴s {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )})))
72fvexi 6657 . . . 4 𝐵 ∈ V
8 rabexg 5207 . . . 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 2821 . . . 4 (𝐴s {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )}) = (𝐴s {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )})
1110, 2ressbas 16533 . . 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 4251 . . 3 ({𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )} ∩ 𝐵) = {𝑚 ∈ (𝐵𝐵) ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )}
14 inidm 4170 . . . 4 (𝐵𝐵) = 𝐵
15 rabeq 3460 . . . 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, 16syl5eq 2868 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → ({𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )} ∩ 𝐵) = {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )})
186, 12, 173eqtr2d 2862 1 ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (Base‘𝐷) = {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  wne 3007  wral 3126  {crab 3130  Vcvv 3471  cin 3909  cfv 6328  (class class class)co 7130  Fincfn 8484  Basecbs 16462  s cress 16463  0gc0g 16692   Mat cmat 20992   DMatALT cdmatalt 44623
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-cnex 10570  ax-1cn 10572  ax-addcl 10574
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 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-om 7556  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-nn 11616  df-ndx 16465  df-slot 16466  df-base 16468  df-sets 16469  df-ress 16470  df-dmatalt 44625
This theorem is referenced by:  dmatALTbasel  44629  dmatbas  44630
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