Theorem fta1 24368

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 2765	. 2 ⊢ (deg‘𝐹) = (deg‘𝐹)
2		dgrcl 24294	. . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
3	2	adantr 472	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐹 ≠ 0𝑝) → (deg‘𝐹) ∈ ℕ0)
4		eqeq2 2776	. . . . . . 7 ⊢ (𝑥 = 0 → ((deg‘𝑓) = 𝑥 ↔ (deg‘𝑓) = 0))
5	4	imbi1d 332	. . . . . 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 3133	. . . . 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 2776	. . . . . . 7 ⊢ (𝑥 = 𝑑 → ((deg‘𝑓) = 𝑥 ↔ (deg‘𝑓) = 𝑑))
8	7	imbi1d 332	. . . . . 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 3133	. . . . 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 2776	. . . . . . 7 ⊢ (𝑥 = (𝑑 + 1) → ((deg‘𝑓) = 𝑥 ↔ (deg‘𝑓) = (𝑑 + 1)))
11	10	imbi1d 332	. . . . . 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 3133	. . . . 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 2776	. . . . . . 7 ⊢ (𝑥 = (deg‘𝐹) → ((deg‘𝑓) = 𝑥 ↔ (deg‘𝑓) = (deg‘𝐹)))
14	13	imbi1d 332	. . . . . 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 3133	. . . . 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 4478	. . . . . . . . . . 11 ⊢ (𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) → 𝑓 ≠ 0𝑝)
17	16	adantr 472	. . . . . . . . . 10 ⊢ ((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) → 𝑓 ≠ 0𝑝)
18		simplr 785	. . . . . . . . . . . . . . 15 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → (deg‘𝑓) = 0)
19		eldifi 3896	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) → 𝑓 ∈ (Poly‘ℂ))
20	19	ad2antrr 717	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → 𝑓 ∈ (Poly‘ℂ))
21		0dgrb 24307	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ (Poly‘ℂ) → ((deg‘𝑓) = 0 ↔ 𝑓 = (ℂ × {(𝑓‘0)})))
22	20, 21	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → ((deg‘𝑓) = 0 ↔ 𝑓 = (ℂ × {(𝑓‘0)})))
23	18, 22	mpbid 223	. . . . . . . . . . . . . 14 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → 𝑓 = (ℂ × {(𝑓‘0)}))
24	23	fveq1d 6381	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → (𝑓‘𝑥) = ((ℂ × {(𝑓‘0)})‘𝑥))
25	19	adantr 472	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) → 𝑓 ∈ (Poly‘ℂ))
26		plyf 24259	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 ∈ (Poly‘ℂ) → 𝑓:ℂ⟶ℂ)
27		ffn 6225	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓:ℂ⟶ℂ → 𝑓 Fn ℂ)
28		fniniseg 6532	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 Fn ℂ → (𝑥 ∈ (◡𝑓 “ {0}) ↔ (𝑥 ∈ ℂ ∧ (𝑓‘𝑥) = 0)))
29	25, 26, 27, 28	4syl 19	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) → (𝑥 ∈ (◡𝑓 “ {0}) ↔ (𝑥 ∈ ℂ ∧ (𝑓‘𝑥) = 0)))
30	29	biimpa 468	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → (𝑥 ∈ ℂ ∧ (𝑓‘𝑥) = 0))
31	30	simprd 489	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → (𝑓‘𝑥) = 0)
32	30	simpld 488	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → 𝑥 ∈ ℂ)
33		fvex 6392	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓‘0) ∈ V
34	33	fvconst2 6666	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℂ → ((ℂ × {(𝑓‘0)})‘𝑥) = (𝑓‘0))
35	32, 34	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → ((ℂ × {(𝑓‘0)})‘𝑥) = (𝑓‘0))
36	24, 31, 35	3eqtr3rd 2808	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → (𝑓‘0) = 0)
37	36	sneqd 4348	. . . . . . . . . . . . . . 15 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → {(𝑓‘0)} = {0})
38	37	xpeq2d 5309	. . . . . . . . . . . . . 14 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → (ℂ × {(𝑓‘0)}) = (ℂ × {0}))
39	23, 38	eqtrd 2799	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → 𝑓 = (ℂ × {0}))
40		df-0p 23742	. . . . . . . . . . . . 13 ⊢ 0𝑝 = (ℂ × {0})
41	39, 40	syl6eqr 2817	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → 𝑓 = 0𝑝)
42	41	ex 401	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) → (𝑥 ∈ (◡𝑓 “ {0}) → 𝑓 = 0𝑝))
43	42	necon3ad 2950	. . . . . . . . . 10 ⊢ ((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) → (𝑓 ≠ 0𝑝 → ¬ 𝑥 ∈ (◡𝑓 “ {0})))
44	17, 43	mpd 15	. . . . . . . . 9 ⊢ ((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) → ¬ 𝑥 ∈ (◡𝑓 “ {0}))
45	44	eq0rdv 4143	. . . . . . . 8 ⊢ ((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) → (◡𝑓 “ {0}) = ∅)
46	45	ex 401	. . . . . . 7 ⊢ (𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) → ((deg‘𝑓) = 0 → (◡𝑓 “ {0}) = ∅))
47		dgrcl 24294	. . . . . . . . 9 ⊢ (𝑓 ∈ (Poly‘ℂ) → (deg‘𝑓) ∈ ℕ0)
48		nn0ge0 11569	. . . . . . . . 9 ⊢ ((deg‘𝑓) ∈ ℕ0 → 0 ≤ (deg‘𝑓))
49	19, 47, 48	3syl 18	. . . . . . . 8 ⊢ (𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) → 0 ≤ (deg‘𝑓))
50		id 22	. . . . . . . . . . 11 ⊢ ((◡𝑓 “ {0}) = ∅ → (◡𝑓 “ {0}) = ∅)
51		0fin 8399	. . . . . . . . . . 11 ⊢ ∅ ∈ Fin
52	50, 51	syl6eqel 2852	. . . . . . . . . 10 ⊢ ((◡𝑓 “ {0}) = ∅ → (◡𝑓 “ {0}) ∈ Fin)
53	52	biantrurd 528	. . . . . . . . 9 ⊢ ((◡𝑓 “ {0}) = ∅ → ((♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓) ↔ ((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓))))
54		fveq2 6379	. . . . . . . . . . 11 ⊢ ((◡𝑓 “ {0}) = ∅ → (♯‘(◡𝑓 “ {0})) = (♯‘∅))
55		hash0 13365	. . . . . . . . . . 11 ⊢ (♯‘∅) = 0
56	54, 55	syl6eq 2815	. . . . . . . . . 10 ⊢ ((◡𝑓 “ {0}) = ∅ → (♯‘(◡𝑓 “ {0})) = 0)
57	56	breq1d 4821	. . . . . . . . 9 ⊢ ((◡𝑓 “ {0}) = ∅ → ((♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓) ↔ 0 ≤ (deg‘𝑓)))
58	53, 57	bitr3d 272	. . . . . . . 8 ⊢ ((◡𝑓 “ {0}) = ∅ → (((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓)) ↔ 0 ≤ (deg‘𝑓)))
59	49, 58	syl5ibrcom 238	. . . . . . 7 ⊢ (𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) → ((◡𝑓 “ {0}) = ∅ → ((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓))))
60	46, 59	syld 47	. . . . . 6 ⊢ (𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) → ((deg‘𝑓) = 0 → ((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓))))
61	60	rgen 3069	. . . . 5 ⊢ ∀𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑓) = 0 → ((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓)))
62		fveqeq2 6388	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → ((deg‘𝑓) = 𝑑 ↔ (deg‘𝑔) = 𝑑))
63		cnveq 5466	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → ◡𝑓 = ◡𝑔)
64	63	imaeq1d 5649	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → (◡𝑓 “ {0}) = (◡𝑔 “ {0}))
65	64	eleq1d 2829	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → ((◡𝑓 “ {0}) ∈ Fin ↔ (◡𝑔 “ {0}) ∈ Fin))
66	64	fveq2d 6383	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → (♯‘(◡𝑓 “ {0})) = (♯‘(◡𝑔 “ {0})))
67		fveq2 6379	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → (deg‘𝑓) = (deg‘𝑔))
68	66, 67	breq12d 4824	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → ((♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓) ↔ (♯‘(◡𝑔 “ {0})) ≤ (deg‘𝑔)))
69	65, 68	anbi12d 624	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → (((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓)) ↔ ((◡𝑔 “ {0}) ∈ Fin ∧ (♯‘(◡𝑔 “ {0})) ≤ (deg‘𝑔))))
70	62, 69	imbi12d 335	. . . . . . 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	cbvralv 3319	. . . . . 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 718	. . . . . . . . . . . 12 ⊢ (((𝑑 ∈ ℕ0 ∧ 𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})) ∧ (deg‘𝑓) = (𝑑 + 1)) → 0 ≤ (deg‘𝑓))
73	72, 58	syl5ibrcom 238	. . . . . . . . . . 11 ⊢ (((𝑑 ∈ ℕ0 ∧ 𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})) ∧ (deg‘𝑓) = (𝑑 + 1)) → ((◡𝑓 “ {0}) = ∅ → ((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓))))
74	73	a1dd 50	. . . . . . . . . 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 4097	. . . . . . . . . . 11 ⊢ ((◡𝑓 “ {0}) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (◡𝑓 “ {0}))
76		eqid 2765	. . . . . . . . . . . . . 14 ⊢ (◡𝑓 “ {0}) = (◡𝑓 “ {0})
77		simplll 791	. . . . . . . . . . . . . 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 793	. . . . . . . . . . . . . 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 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))))) → (deg‘𝑓) = (𝑑 + 1))
80		simprl 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))))) → 𝑥 ∈ (◡𝑓 “ {0}))
81		simprr 789	. . . . . . . . . . . . . 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 24367	. . . . . . . . . . . . 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 411	. . . . . . . . . . . 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 2028	. . . . . . . . . . 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	syl5bi 233	. . . . . . . . . 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 3023	. . . . . . . . 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 401	. . . . . . . 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 86	. . . . . . 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 3116	. . . . . 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	syl5bi 233	. . . . 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 11724	. . . 4 ⊢ ((deg‘𝐹) ∈ ℕ0 → ∀𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑓) = (deg‘𝐹) → ((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓))))
92	3, 91	syl 17	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐹 ≠ 0𝑝) → ∀𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑓) = (deg‘𝐹) → ((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓))))
93		plyssc 24261	. . . . 5 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ)
94	93	sseli 3759	. . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (Poly‘ℂ))
95		eldifsn 4474	. . . . 5 ⊢ (𝐹 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ↔ (𝐹 ∈ (Poly‘ℂ) ∧ 𝐹 ≠ 0𝑝))
96		fveqeq2 6388	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ((deg‘𝑓) = (deg‘𝐹) ↔ (deg‘𝐹) = (deg‘𝐹)))
97		cnveq 5466	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → ◡𝑓 = ◡𝐹)
98	97	imaeq1d 5649	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → (◡𝑓 “ {0}) = (◡𝐹 “ {0}))
99		fta1.1	. . . . . . . . . 10 ⊢ 𝑅 = (◡𝐹 “ {0})
100	98, 99	syl6eqr 2817	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (◡𝑓 “ {0}) = 𝑅)
101	100	eleq1d 2829	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((◡𝑓 “ {0}) ∈ Fin ↔ 𝑅 ∈ Fin))
102	100	fveq2d 6383	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (♯‘(◡𝑓 “ {0})) = (♯‘𝑅))
103		fveq2 6379	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (deg‘𝑓) = (deg‘𝐹))
104	102, 103	breq12d 4824	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓) ↔ (♯‘𝑅) ≤ (deg‘𝐹)))
105	101, 104	anbi12d 624	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓)) ↔ (𝑅 ∈ Fin ∧ (♯‘𝑅) ≤ (deg‘𝐹))))
106	96, 105	imbi12d 335	. . . . . 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 3458	. . . . 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 226	. . . 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 575	. . 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 15	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐹 ≠ 0𝑝) → ((deg‘𝐹) = (deg‘𝐹) → (𝑅 ∈ Fin ∧ (♯‘𝑅) ≤ (deg‘𝐹))))
111	1, 110	mpi 20	1 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐹 ≠ 0𝑝) → (𝑅 ∈ Fin ∧ (♯‘𝑅) ≤ (deg‘𝐹)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 197   ∧ wa 384   = wceq 1652  ∃wex 1874   ∈ wcel 2155   ≠ wne 2937  ∀wral 3055   ∖ cdif 3731  ∅c0 4081  {csn 4336   class class class wbr 4811   × cxp 5277  ^◡ccnv 5278   “ cima 5282   Fn wfn 6065  ⟶wf 6066  ‘cfv 6070  (class class class)co 6846  Fincfn 8164  ℂcc 10191  0cc0 10193  1c1 10194   + caddc 10196   ≤ cle 10333  ℕ₀cn0 11542  ♯chash 13326  0_𝑝c0p 23741  Polycply 24245  degcdgr 24248

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151  ax-inf2 8757  ax-cnex 10249  ax-resscn 10250  ax-1cn 10251  ax-icn 10252  ax-addcl 10253  ax-addrcl 10254  ax-mulcl 10255  ax-mulrcl 10256  ax-mulcom 10257  ax-addass 10258  ax-mulass 10259  ax-distr 10260  ax-i2m1 10261  ax-1ne0 10262  ax-1rid 10263  ax-rnegex 10264  ax-rrecex 10265  ax-cnre 10266  ax-pre-lttri 10267  ax-pre-lttrn 10268  ax-pre-ltadd 10269  ax-pre-mulgt0 10270  ax-pre-sup 10271  ax-addf 10272

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-se 5239  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-isom 6079  df-riota 6807  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-of 7099  df-om 7268  df-1st 7370  df-2nd 7371  df-wrecs 7614  df-recs 7676  df-rdg 7714  df-1o 7768  df-oadd 7772  df-er 7951  df-map 8066  df-pm 8067  df-en 8165  df-dom 8166  df-sdom 8167  df-fin 8168  df-sup 8559  df-inf 8560  df-oi 8626  df-card 9020  df-cda 9247  df-pnf 10334  df-mnf 10335  df-xr 10336  df-ltxr 10337  df-le 10338  df-sub 10526  df-neg 10527  df-div 10943  df-nn 11279  df-2 11339  df-3 11340  df-n0 11543  df-xnn0 11615  df-z 11629  df-uz 11892  df-rp 12034  df-fz 12539  df-fzo 12679  df-fl 12806  df-seq 13014  df-exp 13073  df-hash 13327  df-cj 14138  df-re 14139  df-im 14140  df-sqrt 14274  df-abs 14275  df-clim 14518  df-rlim 14519  df-sum 14716  df-0p 23742  df-ply 24249  df-idp 24250  df-coe 24251  df-dgr 24252  df-quot 24351

This theorem is referenced by:  vieta1lem2  24371  vieta1  24372  plyexmo  24373  aannenlem1  24388  aalioulem2  24393  basellem4  25115  basellem5  25116  dchrfi  25285