Theorem fta1lem 26222

Description: Lemma for fta1 26223. (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 4753	. . . . . . . . . 10 ⊢ (𝐹 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ↔ (𝐹 ∈ (Poly‘ℂ) ∧ 𝐹 ≠ 0𝑝))
3	1, 2	sylib 218	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 ∈ (Poly‘ℂ) ∧ 𝐹 ≠ 0𝑝))
4	3	simpld 494	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (Poly‘ℂ))
5		fta1.5	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ (◡𝐹 “ {0}))
6		plyf 26110	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Poly‘ℂ) → 𝐹:ℂ⟶ℂ)
7		ffn 6691	. . . . . . . . . . 11 ⊢ (𝐹:ℂ⟶ℂ → 𝐹 Fn ℂ)
8		fniniseg 7035	. . . . . . . . . . 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 232	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ∈ ℂ ∧ (𝐹‘𝐴) = 0))
11	10	simpld 494	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℂ)
12	10	simprd 495	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝐴) = 0)
13		eqid 2730	. . . . . . . . 9 ⊢ (Xp ∘f − (ℂ × {𝐴})) = (Xp ∘f − (ℂ × {𝐴}))
14	13	facth 26221	. . . . . . . 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 1373	. . . . . . 7 ⊢ (𝜑 → 𝐹 = ((Xp ∘f − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xp ∘f − (ℂ × {𝐴})))))
16	15	cnveqd 5842	. . . . . 6 ⊢ (𝜑 → ◡𝐹 = ◡((Xp ∘f − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xp ∘f − (ℂ × {𝐴})))))
17	16	imaeq1d 6033	. . . . 5 ⊢ (𝜑 → (◡𝐹 “ {0}) = (◡((Xp ∘f − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xp ∘f − (ℂ × {𝐴})))) “ {0}))
18		cnex 11156	. . . . . . 7 ⊢ ℂ ∈ V
19	18	a1i 11	. . . . . 6 ⊢ (𝜑 → ℂ ∈ V)
20		ssid 3972	. . . . . . . . 9 ⊢ ℂ ⊆ ℂ
21		ax-1cn 11133	. . . . . . . . 9 ⊢ 1 ∈ ℂ
22		plyid 26121	. . . . . . . . 9 ⊢ ((ℂ ⊆ ℂ ∧ 1 ∈ ℂ) → Xp ∈ (Poly‘ℂ))
23	20, 21, 22	mp2an 692	. . . . . . . 8 ⊢ Xp ∈ (Poly‘ℂ)
24		plyconst 26118	. . . . . . . . 9 ⊢ ((ℂ ⊆ ℂ ∧ 𝐴 ∈ ℂ) → (ℂ × {𝐴}) ∈ (Poly‘ℂ))
25	20, 11, 24	sylancr 587	. . . . . . . 8 ⊢ (𝜑 → (ℂ × {𝐴}) ∈ (Poly‘ℂ))
26		plysubcl 26134	. . . . . . . 8 ⊢ ((Xp ∈ (Poly‘ℂ) ∧ (ℂ × {𝐴}) ∈ (Poly‘ℂ)) → (Xp ∘f − (ℂ × {𝐴})) ∈ (Poly‘ℂ))
27	23, 25, 26	sylancr 587	. . . . . . 7 ⊢ (𝜑 → (Xp ∘f − (ℂ × {𝐴})) ∈ (Poly‘ℂ))
28		plyf 26110	. . . . . . 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 26219	. . . . . . . . . . . 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 1143	. . . . . . . . . 10 ⊢ (𝜑 → (deg‘(Xp ∘f − (ℂ × {𝐴}))) = 1)
33		ax-1ne0 11144	. . . . . . . . . . 11 ⊢ 1 ≠ 0
34	33	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 1 ≠ 0)
35	32, 34	eqnetrd 2993	. . . . . . . . 9 ⊢ (𝜑 → (deg‘(Xp ∘f − (ℂ × {𝐴}))) ≠ 0)
36		fveq2 6861	. . . . . . . . . . 11 ⊢ ((Xp ∘f − (ℂ × {𝐴})) = 0𝑝 → (deg‘(Xp ∘f − (ℂ × {𝐴}))) = (deg‘0𝑝))
37		dgr0 26175	. . . . . . . . . . 11 ⊢ (deg‘0𝑝) = 0
38	36, 37	eqtrdi 2781	. . . . . . . . . 10 ⊢ ((Xp ∘f − (ℂ × {𝐴})) = 0𝑝 → (deg‘(Xp ∘f − (ℂ × {𝐴}))) = 0)
39	38	necon3i 2958	. . . . . . . . 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 26217	. . . . . . . 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 1373	. . . . . . 7 ⊢ (𝜑 → (𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) ∈ (Poly‘ℂ))
43		plyf 26110	. . . . . . 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 26196	. . . . . 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 1373	. . . . 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 1144	. . . . . 6 ⊢ (𝜑 → (◡(Xp ∘f − (ℂ × {𝐴})) “ {0}) = {𝐴})
48	47	uneq1d 4133	. . . . 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 2769	. . . 4 ⊢ (𝜑 → (◡𝐹 “ {0}) = ({𝐴} ∪ (◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})))
50		fta1.1	. . . 4 ⊢ 𝑅 = (◡𝐹 “ {0})
51		uncom 4124	. . . 4 ⊢ ((◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴}) = ({𝐴} ∪ (◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}))
52	49, 50, 51	3eqtr4g 2790	. . 3 ⊢ (𝜑 → 𝑅 = ((◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴}))
53	21	a1i 11	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℂ)
54		dgrcl 26145	. . . . . . . . 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 12512	. . . . . . 7 ⊢ (𝜑 → (deg‘(𝐹 quot (Xp ∘f − (ℂ × {𝐴})))) ∈ ℂ)
57		fta1.2	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ ℕ0)
58	57	nn0cnd 12512	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ ℂ)
59		addcom 11367	. . . . . . . . 9 ⊢ ((1 ∈ ℂ ∧ 𝐷 ∈ ℂ) → (1 + 𝐷) = (𝐷 + 1))
60	21, 58, 59	sylancr 587	. . . . . . . 8 ⊢ (𝜑 → (1 + 𝐷) = (𝐷 + 1))
61	15	fveq2d 6865	. . . . . . . . 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 495	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ≠ 0𝑝)
64	15	eqcomd 2736	. . . . . . . . . . . 12 ⊢ (𝜑 → ((Xp ∘f − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xp ∘f − (ℂ × {𝐴})))) = 𝐹)
65		0cnd 11174	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ∈ ℂ)
66		mul01 11360	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → (𝑥 · 0) = 0)
67	66	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝑥 · 0) = 0)
68	19, 29, 65, 65, 67	caofid1 7691	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((Xp ∘f − (ℂ × {𝐴})) ∘f · (ℂ × {0})) = (ℂ × {0}))
69		df-0p 25578	. . . . . . . . . . . . . 14 ⊢ 0𝑝 = (ℂ × {0})
70	69	oveq2i 7401	. . . . . . . . . . . . 13 ⊢ ((Xp ∘f − (ℂ × {𝐴})) ∘f · 0𝑝) = ((Xp ∘f − (ℂ × {𝐴})) ∘f · (ℂ × {0}))
71	68, 70, 69	3eqtr4g 2790	. . . . . . . . . . . 12 ⊢ (𝜑 → ((Xp ∘f − (ℂ × {𝐴})) ∘f · 0𝑝) = 0𝑝)
72	63, 64, 71	3netr4d 3003	. . . . . . . . . . 11 ⊢ (𝜑 → ((Xp ∘f − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xp ∘f − (ℂ × {𝐴})))) ≠ ((Xp ∘f − (ℂ × {𝐴})) ∘f · 0𝑝))
73		oveq2 7398	. . . . . . . . . . . 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 2958	. . . . . . . . . . 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 2730	. . . . . . . . . . 11 ⊢ (deg‘(Xp ∘f − (ℂ × {𝐴}))) = (deg‘(Xp ∘f − (ℂ × {𝐴})))
77		eqid 2730	. . . . . . . . . . 11 ⊢ (deg‘(𝐹 quot (Xp ∘f − (ℂ × {𝐴})))) = (deg‘(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))))
78	76, 77	dgrmul 26183	. . . . . . . . . 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 838	. . . . . . . . 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 2773	. . . . . . . 8 ⊢ (𝜑 → (𝐷 + 1) = ((deg‘(Xp ∘f − (ℂ × {𝐴}))) + (deg‘(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))))))
81	32	oveq1d 7405	. . . . . . . 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 2770	. . . . . . 7 ⊢ (𝜑 → (1 + (deg‘(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))))) = (1 + 𝐷))
83	53, 56, 58, 82	addcanad 11386	. . . . . 6 ⊢ (𝜑 → (deg‘(𝐹 quot (Xp ∘f − (ℂ × {𝐴})))) = 𝐷)
84		fveqeq2 6870	. . . . . . . 8 ⊢ (𝑔 = (𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) → ((deg‘𝑔) = 𝐷 ↔ (deg‘(𝐹 quot (Xp ∘f − (ℂ × {𝐴})))) = 𝐷))
85		cnveq 5840	. . . . . . . . . . 11 ⊢ (𝑔 = (𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) → ◡𝑔 = ◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))))
86	85	imaeq1d 6033	. . . . . . . . . 10 ⊢ (𝑔 = (𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) → (◡𝑔 “ {0}) = (◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}))
87	86	eleq1d 2814	. . . . . . . . 9 ⊢ (𝑔 = (𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) → ((◡𝑔 “ {0}) ∈ Fin ↔ (◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}) ∈ Fin))
88	86	fveq2d 6865	. . . . . . . . . 10 ⊢ (𝑔 = (𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) → (♯‘(◡𝑔 “ {0})) = (♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})))
89		fveq2 6861	. . . . . . . . . 10 ⊢ (𝑔 = (𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) → (deg‘𝑔) = (deg‘(𝐹 quot (Xp ∘f − (ℂ × {𝐴})))))
90	88, 89	breq12d 5123	. . . . . . . . 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 344	. . . . . . 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 4753	. . . . . . . 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 583	. . . . . . 7 ⊢ (𝜑 → (𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) ∈ ((Poly‘ℂ) ∖ {0𝑝}))
96	92, 93, 95	rspcdva 3592	. . . . . 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 494	. . . 4 ⊢ (𝜑 → (◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}) ∈ Fin)
99		snfi 9017	. . . 4 ⊢ {𝐴} ∈ Fin
100		unfi 9141	. . . 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 586	. . 3 ⊢ (𝜑 → ((◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴}) ∈ Fin)
102	52, 101	eqeltrd 2829	. 2 ⊢ (𝜑 → 𝑅 ∈ Fin)
103	52	fveq2d 6865	. . 3 ⊢ (𝜑 → (♯‘𝑅) = (♯‘((◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})))
104		hashcl 14328	. . . . . 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 12511	. . . 4 ⊢ (𝜑 → (♯‘((◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ∈ ℝ)
107		hashcl 14328	. . . . . . 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 12511	. . . . 5 ⊢ (𝜑 → (♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})) ∈ ℝ)
110		peano2re 11354	. . . . 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 26145	. . . . . 6 ⊢ (𝐹 ∈ (Poly‘ℂ) → (deg‘𝐹) ∈ ℕ0)
113	4, 112	syl 17	. . . . 5 ⊢ (𝜑 → (deg‘𝐹) ∈ ℕ0)
114	113	nn0red 12511	. . . 4 ⊢ (𝜑 → (deg‘𝐹) ∈ ℝ)
115		hashun2 14355	. . . . . 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 586	. . . . 5 ⊢ (𝜑 → (♯‘((◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ≤ ((♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})) + (♯‘{𝐴})))
117		hashsng 14341	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (♯‘{𝐴}) = 1)
118	11, 117	syl 17	. . . . . 6 ⊢ (𝜑 → (♯‘{𝐴}) = 1)
119	118	oveq2d 7406	. . . . 5 ⊢ (𝜑 → ((♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})) + (♯‘{𝐴})) = ((♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})) + 1))
120	116, 119	breqtrd 5136	. . . 4 ⊢ (𝜑 → (♯‘((◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ≤ ((♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})) + 1))
121	57	nn0red 12511	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ ℝ)
122		1red 11182	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℝ)
123	97	simprd 495	. . . . . . 7 ⊢ (𝜑 → (♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})) ≤ (deg‘(𝐹 quot (Xp ∘f − (ℂ × {𝐴})))))
124	123, 83	breqtrd 5136	. . . . . 6 ⊢ (𝜑 → (♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})) ≤ 𝐷)
125	109, 121, 122, 124	leadd1dd 11799	. . . . 5 ⊢ (𝜑 → ((♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})) + 1) ≤ (𝐷 + 1))
126	125, 62	breqtrrd 5138	. . . 4 ⊢ (𝜑 → ((♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})) + 1) ≤ (deg‘𝐹))
127	106, 111, 114, 120, 126	letrd 11338	. . 3 ⊢ (𝜑 → (♯‘((◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ≤ (deg‘𝐹))
128	103, 127	eqbrtrd 5132	. 2 ⊢ (𝜑 → (♯‘𝑅) ≤ (deg‘𝐹))
129	102, 128	jca 511	1 ⊢ (𝜑 → (𝑅 ∈ Fin ∧ (♯‘𝑅) ≤ (deg‘𝐹)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2926  ∀wral 3045  Vcvv 3450   ∖ cdif 3914   ∪ cun 3915   ⊆ wss 3917  {csn 4592   class class class wbr 5110   × cxp 5639  ^◡ccnv 5640   “ cima 5644   Fn wfn 6509  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390   ∘_f cof 7654  Fincfn 8921  ℂcc 11073  ℝcr 11074  0cc0 11075  1c1 11076   + caddc 11078   · cmul 11080   ≤ cle 11216   − cmin 11412  ℕ₀cn0 12449  ♯chash 14302  0_𝑝c0p 25577  Polycply 26096  X_pcidp 26097  degcdgr 26099   quot cquot 26205

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 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-inf2 9601  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-pre-sup 11153

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-isom 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-of 7656  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-oadd 8441  df-er 8674  df-map 8804  df-pm 8805  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-sup 9400  df-inf 9401  df-oi 9470  df-dju 9861  df-card 9899  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-div 11843  df-nn 12194  df-2 12256  df-3 12257  df-n0 12450  df-xnn0 12523  df-z 12537  df-uz 12801  df-rp 12959  df-fz 13476  df-fzo 13623  df-fl 13761  df-seq 13974  df-exp 14034  df-hash 14303  df-cj 15072  df-re 15073  df-im 15074  df-sqrt 15208  df-abs 15209  df-clim 15461  df-rlim 15462  df-sum 15660  df-0p 25578  df-ply 26100  df-idp 26101  df-coe 26102  df-dgr 26103  df-quot 26206

This theorem is referenced by:  fta1  26223