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Theorem cpmat 22210
Description: Value of the constructor of the set of all constant polynomial matrices, i.e. the set of all 𝑁 x 𝑁 matrices of polynomials over a ring 𝑅. (Contributed by AV, 15-Nov-2019.)
Hypotheses
Ref Expression
cpmat.s 𝑆 = (𝑁 ConstPolyMat 𝑅)
cpmat.p 𝑃 = (Poly1𝑅)
cpmat.c 𝐶 = (𝑁 Mat 𝑃)
cpmat.b 𝐵 = (Base‘𝐶)
Assertion
Ref Expression
cpmat ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑆 = {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑅)})
Distinct variable groups:   𝐵,𝑚   𝑖,𝑁,𝑗,𝑘,𝑚   𝑅,𝑖,𝑗,𝑘,𝑚
Allowed substitution hints:   𝐵(𝑖,𝑗,𝑘)   𝐶(𝑖,𝑗,𝑘,𝑚)   𝑃(𝑖,𝑗,𝑘,𝑚)   𝑆(𝑖,𝑗,𝑘,𝑚)   𝑉(𝑖,𝑗,𝑘,𝑚)

Proof of Theorem cpmat
Dummy variables 𝑛 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cpmat.s . 2 𝑆 = (𝑁 ConstPolyMat 𝑅)
2 df-cpmat 22207 . . . 4 ConstPolyMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ {𝑚 ∈ (Base‘(𝑛 Mat (Poly1𝑟))) ∣ ∀𝑖𝑛𝑗𝑛𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑟)})
32a1i 11 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → ConstPolyMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ {𝑚 ∈ (Base‘(𝑛 Mat (Poly1𝑟))) ∣ ∀𝑖𝑛𝑗𝑛𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑟)}))
4 simpl 483 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → 𝑛 = 𝑁)
5 fveq2 6891 . . . . . . . . 9 (𝑟 = 𝑅 → (Poly1𝑟) = (Poly1𝑅))
65adantl 482 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (Poly1𝑟) = (Poly1𝑅))
74, 6oveq12d 7426 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat (Poly1𝑟)) = (𝑁 Mat (Poly1𝑅)))
87fveq2d 6895 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(𝑛 Mat (Poly1𝑟))) = (Base‘(𝑁 Mat (Poly1𝑅))))
9 cpmat.b . . . . . . 7 𝐵 = (Base‘𝐶)
10 cpmat.c . . . . . . . . 9 𝐶 = (𝑁 Mat 𝑃)
11 cpmat.p . . . . . . . . . 10 𝑃 = (Poly1𝑅)
1211oveq2i 7419 . . . . . . . . 9 (𝑁 Mat 𝑃) = (𝑁 Mat (Poly1𝑅))
1310, 12eqtri 2760 . . . . . . . 8 𝐶 = (𝑁 Mat (Poly1𝑅))
1413fveq2i 6894 . . . . . . 7 (Base‘𝐶) = (Base‘(𝑁 Mat (Poly1𝑅)))
159, 14eqtri 2760 . . . . . 6 𝐵 = (Base‘(𝑁 Mat (Poly1𝑅)))
168, 15eqtr4di 2790 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(𝑛 Mat (Poly1𝑟))) = 𝐵)
17 fveq2 6891 . . . . . . . . . 10 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
1817adantl 482 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (0g𝑟) = (0g𝑅))
1918eqeq2d 2743 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑟) ↔ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑅)))
2019ralbidv 3177 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑟) ↔ ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑅)))
214, 20raleqbidv 3342 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (∀𝑗𝑛𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑟) ↔ ∀𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑅)))
224, 21raleqbidv 3342 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → (∀𝑖𝑛𝑗𝑛𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑟) ↔ ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑅)))
2316, 22rabeqbidv 3449 . . . 4 ((𝑛 = 𝑁𝑟 = 𝑅) → {𝑚 ∈ (Base‘(𝑛 Mat (Poly1𝑟))) ∣ ∀𝑖𝑛𝑗𝑛𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑟)} = {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑅)})
2423adantl 482 . . 3 (((𝑁 ∈ Fin ∧ 𝑅𝑉) ∧ (𝑛 = 𝑁𝑟 = 𝑅)) → {𝑚 ∈ (Base‘(𝑛 Mat (Poly1𝑟))) ∣ ∀𝑖𝑛𝑗𝑛𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑟)} = {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑅)})
25 simpl 483 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑁 ∈ Fin)
26 elex 3492 . . . 4 (𝑅𝑉𝑅 ∈ V)
2726adantl 482 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑅 ∈ V)
289fvexi 6905 . . . 4 𝐵 ∈ V
29 rabexg 5331 . . . 4 (𝐵 ∈ V → {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑅)} ∈ V)
3028, 29mp1i 13 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑅)} ∈ V)
313, 24, 25, 27, 30ovmpod 7559 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑁 ConstPolyMat 𝑅) = {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑅)})
321, 31eqtrid 2784 1 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑆 = {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑅)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wral 3061  {crab 3432  Vcvv 3474  cfv 6543  (class class class)co 7408  cmpo 7410  Fincfn 8938  cn 12211  Basecbs 17143  0gc0g 17384  Poly1cpl1 21700  coe1cco1 21701   Mat cmat 21906   ConstPolyMat ccpmat 22204
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-cpmat 22207
This theorem is referenced by:  cpmatpmat  22211  cpmatel  22212  cpmatsubgpmat  22221
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