Theorem fta1 26434

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 2769	. 2 ⊢ (deg‘𝐹) = (deg‘𝐹)
2		dgrcl 26355	. . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
3	2	adantr 485	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐹 ≠ 0𝑝) → (deg‘𝐹) ∈ ℕ0)
4		eqeq2 2781	. . . . . . 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 3194	. . . . 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 2781	. . . . . . 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 3194	. . . . 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 2781	. . . . . . 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 3194	. . . . 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 2781	. . . . . . 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 3194	. . . . 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 4759	. . . . . . . . . . 11 ⊢ (𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) → 𝑓 ≠ 0𝑝)
17	16	adantr 485	. . . . . . . . . 10 ⊢ ((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) → 𝑓 ≠ 0𝑝)
18		simplr 780	. . . . . . . . . . . . . . 15 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → (deg‘𝑓) = 0)
19		eldifi 4093	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) → 𝑓 ∈ (Poly‘ℂ))
20	19	ad2antrr 738	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → 𝑓 ∈ (Poly‘ℂ))
21		0dgrb 26368	. . . . . . . . . . . . . . . 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 6881	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → (𝑓‘𝑥) = ((ℂ × {(𝑓‘0)})‘𝑥))
25	19	adantr 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) → 𝑓 ∈ (Poly‘ℂ))
26		plyf 26320	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 ∈ (Poly‘ℂ) → 𝑓:ℂ⟶ℂ)
27		ffn 6703	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓:ℂ⟶ℂ → 𝑓 Fn ℂ)
28		fniniseg 7053	. . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → (𝑥 ∈ ℂ ∧ (𝑓‘𝑥) = 0))
31	30	simprd 500	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → (𝑓‘𝑥) = 0)
32	30	simpld 499	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → 𝑥 ∈ ℂ)
33		fvex 6892	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓‘0) ∈ V
34	33	fvconst2 7200	. . . . . . . . . . . . . . . . . 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 2813	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → (𝑓‘0) = 0)
37	36	sneqd 4603	. . . . . . . . . . . . . . 15 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → {(𝑓‘0)} = {0})
38	37	xpeq2d 5689	. . . . . . . . . . . . . 14 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → (ℂ × {(𝑓‘0)}) = (ℂ × {0}))
39	23, 38	eqtrd 2804	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → 𝑓 = (ℂ × {0}))
40		df-0p 25794	. . . . . . . . . . . . 13 ⊢ 0𝑝 = (ℂ × {0})
41	39, 40	eqtr4di 2822	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → 𝑓 = 0𝑝)
42	41	ex 417	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) → (𝑥 ∈ (◡𝑓 “ {0}) → 𝑓 = 0𝑝))
43	42	necon3ad 2977	. . . . . . . . . 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 4370	. . . . . . . 8 ⊢ ((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) → (◡𝑓 “ {0}) = ∅)
46	45	ex 417	. . . . . . 7 ⊢ (𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) → ((deg‘𝑓) = 0 → (◡𝑓 “ {0}) = ∅))
47		dgrcl 26355	. . . . . . . . 9 ⊢ (𝑓 ∈ (Poly‘ℂ) → (deg‘𝑓) ∈ ℕ0)
48		nn0ge0 12525	. . . . . . . . 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 9035	. . . . . . . . . . 11 ⊢ ∅ ∈ Fin
52	50, 51	eqeltrdi 2877	. . . . . . . . . 10 ⊢ ((◡𝑓 “ {0}) = ∅ → (◡𝑓 “ {0}) ∈ Fin)
53	52	biantrurd 541	. . . . . . . . 9 ⊢ ((◡𝑓 “ {0}) = ∅ → ((♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓) ↔ ((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓))))
54		fveq2 6879	. . . . . . . . . . 11 ⊢ ((◡𝑓 “ {0}) = ∅ → (♯‘(◡𝑓 “ {0})) = (♯‘∅))
55		hash0 14399	. . . . . . . . . . 11 ⊢ (♯‘∅) = 0
56	54, 55	eqtrdi 2820	. . . . . . . . . 10 ⊢ ((◡𝑓 “ {0}) = ∅ → (♯‘(◡𝑓 “ {0})) = 0)
57	56	breq1d 5120	. . . . . . . . 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 3087	. . . . 5 ⊢ ∀𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑓) = 0 → ((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓)))
62		fveqeq2 6888	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → ((deg‘𝑓) = 𝑑 ↔ (deg‘𝑔) = 𝑑))
63		cnveq 5857	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → ◡𝑓 = ◡𝑔)
64	63	imaeq1d 6059	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → (◡𝑓 “ {0}) = (◡𝑔 “ {0}))
65	64	eleq1d 2854	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → ((◡𝑓 “ {0}) ∈ Fin ↔ (◡𝑔 “ {0}) ∈ Fin))
66	64	fveq2d 6883	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → (♯‘(◡𝑓 “ {0})) = (♯‘(◡𝑔 “ {0})))
67		fveq2 6879	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → (deg‘𝑓) = (deg‘𝑔))
68	66, 67	breq12d 5123	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → ((♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓) ↔ (♯‘(◡𝑔 “ {0})) ≤ (deg‘𝑔)))
69	65, 68	anbi12d 643	. . . . . . . 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 3249	. . . . . 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 739	. . . . . . . . . . . 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 4314	. . . . . . . . . . 11 ⊢ ((◡𝑓 “ {0}) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (◡𝑓 “ {0}))
76		eqid 2769	. . . . . . . . . . . . . 14 ⊢ (◡𝑓 “ {0}) = (◡𝑓 “ {0})
77		simplll 786	. . . . . . . . . . . . . 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 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))))) → 𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}))
79		simplr 780	. . . . . . . . . . . . . 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 782	. . . . . . . . . . . . . 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 784	. . . . . . . . . . . . . 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 26433	. . . . . . . . . . . . 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 425	. . . . . . . . . . . 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 1960	. . . . . . . . . . 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 3050	. . . . . . . . 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 417	. . . . . . . 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 3171	. . . . . 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 12687	. . . 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 26322	. . . . 5 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ)
94	93	sseli 3941	. . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (Poly‘ℂ))
95		eldifsn 4755	. . . . 5 ⊢ (𝐹 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ↔ (𝐹 ∈ (Poly‘ℂ) ∧ 𝐹 ≠ 0𝑝))
96		fveqeq2 6888	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ((deg‘𝑓) = (deg‘𝐹) ↔ (deg‘𝐹) = (deg‘𝐹)))
97		cnveq 5857	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → ◡𝑓 = ◡𝐹)
98	97	imaeq1d 6059	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → (◡𝑓 “ {0}) = (◡𝐹 “ {0}))
99		fta1.1	. . . . . . . . . 10 ⊢ 𝑅 = (◡𝐹 “ {0})
100	98, 99	eqtr4di 2822	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (◡𝑓 “ {0}) = 𝑅)
101	100	eleq1d 2854	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((◡𝑓 “ {0}) ∈ Fin ↔ 𝑅 ∈ Fin))
102	100	fveq2d 6883	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (♯‘(◡𝑓 “ {0})) = (♯‘𝑅))
103		fveq2 6879	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (deg‘𝑓) = (deg‘𝐹))
104	102, 103	breq12d 5123	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓) ↔ (♯‘𝑅) ≤ (deg‘𝐹)))
105	101, 104	anbi12d 643	. . . . . . 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 3586	. . . . 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 591	. . 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)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567  ∃wex 1806   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085   ∖ cdif 3910  ∅c0 4294  {csn 4591   class class class wbr 5110   × cxp 5657  ^◡ccnv 5658   “ cima 5662   Fn wfn 6528  ⟶wf 6529  ‘cfv 6533  (class class class)co 7408  Fincfn 8939  ℂcc 11094  0cc0 11096  1c1 11097   + caddc 11099   ≤ cle 11240  ℕ₀cn0 12500  ♯chash 14362  0_𝑝c0p 25793  Polycply 26306  degcdgr 26309

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730  ax-inf2 9606  ax-cnex 11152  ax-resscn 11153  ax-1cn 11154  ax-icn 11155  ax-addcl 11156  ax-addrcl 11157  ax-mulcl 11158  ax-mulrcl 11159  ax-mulcom 11160  ax-addass 11161  ax-mulass 11162  ax-distr 11163  ax-i2m1 11164  ax-1ne0 11165  ax-1rid 11166  ax-rnegex 11167  ax-rrecex 11168  ax-cnre 11169  ax-pre-lttri 11170  ax-pre-lttrn 11171  ax-pre-ltadd 11172  ax-pre-mulgt0 11173  ax-pre-sup 11174

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-of 7672  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-oadd 8453  df-er 8690  df-map 8822  df-pm 8823  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9398  df-inf 9399  df-oi 9468  df-dju 9883  df-card 9921  df-pnf 11241  df-mnf 11242  df-xr 11243  df-ltxr 11244  df-le 11245  df-sub 11439  df-neg 11440  df-div 11868  df-nn 12230  df-2 12299  df-3 12300  df-n0 12501  df-xnn0 12574  df-z 12588  df-uz 12859  df-rp 13013  df-fz 13532  df-fzo 13679  df-fl 13821  df-seq 14034  df-exp 14094  df-hash 14363  df-cj 15146  df-re 15147  df-im 15148  df-sqrt 15282  df-abs 15283  df-clim 15535  df-rlim 15536  df-sum 15734  df-0p 25794  df-ply 26310  df-idp 26311  df-coe 26312  df-dgr 26313  df-quot 26417

This theorem is referenced by:  vieta1lem2  26437  vieta1  26438  plyexmo  26439  aannenlem1  26454  aalioulem2  26459  basellem4  27210  dchrfi  27381  sinnpoly  47510