Theorem mhpmpl

Description: A homogeneous polynomial is a polynomial. (Contributed by Steven Nguyen, 25-Aug-2023)

	Ref	Expression
Hypotheses	mhpmpl.h	⊢ 𝐻 = ( 𝐼 mHomP 𝑅 )
	mhpmpl.p	⊢ 𝑃 = ( 𝐼 mPoly 𝑅 )
	mhpmpl.b	⊢ 𝐵 = ( Base ‘ 𝑃 )
	mhpmpl.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝐻 ‘ 𝑁 ) )
Assertion	mhpmpl	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		mhpmpl.h	⊢ 𝐻 = ( 𝐼 mHomP 𝑅 )
2		mhpmpl.p	⊢ 𝑃 = ( 𝐼 mPoly 𝑅 )
3		mhpmpl.b	⊢ 𝐵 = ( Base ‘ 𝑃 )
4		mhpmpl.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝐻 ‘ 𝑁 ) )
5		eqid	⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 )
6		eqid	⊢ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } = { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin }
7	1 4	mhprcl	⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
8	1 2 3 5 6 7	ismhp	⊢ ( 𝜑 → ( 𝑋 ∈ ( 𝐻 ‘ 𝑁 ) ↔ ( 𝑋 ∈ 𝐵 ∧ ( 𝑋 supp ( 0g ‘ 𝑅 ) ) ⊆ { 𝑔 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } ) ) )
9	8	simprbda	⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐻 ‘ 𝑁 ) ) → 𝑋 ∈ 𝐵 )
10	4 9	mpdan	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )