Theorem fta1lem 24895

Description: Lemma for fta1 24896. (Contributed by Mario Carneiro, 26-Jul-2014.)

Hypotheses

Ref	Expression
fta1.1	⊢ 𝑅 = (◡𝐹 “ {0})
fta1.2	⊢ (𝜑 → 𝐷 ∈ ℕ0)
fta1.3	⊢ (𝜑 → 𝐹 ∈ ((Poly‘ℂ) ∖ {0𝑝}))
fta1.4	⊢ (𝜑 → (deg‘𝐹) = (𝐷 + 1))
fta1.5	⊢ (𝜑 → 𝐴 ∈ (◡𝐹 “ {0}))
fta1.6	⊢ (𝜑 → ∀𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑔) = 𝐷 → ((◡𝑔 “ {0}) ∈ Fin ∧ (♯‘(◡𝑔 “ {0})) ≤ (deg‘𝑔))))

Assertion

Ref	Expression
fta1lem	⊢ (𝜑 → (𝑅 ∈ Fin ∧ (♯‘𝑅) ≤ (deg‘𝐹)))

Distinct variable groups:   𝐴,𝑔   𝐷,𝑔   𝑔,𝐹

Allowed substitution hints:   𝜑(𝑔)   𝑅(𝑔)

Proof of Theorem fta1lem

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fta1.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ ((Poly‘ℂ) ∖ {0𝑝}))
2		eldifsn 4718	. . . . . . . . . 10 ⊢ (𝐹 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ↔ (𝐹 ∈ (Poly‘ℂ) ∧ 𝐹 ≠ 0𝑝))
3	1, 2	sylib 220	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 ∈ (Poly‘ℂ) ∧ 𝐹 ≠ 0𝑝))
4	3	simpld 497	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (Poly‘ℂ))
5		fta1.5	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ (◡𝐹 “ {0}))
6		plyf 24787	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Poly‘ℂ) → 𝐹:ℂ⟶ℂ)
7		ffn 6513	. . . . . . . . . . 11 ⊢ (𝐹:ℂ⟶ℂ → 𝐹 Fn ℂ)
8		fniniseg 6829	. . . . . . . . . . 11 ⊢ (𝐹 Fn ℂ → (𝐴 ∈ (◡𝐹 “ {0}) ↔ (𝐴 ∈ ℂ ∧ (𝐹‘𝐴) = 0)))
9	4, 6, 7, 8	4syl 19	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ∈ (◡𝐹 “ {0}) ↔ (𝐴 ∈ ℂ ∧ (𝐹‘𝐴) = 0)))
10	5, 9	mpbid 234	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ∈ ℂ ∧ (𝐹‘𝐴) = 0))
11	10	simpld 497	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℂ)
12	10	simprd 498	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝐴) = 0)
13		eqid 2821	. . . . . . . . 9 ⊢ (Xp ∘f − (ℂ × {𝐴})) = (Xp ∘f − (ℂ × {𝐴}))
14	13	facth 24894	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐴 ∈ ℂ ∧ (𝐹‘𝐴) = 0) → 𝐹 = ((Xp ∘f − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xp ∘f − (ℂ × {𝐴})))))
15	4, 11, 12, 14	syl3anc 1367	. . . . . . 7 ⊢ (𝜑 → 𝐹 = ((Xp ∘f − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xp ∘f − (ℂ × {𝐴})))))
16	15	cnveqd 5745	. . . . . 6 ⊢ (𝜑 → ◡𝐹 = ◡((Xp ∘f − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xp ∘f − (ℂ × {𝐴})))))
17	16	imaeq1d 5927	. . . . 5 ⊢ (𝜑 → (◡𝐹 “ {0}) = (◡((Xp ∘f − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xp ∘f − (ℂ × {𝐴})))) “ {0}))
18		cnex 10617	. . . . . . 7 ⊢ ℂ ∈ V
19	18	a1i 11	. . . . . 6 ⊢ (𝜑 → ℂ ∈ V)
20		ssid 3988	. . . . . . . . 9 ⊢ ℂ ⊆ ℂ
21		ax-1cn 10594	. . . . . . . . 9 ⊢ 1 ∈ ℂ
22		plyid 24798	. . . . . . . . 9 ⊢ ((ℂ ⊆ ℂ ∧ 1 ∈ ℂ) → Xp ∈ (Poly‘ℂ))
23	20, 21, 22	mp2an 690	. . . . . . . 8 ⊢ Xp ∈ (Poly‘ℂ)
24		plyconst 24795	. . . . . . . . 9 ⊢ ((ℂ ⊆ ℂ ∧ 𝐴 ∈ ℂ) → (ℂ × {𝐴}) ∈ (Poly‘ℂ))
25	20, 11, 24	sylancr 589	. . . . . . . 8 ⊢ (𝜑 → (ℂ × {𝐴}) ∈ (Poly‘ℂ))
26		plysubcl 24811	. . . . . . . 8 ⊢ ((Xp ∈ (Poly‘ℂ) ∧ (ℂ × {𝐴}) ∈ (Poly‘ℂ)) → (Xp ∘f − (ℂ × {𝐴})) ∈ (Poly‘ℂ))
27	23, 25, 26	sylancr 589	. . . . . . 7 ⊢ (𝜑 → (Xp ∘f − (ℂ × {𝐴})) ∈ (Poly‘ℂ))
28		plyf 24787	. . . . . . 7 ⊢ ((Xp ∘f − (ℂ × {𝐴})) ∈ (Poly‘ℂ) → (Xp ∘f − (ℂ × {𝐴})):ℂ⟶ℂ)
29	27, 28	syl 17	. . . . . 6 ⊢ (𝜑 → (Xp ∘f − (ℂ × {𝐴})):ℂ⟶ℂ)
30	13	plyremlem 24892	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℂ → ((Xp ∘f − (ℂ × {𝐴})) ∈ (Poly‘ℂ) ∧ (deg‘(Xp ∘f − (ℂ × {𝐴}))) = 1 ∧ (◡(Xp ∘f − (ℂ × {𝐴})) “ {0}) = {𝐴}))
31	11, 30	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ((Xp ∘f − (ℂ × {𝐴})) ∈ (Poly‘ℂ) ∧ (deg‘(Xp ∘f − (ℂ × {𝐴}))) = 1 ∧ (◡(Xp ∘f − (ℂ × {𝐴})) “ {0}) = {𝐴}))
32	31	simp2d 1139	. . . . . . . . . 10 ⊢ (𝜑 → (deg‘(Xp ∘f − (ℂ × {𝐴}))) = 1)
33		ax-1ne0 10605	. . . . . . . . . . 11 ⊢ 1 ≠ 0
34	33	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 1 ≠ 0)
35	32, 34	eqnetrd 3083	. . . . . . . . 9 ⊢ (𝜑 → (deg‘(Xp ∘f − (ℂ × {𝐴}))) ≠ 0)
36		fveq2 6669	. . . . . . . . . . 11 ⊢ ((Xp ∘f − (ℂ × {𝐴})) = 0𝑝 → (deg‘(Xp ∘f − (ℂ × {𝐴}))) = (deg‘0𝑝))
37		dgr0 24851	. . . . . . . . . . 11 ⊢ (deg‘0𝑝) = 0
38	36, 37	syl6eq 2872	. . . . . . . . . 10 ⊢ ((Xp ∘f − (ℂ × {𝐴})) = 0𝑝 → (deg‘(Xp ∘f − (ℂ × {𝐴}))) = 0)
39	38	necon3i 3048	. . . . . . . . 9 ⊢ ((deg‘(Xp ∘f − (ℂ × {𝐴}))) ≠ 0 → (Xp ∘f − (ℂ × {𝐴})) ≠ 0𝑝)
40	35, 39	syl 17	. . . . . . . 8 ⊢ (𝜑 → (Xp ∘f − (ℂ × {𝐴})) ≠ 0𝑝)
41		quotcl2 24890	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘ℂ) ∧ (Xp ∘f − (ℂ × {𝐴})) ∈ (Poly‘ℂ) ∧ (Xp ∘f − (ℂ × {𝐴})) ≠ 0𝑝) → (𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) ∈ (Poly‘ℂ))
42	4, 27, 40, 41	syl3anc 1367	. . . . . . 7 ⊢ (𝜑 → (𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) ∈ (Poly‘ℂ))
43		plyf 24787	. . . . . . 7 ⊢ ((𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) ∈ (Poly‘ℂ) → (𝐹 quot (Xp ∘f − (ℂ × {𝐴}))):ℂ⟶ℂ)
44	42, 43	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐹 quot (Xp ∘f − (ℂ × {𝐴}))):ℂ⟶ℂ)
45		ofmulrt 24870	. . . . . 6 ⊢ ((ℂ ∈ V ∧ (Xp ∘f − (ℂ × {𝐴})):ℂ⟶ℂ ∧ (𝐹 quot (Xp ∘f − (ℂ × {𝐴}))):ℂ⟶ℂ) → (◡((Xp ∘f − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xp ∘f − (ℂ × {𝐴})))) “ {0}) = ((◡(Xp ∘f − (ℂ × {𝐴})) “ {0}) ∪ (◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})))
46	19, 29, 44, 45	syl3anc 1367	. . . . 5 ⊢ (𝜑 → (◡((Xp ∘f − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xp ∘f − (ℂ × {𝐴})))) “ {0}) = ((◡(Xp ∘f − (ℂ × {𝐴})) “ {0}) ∪ (◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})))
47	31	simp3d 1140	. . . . . 6 ⊢ (𝜑 → (◡(Xp ∘f − (ℂ × {𝐴})) “ {0}) = {𝐴})
48	47	uneq1d 4137	. . . . 5 ⊢ (𝜑 → ((◡(Xp ∘f − (ℂ × {𝐴})) “ {0}) ∪ (◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})) = ({𝐴} ∪ (◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})))
49	17, 46, 48	3eqtrd 2860	. . . 4 ⊢ (𝜑 → (◡𝐹 “ {0}) = ({𝐴} ∪ (◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})))
50		fta1.1	. . . 4 ⊢ 𝑅 = (◡𝐹 “ {0})
51		uncom 4128	. . . 4 ⊢ ((◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴}) = ({𝐴} ∪ (◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}))
52	49, 50, 51	3eqtr4g 2881	. . 3 ⊢ (𝜑 → 𝑅 = ((◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴}))
53	21	a1i 11	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℂ)
54		dgrcl 24822	. . . . . . . . 9 ⊢ ((𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) ∈ (Poly‘ℂ) → (deg‘(𝐹 quot (Xp ∘f − (ℂ × {𝐴})))) ∈ ℕ0)
55	42, 54	syl 17	. . . . . . . 8 ⊢ (𝜑 → (deg‘(𝐹 quot (Xp ∘f − (ℂ × {𝐴})))) ∈ ℕ0)
56	55	nn0cnd 11956	. . . . . . 7 ⊢ (𝜑 → (deg‘(𝐹 quot (Xp ∘f − (ℂ × {𝐴})))) ∈ ℂ)
57		fta1.2	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ ℕ0)
58	57	nn0cnd 11956	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ ℂ)
59		addcom 10825	. . . . . . . . 9 ⊢ ((1 ∈ ℂ ∧ 𝐷 ∈ ℂ) → (1 + 𝐷) = (𝐷 + 1))
60	21, 58, 59	sylancr 589	. . . . . . . 8 ⊢ (𝜑 → (1 + 𝐷) = (𝐷 + 1))
61	15	fveq2d 6673	. . . . . . . . 9 ⊢ (𝜑 → (deg‘𝐹) = (deg‘((Xp ∘f − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xp ∘f − (ℂ × {𝐴}))))))
62		fta1.4	. . . . . . . . 9 ⊢ (𝜑 → (deg‘𝐹) = (𝐷 + 1))
63	3	simprd 498	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ≠ 0𝑝)
64	15	eqcomd 2827	. . . . . . . . . . . 12 ⊢ (𝜑 → ((Xp ∘f − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xp ∘f − (ℂ × {𝐴})))) = 𝐹)
65		0cnd 10633	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ∈ ℂ)
66		mul01 10818	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → (𝑥 · 0) = 0)
67	66	adantl 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝑥 · 0) = 0)
68	19, 29, 65, 65, 67	caofid1 7438	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((Xp ∘f − (ℂ × {𝐴})) ∘f · (ℂ × {0})) = (ℂ × {0}))
69		df-0p 24270	. . . . . . . . . . . . . 14 ⊢ 0𝑝 = (ℂ × {0})
70	69	oveq2i 7166	. . . . . . . . . . . . 13 ⊢ ((Xp ∘f − (ℂ × {𝐴})) ∘f · 0𝑝) = ((Xp ∘f − (ℂ × {𝐴})) ∘f · (ℂ × {0}))
71	68, 70, 69	3eqtr4g 2881	. . . . . . . . . . . 12 ⊢ (𝜑 → ((Xp ∘f − (ℂ × {𝐴})) ∘f · 0𝑝) = 0𝑝)
72	63, 64, 71	3netr4d 3093	. . . . . . . . . . 11 ⊢ (𝜑 → ((Xp ∘f − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xp ∘f − (ℂ × {𝐴})))) ≠ ((Xp ∘f − (ℂ × {𝐴})) ∘f · 0𝑝))
73		oveq2 7163	. . . . . . . . . . . 12 ⊢ ((𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) = 0𝑝 → ((Xp ∘f − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xp ∘f − (ℂ × {𝐴})))) = ((Xp ∘f − (ℂ × {𝐴})) ∘f · 0𝑝))
74	73	necon3i 3048	. . . . . . . . . . 11 ⊢ (((Xp ∘f − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xp ∘f − (ℂ × {𝐴})))) ≠ ((Xp ∘f − (ℂ × {𝐴})) ∘f · 0𝑝) → (𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) ≠ 0𝑝)
75	72, 74	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) ≠ 0𝑝)
76		eqid 2821	. . . . . . . . . . 11 ⊢ (deg‘(Xp ∘f − (ℂ × {𝐴}))) = (deg‘(Xp ∘f − (ℂ × {𝐴})))
77		eqid 2821	. . . . . . . . . . 11 ⊢ (deg‘(𝐹 quot (Xp ∘f − (ℂ × {𝐴})))) = (deg‘(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))))
78	76, 77	dgrmul 24859	. . . . . . . . . 10 ⊢ ((((Xp ∘f − (ℂ × {𝐴})) ∈ (Poly‘ℂ) ∧ (Xp ∘f − (ℂ × {𝐴})) ≠ 0𝑝) ∧ ((𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) ∈ (Poly‘ℂ) ∧ (𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) ≠ 0𝑝)) → (deg‘((Xp ∘f − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xp ∘f − (ℂ × {𝐴}))))) = ((deg‘(Xp ∘f − (ℂ × {𝐴}))) + (deg‘(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))))))
79	27, 40, 42, 75, 78	syl22anc 836	. . . . . . . . 9 ⊢ (𝜑 → (deg‘((Xp ∘f − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xp ∘f − (ℂ × {𝐴}))))) = ((deg‘(Xp ∘f − (ℂ × {𝐴}))) + (deg‘(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))))))
80	61, 62, 79	3eqtr3d 2864	. . . . . . . 8 ⊢ (𝜑 → (𝐷 + 1) = ((deg‘(Xp ∘f − (ℂ × {𝐴}))) + (deg‘(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))))))
81	32	oveq1d 7170	. . . . . . . 8 ⊢ (𝜑 → ((deg‘(Xp ∘f − (ℂ × {𝐴}))) + (deg‘(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))))) = (1 + (deg‘(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))))))
82	60, 80, 81	3eqtrrd 2861	. . . . . . 7 ⊢ (𝜑 → (1 + (deg‘(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))))) = (1 + 𝐷))
83	53, 56, 58, 82	addcanad 10844	. . . . . 6 ⊢ (𝜑 → (deg‘(𝐹 quot (Xp ∘f − (ℂ × {𝐴})))) = 𝐷)
84		fveqeq2 6678	. . . . . . . 8 ⊢ (𝑔 = (𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) → ((deg‘𝑔) = 𝐷 ↔ (deg‘(𝐹 quot (Xp ∘f − (ℂ × {𝐴})))) = 𝐷))
85		cnveq 5743	. . . . . . . . . . 11 ⊢ (𝑔 = (𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) → ◡𝑔 = ◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))))
86	85	imaeq1d 5927	. . . . . . . . . 10 ⊢ (𝑔 = (𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) → (◡𝑔 “ {0}) = (◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}))
87	86	eleq1d 2897	. . . . . . . . 9 ⊢ (𝑔 = (𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) → ((◡𝑔 “ {0}) ∈ Fin ↔ (◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}) ∈ Fin))
88	86	fveq2d 6673	. . . . . . . . . 10 ⊢ (𝑔 = (𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) → (♯‘(◡𝑔 “ {0})) = (♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})))
89		fveq2 6669	. . . . . . . . . 10 ⊢ (𝑔 = (𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) → (deg‘𝑔) = (deg‘(𝐹 quot (Xp ∘f − (ℂ × {𝐴})))))
90	88, 89	breq12d 5078	. . . . . . . . 9 ⊢ (𝑔 = (𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) → ((♯‘(◡𝑔 “ {0})) ≤ (deg‘𝑔) ↔ (♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})) ≤ (deg‘(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))))))
91	87, 90	anbi12d 632	. . . . . . . 8 ⊢ (𝑔 = (𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) → (((◡𝑔 “ {0}) ∈ Fin ∧ (♯‘(◡𝑔 “ {0})) ≤ (deg‘𝑔)) ↔ ((◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}) ∈ Fin ∧ (♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})) ≤ (deg‘(𝐹 quot (Xp ∘f − (ℂ × {𝐴})))))))
92	84, 91	imbi12d 347	. . . . . . 7 ⊢ (𝑔 = (𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) → (((deg‘𝑔) = 𝐷 → ((◡𝑔 “ {0}) ∈ Fin ∧ (♯‘(◡𝑔 “ {0})) ≤ (deg‘𝑔))) ↔ ((deg‘(𝐹 quot (Xp ∘f − (ℂ × {𝐴})))) = 𝐷 → ((◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}) ∈ Fin ∧ (♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})) ≤ (deg‘(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))))))))
93		fta1.6	. . . . . . 7 ⊢ (𝜑 → ∀𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑔) = 𝐷 → ((◡𝑔 “ {0}) ∈ Fin ∧ (♯‘(◡𝑔 “ {0})) ≤ (deg‘𝑔))))
94		eldifsn 4718	. . . . . . . 8 ⊢ ((𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) ∈ ((Poly‘ℂ) ∖ {0𝑝}) ↔ ((𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) ∈ (Poly‘ℂ) ∧ (𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) ≠ 0𝑝))
95	42, 75, 94	sylanbrc 585	. . . . . . 7 ⊢ (𝜑 → (𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) ∈ ((Poly‘ℂ) ∖ {0𝑝}))
96	92, 93, 95	rspcdva 3624	. . . . . 6 ⊢ (𝜑 → ((deg‘(𝐹 quot (Xp ∘f − (ℂ × {𝐴})))) = 𝐷 → ((◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}) ∈ Fin ∧ (♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})) ≤ (deg‘(𝐹 quot (Xp ∘f − (ℂ × {𝐴})))))))
97	83, 96	mpd 15	. . . . 5 ⊢ (𝜑 → ((◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}) ∈ Fin ∧ (♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})) ≤ (deg‘(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))))))
98	97	simpld 497	. . . 4 ⊢ (𝜑 → (◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}) ∈ Fin)
99		snfi 8593	. . . 4 ⊢ {𝐴} ∈ Fin
100		unfi 8784	. . . 4 ⊢ (((◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}) ∈ Fin ∧ {𝐴} ∈ Fin) → ((◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴}) ∈ Fin)
101	98, 99, 100	sylancl 588	. . 3 ⊢ (𝜑 → ((◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴}) ∈ Fin)
102	52, 101	eqeltrd 2913	. 2 ⊢ (𝜑 → 𝑅 ∈ Fin)
103	52	fveq2d 6673	. . 3 ⊢ (𝜑 → (♯‘𝑅) = (♯‘((◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})))
104		hashcl 13716	. . . . . 6 ⊢ (((◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴}) ∈ Fin → (♯‘((◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ∈ ℕ0)
105	101, 104	syl 17	. . . . 5 ⊢ (𝜑 → (♯‘((◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ∈ ℕ0)
106	105	nn0red 11955	. . . 4 ⊢ (𝜑 → (♯‘((◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ∈ ℝ)
107		hashcl 13716	. . . . . . 7 ⊢ ((◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}) ∈ Fin → (♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})) ∈ ℕ0)
108	98, 107	syl 17	. . . . . 6 ⊢ (𝜑 → (♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})) ∈ ℕ0)
109	108	nn0red 11955	. . . . 5 ⊢ (𝜑 → (♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})) ∈ ℝ)
110		peano2re 10812	. . . . 5 ⊢ ((♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})) ∈ ℝ → ((♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})) + 1) ∈ ℝ)
111	109, 110	syl 17	. . . 4 ⊢ (𝜑 → ((♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})) + 1) ∈ ℝ)
112		dgrcl 24822	. . . . . 6 ⊢ (𝐹 ∈ (Poly‘ℂ) → (deg‘𝐹) ∈ ℕ0)
113	4, 112	syl 17	. . . . 5 ⊢ (𝜑 → (deg‘𝐹) ∈ ℕ0)
114	113	nn0red 11955	. . . 4 ⊢ (𝜑 → (deg‘𝐹) ∈ ℝ)
115		hashun2 13743	. . . . . 6 ⊢ (((◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}) ∈ Fin ∧ {𝐴} ∈ Fin) → (♯‘((◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ≤ ((♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})) + (♯‘{𝐴})))
116	98, 99, 115	sylancl 588	. . . . 5 ⊢ (𝜑 → (♯‘((◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ≤ ((♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})) + (♯‘{𝐴})))
117		hashsng 13729	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (♯‘{𝐴}) = 1)
118	11, 117	syl 17	. . . . . 6 ⊢ (𝜑 → (♯‘{𝐴}) = 1)
119	118	oveq2d 7171	. . . . 5 ⊢ (𝜑 → ((♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})) + (♯‘{𝐴})) = ((♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})) + 1))
120	116, 119	breqtrd 5091	. . . 4 ⊢ (𝜑 → (♯‘((◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ≤ ((♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})) + 1))
121	57	nn0red 11955	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ ℝ)
122		1red 10641	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℝ)
123	97	simprd 498	. . . . . . 7 ⊢ (𝜑 → (♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})) ≤ (deg‘(𝐹 quot (Xp ∘f − (ℂ × {𝐴})))))
124	123, 83	breqtrd 5091	. . . . . 6 ⊢ (𝜑 → (♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})) ≤ 𝐷)
125	109, 121, 122, 124	leadd1dd 11253	. . . . 5 ⊢ (𝜑 → ((♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})) + 1) ≤ (𝐷 + 1))
126	125, 62	breqtrrd 5093	. . . 4 ⊢ (𝜑 → ((♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})) + 1) ≤ (deg‘𝐹))
127	106, 111, 114, 120, 126	letrd 10796	. . 3 ⊢ (𝜑 → (♯‘((◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ≤ (deg‘𝐹))
128	103, 127	eqbrtrd 5087	. 2 ⊢ (𝜑 → (♯‘𝑅) ≤ (deg‘𝐹))
129	102, 128	jca 514	1 ⊢ (𝜑 → (𝑅 ∈ Fin ∧ (♯‘𝑅) ≤ (deg‘𝐹)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110   ≠ wne 3016  ∀wral 3138  Vcvv 3494   ∖ cdif 3932   ∪ cun 3933   ⊆ wss 3935  {csn 4566   class class class wbr 5065   × cxp 5552  ^◡ccnv 5553   “ cima 5557   Fn wfn 6349  ⟶wf 6350  ‘cfv 6354  (class class class)co 7155   ∘_f cof 7406  Fincfn 8508  ℂcc 10534  ℝcr 10535  0cc0 10536  1c1 10537   + caddc 10539   · cmul 10541   ≤ cle 10675   − cmin 10869  ℕ₀cn0 11896  ♯chash 13689  0_𝑝c0p 24269  Polycply 24773  X_pcidp 24774  degcdgr 24776   quot cquot 24878

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460  ax-inf2 9103  ax-cnex 10592  ax-resscn 10593  ax-1cn 10594  ax-icn 10595  ax-addcl 10596  ax-addrcl 10597  ax-mulcl 10598  ax-mulrcl 10599  ax-mulcom 10600  ax-addass 10601  ax-mulass 10602  ax-distr 10603  ax-i2m1 10604  ax-1ne0 10605  ax-1rid 10606  ax-rnegex 10607  ax-rrecex 10608  ax-cnre 10609  ax-pre-lttri 10610  ax-pre-lttrn 10611  ax-pre-ltadd 10612  ax-pre-mulgt0 10613  ax-pre-sup 10614  ax-addf 10615

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-fal 1546  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-int 4876  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-se 5514  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-isom 6363  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-of 7408  df-om 7580  df-1st 7688  df-2nd 7689  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-1o 8101  df-oadd 8105  df-er 8288  df-map 8407  df-pm 8408  df-en 8509  df-dom 8510  df-sdom 8511  df-fin 8512  df-sup 8905  df-inf 8906  df-oi 8973  df-dju 9329  df-card 9367  df-pnf 10676  df-mnf 10677  df-xr 10678  df-ltxr 10679  df-le 10680  df-sub 10871  df-neg 10872  df-div 11297  df-nn 11638  df-2 11699  df-3 11700  df-n0 11897  df-xnn0 11967  df-z 11981  df-uz 12243  df-rp 12389  df-fz 12892  df-fzo 13033  df-fl 13161  df-seq 13369  df-exp 13429  df-hash 13690  df-cj 14457  df-re 14458  df-im 14459  df-sqrt 14593  df-abs 14594  df-clim 14844  df-rlim 14845  df-sum 15042  df-0p 24270  df-ply 24777  df-idp 24778  df-coe 24779  df-dgr 24780  df-quot 24879

This theorem is referenced by:  fta1  24896