Theorem mon1pcl 21744

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 2452	. . 3 |- ( 0g ` P ) = ( 0g ` P )
4		eqid 2452	. . 3 |- ( deg1 ` R ) = ( deg1 ` R )
5		mon1pcl.m	. . 3 |- M = (Monic1p ` R )
6		eqid 2452	. . 3 |- ( 1r ` R ) = ( 1r ` R )
7	1, 2, 3, 4, 5, 6	ismon1p 21742	. 2 |- ( F e. M <-> ( F e. B /\ F =/= ( 0g ` P ) /\ ( (coe1 ` F ) ` ( ( deg1 ` R ) ` F ) ) = ( 1r ` R ) ) )
8	7	simp1bi 1003	1 |- ( F e. M -> F e. B )

Syntax hints:   -> wi 4   = wceq 1370   e. wcel 1758   =/= wne 2645   ` cfv 5521   Basecbs 14287   0gc0g 14492   1rcur 16720  Poly₁cpl1 17752  coe₁cco1 17753   deg₁ cdg1 21651  Monic_1pcmn1 21725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-iota 5484  df-fun 5523  df-fv 5529  df-slot 14291  df-base 14292  df-mon1 21730

This theorem is referenced by:  mon1puc1p  21750  deg1submon1p  21752  ply1rem  21763  fta1glem1  21765  fta1glem2  21766  mon1psubm  29717