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Theorem mon1pcl 24743
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 2822 . . 3 (0g𝑃) = (0g𝑃)
4 eqid 2822 . . 3 ( deg1𝑅) = ( deg1𝑅)
5 mon1pcl.m . . 3 𝑀 = (Monic1p𝑅)
6 eqid 2822 . . 3 (1r𝑅) = (1r𝑅)
71, 2, 3, 4, 5, 6ismon1p 24741 . 2 (𝐹𝑀 ↔ (𝐹𝐵𝐹 ≠ (0g𝑃) ∧ ((coe1𝐹)‘(( deg1𝑅)‘𝐹)) = (1r𝑅)))
87simp1bi 1142 1 (𝐹𝑀𝐹𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2114  wne 3011  cfv 6334  Basecbs 16474  0gc0g 16704  1rcur 19242  Poly1cpl1 20804  coe1cco1 20805   deg1 cdg1 24653  Monic1pcmn1 24724
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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307
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 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-iota 6293  df-fun 6336  df-fv 6342  df-slot 16478  df-base 16480  df-mon1 24729
This theorem is referenced by:  mon1puc1p  24749  deg1submon1p  24751  ply1rem  24762  fta1glem1  24764  fta1glem2  24765  mon1psubm  40081
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