Theorem ismhp3

Description: A polynomial is homogeneous iff the degree of every nonzero term is the same. (Contributed by SN, 22-Jul-2024)

	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 )
	ismhp2.1	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
Assertion	ismhp3	⊢ ( 𝜑 → ( 𝑋 ∈ ( 𝐻 ‘ 𝑁 ) ↔ ∀ 𝑑 ∈ 𝐷 ( ( 𝑋 ‘ 𝑑 ) ≠ 0 → ( ( ℂfld ↾s ℕ0 ) Σg 𝑑 ) = 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		ismhp2.1	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
10	1 2 3 4 5 6 7 8	ismhp	⊢ ( 𝜑 → ( 𝑋 ∈ ( 𝐻 ‘ 𝑁 ) ↔ ( 𝑋 ∈ 𝐵 ∧ ( 𝑋 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } ) ) )
11	9	biantrurd	⊢ ( 𝜑 → ( ( 𝑋 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } ↔ ( 𝑋 ∈ 𝐵 ∧ ( 𝑋 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } ) ) )
12		eqid	⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
13	2 12 3 5 9	mplelf	⊢ ( 𝜑 → 𝑋 : 𝐷 ⟶ ( Base ‘ 𝑅 ) )
14	13	ffnd	⊢ ( 𝜑 → 𝑋 Fn 𝐷 )
15	4	fvexi	⊢ 0 ∈ V
16	15	a1i	⊢ ( 𝜑 → 0 ∈ V )
17		elsuppfng	⊢ ( ( 𝑋 Fn 𝐷 ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ V ) → ( 𝑑 ∈ ( 𝑋 supp 0 ) ↔ ( 𝑑 ∈ 𝐷 ∧ ( 𝑋 ‘ 𝑑 ) ≠ 0 ) ) )
18	14 9 16 17	syl3anc	⊢ ( 𝜑 → ( 𝑑 ∈ ( 𝑋 supp 0 ) ↔ ( 𝑑 ∈ 𝐷 ∧ ( 𝑋 ‘ 𝑑 ) ≠ 0 ) ) )
19		oveq2	⊢ ( 𝑔 = 𝑑 → ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = ( ( ℂfld ↾s ℕ0 ) Σg 𝑑 ) )
20	19	eqeq1d	⊢ ( 𝑔 = 𝑑 → ( ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 ↔ ( ( ℂfld ↾s ℕ0 ) Σg 𝑑 ) = 𝑁 ) )
21	20	elrab	⊢ ( 𝑑 ∈ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } ↔ ( 𝑑 ∈ 𝐷 ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑑 ) = 𝑁 ) )
22	21	a1i	⊢ ( 𝜑 → ( 𝑑 ∈ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } ↔ ( 𝑑 ∈ 𝐷 ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑑 ) = 𝑁 ) ) )
23	18 22	imbi12d	⊢ ( 𝜑 → ( ( 𝑑 ∈ ( 𝑋 supp 0 ) → 𝑑 ∈ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } ) ↔ ( ( 𝑑 ∈ 𝐷 ∧ ( 𝑋 ‘ 𝑑 ) ≠ 0 ) → ( 𝑑 ∈ 𝐷 ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑑 ) = 𝑁 ) ) ) )
24		imdistan	⊢ ( ( 𝑑 ∈ 𝐷 → ( ( 𝑋 ‘ 𝑑 ) ≠ 0 → ( ( ℂfld ↾s ℕ0 ) Σg 𝑑 ) = 𝑁 ) ) ↔ ( ( 𝑑 ∈ 𝐷 ∧ ( 𝑋 ‘ 𝑑 ) ≠ 0 ) → ( 𝑑 ∈ 𝐷 ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑑 ) = 𝑁 ) ) )
25	23 24	bitr4di	⊢ ( 𝜑 → ( ( 𝑑 ∈ ( 𝑋 supp 0 ) → 𝑑 ∈ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } ) ↔ ( 𝑑 ∈ 𝐷 → ( ( 𝑋 ‘ 𝑑 ) ≠ 0 → ( ( ℂfld ↾s ℕ0 ) Σg 𝑑 ) = 𝑁 ) ) ) )
26	25	albidv	⊢ ( 𝜑 → ( ∀ 𝑑 ( 𝑑 ∈ ( 𝑋 supp 0 ) → 𝑑 ∈ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } ) ↔ ∀ 𝑑 ( 𝑑 ∈ 𝐷 → ( ( 𝑋 ‘ 𝑑 ) ≠ 0 → ( ( ℂfld ↾s ℕ0 ) Σg 𝑑 ) = 𝑁 ) ) ) )
27		dfss2	⊢ ( ( 𝑋 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } ↔ ∀ 𝑑 ( 𝑑 ∈ ( 𝑋 supp 0 ) → 𝑑 ∈ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } ) )
28		df-ral	⊢ ( ∀ 𝑑 ∈ 𝐷 ( ( 𝑋 ‘ 𝑑 ) ≠ 0 → ( ( ℂfld ↾s ℕ0 ) Σg 𝑑 ) = 𝑁 ) ↔ ∀ 𝑑 ( 𝑑 ∈ 𝐷 → ( ( 𝑋 ‘ 𝑑 ) ≠ 0 → ( ( ℂfld ↾s ℕ0 ) Σg 𝑑 ) = 𝑁 ) ) )
29	26 27 28	3bitr4g	⊢ ( 𝜑 → ( ( 𝑋 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } ↔ ∀ 𝑑 ∈ 𝐷 ( ( 𝑋 ‘ 𝑑 ) ≠ 0 → ( ( ℂfld ↾s ℕ0 ) Σg 𝑑 ) = 𝑁 ) ) )
30	10 11 29	3bitr2d	⊢ ( 𝜑 → ( 𝑋 ∈ ( 𝐻 ‘ 𝑁 ) ↔ ∀ 𝑑 ∈ 𝐷 ( ( 𝑋 ‘ 𝑑 ) ≠ 0 → ( ( ℂfld ↾s ℕ0 ) Σg 𝑑 ) = 𝑁 ) ) )