Theorem mhpdeg

Description: All nonzero terms of a homogeneous polynomial have degree N . (Contributed by Steven Nguyen, 25-Aug-2023)

	Ref	Expression
Hypotheses	mhpdeg.h	|- H = ( I mHomP R )
	mhpdeg.0	|- .0. = ( 0g ` R )
	mhpdeg.d	|- D = { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin }
	mhpdeg.x	|- ( ph -> X e. ( H ` N ) )
Assertion	mhpdeg	|- ( ph -> ( X supp .0. ) C_ { g e. D | ( ( CCfld |`s NN0 ) gsum g ) = N } )

Proof

Step	Hyp	Ref	Expression
1		mhpdeg.h	|- H = ( I mHomP R )
2		mhpdeg.0	|- .0. = ( 0g ` R )
3		mhpdeg.d	|- D = { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin }
4		mhpdeg.x	|- ( ph -> X e. ( H ` N ) )
5		eqid	|- ( I mPoly R ) = ( I mPoly R )
6		eqid	|- ( Base ` ( I mPoly R ) ) = ( Base ` ( I mPoly R ) )
7	1 4	mhprcl	|- ( ph -> N e. NN0 )
8	1 5 6 2 3 7	ismhp	|- ( ph -> ( X e. ( H ` N ) <-> ( X e. ( Base ` ( I mPoly R ) ) /\ ( X supp .0. ) C_ { g e. D | ( ( CCfld |`s NN0 ) gsum g ) = N } ) ) )
9	8	simplbda	|- ( ( ph /\ X e. ( H ` N ) ) -> ( X supp .0. ) C_ { g e. D | ( ( CCfld |`s NN0 ) gsum g ) = N } )
10	4 9	mpdan	|- ( ph -> ( X supp .0. ) C_ { g e. D | ( ( CCfld |`s NN0 ) gsum g ) = N } )