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Theorem cpmatel 22731
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 22729 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑆 = {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑅)})
653adant3 1132 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) → 𝑆 = {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑅)})
76eleq2d 2824 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) → (𝑀𝑆𝑀 ∈ {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑅)}))
8 oveq 7451 . . . . . . . . 9 (𝑚 = 𝑀 → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
98fveq2d 6923 . . . . . . . 8 (𝑚 = 𝑀 → (coe1‘(𝑖𝑚𝑗)) = (coe1‘(𝑖𝑀𝑗)))
109fveq1d 6921 . . . . . . 7 (𝑚 = 𝑀 → ((coe1‘(𝑖𝑚𝑗))‘𝑘) = ((coe1‘(𝑖𝑀𝑗))‘𝑘))
1110eqeq1d 2736 . . . . . 6 (𝑚 = 𝑀 → (((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑅) ↔ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g𝑅)))
1211ralbidv 3180 . . . . 5 (𝑚 = 𝑀 → (∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑅) ↔ ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g𝑅)))
13122ralbidv 3222 . . . 4 (𝑚 = 𝑀 → (∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑅) ↔ ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g𝑅)))
1413elrab 3703 . . 3 (𝑀 ∈ {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑅)} ↔ (𝑀𝐵 ∧ ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g𝑅)))
157, 14bitrdi 287 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) → (𝑀𝑆 ↔ (𝑀𝐵 ∧ ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g𝑅))))
16153anibar 1329 1 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) → (𝑀𝑆 ↔ ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1537  wcel 2103  wral 3063  {crab 3438  cfv 6572  (class class class)co 7445  Fincfn 8999  cn 12289  Basecbs 17253  0gc0g 17494  Poly1cpl1 22192  coe1cco1 22193   Mat cmat 22425   ConstPolyMat ccpmat 22723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pr 5450
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-sbc 3799  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5170  df-opab 5232  df-id 5597  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-iota 6524  df-fun 6574  df-fv 6580  df-ov 7448  df-oprab 7449  df-mpo 7450  df-cpmat 22726
This theorem is referenced by:  cpmatelimp  22732  cpmatel2  22733  1elcpmat  22735  cpmatmcl  22739  m2cpm  22761
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