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 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin }
	mhpdeg.i	⊢ ( 𝜑 → 𝐼 ∈ 𝑉 )
	mhpdeg.r	⊢ ( 𝜑 → 𝑅 ∈ 𝑊 )
	mhpdeg.n	⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
	mhpdeg.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝐻 ‘ 𝑁 ) )
Assertion	mhpdeg	⊢ ( 𝜑 → ( 𝑋 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 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 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin }
4		mhpdeg.i	⊢ ( 𝜑 → 𝐼 ∈ 𝑉 )
5		mhpdeg.r	⊢ ( 𝜑 → 𝑅 ∈ 𝑊 )
6		mhpdeg.n	⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
7		mhpdeg.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝐻 ‘ 𝑁 ) )
8		eqid	⊢ ( 𝐼 mPoly 𝑅 ) = ( 𝐼 mPoly 𝑅 )
9		eqid	⊢ ( Base ‘ ( 𝐼 mPoly 𝑅 ) ) = ( Base ‘ ( 𝐼 mPoly 𝑅 ) )
10	1 8 9 2 3 4 5 6	ismhp	⊢ ( 𝜑 → ( 𝑋 ∈ ( 𝐻 ‘ 𝑁 ) ↔ ( 𝑋 ∈ ( Base ‘ ( 𝐼 mPoly 𝑅 ) ) ∧ ( 𝑋 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } ) ) )
11	10	simplbda	⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐻 ‘ 𝑁 ) ) → ( 𝑋 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } )
12	7 11	mpdan	⊢ ( 𝜑 → ( 𝑋 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } )