Theorem mon1pn0 19532

Description: Monic polynomials are not zero. (Contributed by Stefan O'Rear, 28-Mar-2015.)

Hypotheses

Ref	Expression
uc1pn0.p	|- P = (Poly1 ` R )
uc1pn0.z	|- .0. = ( 0g ` P )
mon1pn0.m	|- M = (Monic1p ` R )

Assertion

Ref	Expression
mon1pn0	|- ( F e. M -> F =/= .0. )

Proof of Theorem mon1pn0

Step	Hyp	Ref	Expression
1		uc1pn0.p	. . 3 |- P = (Poly1 ` R )
2		eqid 2283	. . 3 |- ( Base ` P ) = ( Base ` P )
3		uc1pn0.z	. . 3 |- .0. = ( 0g ` P )
4		eqid 2283	. . 3 |- ( deg1 ` R ) = ( deg1 ` R )
5		mon1pn0.m	. . 3 |- M = (Monic1p ` R )
6		eqid 2283	. . 3 |- ( 1r ` R ) = ( 1r ` R )
7	1, 2, 3, 4, 5, 6	ismon1p 19528	. 2 |- ( F e. M <-> ( F e. ( Base ` P ) /\ F =/= .0. /\ ( (coe1 ` F ) ` ( ( deg1 ` R ) ` F ) ) = ( 1r ` R ) ) )
8	7	simp2bi 971	1 |- ( F e. M -> F =/= .0. )

Syntax hints:   -> wi 4   = wceq 1623   e. wcel 1684   =/= wne 2446   ` cfv 5255   Basecbs 13148   0gc0g 13400   1rcur 15339  Poly₁cpl1 16252  coe₁cco1 16255   deg₁ cdg1 19440  Monic_1pcmn1 19511

This theorem is referenced by:  mon1puc1p  19536  deg1submon1p  19538  mon1psubm  27525

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-slot 13152  df-base 13153  df-mon1 19516