This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 )