Theorem fta1 26622

Description: The easy direction of the Fundamental Theorem of Algebra: A nonzero polynomial has at most deg(𝐹) roots. (Contributed by Mario Carneiro, 26-Jul-2014.)

Hypothesis

Ref	Expression
fta1.1	⊢ 𝑅 = (◡𝐹 “ {0})

Assertion

Ref	Expression
fta1	⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐹 ≠ 0𝑝) → (𝑅 ∈ Fin ∧ (♯‘𝑅) ≤ (deg‘𝐹)))

Proof of Theorem fta1

Dummy variables 𝑥 𝑔 𝑓 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2761	. 2 ⊢ (deg‘𝐹) = (deg‘𝐹)
2		dgrcl 26545	. . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
3	2	adantr 486	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐹 ≠ 0𝑝) → (deg‘𝐹) ∈ ℕ0)
4		eqeq2 2773	. . . . . . 7 ⊢ (𝑥 = 0 → ((deg‘𝑓) = 𝑥 ↔ (deg‘𝑓) = 0))
5	4	imbi1d 344	. . . . . 6 ⊢ (𝑥 = 0 → (((deg‘𝑓) = 𝑥 → ((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓))) ↔ ((deg‘𝑓) = 0 → ((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓)))))
6	5	ralbidv 3186	. . . . 5 ⊢ (𝑥 = 0 → (∀𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑓) = 𝑥 → ((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓))) ↔ ∀𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑓) = 0 → ((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓)))))
7		eqeq2 2773	. . . . . . 7 ⊢ (𝑥 = 𝑑 → ((deg‘𝑓) = 𝑥 ↔ (deg‘𝑓) = 𝑑))
8	7	imbi1d 344	. . . . . 6 ⊢ (𝑥 = 𝑑 → (((deg‘𝑓) = 𝑥 → ((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓))) ↔ ((deg‘𝑓) = 𝑑 → ((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓)))))
9	8	ralbidv 3186	. . . . 5 ⊢ (𝑥 = 𝑑 → (∀𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑓) = 𝑥 → ((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓))) ↔ ∀𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑓) = 𝑑 → ((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓)))))
10		eqeq2 2773	. . . . . . 7 ⊢ (𝑥 = (𝑑 + 1) → ((deg‘𝑓) = 𝑥 ↔ (deg‘𝑓) = (𝑑 + 1)))
11	10	imbi1d 344	. . . . . 6 ⊢ (𝑥 = (𝑑 + 1) → (((deg‘𝑓) = 𝑥 → ((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓))) ↔ ((deg‘𝑓) = (𝑑 + 1) → ((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓)))))
12	11	ralbidv 3186	. . . . 5 ⊢ (𝑥 = (𝑑 + 1) → (∀𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑓) = 𝑥 → ((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓))) ↔ ∀𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑓) = (𝑑 + 1) → ((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓)))))
13		eqeq2 2773	. . . . . . 7 ⊢ (𝑥 = (deg‘𝐹) → ((deg‘𝑓) = 𝑥 ↔ (deg‘𝑓) = (deg‘𝐹)))
14	13	imbi1d 344	. . . . . 6 ⊢ (𝑥 = (deg‘𝐹) → (((deg‘𝑓) = 𝑥 → ((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓))) ↔ ((deg‘𝑓) = (deg‘𝐹) → ((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓)))))
15	14	ralbidv 3186	. . . . 5 ⊢ (𝑥 = (deg‘𝐹) → (∀𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑓) = 𝑥 → ((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓))) ↔ ∀𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑓) = (deg‘𝐹) → ((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓)))))
16		eldifsni 4753	. . . . . . . . . . 11 ⊢ (𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) → 𝑓 ≠ 0𝑝)
17	16	adantr 486	. . . . . . . . . 10 ⊢ ((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) → 𝑓 ≠ 0𝑝)
18		simplr 781	. . . . . . . . . . . . . . 15 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → (deg‘𝑓) = 0)
19		eldifi 4078	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) → 𝑓 ∈ (Poly‘ℂ))
20	19	ad2antrr 739	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → 𝑓 ∈ (Poly‘ℂ))
21		0dgrb 26558	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ (Poly‘ℂ) → ((deg‘𝑓) = 0 ↔ 𝑓 = (ℂ × {(𝑓‘0)})))
22	20, 21	syl 18	. . . . . . . . . . . . . . 15 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → ((deg‘𝑓) = 0 ↔ 𝑓 = (ℂ × {(𝑓‘0)})))
23	18, 22	mpbid 235	. . . . . . . . . . . . . 14 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → 𝑓 = (ℂ × {(𝑓‘0)}))
24	23	fveq1d 6885	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → (𝑓‘𝑥) = ((ℂ × {(𝑓‘0)})‘𝑥))
25	19	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) → 𝑓 ∈ (Poly‘ℂ))
26		plyf 26509	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 ∈ (Poly‘ℂ) → 𝑓:ℂ⟶ℂ)
27		ffn 6707	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓:ℂ⟶ℂ → 𝑓 Fn ℂ)
28		fniniseg 7057	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 Fn ℂ → (𝑥 ∈ (◡𝑓 “ {0}) ↔ (𝑥 ∈ ℂ ∧ (𝑓‘𝑥) = 0)))
29	25, 26, 27, 28	4syl 20	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) → (𝑥 ∈ (◡𝑓 “ {0}) ↔ (𝑥 ∈ ℂ ∧ (𝑓‘𝑥) = 0)))
30	29	biimpa 482	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → (𝑥 ∈ ℂ ∧ (𝑓‘𝑥) = 0))
31	30	simprd 501	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → (𝑓‘𝑥) = 0)
32	30	simpld 500	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → 𝑥 ∈ ℂ)
33		fvex 6896	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓‘0) ∈ V
34	33	fvconst2 7208	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℂ → ((ℂ × {(𝑓‘0)})‘𝑥) = (𝑓‘0))
35	32, 34	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → ((ℂ × {(𝑓‘0)})‘𝑥) = (𝑓‘0))
36	24, 31, 35	3eqtr3rd 2805	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → (𝑓‘0) = 0)
37	36	sneqd 4596	. . . . . . . . . . . . . . 15 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → {(𝑓‘0)} = {0})
38	37	xpeq2d 5681	. . . . . . . . . . . . . 14 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → (ℂ × {(𝑓‘0)}) = (ℂ × {0}))
39	23, 38	eqtrd 2796	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → 𝑓 = (ℂ × {0}))
40		df-0p 25984	. . . . . . . . . . . . 13 ⊢ 0𝑝 = (ℂ × {0})
41	39, 40	eqtr4di 2814	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → 𝑓 = 0𝑝)
42	41	ex 418	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) → (𝑥 ∈ (◡𝑓 “ {0}) → 𝑓 = 0𝑝))
43	42	necon3ad 2969	. . . . . . . . . 10 ⊢ ((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) → (𝑓 ≠ 0𝑝 → ¬ 𝑥 ∈ (◡𝑓 “ {0})))
44	17, 43	mpd 16	. . . . . . . . 9 ⊢ ((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) → ¬ 𝑥 ∈ (◡𝑓 “ {0}))
45	44	eq0rdv 4365	. . . . . . . 8 ⊢ ((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) → (◡𝑓 “ {0}) = ∅)
46	45	ex 418	. . . . . . 7 ⊢ (𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) → ((deg‘𝑓) = 0 → (◡𝑓 “ {0}) = ∅))
47		dgrcl 26545	. . . . . . . . 9 ⊢ (𝑓 ∈ (Poly‘ℂ) → (deg‘𝑓) ∈ ℕ0)
48		nn0ge0 12624	. . . . . . . . 9 ⊢ ((deg‘𝑓) ∈ ℕ0 → 0 ≤ (deg‘𝑓))
49	19, 47, 48	3syl 19	. . . . . . . 8 ⊢ (𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) → 0 ≤ (deg‘𝑓))
50		id 23	. . . . . . . . . . 11 ⊢ ((◡𝑓 “ {0}) = ∅ → (◡𝑓 “ {0}) = ∅)
51		0fi 9063	. . . . . . . . . . 11 ⊢ ∅ ∈ Fin
52	50, 51	eqeltrdi 2869	. . . . . . . . . 10 ⊢ ((◡𝑓 “ {0}) = ∅ → (◡𝑓 “ {0}) ∈ Fin)
53	52	biantrurd 542	. . . . . . . . 9 ⊢ ((◡𝑓 “ {0}) = ∅ → ((♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓) ↔ ((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓))))
54		fveq2 6883	. . . . . . . . . . 11 ⊢ ((◡𝑓 “ {0}) = ∅ → (♯‘(◡𝑓 “ {0})) = (♯‘∅))
55		hash0 14504	. . . . . . . . . . 11 ⊢ (♯‘∅) = 0
56	54, 55	eqtrdi 2812	. . . . . . . . . 10 ⊢ ((◡𝑓 “ {0}) = ∅ → (♯‘(◡𝑓 “ {0})) = 0)
57	56	breq1d 5113	. . . . . . . . 9 ⊢ ((◡𝑓 “ {0}) = ∅ → ((♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓) ↔ 0 ≤ (deg‘𝑓)))
58	53, 57	bitr3d 284	. . . . . . . 8 ⊢ ((◡𝑓 “ {0}) = ∅ → (((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓)) ↔ 0 ≤ (deg‘𝑓)))
59	49, 58	syl5ibrcom 250	. . . . . . 7 ⊢ (𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) → ((◡𝑓 “ {0}) = ∅ → ((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓))))
60	46, 59	syld 48	. . . . . 6 ⊢ (𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) → ((deg‘𝑓) = 0 → ((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓))))
61	60	rgen 3079	. . . . 5 ⊢ ∀𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑓) = 0 → ((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓)))
62		fveqeq2 6892	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → ((deg‘𝑓) = 𝑑 ↔ (deg‘𝑔) = 𝑑))
63		cnveq 5851	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → ◡𝑓 = ◡𝑔)
64	63	imaeq1d 6051	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → (◡𝑓 “ {0}) = (◡𝑔 “ {0}))
65	64	eleq1d 2846	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → ((◡𝑓 “ {0}) ∈ Fin ↔ (◡𝑔 “ {0}) ∈ Fin))
66	64	fveq2d 6887	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → (♯‘(◡𝑓 “ {0})) = (♯‘(◡𝑔 “ {0})))
67		fveq2 6883	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → (deg‘𝑓) = (deg‘𝑔))
68	66, 67	breq12d 5116	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → ((♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓) ↔ (♯‘(◡𝑔 “ {0})) ≤ (deg‘𝑔)))
69	65, 68	anbi12d 644	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → (((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓)) ↔ ((◡𝑔 “ {0}) ∈ Fin ∧ (♯‘(◡𝑔 “ {0})) ≤ (deg‘𝑔))))
70	62, 69	imbi12d 347	. . . . . . 7 ⊢ (𝑓 = 𝑔 → (((deg‘𝑓) = 𝑑 → ((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓))) ↔ ((deg‘𝑔) = 𝑑 → ((◡𝑔 “ {0}) ∈ Fin ∧ (♯‘(◡𝑔 “ {0})) ≤ (deg‘𝑔)))))
71	70	cbvralvw 3241	. . . . . 6 ⊢ (∀𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑓) = 𝑑 → ((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓))) ↔ ∀𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑔) = 𝑑 → ((◡𝑔 “ {0}) ∈ Fin ∧ (♯‘(◡𝑔 “ {0})) ≤ (deg‘𝑔))))
72	49	ad2antlr 740	. . . . . . . . . . . 12 ⊢ (((𝑑 ∈ ℕ0 ∧ 𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})) ∧ (deg‘𝑓) = (𝑑 + 1)) → 0 ≤ (deg‘𝑓))
73	72, 58	syl5ibrcom 250	. . . . . . . . . . 11 ⊢ (((𝑑 ∈ ℕ0 ∧ 𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})) ∧ (deg‘𝑓) = (𝑑 + 1)) → ((◡𝑓 “ {0}) = ∅ → ((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓))))
74	73	a1dd 51	. . . . . . . . . 10 ⊢ (((𝑑 ∈ ℕ0 ∧ 𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})) ∧ (deg‘𝑓) = (𝑑 + 1)) → ((◡𝑓 “ {0}) = ∅ → (∀𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑔) = 𝑑 → ((◡𝑔 “ {0}) ∈ Fin ∧ (♯‘(◡𝑔 “ {0})) ≤ (deg‘𝑔))) → ((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓)))))
75		n0 4300	. . . . . . . . . . 11 ⊢ ((◡𝑓 “ {0}) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (◡𝑓 “ {0}))
76		eqid 2761	. . . . . . . . . . . . . 14 ⊢ (◡𝑓 “ {0}) = (◡𝑓 “ {0})
77		simplll 787	. . . . . . . . . . . . . 14 ⊢ ((((𝑑 ∈ ℕ0 ∧ 𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})) ∧ (deg‘𝑓) = (𝑑 + 1)) ∧ (𝑥 ∈ (◡𝑓 “ {0}) ∧ ∀𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑔) = 𝑑 → ((◡𝑔 “ {0}) ∈ Fin ∧ (♯‘(◡𝑔 “ {0})) ≤ (deg‘𝑔))))) → 𝑑 ∈ ℕ0)
78		simpllr 788	. . . . . . . . . . . . . 14 ⊢ ((((𝑑 ∈ ℕ0 ∧ 𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})) ∧ (deg‘𝑓) = (𝑑 + 1)) ∧ (𝑥 ∈ (◡𝑓 “ {0}) ∧ ∀𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑔) = 𝑑 → ((◡𝑔 “ {0}) ∈ Fin ∧ (♯‘(◡𝑔 “ {0})) ≤ (deg‘𝑔))))) → 𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}))
79		simplr 781	. . . . . . . . . . . . . 14 ⊢ ((((𝑑 ∈ ℕ0 ∧ 𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})) ∧ (deg‘𝑓) = (𝑑 + 1)) ∧ (𝑥 ∈ (◡𝑓 “ {0}) ∧ ∀𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑔) = 𝑑 → ((◡𝑔 “ {0}) ∈ Fin ∧ (♯‘(◡𝑔 “ {0})) ≤ (deg‘𝑔))))) → (deg‘𝑓) = (𝑑 + 1))
80		simprl 783	. . . . . . . . . . . . . 14 ⊢ ((((𝑑 ∈ ℕ0 ∧ 𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})) ∧ (deg‘𝑓) = (𝑑 + 1)) ∧ (𝑥 ∈ (◡𝑓 “ {0}) ∧ ∀𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑔) = 𝑑 → ((◡𝑔 “ {0}) ∈ Fin ∧ (♯‘(◡𝑔 “ {0})) ≤ (deg‘𝑔))))) → 𝑥 ∈ (◡𝑓 “ {0}))
81		simprr 785	. . . . . . . . . . . . . 14 ⊢ ((((𝑑 ∈ ℕ0 ∧ 𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})) ∧ (deg‘𝑓) = (𝑑 + 1)) ∧ (𝑥 ∈ (◡𝑓 “ {0}) ∧ ∀𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑔) = 𝑑 → ((◡𝑔 “ {0}) ∈ Fin ∧ (♯‘(◡𝑔 “ {0})) ≤ (deg‘𝑔))))) → ∀𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑔) = 𝑑 → ((◡𝑔 “ {0}) ∈ Fin ∧ (♯‘(◡𝑔 “ {0})) ≤ (deg‘𝑔))))
82	76, 77, 78, 79, 80, 81	fta1lem 26621	. . . . . . . . . . . . 13 ⊢ ((((𝑑 ∈ ℕ0 ∧ 𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})) ∧ (deg‘𝑓) = (𝑑 + 1)) ∧ (𝑥 ∈ (◡𝑓 “ {0}) ∧ ∀𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑔) = 𝑑 → ((◡𝑔 “ {0}) ∈ Fin ∧ (♯‘(◡𝑔 “ {0})) ≤ (deg‘𝑔))))) → ((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓)))
83	82	exp32 426	. . . . . . . . . . . 12 ⊢ (((𝑑 ∈ ℕ0 ∧ 𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})) ∧ (deg‘𝑓) = (𝑑 + 1)) → (𝑥 ∈ (◡𝑓 “ {0}) → (∀𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑔) = 𝑑 → ((◡𝑔 “ {0}) ∈ Fin ∧ (♯‘(◡𝑔 “ {0})) ≤ (deg‘𝑔))) → ((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓)))))
84	83	exlimdv 1966	. . . . . . . . . . 11 ⊢ (((𝑑 ∈ ℕ0 ∧ 𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})) ∧ (deg‘𝑓) = (𝑑 + 1)) → (∃𝑥 𝑥 ∈ (◡𝑓 “ {0}) → (∀𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑔) = 𝑑 → ((◡𝑔 “ {0}) ∈ Fin ∧ (♯‘(◡𝑔 “ {0})) ≤ (deg‘𝑔))) → ((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓)))))
85	75, 84	biimtrid 245	. . . . . . . . . 10 ⊢ (((𝑑 ∈ ℕ0 ∧ 𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})) ∧ (deg‘𝑓) = (𝑑 + 1)) → ((◡𝑓 “ {0}) ≠ ∅ → (∀𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑔) = 𝑑 → ((◡𝑔 “ {0}) ∈ Fin ∧ (♯‘(◡𝑔 “ {0})) ≤ (deg‘𝑔))) → ((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓)))))
86	74, 85	pm2.61dne 3042	. . . . . . . . 9 ⊢ (((𝑑 ∈ ℕ0 ∧ 𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})) ∧ (deg‘𝑓) = (𝑑 + 1)) → (∀𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑔) = 𝑑 → ((◡𝑔 “ {0}) ∈ Fin ∧ (♯‘(◡𝑔 “ {0})) ≤ (deg‘𝑔))) → ((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓))))
87	86	ex 418	. . . . . . . 8 ⊢ ((𝑑 ∈ ℕ0 ∧ 𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})) → ((deg‘𝑓) = (𝑑 + 1) → (∀𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑔) = 𝑑 → ((◡𝑔 “ {0}) ∈ Fin ∧ (♯‘(◡𝑔 “ {0})) ≤ (deg‘𝑔))) → ((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓)))))
88	87	com23 87	. . . . . . 7 ⊢ ((𝑑 ∈ ℕ0 ∧ 𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})) → (∀𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑔) = 𝑑 → ((◡𝑔 “ {0}) ∈ Fin ∧ (♯‘(◡𝑔 “ {0})) ≤ (deg‘𝑔))) → ((deg‘𝑓) = (𝑑 + 1) → ((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓)))))
89	88	ralrimdva 3163	. . . . . 6 ⊢ (𝑑 ∈ ℕ0 → (∀𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑔) = 𝑑 → ((◡𝑔 “ {0}) ∈ Fin ∧ (♯‘(◡𝑔 “ {0})) ≤ (deg‘𝑔))) → ∀𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑓) = (𝑑 + 1) → ((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓)))))
90	71, 89	biimtrid 245	. . . . 5 ⊢ (𝑑 ∈ ℕ0 → (∀𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑓) = 𝑑 → ((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓))) → ∀𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑓) = (𝑑 + 1) → ((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓)))))
91	6, 9, 12, 15, 61, 90	nn0ind 12787	. . . 4 ⊢ ((deg‘𝐹) ∈ ℕ0 → ∀𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑓) = (deg‘𝐹) → ((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓))))
92	3, 91	syl 18	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐹 ≠ 0𝑝) → ∀𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑓) = (deg‘𝐹) → ((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓))))
93		plyssc 26511	. . . . 5 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ)
94	93	sseli 3927	. . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (Poly‘ℂ))
95		eldifsn 4748	. . . . 5 ⊢ (𝐹 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ↔ (𝐹 ∈ (Poly‘ℂ) ∧ 𝐹 ≠ 0𝑝))
96		fveqeq2 6892	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ((deg‘𝑓) = (deg‘𝐹) ↔ (deg‘𝐹) = (deg‘𝐹)))
97		cnveq 5851	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → ◡𝑓 = ◡𝐹)
98	97	imaeq1d 6051	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → (◡𝑓 “ {0}) = (◡𝐹 “ {0}))
99		fta1.1	. . . . . . . . . 10 ⊢ 𝑅 = (◡𝐹 “ {0})
100	98, 99	eqtr4di 2814	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (◡𝑓 “ {0}) = 𝑅)
101	100	eleq1d 2846	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((◡𝑓 “ {0}) ∈ Fin ↔ 𝑅 ∈ Fin))
102	100	fveq2d 6887	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (♯‘(◡𝑓 “ {0})) = (♯‘𝑅))
103		fveq2 6883	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (deg‘𝑓) = (deg‘𝐹))
104	102, 103	breq12d 5116	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓) ↔ (♯‘𝑅) ≤ (deg‘𝐹)))
105	101, 104	anbi12d 644	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓)) ↔ (𝑅 ∈ Fin ∧ (♯‘𝑅) ≤ (deg‘𝐹))))
106	96, 105	imbi12d 347	. . . . . 6 ⊢ (𝑓 = 𝐹 → (((deg‘𝑓) = (deg‘𝐹) → ((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓))) ↔ ((deg‘𝐹) = (deg‘𝐹) → (𝑅 ∈ Fin ∧ (♯‘𝑅) ≤ (deg‘𝐹)))))
107	106	rspcv 3573	. . . . 5 ⊢ (𝐹 ∈ ((Poly‘ℂ) ∖ {0𝑝}) → (∀𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑓) = (deg‘𝐹) → ((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓))) → ((deg‘𝐹) = (deg‘𝐹) → (𝑅 ∈ Fin ∧ (♯‘𝑅) ≤ (deg‘𝐹)))))
108	95, 107	sylbir 238	. . . 4 ⊢ ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐹 ≠ 0𝑝) → (∀𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑓) = (deg‘𝐹) → ((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓))) → ((deg‘𝐹) = (deg‘𝐹) → (𝑅 ∈ Fin ∧ (♯‘𝑅) ≤ (deg‘𝐹)))))
109	94, 108	sylan 592	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐹 ≠ 0𝑝) → (∀𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑓) = (deg‘𝐹) → ((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓))) → ((deg‘𝐹) = (deg‘𝐹) → (𝑅 ∈ Fin ∧ (♯‘𝑅) ≤ (deg‘𝐹)))))
110	92, 109	mpd 16	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐹 ≠ 0𝑝) → ((deg‘𝐹) = (deg‘𝐹) → (𝑅 ∈ Fin ∧ (♯‘𝑅) ≤ (deg‘𝐹))))
111	1, 110	mpi 21	1 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐹 ≠ 0𝑝) → (𝑅 ∈ Fin ∧ (♯‘𝑅) ≤ (deg‘𝐹)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077   ∖ cdif 3896  ∅c0 4279  {csn 4584   class class class wbr 5103   × cxp 5649  ^◡ccnv 5650   “ cima 5654   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  (class class class)co 7418  Fincfn 8966  ℂcc 11191  0cc0 11193  1c1 11194   + caddc 11196   ≤ cle 11337  ℕ₀cn0 12599  ♯chash 14467  0_𝑝c0p 25983  Polycply 26495  degcdgr 26498

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-inf2 9635  ax-cnex 11249  ax-resscn 11250  ax-1cn 11251  ax-icn 11252  ax-addcl 11253  ax-addrcl 11254  ax-mulcl 11255  ax-mulrcl 11256  ax-mulcom 11257  ax-addass 11258  ax-mulass 11259  ax-distr 11260  ax-i2m1 11261  ax-1ne0 11262  ax-1rid 11263  ax-rnegex 11264  ax-rrecex 11265  ax-cnre 11266  ax-pre-lttri 11267  ax-pre-lttrn 11268  ax-pre-ltadd 11269  ax-pre-mulgt0 11270  ax-pre-sup 11271

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-of 7691  df-om 7876  df-1st 7999  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-1o 8469  df-oadd 8473  df-er 8710  df-map 8842  df-pm 8843  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-sup 9427  df-inf 9428  df-oi 9497  df-dju 9975  df-card 10013  df-pnf 11338  df-mnf 11339  df-xr 11340  df-ltxr 11341  df-le 11342  df-sub 11536  df-neg 11537  df-div 11967  df-nn 12329  df-2 12398  df-3 12399  df-n0 12600  df-xnn0 12673  df-z 12687  df-uz 12959  df-rp 13114  df-fz 13633  df-fzo 13782  df-fl 13925  df-seq 14138  df-exp 14198  df-hash 14468  df-cj 15259  df-re 15260  df-im 15261  df-sqrt 15395  df-abs 15396  df-clim 15648  df-rlim 15649  df-sum 15847  df-0p 25984  df-ply 26499  df-idp 26500  df-coe 26501  df-dgr 26502  df-quot 26605

This theorem is used by:  rnplynfin  26623  plyconz  26624  vieta1lem2  26627  vieta1  26628  plyexmo  26629  aannenlem1  26648  aalioulem2  26653  basellem4  27404  dchrfi  27575  sinnpoly  47910