Theorem mon1pcl 22277

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 2467	. . 3 |- ( 0g ` P ) = ( 0g ` P )
4		eqid 2467	. . 3 |- ( deg1 ` R ) = ( deg1 ` R )
5		mon1pcl.m	. . 3 |- M = (Monic1p ` R )
6		eqid 2467	. . 3 |- ( 1r ` R ) = ( 1r ` R )
7	1, 2, 3, 4, 5, 6	ismon1p 22275	. 2 |- ( F e. M <-> ( F e. B /\ F =/= ( 0g ` P ) /\ ( (coe1 ` F ) ` ( ( deg1 ` R ) ` F ) ) = ( 1r ` R ) ) )
8	7	simp1bi 1011	1 |- ( F e. M -> F e. B )

Syntax hints:   -> wi 4   = wceq 1379   e. wcel 1767   =/= wne 2662   ` cfv 5586   Basecbs 14483   0gc0g 14688   1rcur 16940  Poly₁cpl1 17984  coe₁cco1 17985   deg₁ cdg1 22184  Monic_1pcmn1 22258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fv 5594  df-slot 14487  df-base 14488  df-mon1 22263

This theorem is referenced by:  mon1puc1p  22283  deg1submon1p  22285  ply1rem  22296  fta1glem1  22298  fta1glem2  22299  mon1psubm  30771