Theorem mon1pldg

Description: Unitic polynomials have one leading coefficients. (Contributed by Stefan O'Rear, 28-Mar-2015)

	Ref	Expression
Hypotheses	mon1pldg.d	|- D = ( deg1 ` R )
	mon1pldg.o	|- .1. = ( 1r ` R )
	mon1pldg.m	|- M = ( Monic1p ` R )
Assertion	mon1pldg	|- ( F e. M -> ( ( coe1 ` F ) ` ( D ` F ) ) = .1. )

Proof

Step	Hyp	Ref	Expression
1		mon1pldg.d	|- D = ( deg1 ` R )
2		mon1pldg.o	|- .1. = ( 1r ` R )
3		mon1pldg.m	|- M = ( Monic1p ` R )
4		eqid	|- ( Poly1 ` R ) = ( Poly1 ` R )
5		eqid	|- ( Base ` ( Poly1 ` R ) ) = ( Base ` ( Poly1 ` R ) )
6		eqid	|- ( 0g ` ( Poly1 ` R ) ) = ( 0g ` ( Poly1 ` R ) )
7	4 5 6 1 3 2	ismon1p	|- ( F e. M <-> ( F e. ( Base ` ( Poly1 ` R ) ) /\ F =/= ( 0g ` ( Poly1 ` R ) ) /\ ( ( coe1 ` F ) ` ( D ` F ) ) = .1. ) )
8	7	simp3bi	|- ( F e. M -> ( ( coe1 ` F ) ` ( D ` F ) ) = .1. )