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Metamath Proof Explorer

Theorem mhp0cl

Description: The zero polynomial is homogeneous. Under df-mhp , it has any (nonnegative integer) degree which loosely corresponds to the value "undefined". The values -oo and 0 are also used in Metamath (by df-mdeg and df-dgr respectively) and the literature: https://math.stackexchange.com/a/1796314/593843 . (Contributed by SN, 12-Sep-2023)

Ref Expression
Hypotheses mhp0cl.h ⊢ 𝐻 = ( 𝐼 mHomP 𝑅 )
mhp0cl.0 ⊢ 0 = ( 0g ‘ 𝑅 )
mhp0cl.d ⊢ 𝐷 = { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin }
mhp0cl.i ⊢ ( 𝜑 → 𝐼 ∈ 𝑉 )
mhp0cl.r ⊢ ( 𝜑 → 𝑅 ∈ Grp )
mhp0cl.n ⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
Assertion mhp0cl ( 𝜑 → ( 𝐷 × { 0 } ) ∈ ( 𝐻 ‘ 𝑁 ) )

Proof

Step Hyp Ref Expression
1 mhp0cl.h ⊢ 𝐻 = ( 𝐼 mHomP 𝑅 )
2 mhp0cl.0 ⊢ 0 = ( 0g ‘ 𝑅 )
3 mhp0cl.d ⊢ 𝐷 = { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin }
4 mhp0cl.i ⊢ ( 𝜑 → 𝐼 ∈ 𝑉 )
5 mhp0cl.r ⊢ ( 𝜑 → 𝑅 ∈ Grp )
6 mhp0cl.n ⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
7 eqid ⊢ ( 𝐼 mPoly 𝑅 ) = ( 𝐼 mPoly 𝑅 )
8 eqid ⊢ ( Base ‘ ( 𝐼 mPoly 𝑅 ) ) = ( Base ‘ ( 𝐼 mPoly 𝑅 ) )
9 eqid ⊢ ( 0g ‘ ( 𝐼 mPoly 𝑅 ) ) = ( 0g ‘ ( 𝐼 mPoly 𝑅 ) )
10 7 3 2 9 4 5 mpl0 ⊢ ( 𝜑 → ( 0g ‘ ( 𝐼 mPoly 𝑅 ) ) = ( 𝐷 × { 0 } ) )
11 7 mplgrp ⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Grp ) → ( 𝐼 mPoly 𝑅 ) ∈ Grp )
12 4 5 11 syl2anc ⊢ ( 𝜑 → ( 𝐼 mPoly 𝑅 ) ∈ Grp )
13 8 9 grpidcl ⊢ ( ( 𝐼 mPoly 𝑅 ) ∈ Grp → ( 0g ‘ ( 𝐼 mPoly 𝑅 ) ) ∈ ( Base ‘ ( 𝐼 mPoly 𝑅 ) ) )
14 12 13 syl ⊢ ( 𝜑 → ( 0g ‘ ( 𝐼 mPoly 𝑅 ) ) ∈ ( Base ‘ ( 𝐼 mPoly 𝑅 ) ) )
15 10 14 eqeltrrd ⊢ ( 𝜑 → ( 𝐷 × { 0 } ) ∈ ( Base ‘ ( 𝐼 mPoly 𝑅 ) ) )
16 fczsupp0 ⊢ ( ( 𝐷 × { 0 } ) supp 0 ) = ∅
17 0ss ⊢ ∅ ⊆ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 }
18 16 17 eqsstri ⊢ ( ( 𝐷 × { 0 } ) supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 }
19 18 a1i ⊢ ( 𝜑 → ( ( 𝐷 × { 0 } ) supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } )
20 1 7 8 2 3 6 15 19 ismhp2 ⊢ ( 𝜑 → ( 𝐷 × { 0 } ) ∈ ( 𝐻 ‘ 𝑁 ) )