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 = ( 0g𝑅 )
mhpdeg.d 𝐷 = { ∈ ( ℕ0m 𝐼 ) ∣ ( “ ℕ ) ∈ Fin }
mhpdeg.x ( 𝜑𝑋 ∈ ( 𝐻𝑁 ) )
Assertion mhpdeg ( 𝜑 → ( 𝑋 supp 0 ) ⊆ { 𝑔𝐷 ∣ ( ( ℂflds0 ) Σg 𝑔 ) = 𝑁 } )

Proof

Step Hyp Ref Expression
1 mhpdeg.h 𝐻 = ( 𝐼 mHomP 𝑅 )
2 mhpdeg.0 0 = ( 0g𝑅 )
3 mhpdeg.d 𝐷 = { ∈ ( ℕ0m 𝐼 ) ∣ ( “ ℕ ) ∈ Fin }
4 mhpdeg.x ( 𝜑𝑋 ∈ ( 𝐻𝑁 ) )
5 eqid ( 𝐼 mPoly 𝑅 ) = ( 𝐼 mPoly 𝑅 )
6 eqid ( Base ‘ ( 𝐼 mPoly 𝑅 ) ) = ( Base ‘ ( 𝐼 mPoly 𝑅 ) )
7 1 4 mhprcl ( 𝜑𝑁 ∈ ℕ0 )
8 1 5 6 2 3 7 ismhp ( 𝜑 → ( 𝑋 ∈ ( 𝐻𝑁 ) ↔ ( 𝑋 ∈ ( Base ‘ ( 𝐼 mPoly 𝑅 ) ) ∧ ( 𝑋 supp 0 ) ⊆ { 𝑔𝐷 ∣ ( ( ℂflds0 ) Σg 𝑔 ) = 𝑁 } ) ) )
9 8 simplbda ( ( 𝜑𝑋 ∈ ( 𝐻𝑁 ) ) → ( 𝑋 supp 0 ) ⊆ { 𝑔𝐷 ∣ ( ( ℂflds0 ) Σg 𝑔 ) = 𝑁 } )
10 4 9 mpdan ( 𝜑 → ( 𝑋 supp 0 ) ⊆ { 𝑔𝐷 ∣ ( ( ℂflds0 ) Σg 𝑔 ) = 𝑁 } )