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Theorem cpmat 20333
 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 20330 . . . 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 472 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → 𝑛 = 𝑁)
5 fveq2 6103 . . . . . . . . 9 (𝑟 = 𝑅 → (Poly1𝑟) = (Poly1𝑅))
65adantl 481 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (Poly1𝑟) = (Poly1𝑅))
74, 6oveq12d 6567 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat (Poly1𝑟)) = (𝑁 Mat (Poly1𝑅)))
87fveq2d 6107 . . . . . 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 6560 . . . . . . . . 9 (𝑁 Mat 𝑃) = (𝑁 Mat (Poly1𝑅))
1310, 12eqtri 2632 . . . . . . . 8 𝐶 = (𝑁 Mat (Poly1𝑅))
1413fveq2i 6106 . . . . . . 7 (Base‘𝐶) = (Base‘(𝑁 Mat (Poly1𝑅)))
159, 14eqtri 2632 . . . . . 6 𝐵 = (Base‘(𝑁 Mat (Poly1𝑅)))
168, 15syl6eqr 2662 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(𝑛 Mat (Poly1𝑟))) = 𝐵)
17 fveq2 6103 . . . . . . . . . 10 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
1817adantl 481 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (0g𝑟) = (0g𝑅))
1918eqeq2d 2620 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑟) ↔ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑅)))
2019ralbidv 2969 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑟) ↔ ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑅)))
214, 20raleqbidv 3129 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (∀𝑗𝑛𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑟) ↔ ∀𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑅)))
224, 21raleqbidv 3129 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → (∀𝑖𝑛𝑗𝑛𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑟) ↔ ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑅)))
2316, 22rabeqbidv 3168 . . . 4 ((𝑛 = 𝑁𝑟 = 𝑅) → {𝑚 ∈ (Base‘(𝑛 Mat (Poly1𝑟))) ∣ ∀𝑖𝑛𝑗𝑛𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑟)} = {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑅)})
2423adantl 481 . . 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 472 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑁 ∈ Fin)
26 elex 3185 . . . 4 (𝑅𝑉𝑅 ∈ V)
2726adantl 481 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑅 ∈ V)
28 fvex 6113 . . . . 5 (Base‘𝐶) ∈ V
299, 28eqeltri 2684 . . . 4 𝐵 ∈ V
30 rabexg 4739 . . . 4 (𝐵 ∈ V → {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑅)} ∈ V)
3129, 30mp1i 13 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑅)} ∈ V)
323, 24, 25, 27, 31ovmpt2d 6686 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑁 ConstPolyMat 𝑅) = {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑅)})
331, 32syl5eq 2656 1 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑆 = {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑅)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  Fincfn 7841  ℕcn 10897  Basecbs 15695  0gc0g 15923  Poly1cpl1 19368  coe1cco1 19369   Mat cmat 20032   ConstPolyMat ccpmat 20327 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-cpmat 20330 This theorem is referenced by:  cpmatpmat  20334  cpmatel  20335  cpmatsubgpmat  20344
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