Theorem fta1lem 26610

Description: Lemma for fta1 26611. (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 4748	. . . . . . . . . 10 ⊢ (𝐹 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ↔ (𝐹 ∈ (Poly‘ℂ) ∧ 𝐹 ≠ 0𝑝))
3	1, 2	sylib 221	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 ∈ (Poly‘ℂ) ∧ 𝐹 ≠ 0𝑝))
4	3	simpld 500	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (Poly‘ℂ))
5		fta1.5	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ (◡𝐹 “ {0}))
6		plyf 26496	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Poly‘ℂ) → 𝐹:ℂ⟶ℂ)
7		ffn 6701	. . . . . . . . . . 11 ⊢ (𝐹:ℂ⟶ℂ → 𝐹 Fn ℂ)
8		fniniseg 7051	. . . . . . . . . . 11 ⊢ (𝐹 Fn ℂ → (𝐴 ∈ (◡𝐹 “ {0}) ↔ (𝐴 ∈ ℂ ∧ (𝐹‘𝐴) = 0)))
9	4, 6, 7, 8	4syl 20	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ∈ (◡𝐹 “ {0}) ↔ (𝐴 ∈ ℂ ∧ (𝐹‘𝐴) = 0)))
10	5, 9	mpbid 235	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ∈ ℂ ∧ (𝐹‘𝐴) = 0))
11	10	simpld 500	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℂ)
12	10	simprd 501	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝐴) = 0)
13		eqid 2761	. . . . . . . . 9 ⊢ (Xp ∘f − (ℂ × {𝐴})) = (Xp ∘f − (ℂ × {𝐴}))
14	13	facth 26609	. . . . . . . 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 1398	. . . . . . 7 ⊢ (𝜑 → 𝐹 = ((Xp ∘f − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xp ∘f − (ℂ × {𝐴})))))
16	15	cnveqd 5853	. . . . . 6 ⊢ (𝜑 → ◡𝐹 = ◡((Xp ∘f − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xp ∘f − (ℂ × {𝐴})))))
17	16	imaeq1d 6053	. . . . 5 ⊢ (𝜑 → (◡𝐹 “ {0}) = (◡((Xp ∘f − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xp ∘f − (ℂ × {𝐴})))) “ {0}))
18		cnex 11262	. . . . . . 7 ⊢ ℂ ∈ V
19	18	a1i 11	. . . . . 6 ⊢ (𝜑 → ℂ ∈ V)
20		ssid 3953	. . . . . . . . 9 ⊢ ℂ ⊆ ℂ
21		ax-1cn 11239	. . . . . . . . 9 ⊢ 1 ∈ ℂ
22		plyid 26507	. . . . . . . . 9 ⊢ ((ℂ ⊆ ℂ ∧ 1 ∈ ℂ) → Xp ∈ (Poly‘ℂ))
23	20, 21, 22	mp2an 705	. . . . . . . 8 ⊢ Xp ∈ (Poly‘ℂ)
24		plyconst 26504	. . . . . . . . 9 ⊢ ((ℂ ⊆ ℂ ∧ 𝐴 ∈ ℂ) → (ℂ × {𝐴}) ∈ (Poly‘ℂ))
25	20, 11, 24	sylancr 599	. . . . . . . 8 ⊢ (𝜑 → (ℂ × {𝐴}) ∈ (Poly‘ℂ))
26		plysubcl 26521	. . . . . . . 8 ⊢ ((Xp ∈ (Poly‘ℂ) ∧ (ℂ × {𝐴}) ∈ (Poly‘ℂ)) → (Xp ∘f − (ℂ × {𝐴})) ∈ (Poly‘ℂ))
27	23, 25, 26	sylancr 599	. . . . . . 7 ⊢ (𝜑 → (Xp ∘f − (ℂ × {𝐴})) ∈ (Poly‘ℂ))
28		plyf 26496	. . . . . . 7 ⊢ ((Xp ∘f − (ℂ × {𝐴})) ∈ (Poly‘ℂ) → (Xp ∘f − (ℂ × {𝐴})):ℂ⟶ℂ)
29	27, 28	syl 18	. . . . . 6 ⊢ (𝜑 → (Xp ∘f − (ℂ × {𝐴})):ℂ⟶ℂ)
30	13	plyremlem 26607	. . . . . . . . . . . 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 18	. . . . . . . . . . 11 ⊢ (𝜑 → ((Xp ∘f − (ℂ × {𝐴})) ∈ (Poly‘ℂ) ∧ (deg‘(Xp ∘f − (ℂ × {𝐴}))) = 1 ∧ (◡(Xp ∘f − (ℂ × {𝐴})) “ {0}) = {𝐴}))
32	31	simp2d 1161	. . . . . . . . . 10 ⊢ (𝜑 → (deg‘(Xp ∘f − (ℂ × {𝐴}))) = 1)
33		ax-1ne0 11250	. . . . . . . . . . 11 ⊢ 1 ≠ 0
34	33	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 1 ≠ 0)
35	32, 34	eqnetrd 3023	. . . . . . . . 9 ⊢ (𝜑 → (deg‘(Xp ∘f − (ℂ × {𝐴}))) ≠ 0)
36		fveq2 6877	. . . . . . . . . . 11 ⊢ ((Xp ∘f − (ℂ × {𝐴})) = 0𝑝 → (deg‘(Xp ∘f − (ℂ × {𝐴}))) = (deg‘0𝑝))
37		dgr0 26561	. . . . . . . . . . 11 ⊢ (deg‘0𝑝) = 0
38	36, 37	eqtrdi 2812	. . . . . . . . . 10 ⊢ ((Xp ∘f − (ℂ × {𝐴})) = 0𝑝 → (deg‘(Xp ∘f − (ℂ × {𝐴}))) = 0)
39	38	necon3i 2988	. . . . . . . . 9 ⊢ ((deg‘(Xp ∘f − (ℂ × {𝐴}))) ≠ 0 → (Xp ∘f − (ℂ × {𝐴})) ≠ 0𝑝)
40	35, 39	syl 18	. . . . . . . 8 ⊢ (𝜑 → (Xp ∘f − (ℂ × {𝐴})) ≠ 0𝑝)
41		quotcl2 26605	. . . . . . . 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 1398	. . . . . . 7 ⊢ (𝜑 → (𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) ∈ (Poly‘ℂ))
43		plyf 26496	. . . . . . 7 ⊢ ((𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) ∈ (Poly‘ℂ) → (𝐹 quot (Xp ∘f − (ℂ × {𝐴}))):ℂ⟶ℂ)
44	42, 43	syl 18	. . . . . 6 ⊢ (𝜑 → (𝐹 quot (Xp ∘f − (ℂ × {𝐴}))):ℂ⟶ℂ)
45		ofmulrt 26582	. . . . . 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 1398	. . . . 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 1162	. . . . . 6 ⊢ (𝜑 → (◡(Xp ∘f − (ℂ × {𝐴})) “ {0}) = {𝐴})
48	47	uneq1d 4114	. . . . 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 2800	. . . 4 ⊢ (𝜑 → (◡𝐹 “ {0}) = ({𝐴} ∪ (◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})))
50		fta1.1	. . . 4 ⊢ 𝑅 = (◡𝐹 “ {0})
51		uncom 4105	. . . 4 ⊢ ((◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴}) = ({𝐴} ∪ (◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}))
52	49, 50, 51	3eqtr4g 2821	. . 3 ⊢ (𝜑 → 𝑅 = ((◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴}))
53	21	a1i 11	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℂ)
54		dgrcl 26532	. . . . . . . . 9 ⊢ ((𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) ∈ (Poly‘ℂ) → (deg‘(𝐹 quot (Xp ∘f − (ℂ × {𝐴})))) ∈ ℕ0)
55	42, 54	syl 18	. . . . . . . 8 ⊢ (𝜑 → (deg‘(𝐹 quot (Xp ∘f − (ℂ × {𝐴})))) ∈ ℕ0)
56	55	nn0cnd 12650	. . . . . . 7 ⊢ (𝜑 → (deg‘(𝐹 quot (Xp ∘f − (ℂ × {𝐴})))) ∈ ℂ)
57		fta1.2	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ ℕ0)
58	57	nn0cnd 12650	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ ℂ)
59		addcom 11477	. . . . . . . . 9 ⊢ ((1 ∈ ℂ ∧ 𝐷 ∈ ℂ) → (1 + 𝐷) = (𝐷 + 1))
60	21, 58, 59	sylancr 599	. . . . . . . 8 ⊢ (𝜑 → (1 + 𝐷) = (𝐷 + 1))
61	15	fveq2d 6881	. . . . . . . . 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 501	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ≠ 0𝑝)
64	15	eqcomd 2767	. . . . . . . . . . . 12 ⊢ (𝜑 → ((Xp ∘f − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xp ∘f − (ℂ × {𝐴})))) = 𝐹)
65		0cnd 11280	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ∈ ℂ)
66		mul01 11470	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → (𝑥 · 0) = 0)
67	66	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝑥 · 0) = 0)
68	19, 29, 65, 65, 67	caofid1 7717	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((Xp ∘f − (ℂ × {𝐴})) ∘f · (ℂ × {0})) = (ℂ × {0}))
69		df-0p 25971	. . . . . . . . . . . . . 14 ⊢ 0𝑝 = (ℂ × {0})
70	69	oveq2i 7423	. . . . . . . . . . . . 13 ⊢ ((Xp ∘f − (ℂ × {𝐴})) ∘f · 0𝑝) = ((Xp ∘f − (ℂ × {𝐴})) ∘f · (ℂ × {0}))
71	68, 70, 69	3eqtr4g 2821	. . . . . . . . . . . 12 ⊢ (𝜑 → ((Xp ∘f − (ℂ × {𝐴})) ∘f · 0𝑝) = 0𝑝)
72	63, 64, 71	3netr4d 3033	. . . . . . . . . . 11 ⊢ (𝜑 → ((Xp ∘f − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xp ∘f − (ℂ × {𝐴})))) ≠ ((Xp ∘f − (ℂ × {𝐴})) ∘f · 0𝑝))
73		oveq2 7420	. . . . . . . . . . . 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 2988	. . . . . . . . . . 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 18	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) ≠ 0𝑝)
76		eqid 2761	. . . . . . . . . . 11 ⊢ (deg‘(Xp ∘f − (ℂ × {𝐴}))) = (deg‘(Xp ∘f − (ℂ × {𝐴})))
77		eqid 2761	. . . . . . . . . . 11 ⊢ (deg‘(𝐹 quot (Xp ∘f − (ℂ × {𝐴})))) = (deg‘(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))))
78	76, 77	dgrmul 26569	. . . . . . . . . 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 852	. . . . . . . . 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 2804	. . . . . . . 8 ⊢ (𝜑 → (𝐷 + 1) = ((deg‘(Xp ∘f − (ℂ × {𝐴}))) + (deg‘(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))))))
81	32	oveq1d 7427	. . . . . . . 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 2801	. . . . . . 7 ⊢ (𝜑 → (1 + (deg‘(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))))) = (1 + 𝐷))
83	53, 56, 58, 82	addcanad 11496	. . . . . 6 ⊢ (𝜑 → (deg‘(𝐹 quot (Xp ∘f − (ℂ × {𝐴})))) = 𝐷)
84		fveqeq2 6886	. . . . . . . 8 ⊢ (𝑔 = (𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) → ((deg‘𝑔) = 𝐷 ↔ (deg‘(𝐹 quot (Xp ∘f − (ℂ × {𝐴})))) = 𝐷))
85		cnveq 5851	. . . . . . . . . . 11 ⊢ (𝑔 = (𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) → ◡𝑔 = ◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))))
86	85	imaeq1d 6053	. . . . . . . . . 10 ⊢ (𝑔 = (𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) → (◡𝑔 “ {0}) = (◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}))
87	86	eleq1d 2846	. . . . . . . . 9 ⊢ (𝑔 = (𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) → ((◡𝑔 “ {0}) ∈ Fin ↔ (◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}) ∈ Fin))
88	86	fveq2d 6881	. . . . . . . . . 10 ⊢ (𝑔 = (𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) → (♯‘(◡𝑔 “ {0})) = (♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})))
89		fveq2 6877	. . . . . . . . . 10 ⊢ (𝑔 = (𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) → (deg‘𝑔) = (deg‘(𝐹 quot (Xp ∘f − (ℂ × {𝐴})))))
90	88, 89	breq12d 5116	. . . . . . . . 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 644	. . . . . . . 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 4748	. . . . . . . 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 595	. . . . . . 7 ⊢ (𝜑 → (𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) ∈ ((Poly‘ℂ) ∖ {0𝑝}))
96	92, 93, 95	rspcdva 3578	. . . . . 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 16	. . . . 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 500	. . . 4 ⊢ (𝜑 → (◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}) ∈ Fin)
99		snfi 9055	. . . 4 ⊢ {𝐴} ∈ Fin
100		unfi 9170	. . . 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 598	. . 3 ⊢ (𝜑 → ((◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴}) ∈ Fin)
102	52, 101	eqeltrd 2861	. 2 ⊢ (𝜑 → 𝑅 ∈ Fin)
103	52	fveq2d 6881	. . 3 ⊢ (𝜑 → (♯‘𝑅) = (♯‘((◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})))
104		hashcl 14480	. . . . . 6 ⊢ (((◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴}) ∈ Fin → (♯‘((◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ∈ ℕ0)
105	101, 104	syl 18	. . . . 5 ⊢ (𝜑 → (♯‘((◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ∈ ℕ0)
106	105	nn0red 12649	. . . 4 ⊢ (𝜑 → (♯‘((◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ∈ ℝ)
107		hashcl 14480	. . . . . . 7 ⊢ ((◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}) ∈ Fin → (♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})) ∈ ℕ0)
108	98, 107	syl 18	. . . . . 6 ⊢ (𝜑 → (♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})) ∈ ℕ0)
109	108	nn0red 12649	. . . . 5 ⊢ (𝜑 → (♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})) ∈ ℝ)
110		peano2re 11464	. . . . 5 ⊢ ((♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})) ∈ ℝ → ((♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})) + 1) ∈ ℝ)
111	109, 110	syl 18	. . . 4 ⊢ (𝜑 → ((♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})) + 1) ∈ ℝ)
112		dgrcl 26532	. . . . . 6 ⊢ (𝐹 ∈ (Poly‘ℂ) → (deg‘𝐹) ∈ ℕ0)
113	4, 112	syl 18	. . . . 5 ⊢ (𝜑 → (deg‘𝐹) ∈ ℕ0)
114	113	nn0red 12649	. . . 4 ⊢ (𝜑 → (deg‘𝐹) ∈ ℝ)
115		hashun2 14507	. . . . . 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 598	. . . . 5 ⊢ (𝜑 → (♯‘((◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ≤ ((♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})) + (♯‘{𝐴})))
117		hashsng 14493	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (♯‘{𝐴}) = 1)
118	11, 117	syl 18	. . . . . 6 ⊢ (𝜑 → (♯‘{𝐴}) = 1)
119	118	oveq2d 7428	. . . . 5 ⊢ (𝜑 → ((♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})) + (♯‘{𝐴})) = ((♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})) + 1))
120	116, 119	breqtrd 5131	. . . 4 ⊢ (𝜑 → (♯‘((◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ≤ ((♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})) + 1))
121	57	nn0red 12649	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ ℝ)
122		1red 11290	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℝ)
123	97	simprd 501	. . . . . . 7 ⊢ (𝜑 → (♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})) ≤ (deg‘(𝐹 quot (Xp ∘f − (ℂ × {𝐴})))))
124	123, 83	breqtrd 5131	. . . . . 6 ⊢ (𝜑 → (♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})) ≤ 𝐷)
125	109, 121, 122, 124	leadd1dd 11911	. . . . 5 ⊢ (𝜑 → ((♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})) + 1) ≤ (𝐷 + 1))
126	125, 62	breqtrrd 5133	. . . 4 ⊢ (𝜑 → ((♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})) + 1) ≤ (deg‘𝐹))
127	106, 111, 114, 120, 126	letrd 11448	. . 3 ⊢ (𝜑 → (♯‘((◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ≤ (deg‘𝐹))
128	103, 127	eqbrtrd 5127	. 2 ⊢ (𝜑 → (♯‘𝑅) ≤ (deg‘𝐹))
129	102, 128	jca 521	1 ⊢ (𝜑 → (𝑅 ∈ Fin ∧ (♯‘𝑅) ≤ (deg‘𝐹)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077  Vcvv 3451   ∖ cdif 3896   ∪ cun 3897   ⊆ wss 3899  {csn 4584   class class class wbr 5103   × cxp 5649  ^◡ccnv 5650   “ cima 5654   Fn wfn 6526  ⟶wf 6527  ‘cfv 6531  (class class class)co 7412   ∘_f cof 7680  Fincfn 8957  ℂcc 11179  ℝcr 11180  0cc0 11181  1c1 11182   + caddc 11184   · cmul 11186   ≤ cle 11325   − cmin 11522  ℕ₀cn0 12587  ♯chash 14454  0_𝑝c0p 25970  Polycply 26482  X_pcidp 26483  degcdgr 26485   quot cquot 26593

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-inf2 9626  ax-cnex 11237  ax-resscn 11238  ax-1cn 11239  ax-icn 11240  ax-addcl 11241  ax-addrcl 11242  ax-mulcl 11243  ax-mulrcl 11244  ax-mulcom 11245  ax-addass 11246  ax-mulass 11247  ax-distr 11248  ax-i2m1 11249  ax-1ne0 11250  ax-1rid 11251  ax-rnegex 11252  ax-rrecex 11253  ax-cnre 11254  ax-pre-lttri 11255  ax-pre-lttrn 11256  ax-pre-ltadd 11257  ax-pre-mulgt0 11258  ax-pre-sup 11259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-isom 6540  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7682  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8460  df-oadd 8464  df-er 8701  df-map 8833  df-pm 8834  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-sup 9418  df-inf 9419  df-oi 9488  df-dju 9963  df-card 10001  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11524  df-neg 11525  df-div 11955  df-nn 12317  df-2 12386  df-3 12387  df-n0 12588  df-xnn0 12661  df-z 12675  df-uz 12947  df-rp 13102  df-fz 13621  df-fzo 13769  df-fl 13912  df-seq 14125  df-exp 14185  df-hash 14455  df-cj 15246  df-re 15247  df-im 15248  df-sqrt 15382  df-abs 15383  df-clim 15635  df-rlim 15636  df-sum 15834  df-0p 25971  df-ply 26486  df-idp 26487  df-coe 26488  df-dgr 26489  df-quot 26594

This theorem is used by:  fta1  26611