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Theorem mdegle0 26033

Description: A polynomial has nonpositive degree iff it is a constant. (Contributed by Stefan O'Rear, 29-Mar-2015.)

**Hypotheses**
Ref	Expression
mdegaddle.y	⊢ 𝑌 = (𝐼 mPoly 𝑅)
mdegaddle.d	⊢ 𝐷 = (𝐼 mDeg 𝑅)
mdegaddle.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
mdegaddle.r	⊢ (𝜑 → 𝑅 ∈ Ring)
mdegle0.b	⊢ 𝐵 = (Base‘𝑌)
mdegle0.a	⊢ 𝐴 = (algSc‘𝑌)
mdegle0.f	⊢ (𝜑 → 𝐹 ∈ 𝐵)

**Assertion**
Ref	Expression
mdegle0	⊢ (𝜑 → ((𝐷‘𝐹) ≤ 0 ↔ 𝐹 = (𝐴‘(𝐹‘(𝐼 × {0})))))

Proof of Theorem mdegle0 Dummy variables 𝑥 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mdegle0.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
2		0xr 11299	. . 3 ⊢ 0 ∈ ℝ^*
3		mdegaddle.d	. . . 4 ⊢ 𝐷 = (𝐼 mDeg 𝑅)
4		mdegaddle.y	. . . 4 ⊢ 𝑌 = (𝐼 mPoly 𝑅)
5		mdegle0.b	. . . 4 ⊢ 𝐵 = (Base‘𝑌)
6		eqid 2728	. . . 4 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
7		eqid 2728	. . . 4 ⊢ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} = {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}
8		eqid 2728	. . . 4 ⊢ (𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ↦ (ℂ_fld Σ_g 𝑏)) = (𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ↦ (ℂ_fld Σ_g 𝑏))
9	3, 4, 5, 6, 7, 8	mdegleb 26020	. . 3 ⊢ ((𝐹 ∈ 𝐵 ∧ 0 ∈ ℝ^*) → ((𝐷‘𝐹) ≤ 0 ↔ ∀𝑥 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} (0 < ((𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ↦ (ℂ_fld Σ_g 𝑏))‘𝑥) → (𝐹‘𝑥) = (0_g‘𝑅))))
10	1, 2, 9	sylancl 584	. 2 ⊢ (𝜑 → ((𝐷‘𝐹) ≤ 0 ↔ ∀𝑥 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} (0 < ((𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ↦ (ℂ_fld Σ_g 𝑏))‘𝑥) → (𝐹‘𝑥) = (0_g‘𝑅))))
11	7, 8	tdeglem1 26011	. . . . . . . . . 10 ⊢ (𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ↦ (ℂ_fld Σ_g 𝑏)):{𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}⟶ℕ₀
12	11	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ↦ (ℂ_fld Σ_g 𝑏)):{𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}⟶ℕ₀)
13	12	ffvelcdmda 7099	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}) → ((𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ↦ (ℂ_fld Σ_g 𝑏))‘𝑥) ∈ ℕ₀)
14		nn0re 12519	. . . . . . . . 9 ⊢ (((𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ↦ (ℂ_fld Σ_g 𝑏))‘𝑥) ∈ ℕ₀ → ((𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ↦ (ℂ_fld Σ_g 𝑏))‘𝑥) ∈ ℝ)
15		nn0ge0 12535	. . . . . . . . 9 ⊢ (((𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ↦ (ℂ_fld Σ_g 𝑏))‘𝑥) ∈ ℕ₀ → 0 ≤ ((𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ↦ (ℂ_fld Σ_g 𝑏))‘𝑥))
16	14, 15	jca 510	. . . . . . . 8 ⊢ (((𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ↦ (ℂ_fld Σ_g 𝑏))‘𝑥) ∈ ℕ₀ → (((𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ↦ (ℂ_fld Σ_g 𝑏))‘𝑥) ∈ ℝ ∧ 0 ≤ ((𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ↦ (ℂ_fld Σ_g 𝑏))‘𝑥)))
17		ne0gt0 11357	. . . . . . . 8 ⊢ ((((𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ↦ (ℂ_fld Σ_g 𝑏))‘𝑥) ∈ ℝ ∧ 0 ≤ ((𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ↦ (ℂ_fld Σ_g 𝑏))‘𝑥)) → (((𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ↦ (ℂ_fld Σ_g 𝑏))‘𝑥) ≠ 0 ↔ 0 < ((𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ↦ (ℂ_fld Σ_g 𝑏))‘𝑥)))
18	13, 16, 17	3syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}) → (((𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ↦ (ℂ_fld Σ_g 𝑏))‘𝑥) ≠ 0 ↔ 0 < ((𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ↦ (ℂ_fld Σ_g 𝑏))‘𝑥)))
19	7, 8	tdeglem4 26015	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} → (((𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ↦ (ℂ_fld Σ_g 𝑏))‘𝑥) = 0 ↔ 𝑥 = (𝐼 × {0})))
20	19	adantl 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}) → (((𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ↦ (ℂ_fld Σ_g 𝑏))‘𝑥) = 0 ↔ 𝑥 = (𝐼 × {0})))
21	20	necon3abid 2974	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}) → (((𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ↦ (ℂ_fld Σ_g 𝑏))‘𝑥) ≠ 0 ↔ ¬ 𝑥 = (𝐼 × {0})))
22	18, 21	bitr3d 280	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}) → (0 < ((𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ↦ (ℂ_fld Σ_g 𝑏))‘𝑥) ↔ ¬ 𝑥 = (𝐼 × {0})))
23	22	imbi1d 340	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}) → ((0 < ((𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ↦ (ℂ_fld Σ_g 𝑏))‘𝑥) → (𝐹‘𝑥) = (0_g‘𝑅)) ↔ (¬ 𝑥 = (𝐼 × {0}) → (𝐹‘𝑥) = (0_g‘𝑅))))
24		eqeq2 2740	. . . . . . . 8 ⊢ ((𝐹‘(𝐼 × {0})) = if(𝑥 = (𝐼 × {0}), (𝐹‘(𝐼 × {0})), (0_g‘𝑅)) → ((𝐹‘𝑥) = (𝐹‘(𝐼 × {0})) ↔ (𝐹‘𝑥) = if(𝑥 = (𝐼 × {0}), (𝐹‘(𝐼 × {0})), (0_g‘𝑅))))
25	24	bibi1d 342	. . . . . . 7 ⊢ ((𝐹‘(𝐼 × {0})) = if(𝑥 = (𝐼 × {0}), (𝐹‘(𝐼 × {0})), (0_g‘𝑅)) → (((𝐹‘𝑥) = (𝐹‘(𝐼 × {0})) ↔ (¬ 𝑥 = (𝐼 × {0}) → (𝐹‘𝑥) = (0_g‘𝑅))) ↔ ((𝐹‘𝑥) = if(𝑥 = (𝐼 × {0}), (𝐹‘(𝐼 × {0})), (0_g‘𝑅)) ↔ (¬ 𝑥 = (𝐼 × {0}) → (𝐹‘𝑥) = (0_g‘𝑅)))))
26		eqeq2 2740	. . . . . . . 8 ⊢ ((0_g‘𝑅) = if(𝑥 = (𝐼 × {0}), (𝐹‘(𝐼 × {0})), (0_g‘𝑅)) → ((𝐹‘𝑥) = (0_g‘𝑅) ↔ (𝐹‘𝑥) = if(𝑥 = (𝐼 × {0}), (𝐹‘(𝐼 × {0})), (0_g‘𝑅))))
27	26	bibi1d 342	. . . . . . 7 ⊢ ((0_g‘𝑅) = if(𝑥 = (𝐼 × {0}), (𝐹‘(𝐼 × {0})), (0_g‘𝑅)) → (((𝐹‘𝑥) = (0_g‘𝑅) ↔ (¬ 𝑥 = (𝐼 × {0}) → (𝐹‘𝑥) = (0_g‘𝑅))) ↔ ((𝐹‘𝑥) = if(𝑥 = (𝐼 × {0}), (𝐹‘(𝐼 × {0})), (0_g‘𝑅)) ↔ (¬ 𝑥 = (𝐼 × {0}) → (𝐹‘𝑥) = (0_g‘𝑅)))))
28		fveq2 6902	. . . . . . . . 9 ⊢ (𝑥 = (𝐼 × {0}) → (𝐹‘𝑥) = (𝐹‘(𝐼 × {0})))
29		pm2.24 124	. . . . . . . . 9 ⊢ (𝑥 = (𝐼 × {0}) → (¬ 𝑥 = (𝐼 × {0}) → (𝐹‘𝑥) = (0_g‘𝑅)))
30	28, 29	2thd 264	. . . . . . . 8 ⊢ (𝑥 = (𝐼 × {0}) → ((𝐹‘𝑥) = (𝐹‘(𝐼 × {0})) ↔ (¬ 𝑥 = (𝐼 × {0}) → (𝐹‘𝑥) = (0_g‘𝑅))))
31	30	adantl 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = (𝐼 × {0})) → ((𝐹‘𝑥) = (𝐹‘(𝐼 × {0})) ↔ (¬ 𝑥 = (𝐼 × {0}) → (𝐹‘𝑥) = (0_g‘𝑅))))
32		biimt 359	. . . . . . . 8 ⊢ (¬ 𝑥 = (𝐼 × {0}) → ((𝐹‘𝑥) = (0_g‘𝑅) ↔ (¬ 𝑥 = (𝐼 × {0}) → (𝐹‘𝑥) = (0_g‘𝑅))))
33	32	adantl 480	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑥 = (𝐼 × {0})) → ((𝐹‘𝑥) = (0_g‘𝑅) ↔ (¬ 𝑥 = (𝐼 × {0}) → (𝐹‘𝑥) = (0_g‘𝑅))))
34	25, 27, 31, 33	ifbothda 4570	. . . . . 6 ⊢ (𝜑 → ((𝐹‘𝑥) = if(𝑥 = (𝐼 × {0}), (𝐹‘(𝐼 × {0})), (0_g‘𝑅)) ↔ (¬ 𝑥 = (𝐼 × {0}) → (𝐹‘𝑥) = (0_g‘𝑅))))
35	34	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}) → ((𝐹‘𝑥) = if(𝑥 = (𝐼 × {0}), (𝐹‘(𝐼 × {0})), (0_g‘𝑅)) ↔ (¬ 𝑥 = (𝐼 × {0}) → (𝐹‘𝑥) = (0_g‘𝑅))))
36	23, 35	bitr4d 281	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}) → ((0 < ((𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ↦ (ℂ_fld Σ_g 𝑏))‘𝑥) → (𝐹‘𝑥) = (0_g‘𝑅)) ↔ (𝐹‘𝑥) = if(𝑥 = (𝐼 × {0}), (𝐹‘(𝐼 × {0})), (0_g‘𝑅))))
37	36	ralbidva 3173	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} (0 < ((𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ↦ (ℂ_fld Σ_g 𝑏))‘𝑥) → (𝐹‘𝑥) = (0_g‘𝑅)) ↔ ∀𝑥 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} (𝐹‘𝑥) = if(𝑥 = (𝐼 × {0}), (𝐹‘(𝐼 × {0})), (0_g‘𝑅))))
38		eqid 2728	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
39	4, 38, 5, 7, 1	mplelf 21947	. . . . . 6 ⊢ (𝜑 → 𝐹:{𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
40	39	feqmptd 6972	. . . . 5 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ↦ (𝐹‘𝑥)))
41		mdegle0.a	. . . . . 6 ⊢ 𝐴 = (algSc‘𝑌)
42		mdegaddle.i	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
43		mdegaddle.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Ring)
44	7	psrbag0 22013	. . . . . . . 8 ⊢ (𝐼 ∈ 𝑉 → (𝐼 × {0}) ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin})
45	42, 44	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐼 × {0}) ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin})
46	39, 45	ffvelcdmd 7100	. . . . . 6 ⊢ (𝜑 → (𝐹‘(𝐼 × {0})) ∈ (Base‘𝑅))
47	4, 7, 6, 38, 41, 42, 43, 46	mplascl 22015	. . . . 5 ⊢ (𝜑 → (𝐴‘(𝐹‘(𝐼 × {0}))) = (𝑥 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ↦ if(𝑥 = (𝐼 × {0}), (𝐹‘(𝐼 × {0})), (0_g‘𝑅))))
48	40, 47	eqeq12d 2744	. . . 4 ⊢ (𝜑 → (𝐹 = (𝐴‘(𝐹‘(𝐼 × {0}))) ↔ (𝑥 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ↦ (𝐹‘𝑥)) = (𝑥 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ↦ if(𝑥 = (𝐼 × {0}), (𝐹‘(𝐼 × {0})), (0_g‘𝑅)))))
49		fvex 6915	. . . . . 6 ⊢ (𝐹‘𝑥) ∈ V
50	49	rgenw 3062	. . . . 5 ⊢ ∀𝑥 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} (𝐹‘𝑥) ∈ V
51		mpteqb 7029	. . . . 5 ⊢ (∀𝑥 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} (𝐹‘𝑥) ∈ V → ((𝑥 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ↦ (𝐹‘𝑥)) = (𝑥 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ↦ if(𝑥 = (𝐼 × {0}), (𝐹‘(𝐼 × {0})), (0_g‘𝑅))) ↔ ∀𝑥 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} (𝐹‘𝑥) = if(𝑥 = (𝐼 × {0}), (𝐹‘(𝐼 × {0})), (0_g‘𝑅))))
52	50, 51	mp1i 13	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ↦ (𝐹‘𝑥)) = (𝑥 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ↦ if(𝑥 = (𝐼 × {0}), (𝐹‘(𝐼 × {0})), (0_g‘𝑅))) ↔ ∀𝑥 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} (𝐹‘𝑥) = if(𝑥 = (𝐼 × {0}), (𝐹‘(𝐼 × {0})), (0_g‘𝑅))))
53	48, 52	bitrd 278	. . 3 ⊢ (𝜑 → (𝐹 = (𝐴‘(𝐹‘(𝐼 × {0}))) ↔ ∀𝑥 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} (𝐹‘𝑥) = if(𝑥 = (𝐼 × {0}), (𝐹‘(𝐼 × {0})), (0_g‘𝑅))))
54	37, 53	bitr4d 281	. 2 ⊢ (𝜑 → (∀𝑥 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} (0 < ((𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ↦ (ℂ_fld Σ_g 𝑏))‘𝑥) → (𝐹‘𝑥) = (0_g‘𝑅)) ↔ 𝐹 = (𝐴‘(𝐹‘(𝐼 × {0})))))
55	10, 54	bitrd 278	1 ⊢ (𝜑 → ((𝐷‘𝐹) ≤ 0 ↔ 𝐹 = (𝐴‘(𝐹‘(𝐼 × {0})))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 ∀wral 3058 {crab 3430 Vcvv 3473 ifcif 4532 {csn 4632 class class class wbr 5152 ↦ cmpt 5235 × cxp 5680 ^◡ccnv 5681 “ cima 5685 ⟶wf 6549 ‘cfv 6553 (class class class)co 7426 ↑_m cmap 8851 Fincfn 8970 ℝcr 11145 0cc0 11146 ℝ^*cxr 11285 < clt 11286 ≤ cle 11287 ℕcn 12250 ℕ₀cn0 12510 Basecbs 17187 0_gc0g 17428 Σ_g cgsu 17429 Ringcrg 20180 ℂ_fldccnfld 21286 algSccascl 21793 mPoly cmpl 21846 mDeg cmdg 26006

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 ax-addf 11225

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-ofr 7692 df-om 7877 df-1st 7999 df-2nd 8000 df-supp 8172 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-pm 8854 df-ixp 8923 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fsupp 9394 df-sup 9473 df-oi 9541 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-fz 13525 df-fzo 13668 df-seq 14007 df-hash 14330 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-starv 17255 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-unif 17263 df-hom 17264 df-cco 17265 df-0g 17430 df-gsum 17431 df-prds 17436 df-pws 17438 df-mre 17573 df-mrc 17574 df-acs 17576 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-mhm 18747 df-submnd 18748 df-grp 18900 df-minusg 18901 df-mulg 19031 df-subg 19085 df-ghm 19175 df-cntz 19275 df-cmn 19744 df-abl 19745 df-mgp 20082 df-rng 20100 df-ur 20129 df-ring 20182 df-cring 20183 df-subrng 20490 df-subrg 20515 df-cnfld 21287 df-ascl 21796 df-psr 21849 df-mpl 21851 df-mdeg 26008

This theorem is referenced by: deg1le0 26067

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