Theorem ismhp

Description: Property of being a homogeneous polynomial. (Contributed by Steven Nguyen, 25-Aug-2023)

	Ref	Expression
Hypotheses	mhpfval.h	⊢ 𝐻 = ( 𝐼 mHomP 𝑅 )
	mhpfval.p	⊢ 𝑃 = ( 𝐼 mPoly 𝑅 )
	mhpfval.b	⊢ 𝐵 = ( Base ‘ 𝑃 )
	mhpfval.0	⊢ 0 = ( 0g ‘ 𝑅 )
	mhpfval.d	⊢ 𝐷 = { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin }
	mhpfval.i	⊢ ( 𝜑 → 𝐼 ∈ 𝑉 )
	mhpfval.r	⊢ ( 𝜑 → 𝑅 ∈ 𝑊 )
	mhpval.n	⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
Assertion	ismhp	⊢ ( 𝜑 → ( 𝑋 ∈ ( 𝐻 ‘ 𝑁 ) ↔ ( 𝑋 ∈ 𝐵 ∧ ( 𝑋 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } ) ) )

Proof

Step	Hyp	Ref	Expression
1		mhpfval.h	⊢ 𝐻 = ( 𝐼 mHomP 𝑅 )
2		mhpfval.p	⊢ 𝑃 = ( 𝐼 mPoly 𝑅 )
3		mhpfval.b	⊢ 𝐵 = ( Base ‘ 𝑃 )
4		mhpfval.0	⊢ 0 = ( 0g ‘ 𝑅 )
5		mhpfval.d	⊢ 𝐷 = { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin }
6		mhpfval.i	⊢ ( 𝜑 → 𝐼 ∈ 𝑉 )
7		mhpfval.r	⊢ ( 𝜑 → 𝑅 ∈ 𝑊 )
8		mhpval.n	⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
9	1 2 3 4 5 6 7 8	mhpval	⊢ ( 𝜑 → ( 𝐻 ‘ 𝑁 ) = { 𝑓 ∈ 𝐵 ∣ ( 𝑓 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } } )
10	9	eleq2d	⊢ ( 𝜑 → ( 𝑋 ∈ ( 𝐻 ‘ 𝑁 ) ↔ 𝑋 ∈ { 𝑓 ∈ 𝐵 ∣ ( 𝑓 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } } ) )
11		oveq1	⊢ ( 𝑓 = 𝑋 → ( 𝑓 supp 0 ) = ( 𝑋 supp 0 ) )
12	11	sseq1d	⊢ ( 𝑓 = 𝑋 → ( ( 𝑓 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } ↔ ( 𝑋 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } ) )
13	12	elrab	⊢ ( 𝑋 ∈ { 𝑓 ∈ 𝐵 ∣ ( 𝑓 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } } ↔ ( 𝑋 ∈ 𝐵 ∧ ( 𝑋 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } ) )
14	10 13	bitrdi	⊢ ( 𝜑 → ( 𝑋 ∈ ( 𝐻 ‘ 𝑁 ) ↔ ( 𝑋 ∈ 𝐵 ∧ ( 𝑋 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } ) ) )