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