Theorem fta1 24900

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 2824	. 2 ⊢ (deg‘𝐹) = (deg‘𝐹)
2		dgrcl 24826	. . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
3	2	adantr 483	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐹 ≠ 0𝑝) → (deg‘𝐹) ∈ ℕ0)
4		eqeq2 2836	. . . . . . 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 3200	. . . . 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 2836	. . . . . . 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 3200	. . . . 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 2836	. . . . . . 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 3200	. . . . 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 2836	. . . . . . 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 3200	. . . . 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 4725	. . . . . . . . . . 11 ⊢ (𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) → 𝑓 ≠ 0𝑝)
17	16	adantr 483	. . . . . . . . . 10 ⊢ ((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) → 𝑓 ≠ 0𝑝)
18		simplr 767	. . . . . . . . . . . . . . 15 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → (deg‘𝑓) = 0)
19		eldifi 4106	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) → 𝑓 ∈ (Poly‘ℂ))
20	19	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → 𝑓 ∈ (Poly‘ℂ))
21		0dgrb 24839	. . . . . . . . . . . . . . . 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 234	. . . . . . . . . . . . . 14 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → 𝑓 = (ℂ × {(𝑓‘0)}))
24	23	fveq1d 6675	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → (𝑓‘𝑥) = ((ℂ × {(𝑓‘0)})‘𝑥))
25	19	adantr 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) → 𝑓 ∈ (Poly‘ℂ))
26		plyf 24791	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 ∈ (Poly‘ℂ) → 𝑓:ℂ⟶ℂ)
27		ffn 6517	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓:ℂ⟶ℂ → 𝑓 Fn ℂ)
28		fniniseg 6833	. . . . . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → (𝑥 ∈ ℂ ∧ (𝑓‘𝑥) = 0))
31	30	simprd 498	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → (𝑓‘𝑥) = 0)
32	30	simpld 497	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → 𝑥 ∈ ℂ)
33		fvex 6686	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓‘0) ∈ V
34	33	fvconst2 6969	. . . . . . . . . . . . . . . . . 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 2868	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → (𝑓‘0) = 0)
37	36	sneqd 4582	. . . . . . . . . . . . . . 15 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → {(𝑓‘0)} = {0})
38	37	xpeq2d 5588	. . . . . . . . . . . . . 14 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → (ℂ × {(𝑓‘0)}) = (ℂ × {0}))
39	23, 38	eqtrd 2859	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → 𝑓 = (ℂ × {0}))
40		df-0p 24274	. . . . . . . . . . . . 13 ⊢ 0𝑝 = (ℂ × {0})
41	39, 40	syl6eqr 2877	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (◡𝑓 “ {0})) → 𝑓 = 0𝑝)
42	41	ex 415	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) → (𝑥 ∈ (◡𝑓 “ {0}) → 𝑓 = 0𝑝))
43	42	necon3ad 3032	. . . . . . . . . 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 4360	. . . . . . . 8 ⊢ ((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) → (◡𝑓 “ {0}) = ∅)
46	45	ex 415	. . . . . . 7 ⊢ (𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) → ((deg‘𝑓) = 0 → (◡𝑓 “ {0}) = ∅))
47		dgrcl 24826	. . . . . . . . 9 ⊢ (𝑓 ∈ (Poly‘ℂ) → (deg‘𝑓) ∈ ℕ0)
48		nn0ge0 11925	. . . . . . . . 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 8749	. . . . . . . . . . 11 ⊢ ∅ ∈ Fin
52	50, 51	eqeltrdi 2924	. . . . . . . . . 10 ⊢ ((◡𝑓 “ {0}) = ∅ → (◡𝑓 “ {0}) ∈ Fin)
53	52	biantrurd 535	. . . . . . . . 9 ⊢ ((◡𝑓 “ {0}) = ∅ → ((♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓) ↔ ((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓))))
54		fveq2 6673	. . . . . . . . . . 11 ⊢ ((◡𝑓 “ {0}) = ∅ → (♯‘(◡𝑓 “ {0})) = (♯‘∅))
55		hash0 13731	. . . . . . . . . . 11 ⊢ (♯‘∅) = 0
56	54, 55	syl6eq 2875	. . . . . . . . . 10 ⊢ ((◡𝑓 “ {0}) = ∅ → (♯‘(◡𝑓 “ {0})) = 0)
57	56	breq1d 5079	. . . . . . . . 9 ⊢ ((◡𝑓 “ {0}) = ∅ → ((♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓) ↔ 0 ≤ (deg‘𝑓)))
58	53, 57	bitr3d 283	. . . . . . . 8 ⊢ ((◡𝑓 “ {0}) = ∅ → (((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓)) ↔ 0 ≤ (deg‘𝑓)))
59	49, 58	syl5ibrcom 249	. . . . . . 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 3151	. . . . 5 ⊢ ∀𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑓) = 0 → ((◡𝑓 “ {0}) ∈ Fin ∧ (♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓)))
62		fveqeq2 6682	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → ((deg‘𝑓) = 𝑑 ↔ (deg‘𝑔) = 𝑑))
63		cnveq 5747	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → ◡𝑓 = ◡𝑔)
64	63	imaeq1d 5931	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → (◡𝑓 “ {0}) = (◡𝑔 “ {0}))
65	64	eleq1d 2900	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → ((◡𝑓 “ {0}) ∈ Fin ↔ (◡𝑔 “ {0}) ∈ Fin))
66	64	fveq2d 6677	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → (♯‘(◡𝑓 “ {0})) = (♯‘(◡𝑔 “ {0})))
67		fveq2 6673	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → (deg‘𝑓) = (deg‘𝑔))
68	66, 67	breq12d 5082	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → ((♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓) ↔ (♯‘(◡𝑔 “ {0})) ≤ (deg‘𝑔)))
69	65, 68	anbi12d 632	. . . . . . . 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 3452	. . . . . 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 725	. . . . . . . . . . . 12 ⊢ (((𝑑 ∈ ℕ0 ∧ 𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})) ∧ (deg‘𝑓) = (𝑑 + 1)) → 0 ≤ (deg‘𝑓))
73	72, 58	syl5ibrcom 249	. . . . . . . . . . 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 4313	. . . . . . . . . . 11 ⊢ ((◡𝑓 “ {0}) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (◡𝑓 “ {0}))
76		eqid 2824	. . . . . . . . . . . . . 14 ⊢ (◡𝑓 “ {0}) = (◡𝑓 “ {0})
77		simplll 773	. . . . . . . . . . . . . 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 774	. . . . . . . . . . . . . 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 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))))) → (deg‘𝑓) = (𝑑 + 1))
80		simprl 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))))) → 𝑥 ∈ (◡𝑓 “ {0}))
81		simprr 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))))) → ∀𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑔) = 𝑑 → ((◡𝑔 “ {0}) ∈ Fin ∧ (♯‘(◡𝑔 “ {0})) ≤ (deg‘𝑔))))
82	76, 77, 78, 79, 80, 81	fta1lem 24899	. . . . . . . . . . . . 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 423	. . . . . . . . . . . 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 1933	. . . . . . . . . . 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 244	. . . . . . . . . 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 3106	. . . . . . . . 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 415	. . . . . . . 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 3192	. . . . . 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 244	. . . . 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 12080	. . . 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 24793	. . . . 5 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ)
94	93	sseli 3966	. . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (Poly‘ℂ))
95		eldifsn 4722	. . . . 5 ⊢ (𝐹 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ↔ (𝐹 ∈ (Poly‘ℂ) ∧ 𝐹 ≠ 0𝑝))
96		fveqeq2 6682	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ((deg‘𝑓) = (deg‘𝐹) ↔ (deg‘𝐹) = (deg‘𝐹)))
97		cnveq 5747	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → ◡𝑓 = ◡𝐹)
98	97	imaeq1d 5931	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → (◡𝑓 “ {0}) = (◡𝐹 “ {0}))
99		fta1.1	. . . . . . . . . 10 ⊢ 𝑅 = (◡𝐹 “ {0})
100	98, 99	syl6eqr 2877	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (◡𝑓 “ {0}) = 𝑅)
101	100	eleq1d 2900	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((◡𝑓 “ {0}) ∈ Fin ↔ 𝑅 ∈ Fin))
102	100	fveq2d 6677	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (♯‘(◡𝑓 “ {0})) = (♯‘𝑅))
103		fveq2 6673	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (deg‘𝑓) = (deg‘𝐹))
104	102, 103	breq12d 5082	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((♯‘(◡𝑓 “ {0})) ≤ (deg‘𝑓) ↔ (♯‘𝑅) ≤ (deg‘𝐹)))
105	101, 104	anbi12d 632	. . . . . . 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 3621	. . . . 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 237	. . . 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 582	. . 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 208   ∧ wa 398   = wceq 1536  ∃wex 1779   ∈ wcel 2113   ≠ wne 3019  ∀wral 3141   ∖ cdif 3936  ∅c0 4294  {csn 4570   class class class wbr 5069   × cxp 5556  ^◡ccnv 5557   “ cima 5561   Fn wfn 6353  ⟶wf 6354  ‘cfv 6358  (class class class)co 7159  Fincfn 8512  ℂcc 10538  0cc0 10540  1c1 10541   + caddc 10543   ≤ cle 10679  ℕ₀cn0 11900  ♯chash 13693  0_𝑝c0p 24273  Polycply 24777  degcdgr 24780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464  ax-inf2 9107  ax-cnex 10596  ax-resscn 10597  ax-1cn 10598  ax-icn 10599  ax-addcl 10600  ax-addrcl 10601  ax-mulcl 10602  ax-mulrcl 10603  ax-mulcom 10604  ax-addass 10605  ax-mulass 10606  ax-distr 10607  ax-i2m1 10608  ax-1ne0 10609  ax-1rid 10610  ax-rnegex 10611  ax-rrecex 10612  ax-cnre 10613  ax-pre-lttri 10614  ax-pre-lttrn 10615  ax-pre-ltadd 10616  ax-pre-mulgt0 10617  ax-pre-sup 10618  ax-addf 10619

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-fal 1549  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-nel 3127  df-ral 3146  df-rex 3147  df-reu 3148  df-rmo 3149  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-int 4880  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-se 5518  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-isom 6367  df-riota 7117  df-ov 7162  df-oprab 7163  df-mpo 7164  df-of 7412  df-om 7584  df-1st 7692  df-2nd 7693  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-1o 8105  df-oadd 8109  df-er 8292  df-map 8411  df-pm 8412  df-en 8513  df-dom 8514  df-sdom 8515  df-fin 8516  df-sup 8909  df-inf 8910  df-oi 8977  df-dju 9333  df-card 9371  df-pnf 10680  df-mnf 10681  df-xr 10682  df-ltxr 10683  df-le 10684  df-sub 10875  df-neg 10876  df-div 11301  df-nn 11642  df-2 11703  df-3 11704  df-n0 11901  df-xnn0 11971  df-z 11985  df-uz 12247  df-rp 12393  df-fz 12896  df-fzo 13037  df-fl 13165  df-seq 13373  df-exp 13433  df-hash 13694  df-cj 14461  df-re 14462  df-im 14463  df-sqrt 14597  df-abs 14598  df-clim 14848  df-rlim 14849  df-sum 15046  df-0p 24274  df-ply 24781  df-idp 24782  df-coe 24783  df-dgr 24784  df-quot 24883

This theorem is referenced by:  vieta1lem2  24903  vieta1  24904  plyexmo  24905  aannenlem1  24920  aalioulem2  24925  basellem4  25664  dchrfi  25834