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 ∣ Σ 𝑗 ∈ ℕ0 ( 𝑔 ‘ 𝑗 ) = 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 ∣ Σ 𝑗 ∈ ℕ0 ( 𝑔 ‘ 𝑗 ) = 𝑛 } } ) )
10		eqeq2	⊢ ( 𝑛 = 𝑁 → ( Σ 𝑗 ∈ ℕ0 ( 𝑔 ‘ 𝑗 ) = 𝑛 ↔ Σ 𝑗 ∈ ℕ0 ( 𝑔 ‘ 𝑗 ) = 𝑁 ) )
11	10	rabbidv	⊢ ( 𝑛 = 𝑁 → { 𝑔 ∈ 𝐷 ∣ Σ 𝑗 ∈ ℕ0 ( 𝑔 ‘ 𝑗 ) = 𝑛 } = { 𝑔 ∈ 𝐷 ∣ Σ 𝑗 ∈ ℕ0 ( 𝑔 ‘ 𝑗 ) = 𝑁 } )
12	11	sseq2d	⊢ ( 𝑛 = 𝑁 → ( ( 𝑓 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ Σ 𝑗 ∈ ℕ0 ( 𝑔 ‘ 𝑗 ) = 𝑛 } ↔ ( 𝑓 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ Σ 𝑗 ∈ ℕ0 ( 𝑔 ‘ 𝑗 ) = 𝑁 } ) )
13	12	rabbidv	⊢ ( 𝑛 = 𝑁 → { 𝑓 ∈ 𝐵 ∣ ( 𝑓 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ Σ 𝑗 ∈ ℕ0 ( 𝑔 ‘ 𝑗 ) = 𝑛 } } = { 𝑓 ∈ 𝐵 ∣ ( 𝑓 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ Σ 𝑗 ∈ ℕ0 ( 𝑔 ‘ 𝑗 ) = 𝑁 } } )
14	13	adantl	⊢ ( ( 𝜑 ∧ 𝑛 = 𝑁 ) → { 𝑓 ∈ 𝐵 ∣ ( 𝑓 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ Σ 𝑗 ∈ ℕ0 ( 𝑔 ‘ 𝑗 ) = 𝑛 } } = { 𝑓 ∈ 𝐵 ∣ ( 𝑓 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ Σ 𝑗 ∈ ℕ0 ( 𝑔 ‘ 𝑗 ) = 𝑁 } } )
15	3	fvexi	⊢ 𝐵 ∈ V
16	15	rabex	⊢ { 𝑓 ∈ 𝐵 ∣ ( 𝑓 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ Σ 𝑗 ∈ ℕ0 ( 𝑔 ‘ 𝑗 ) = 𝑁 } } ∈ V
17	16	a1i	⊢ ( 𝜑 → { 𝑓 ∈ 𝐵 ∣ ( 𝑓 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ Σ 𝑗 ∈ ℕ0 ( 𝑔 ‘ 𝑗 ) = 𝑁 } } ∈ V )
18	9 14 8 17	fvmptd	⊢ ( 𝜑 → ( 𝐻 ‘ 𝑁 ) = { 𝑓 ∈ 𝐵 ∣ ( 𝑓 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ Σ 𝑗 ∈ ℕ0 ( 𝑔 ‘ 𝑗 ) = 𝑁 } } )