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Theorem mon1pcl 24665
Description: Monic polynomials are polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Hypotheses
Ref Expression
uc1pcl.p 𝑃 = (Poly1𝑅)
uc1pcl.b 𝐵 = (Base‘𝑃)
mon1pcl.m 𝑀 = (Monic1p𝑅)
Assertion
Ref Expression
mon1pcl (𝐹𝑀𝐹𝐵)

Proof of Theorem mon1pcl
StepHypRef Expression
1 uc1pcl.p . . 3 𝑃 = (Poly1𝑅)
2 uc1pcl.b . . 3 𝐵 = (Base‘𝑃)
3 eqid 2818 . . 3 (0g𝑃) = (0g𝑃)
4 eqid 2818 . . 3 ( deg1𝑅) = ( deg1𝑅)
5 mon1pcl.m . . 3 𝑀 = (Monic1p𝑅)
6 eqid 2818 . . 3 (1r𝑅) = (1r𝑅)
71, 2, 3, 4, 5, 6ismon1p 24663 . 2 (𝐹𝑀 ↔ (𝐹𝐵𝐹 ≠ (0g𝑃) ∧ ((coe1𝐹)‘(( deg1𝑅)‘𝐹)) = (1r𝑅)))
87simp1bi 1137 1 (𝐹𝑀𝐹𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  wne 3013  cfv 6348  Basecbs 16471  0gc0g 16701  1rcur 19180  Poly1cpl1 20273  coe1cco1 20274   deg1 cdg1 24575  Monic1pcmn1 24646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-slot 16475  df-base 16477  df-mon1 24651
This theorem is referenced by:  mon1puc1p  24671  deg1submon1p  24673  ply1rem  24684  fta1glem1  24686  fta1glem2  24687  mon1psubm  39684
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