Metamath Proof Explorer


Theorem ismhp2

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

Ref Expression
Hypotheses ismhp.h 𝐻 = ( 𝐼 mHomP 𝑅 )
ismhp.p 𝑃 = ( 𝐼 mPoly 𝑅 )
ismhp.b 𝐵 = ( Base ‘ 𝑃 )
ismhp.0 0 = ( 0g𝑅 )
ismhp.d 𝐷 = { ∈ ( ℕ0m 𝐼 ) ∣ ( “ ℕ ) ∈ Fin }
ismhp.n ( 𝜑𝑁 ∈ ℕ0 )
ismhp2.1 ( 𝜑𝑋𝐵 )
ismhp2.2 ( 𝜑 → ( 𝑋 supp 0 ) ⊆ { 𝑔𝐷 ∣ ( ( ℂflds0 ) Σg 𝑔 ) = 𝑁 } )
Assertion ismhp2 ( 𝜑𝑋 ∈ ( 𝐻𝑁 ) )

Proof

Step Hyp Ref Expression
1 ismhp.h 𝐻 = ( 𝐼 mHomP 𝑅 )
2 ismhp.p 𝑃 = ( 𝐼 mPoly 𝑅 )
3 ismhp.b 𝐵 = ( Base ‘ 𝑃 )
4 ismhp.0 0 = ( 0g𝑅 )
5 ismhp.d 𝐷 = { ∈ ( ℕ0m 𝐼 ) ∣ ( “ ℕ ) ∈ Fin }
6 ismhp.n ( 𝜑𝑁 ∈ ℕ0 )
7 ismhp2.1 ( 𝜑𝑋𝐵 )
8 ismhp2.2 ( 𝜑 → ( 𝑋 supp 0 ) ⊆ { 𝑔𝐷 ∣ ( ( ℂflds0 ) Σg 𝑔 ) = 𝑁 } )
9 1 2 3 4 5 6 ismhp ( 𝜑 → ( 𝑋 ∈ ( 𝐻𝑁 ) ↔ ( 𝑋𝐵 ∧ ( 𝑋 supp 0 ) ⊆ { 𝑔𝐷 ∣ ( ( ℂflds0 ) Σg 𝑔 ) = 𝑁 } ) ) )
10 7 8 9 mpbir2and ( 𝜑𝑋 ∈ ( 𝐻𝑁 ) )