Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  uc1pcl Structured version   Visualization version   GIF version

Theorem uc1pcl 24747
 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 2801 . . 3 (0g𝑃) = (0g𝑃)
4 eqid 2801 . . 3 ( deg1𝑅) = ( deg1𝑅)
5 uc1pcl.c . . 3 𝐶 = (Unic1p𝑅)
6 eqid 2801 . . 3 (Unit‘𝑅) = (Unit‘𝑅)
71, 2, 3, 4, 5, 6isuc1p 24744 . 2 (𝐹𝐶 ↔ (𝐹𝐵𝐹 ≠ (0g𝑃) ∧ ((coe1𝐹)‘(( deg1𝑅)‘𝐹)) ∈ (Unit‘𝑅)))
87simp1bi 1142 1 (𝐹𝐶𝐹𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2112   ≠ wne 2990  ‘cfv 6328  Basecbs 16478  0gc0g 16708  Unitcui 19388  Poly1cpl1 20809  coe1cco1 20810   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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298 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 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6287  df-fun 6330  df-fv 6336  df-slot 16482  df-base 16484  df-uc1p 24735 This theorem is referenced by:  uc1pdeg  24751  uc1pmon1p  24755  q1peqb  24758  r1pcl  24761  r1pdeglt  24762  r1pid  24763  dvdsq1p  24764  dvdsr1p  24765
 Copyright terms: Public domain W3C validator