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Theorem uc1pn0 24749
Description: Unitic polynomials are not zero. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Hypotheses
Ref Expression
uc1pn0.p 𝑃 = (Poly1𝑅)
uc1pn0.z 0 = (0g𝑃)
uc1pn0.c 𝐶 = (Unic1p𝑅)
Assertion
Ref Expression
uc1pn0 (𝐹𝐶𝐹0 )

Proof of Theorem uc1pn0
StepHypRef Expression
1 uc1pn0.p . . 3 𝑃 = (Poly1𝑅)
2 eqid 2824 . . 3 (Base‘𝑃) = (Base‘𝑃)
3 uc1pn0.z . . 3 0 = (0g𝑃)
4 eqid 2824 . . 3 ( deg1𝑅) = ( deg1𝑅)
5 uc1pn0.c . . 3 𝐶 = (Unic1p𝑅)
6 eqid 2824 . . 3 (Unit‘𝑅) = (Unit‘𝑅)
71, 2, 3, 4, 5, 6isuc1p 24744 . 2 (𝐹𝐶 ↔ (𝐹 ∈ (Base‘𝑃) ∧ 𝐹0 ∧ ((coe1𝐹)‘(( deg1𝑅)‘𝐹)) ∈ (Unit‘𝑅)))
87simp2bi 1143 1 (𝐹𝐶𝐹0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  wne 3014  cfv 6343  Basecbs 16483  0gc0g 16713  Unitcui 19392  Poly1cpl1 20345  coe1cco1 20346   deg1 cdg1 24658  Unic1pcuc1p 24730
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-iota 6302  df-fun 6345  df-fv 6351  df-slot 16487  df-base 16489  df-uc1p 24735
This theorem is referenced by:  uc1pdeg  24751  q1peqb  24758
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