Theorem uc1pn0 23905

Description: Unitic 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 )
uc1pn0.c	|- C = (Unic1p ` R )

Assertion

Ref	Expression
uc1pn0	|- ( F e. C -> F =/= .0. )

Proof of Theorem uc1pn0

Step	Hyp	Ref	Expression
1		uc1pn0.p	. . 3 |- P = (Poly1 ` R )
2		eqid 2622	. . 3 |- ( Base ` P ) = ( Base ` P )
3		uc1pn0.z	. . 3 |- .0. = ( 0g ` P )
4		eqid 2622	. . 3 |- ( deg1 ` R ) = ( deg1 ` R )
5		uc1pn0.c	. . 3 |- C = (Unic1p ` R )
6		eqid 2622	. . 3 |- (Unit ` R ) = (Unit ` R )
7	1, 2, 3, 4, 5, 6	isuc1p 23900	. 2 |- ( F e. C <-> ( F e. ( Base ` P ) /\ F =/= .0. /\ ( (coe1 ` F ) ` ( ( deg1 ` R ) ` F ) ) e. (Unit ` R ) ) )
8	7	simp2bi 1077	1 |- ( F e. C -> F =/= .0. )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   =/= wne 2794   ` cfv 5888   Basecbs 15857   0gc0g 16100  Unitcui 18639  Poly₁cpl1 19547  coe₁cco1 19548   deg₁ cdg1 23814  Unic_1pcuc1p 23886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-slot 15861  df-base 15863  df-uc1p 23891

This theorem is referenced by:  uc1pdeg  23907  q1peqb  23914