Theorem fta1 25373

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 2738	. 2 ⊢ (deg‘𝐹) = (deg‘𝐹)
2		dgrcl 25299	. . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
3	2	adantr 480	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐹 ≠ 0𝑝) → (deg‘𝐹) ∈ ℕ0)
4		eqeq2 2750	. . . . . . 7 ⊢ (𝑥 = 0 → ((deg‘𝑓) = 𝑥 ↔ (deg‘𝑓) = 0))
5	4	imbi1d 341	. . . . . 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 3120	. . . . 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 2750	. . . . . . 7 ⊢ (𝑥 = 𝑑 → ((deg‘𝑓) = 𝑥 ↔ (deg‘𝑓) = 𝑑))
8	7	imbi1d 341	. . . . . 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 3120	. . . . 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 2750	. . . . . . 7 ⊢ (𝑥 = (𝑑 + 1) → ((deg‘𝑓) = 𝑥 ↔ (deg‘𝑓) = (𝑑 + 1)))
11	10	imbi1d 341	. . . . . 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 3120	. . . . 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 2750	. . . . . . 7 ⊢ (𝑥 = (deg‘𝐹) → ((deg‘𝑓) = 𝑥 ↔ (deg‘𝑓) = (deg‘𝐹)))
14	13	imbi1d 341	. . . . . 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 3120	. . . . 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 4720	. . . . . . . . . . 11 ⊢ (𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) → 𝑓 ≠ 0𝑝)
17	16	adantr 480	. . . . . . . . . 10 ⊢ ((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) → 𝑓 ≠ 0𝑝)
18		simplr 765	. . . . . . . . . . . . . . 15 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → (deg‘𝑓) = 0)
19		eldifi 4057	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) → 𝑓 ∈ (Poly‘ℂ))
20	19	ad2antrr 722	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → 𝑓 ∈ (Poly‘ℂ))
21		0dgrb 25312	. . . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . 14 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → 𝑓 = (ℂ × {(𝑓‘0)}))
24	23	fveq1d 6758	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → (𝑓‘𝑥) = ((ℂ × {(𝑓‘0)})‘𝑥))
25	19	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) → 𝑓 ∈ (Poly‘ℂ))
26		plyf 25264	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 ∈ (Poly‘ℂ) → 𝑓:ℂ⟶ℂ)
27		ffn 6584	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓:ℂ⟶ℂ → 𝑓 Fn ℂ)
28		fniniseg 6919	. . . . . . . . . . . . . . . . . . . 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 476	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → (𝑥 ∈ ℂ ∧ (𝑓‘𝑥) = 0))
31	30	simprd 495	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → (𝑓‘𝑥) = 0)
32	30	simpld 494	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → 𝑥 ∈ ℂ)
33		fvex 6769	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓‘0) ∈ V
34	33	fvconst2 7061	. . . . . . . . . . . . . . . . . 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 2787	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → (𝑓‘0) = 0)
37	36	sneqd 4570	. . . . . . . . . . . . . . 15 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → {(𝑓‘0)} = {0})
38	37	xpeq2d 5610	. . . . . . . . . . . . . 14 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → (ℂ × {(𝑓‘0)}) = (ℂ × {0}))
39	23, 38	eqtrd 2778	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → 𝑓 = (ℂ × {0}))
40		df-0p 24739	. . . . . . . . . . . . 13 ⊢ 0𝑝 = (ℂ × {0})
41	39, 40	eqtr4di 2797	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → 𝑓 = 0𝑝)
42	41	ex 412	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) → (𝑥 ∈ (◡𝑓 “ {0}) → 𝑓 = 0𝑝))
43	42	necon3ad 2955	. . . . . . . . . 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 4335	. . . . . . . 8 ⊢ ((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) → (◡𝑓 “ {0}) = ∅)
46	45	ex 412	. . . . . . 7 ⊢ (𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) → ((deg‘𝑓) = 0 → (◡𝑓 “ {0}) = ∅))
47		dgrcl 25299	. . . . . . . . 9 ⊢ (𝑓 ∈ (Poly‘ℂ) → (deg‘𝑓) ∈ ℕ0)
48		nn0ge0 12188	. . . . . . . . 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 8916	. . . . . . . . . . 11 ⊢ ∅ ∈ Fin
52	50, 51	eqeltrdi 2847	. . . . . . . . . 10 ⊢ ((◡𝑓 “ {0}) = ∅ → (◡𝑓 “ {0}) ∈ Fin)
53	52	biantrurd 532	. . . . . . . . 9 ⊢ ((◡𝑓 “ {0}) = ∅ → ((♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓) ↔ ((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓))))
54		fveq2 6756	. . . . . . . . . . 11 ⊢ ((◡𝑓 “ {0}) = ∅ → (♯‘(◡𝑓 “ {0})) = (♯‘∅))
55		hash0 14010	. . . . . . . . . . 11 ⊢ (♯‘∅) = 0
56	54, 55	eqtrdi 2795	. . . . . . . . . 10 ⊢ ((◡𝑓 “ {0}) = ∅ → (♯‘(◡𝑓 “ {0})) = 0)
57	56	breq1d 5080	. . . . . . . . 9 ⊢ ((◡𝑓 “ {0}) = ∅ → ((♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓) ↔ 0 ≤ (deg‘𝑓)))
58	53, 57	bitr3d 280	. . . . . . . 8 ⊢ ((◡𝑓 “ {0}) = ∅ → (((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓)) ↔ 0 ≤ (deg‘𝑓)))
59	49, 58	syl5ibrcom 246	. . . . . . 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 3073	. . . . 5 ⊢ ∀𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑓) = 0 → ((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓)))
62		fveqeq2 6765	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → ((deg‘𝑓) = 𝑑 ↔ (deg‘𝑔) = 𝑑))
63		cnveq 5771	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → ◡𝑓 = ◡𝑔)
64	63	imaeq1d 5957	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → (◡𝑓 “ {0}) = (◡𝑔 “ {0}))
65	64	eleq1d 2823	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → ((◡𝑓 “ {0}) ∈ Fin ↔ (◡𝑔 “ {0}) ∈ Fin))
66	64	fveq2d 6760	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → (♯‘(◡𝑓 “ {0})) = (♯‘(◡𝑔 “ {0})))
67		fveq2 6756	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → (deg‘𝑓) = (deg‘𝑔))
68	66, 67	breq12d 5083	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → ((♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓) ↔ (♯‘(◡𝑔 “ {0})) ≤ (deg‘𝑔)))
69	65, 68	anbi12d 630	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → (((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓)) ↔ ((◡𝑔 “ {0}) ∈ Fin ∧ (♯‘(◡𝑔 “ {0})) ≤ (deg‘𝑔))))
70	62, 69	imbi12d 344	. . . . . . 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 3372	. . . . . 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 723	. . . . . . . . . . . 12 ⊢ (((𝑑 ∈ ℕ0 ∧ 𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})) ∧ (deg‘𝑓) = (𝑑 + 1)) → 0 ≤ (deg‘𝑓))
73	72, 58	syl5ibrcom 246	. . . . . . . . . . 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 4277	. . . . . . . . . . 11 ⊢ ((◡𝑓 “ {0}) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (◡𝑓 “ {0}))
76		eqid 2738	. . . . . . . . . . . . . 14 ⊢ (◡𝑓 “ {0}) = (◡𝑓 “ {0})
77		simplll 771	. . . . . . . . . . . . . 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 772	. . . . . . . . . . . . . 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 765	. . . . . . . . . . . . . 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 767	. . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . 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 25372	. . . . . . . . . . . . 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 420	. . . . . . . . . . . 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 1937	. . . . . . . . . . 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 241	. . . . . . . . . 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 3030	. . . . . . . . 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 412	. . . . . . . 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 3112	. . . . . 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 241	. . . . 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 12345	. . . 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 25266	. . . . 5 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ)
94	93	sseli 3913	. . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (Poly‘ℂ))
95		eldifsn 4717	. . . . 5 ⊢ (𝐹 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ↔ (𝐹 ∈ (Poly‘ℂ) ∧ 𝐹 ≠ 0𝑝))
96		fveqeq2 6765	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ((deg‘𝑓) = (deg‘𝐹) ↔ (deg‘𝐹) = (deg‘𝐹)))
97		cnveq 5771	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → ◡𝑓 = ◡𝐹)
98	97	imaeq1d 5957	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → (◡𝑓 “ {0}) = (◡𝐹 “ {0}))
99		fta1.1	. . . . . . . . . 10 ⊢ 𝑅 = (◡𝐹 “ {0})
100	98, 99	eqtr4di 2797	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (◡𝑓 “ {0}) = 𝑅)
101	100	eleq1d 2823	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((◡𝑓 “ {0}) ∈ Fin ↔ 𝑅 ∈ Fin))
102	100	fveq2d 6760	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (♯‘(◡𝑓 “ {0})) = (♯‘𝑅))
103		fveq2 6756	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (deg‘𝑓) = (deg‘𝐹))
104	102, 103	breq12d 5083	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓) ↔ (♯‘𝑅) ≤ (deg‘𝐹)))
105	101, 104	anbi12d 630	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓)) ↔ (𝑅 ∈ Fin ∧ (♯‘𝑅) ≤ (deg‘𝐹))))
106	96, 105	imbi12d 344	. . . . . 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 3547	. . . . 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 234	. . . 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 579	. . 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 205   ∧ wa 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063   ∖ cdif 3880  ∅c0 4253  {csn 4558   class class class wbr 5070   × cxp 5578  ^◡ccnv 5579   “ cima 5583   Fn wfn 6413  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255  Fincfn 8691  ℂcc 10800  0cc0 10802  1c1 10803   + caddc 10805   ≤ cle 10941  ℕ₀cn0 12163  ♯chash 13972  0_𝑝c0p 24738  Polycply 25250  degcdgr 25253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-pre-sup 10880

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-of 7511  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-oadd 8271  df-er 8456  df-map 8575  df-pm 8576  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-sup 9131  df-inf 9132  df-oi 9199  df-dju 9590  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-nn 11904  df-2 11966  df-3 11967  df-n0 12164  df-xnn0 12236  df-z 12250  df-uz 12512  df-rp 12660  df-fz 13169  df-fzo 13312  df-fl 13440  df-seq 13650  df-exp 13711  df-hash 13973  df-cj 14738  df-re 14739  df-im 14740  df-sqrt 14874  df-abs 14875  df-clim 15125  df-rlim 15126  df-sum 15326  df-0p 24739  df-ply 25254  df-idp 25255  df-coe 25256  df-dgr 25257  df-quot 25356

This theorem is referenced by:  vieta1lem2  25376  vieta1  25377  plyexmo  25378  aannenlem1  25393  aalioulem2  25398  basellem4  26138  dchrfi  26308