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Theorem uc1pcl 24116
Description: Unitic polynomials are polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Hypotheses
Ref Expression
uc1pcl.p 𝑃 = (Poly1𝑅)
uc1pcl.b 𝐵 = (Base‘𝑃)
uc1pcl.c 𝐶 = (Unic1p𝑅)
Assertion
Ref Expression
uc1pcl (𝐹𝐶𝐹𝐵)

Proof of Theorem uc1pcl
StepHypRef Expression
1 uc1pcl.p . . 3 𝑃 = (Poly1𝑅)
2 uc1pcl.b . . 3 𝐵 = (Base‘𝑃)
3 eqid 2771 . . 3 (0g𝑃) = (0g𝑃)
4 eqid 2771 . . 3 ( deg1𝑅) = ( deg1𝑅)
5 uc1pcl.c . . 3 𝐶 = (Unic1p𝑅)
6 eqid 2771 . . 3 (Unit‘𝑅) = (Unit‘𝑅)
71, 2, 3, 4, 5, 6isuc1p 24113 . 2 (𝐹𝐶 ↔ (𝐹𝐵𝐹 ≠ (0g𝑃) ∧ ((coe1𝐹)‘(( deg1𝑅)‘𝐹)) ∈ (Unit‘𝑅)))
87simp1bi 1139 1 (𝐹𝐶𝐹𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  wne 2943  cfv 6029  Basecbs 16057  0gc0g 16301  Unitcui 18840  Poly1cpl1 19755  coe1cco1 19756   deg1 cdg1 24027  Unic1pcuc1p 24099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5992  df-fun 6031  df-fv 6037  df-slot 16061  df-base 16063  df-uc1p 24104
This theorem is referenced by:  uc1pdeg  24120  uc1pmon1p  24124  q1peqb  24127  r1pcl  24130  r1pdeglt  24131  r1pid  24132  dvdsq1p  24133  dvdsr1p  24134
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