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Metamath Proof Explorer

Theorem mhpval

Description: Value of the "homogeneous polynomial" function. (Contributed by Steven Nguyen, 25-Aug-2023)

Ref Expression
Hypotheses mhpfval.h ⊢ 𝐻 = ( 𝐼 mHomP 𝑅 )
mhpfval.p ⊢ 𝑃 = ( 𝐼 mPoly 𝑅 )
mhpfval.b ⊢ 𝐵 = ( Base ‘ 𝑃 )
mhpfval.0 ⊢ 0 = ( 0g ‘ 𝑅 )
mhpfval.d ⊢ 𝐷 = { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin }
mhpfval.i ⊢ ( 𝜑 → 𝐼 ∈ 𝑉 )
mhpfval.r ⊢ ( 𝜑 → 𝑅 ∈ 𝑊 )
mhpval.n ⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
Assertion mhpval ( 𝜑 → ( 𝐻 ‘ 𝑁 ) = { 𝑓 ∈ 𝐵 ∣ ( 𝑓 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } } )

Proof

Step Hyp Ref Expression
1 mhpfval.h ⊢ 𝐻 = ( 𝐼 mHomP 𝑅 )
2 mhpfval.p ⊢ 𝑃 = ( 𝐼 mPoly 𝑅 )
3 mhpfval.b ⊢ 𝐵 = ( Base ‘ 𝑃 )
4 mhpfval.0 ⊢ 0 = ( 0g ‘ 𝑅 )
5 mhpfval.d ⊢ 𝐷 = { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin }
6 mhpfval.i ⊢ ( 𝜑 → 𝐼 ∈ 𝑉 )
7 mhpfval.r ⊢ ( 𝜑 → 𝑅 ∈ 𝑊 )
8 mhpval.n ⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
9 1 2 3 4 5 6 7 mhpfval ⊢ ( 𝜑 → 𝐻 = ( 𝑛 ∈ ℕ0 ↦ { 𝑓 ∈ 𝐵 ∣ ( 𝑓 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑛 } } ) )
10 eqeq2 ⊢ ( 𝑛 = 𝑁 → ( ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑛 ↔ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 ) )
11 10 rabbidv ⊢ ( 𝑛 = 𝑁 → { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑛 } = { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } )
12 11 sseq2d ⊢ ( 𝑛 = 𝑁 → ( ( 𝑓 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑛 } ↔ ( 𝑓 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } ) )
13 12 rabbidv ⊢ ( 𝑛 = 𝑁 → { 𝑓 ∈ 𝐵 ∣ ( 𝑓 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑛 } } = { 𝑓 ∈ 𝐵 ∣ ( 𝑓 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } } )
14 13 adantl ⊢ ( ( 𝜑 ∧ 𝑛 = 𝑁 ) → { 𝑓 ∈ 𝐵 ∣ ( 𝑓 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑛 } } = { 𝑓 ∈ 𝐵 ∣ ( 𝑓 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } } )
15 3 fvexi ⊢ 𝐵 ∈ V
16 15 rabex ⊢ { 𝑓 ∈ 𝐵 ∣ ( 𝑓 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } } ∈ V
17 16 a1i ⊢ ( 𝜑 → { 𝑓 ∈ 𝐵 ∣ ( 𝑓 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } } ∈ V )
18 9 14 8 17 fvmptd ⊢ ( 𝜑 → ( 𝐻 ‘ 𝑁 ) = { 𝑓 ∈ 𝐵 ∣ ( 𝑓 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } } )