Theorem uc1pldg

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

	Ref	Expression
Hypotheses	uc1pldg.d	|- D = ( deg1 ` R )
	uc1pldg.u	|- U = ( Unit ` R )
	uc1pldg.c	|- C = ( Unic1p ` R )
Assertion	uc1pldg	|- ( F e. C -> ( ( coe1 ` F ) ` ( D ` F ) ) e. U )

Proof

Step	Hyp	Ref	Expression
1		uc1pldg.d	|- D = ( deg1 ` R )
2		uc1pldg.u	|- U = ( Unit ` R )
3		uc1pldg.c	|- C = ( Unic1p ` 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	isuc1p	|- ( F e. C <-> ( F e. ( Base ` ( Poly1 ` R ) ) /\ F =/= ( 0g ` ( Poly1 ` R ) ) /\ ( ( coe1 ` F ) ` ( D ` F ) ) e. U ) )
8	7	simp3bi	|- ( F e. C -> ( ( coe1 ` F ) ` ( D ` F ) ) e. U )