Theorem ismhp

Description: Property of being a homogeneous polynomial. (Contributed by Steven Nguyen, 25-Aug-2023)

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

Proof

Step	Hyp	Ref	Expression
1		ismhp.h	⊢ 𝐻 = ( 𝐼 mHomP 𝑅 )
2		ismhp.p	⊢ 𝑃 = ( 𝐼 mPoly 𝑅 )
3		ismhp.b	⊢ 𝐵 = ( Base ‘ 𝑃 )
4		ismhp.0	⊢ 0 = ( 0g ‘ 𝑅 )
5		ismhp.d	⊢ 𝐷 = { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin }
6		ismhp.n	⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
7		reldmmhp	⊢ Rel dom mHomP
8		id	⊢ ( 𝑋 ∈ ( 𝐻 ‘ 𝑁 ) → 𝑋 ∈ ( 𝐻 ‘ 𝑁 ) )
9	7 1 8	elfvov1	⊢ ( 𝑋 ∈ ( 𝐻 ‘ 𝑁 ) → 𝐼 ∈ V )
10	7 1 8	elfvov2	⊢ ( 𝑋 ∈ ( 𝐻 ‘ 𝑁 ) → 𝑅 ∈ V )
11	9 10	jca	⊢ ( 𝑋 ∈ ( 𝐻 ‘ 𝑁 ) → ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) )
12	11	anim2i	⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐻 ‘ 𝑁 ) ) → ( 𝜑 ∧ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) ) )
13		reldmmpl	⊢ Rel dom mPoly
14	13 2 3	elbasov	⊢ ( 𝑋 ∈ 𝐵 → ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) )
15	14	adantr	⊢ ( ( 𝑋 ∈ 𝐵 ∧ ( 𝑋 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } ) → ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) )
16	15	anim2i	⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ ( 𝑋 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } ) ) → ( 𝜑 ∧ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) ) )
17		simprl	⊢ ( ( 𝜑 ∧ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) ) → 𝐼 ∈ V )
18		simprr	⊢ ( ( 𝜑 ∧ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) ) → 𝑅 ∈ V )
19	6	adantr	⊢ ( ( 𝜑 ∧ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) ) → 𝑁 ∈ ℕ0 )
20	1 2 3 4 5 17 18 19	mhpval	⊢ ( ( 𝜑 ∧ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) ) → ( 𝐻 ‘ 𝑁 ) = { 𝑓 ∈ 𝐵 ∣ ( 𝑓 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } } )
21	20	eleq2d	⊢ ( ( 𝜑 ∧ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) ) → ( 𝑋 ∈ ( 𝐻 ‘ 𝑁 ) ↔ 𝑋 ∈ { 𝑓 ∈ 𝐵 ∣ ( 𝑓 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } } ) )
22		oveq1	⊢ ( 𝑓 = 𝑋 → ( 𝑓 supp 0 ) = ( 𝑋 supp 0 ) )
23	22	sseq1d	⊢ ( 𝑓 = 𝑋 → ( ( 𝑓 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } ↔ ( 𝑋 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } ) )
24	23	elrab	⊢ ( 𝑋 ∈ { 𝑓 ∈ 𝐵 ∣ ( 𝑓 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } } ↔ ( 𝑋 ∈ 𝐵 ∧ ( 𝑋 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } ) )
25	21 24	bitrdi	⊢ ( ( 𝜑 ∧ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) ) → ( 𝑋 ∈ ( 𝐻 ‘ 𝑁 ) ↔ ( 𝑋 ∈ 𝐵 ∧ ( 𝑋 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } ) ) )
26	12 16 25	pm5.21nd	⊢ ( 𝜑 → ( 𝑋 ∈ ( 𝐻 ‘ 𝑁 ) ↔ ( 𝑋 ∈ 𝐵 ∧ ( 𝑋 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } ) ) )