Theorem ismhp2

Description: Deduce a homogeneous polynomial from its properties. (Contributed by SN, 25-May-2024)

	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 )
	ismhp2.1	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	ismhp2.2	⊢ ( 𝜑 → ( 𝑋 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } )
Assertion	ismhp2	⊢ ( 𝜑 → 𝑋 ∈ ( 𝐻 ‘ 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		ismhp2.1	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
10		ismhp2.2	⊢ ( 𝜑 → ( 𝑋 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } )
11	1 2 3 4 5 6 7 8	ismhp	⊢ ( 𝜑 → ( 𝑋 ∈ ( 𝐻 ‘ 𝑁 ) ↔ ( 𝑋 ∈ 𝐵 ∧ ( 𝑋 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } ) ) )
12	9 10 11	mpbir2and	⊢ ( 𝜑 → 𝑋 ∈ ( 𝐻 ‘ 𝑁 ) )