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Theorem cpmatel 20923
Description: Property of a constant polynomial matrix. (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
cpmatel ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) → (𝑀𝑆 ↔ ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g𝑅)))
Distinct variable groups:   𝑖,𝑁,𝑗,𝑘   𝑅,𝑖,𝑗,𝑘   𝑖,𝑀,𝑗,𝑘
Allowed substitution hints:   𝐵(𝑖,𝑗,𝑘)   𝐶(𝑖,𝑗,𝑘)   𝑃(𝑖,𝑗,𝑘)   𝑆(𝑖,𝑗,𝑘)   𝑉(𝑖,𝑗,𝑘)

Proof of Theorem cpmatel
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 cpmat.s . . . . . 6 𝑆 = (𝑁 ConstPolyMat 𝑅)
2 cpmat.p . . . . . 6 𝑃 = (Poly1𝑅)
3 cpmat.c . . . . . 6 𝐶 = (𝑁 Mat 𝑃)
4 cpmat.b . . . . . 6 𝐵 = (Base‘𝐶)
51, 2, 3, 4cpmat 20921 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑆 = {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑅)})
653adant3 1123 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) → 𝑆 = {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑅)})
76eleq2d 2844 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) → (𝑀𝑆𝑀 ∈ {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑅)}))
8 oveq 6928 . . . . . . . . 9 (𝑚 = 𝑀 → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
98fveq2d 6450 . . . . . . . 8 (𝑚 = 𝑀 → (coe1‘(𝑖𝑚𝑗)) = (coe1‘(𝑖𝑀𝑗)))
109fveq1d 6448 . . . . . . 7 (𝑚 = 𝑀 → ((coe1‘(𝑖𝑚𝑗))‘𝑘) = ((coe1‘(𝑖𝑀𝑗))‘𝑘))
1110eqeq1d 2779 . . . . . 6 (𝑚 = 𝑀 → (((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑅) ↔ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g𝑅)))
1211ralbidv 3167 . . . . 5 (𝑚 = 𝑀 → (∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑅) ↔ ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g𝑅)))
13122ralbidv 3170 . . . 4 (𝑚 = 𝑀 → (∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑅) ↔ ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g𝑅)))
1413elrab 3571 . . 3 (𝑀 ∈ {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑅)} ↔ (𝑀𝐵 ∧ ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g𝑅)))
157, 14syl6bb 279 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) → (𝑀𝑆 ↔ (𝑀𝐵 ∧ ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g𝑅))))
16153anibar 1385 1 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) → (𝑀𝑆 ↔ ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1071   = wceq 1601  wcel 2106  wral 3089  {crab 3093  cfv 6135  (class class class)co 6922  Fincfn 8241  cn 11374  Basecbs 16255  0gc0g 16486  Poly1cpl1 19943  coe1cco1 19944   Mat cmat 20617   ConstPolyMat ccpmat 20915
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-sbc 3652  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-opab 4949  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-iota 6099  df-fun 6137  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-cpmat 20918
This theorem is referenced by:  cpmatelimp  20924  cpmatel2  20925  1elcpmat  20927  cpmatmcl  20931  m2cpm  20953
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