Theorem mon1pcl 20020

Description: Monic polynomials are polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)

Hypotheses

Ref	Expression
uc1pcl.p	|- P = (Poly1 ` R )
uc1pcl.b	|- B = ( Base ` P )
mon1pcl.m	|- M = (Monic1p ` R )

Assertion

Ref	Expression
mon1pcl	|- ( F e. M -> F e. B )

Proof of Theorem mon1pcl

Step	Hyp	Ref	Expression
1		uc1pcl.p	. . 3 |- P = (Poly1 ` R )
2		uc1pcl.b	. . 3 |- B = ( Base ` P )
3		eqid 2404	. . 3 |- ( 0g ` P ) = ( 0g ` P )
4		eqid 2404	. . 3 |- ( deg1 ` R ) = ( deg1 ` R )
5		mon1pcl.m	. . 3 |- M = (Monic1p ` R )
6		eqid 2404	. . 3 |- ( 1r ` R ) = ( 1r ` R )
7	1, 2, 3, 4, 5, 6	ismon1p 20018	. 2 |- ( F e. M <-> ( F e. B /\ F =/= ( 0g ` P ) /\ ( (coe1 ` F ) ` ( ( deg1 ` R ) ` F ) ) = ( 1r ` R ) ) )
8	7	simp1bi 972	1 |- ( F e. M -> F e. B )

Syntax hints:   -> wi 4   = wceq 1649   e. wcel 1721   =/= wne 2567   ` cfv 5413   Basecbs 13424   0gc0g 13678   1rcur 15617  Poly₁cpl1 16526  coe₁cco1 16529   deg₁ cdg1 19930  Monic_1pcmn1 20001

This theorem is referenced by:  mon1puc1p  20026  deg1submon1p  20028  ply1rem  20039  fta1glem1  20041  fta1glem2  20042  mon1psubm  27393

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-slot 13428  df-base 13429  df-mon1 20006