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Theorem dmatALTval 42714
Description: The algebra of 𝑁 x 𝑁 diagonal matrices over a 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
dmatALTval ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → 𝐷 = (𝐴s {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 dmatALTval
Dummy variables 𝑛 𝑟 𝑎 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmatALTval.d . 2 𝐷 = (𝑁 DMatALT 𝑅)
2 ovexd 6829 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat 𝑟) ∈ V)
3 id 22 . . . . . . 7 (𝑎 = (𝑛 Mat 𝑟) → 𝑎 = (𝑛 Mat 𝑟))
4 fveq2 6333 . . . . . . . 8 (𝑎 = (𝑛 Mat 𝑟) → (Base‘𝑎) = (Base‘(𝑛 Mat 𝑟)))
5 rabeq 3342 . . . . . . . 8 ((Base‘𝑎) = (Base‘(𝑛 Mat 𝑟)) → {𝑚 ∈ (Base‘𝑎) ∣ ∀𝑖𝑛𝑗𝑛 (𝑖𝑗 → (𝑖𝑚𝑗) = (0g𝑟))} = {𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ∣ ∀𝑖𝑛𝑗𝑛 (𝑖𝑗 → (𝑖𝑚𝑗) = (0g𝑟))})
64, 5syl 17 . . . . . . 7 (𝑎 = (𝑛 Mat 𝑟) → {𝑚 ∈ (Base‘𝑎) ∣ ∀𝑖𝑛𝑗𝑛 (𝑖𝑗 → (𝑖𝑚𝑗) = (0g𝑟))} = {𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ∣ ∀𝑖𝑛𝑗𝑛 (𝑖𝑗 → (𝑖𝑚𝑗) = (0g𝑟))})
73, 6oveq12d 6814 . . . . . 6 (𝑎 = (𝑛 Mat 𝑟) → (𝑎s {𝑚 ∈ (Base‘𝑎) ∣ ∀𝑖𝑛𝑗𝑛 (𝑖𝑗 → (𝑖𝑚𝑗) = (0g𝑟))}) = ((𝑛 Mat 𝑟) ↾s {𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ∣ ∀𝑖𝑛𝑗𝑛 (𝑖𝑗 → (𝑖𝑚𝑗) = (0g𝑟))}))
87adantl 467 . . . . 5 (((𝑛 = 𝑁𝑟 = 𝑅) ∧ 𝑎 = (𝑛 Mat 𝑟)) → (𝑎s {𝑚 ∈ (Base‘𝑎) ∣ ∀𝑖𝑛𝑗𝑛 (𝑖𝑗 → (𝑖𝑚𝑗) = (0g𝑟))}) = ((𝑛 Mat 𝑟) ↾s {𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ∣ ∀𝑖𝑛𝑗𝑛 (𝑖𝑗 → (𝑖𝑚𝑗) = (0g𝑟))}))
92, 8csbied 3709 . . . 4 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat 𝑟) / 𝑎(𝑎s {𝑚 ∈ (Base‘𝑎) ∣ ∀𝑖𝑛𝑗𝑛 (𝑖𝑗 → (𝑖𝑚𝑗) = (0g𝑟))}) = ((𝑛 Mat 𝑟) ↾s {𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ∣ ∀𝑖𝑛𝑗𝑛 (𝑖𝑗 → (𝑖𝑚𝑗) = (0g𝑟))}))
10 oveq12 6805 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat 𝑟) = (𝑁 Mat 𝑅))
11 dmatALTval.a . . . . . 6 𝐴 = (𝑁 Mat 𝑅)
1210, 11syl6eqr 2823 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat 𝑟) = 𝐴)
1312fveq2d 6337 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = (Base‘𝐴))
14 dmatALTval.b . . . . . . 7 𝐵 = (Base‘𝐴)
1513, 14syl6eqr 2823 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = 𝐵)
16 simpl 468 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → 𝑛 = 𝑁)
17 fveq2 6333 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
18 dmatALTval.0 . . . . . . . . . . . 12 0 = (0g𝑅)
1917, 18syl6eqr 2823 . . . . . . . . . . 11 (𝑟 = 𝑅 → (0g𝑟) = 0 )
2019adantl 467 . . . . . . . . . 10 ((𝑛 = 𝑁𝑟 = 𝑅) → (0g𝑟) = 0 )
2120eqeq2d 2781 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → ((𝑖𝑚𝑗) = (0g𝑟) ↔ (𝑖𝑚𝑗) = 0 ))
2221imbi2d 329 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → ((𝑖𝑗 → (𝑖𝑚𝑗) = (0g𝑟)) ↔ (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )))
2316, 22raleqbidv 3301 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (∀𝑗𝑛 (𝑖𝑗 → (𝑖𝑚𝑗) = (0g𝑟)) ↔ ∀𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )))
2416, 23raleqbidv 3301 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (∀𝑖𝑛𝑗𝑛 (𝑖𝑗 → (𝑖𝑚𝑗) = (0g𝑟)) ↔ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )))
2515, 24rabeqbidv 3345 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → {𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ∣ ∀𝑖𝑛𝑗𝑛 (𝑖𝑗 → (𝑖𝑚𝑗) = (0g𝑟))} = {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )})
2612, 25oveq12d 6814 . . . 4 ((𝑛 = 𝑁𝑟 = 𝑅) → ((𝑛 Mat 𝑟) ↾s {𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ∣ ∀𝑖𝑛𝑗𝑛 (𝑖𝑗 → (𝑖𝑚𝑗) = (0g𝑟))}) = (𝐴s {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )}))
279, 26eqtrd 2805 . . 3 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat 𝑟) / 𝑎(𝑎s {𝑚 ∈ (Base‘𝑎) ∣ ∀𝑖𝑛𝑗𝑛 (𝑖𝑗 → (𝑖𝑚𝑗) = (0g𝑟))}) = (𝐴s {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )}))
28 df-dmatalt 42712 . . 3 DMatALT = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑛 Mat 𝑟) / 𝑎(𝑎s {𝑚 ∈ (Base‘𝑎) ∣ ∀𝑖𝑛𝑗𝑛 (𝑖𝑗 → (𝑖𝑚𝑗) = (0g𝑟))}))
29 ovex 6827 . . 3 (𝐴s {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )}) ∈ V
3027, 28, 29ovmpt2a 6942 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (𝑁 DMatALT 𝑅) = (𝐴s {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )}))
311, 30syl5eq 2817 1 ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → 𝐷 = (𝐴s {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wne 2943  wral 3061  {crab 3065  Vcvv 3351  csb 3682  cfv 6030  (class class class)co 6796  Fincfn 8113  Basecbs 16064  s cress 16065  0gc0g 16308   Mat cmat 20430   DMatALT cdmatalt 42710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-iota 5993  df-fun 6032  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-dmatalt 42712
This theorem is referenced by:  dmatALTbas  42715
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