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Metamath Proof Explorer

Theorem mhpdeg

Description: All nonzero terms of a homogeneous polynomial have degree N . (Contributed by Steven Nguyen, 25-Aug-2023)

Ref Expression
Hypotheses mhpdeg.h ⊢ 𝐻 = ( 𝐼 mHomP 𝑅 )
mhpdeg.0 ⊢ 0 = ( 0g ‘ 𝑅 )
mhpdeg.d ⊢ 𝐷 = { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin }
mhpdeg.x ⊢ ( 𝜑 → 𝑋 ∈ ( 𝐻 ‘ 𝑁 ) )
Assertion mhpdeg ( 𝜑 → ( 𝑋 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } )

Proof

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