Theorem fta1lem 24062

Description: Lemma for fta1 24063. (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 4317	. . . . . . . . . 10 ⊢ (𝐹 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ↔ (𝐹 ∈ (Poly‘ℂ) ∧ 𝐹 ≠ 0𝑝))
3	1, 2	sylib 208	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 ∈ (Poly‘ℂ) ∧ 𝐹 ≠ 0𝑝))
4	3	simpld 475	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (Poly‘ℂ))
5		fta1.5	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ (◡𝐹 “ {0}))
6		plyf 23954	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Poly‘ℂ) → 𝐹:ℂ⟶ℂ)
7		ffn 6045	. . . . . . . . . . 11 ⊢ (𝐹:ℂ⟶ℂ → 𝐹 Fn ℂ)
8		fniniseg 6338	. . . . . . . . . . 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 222	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ∈ ℂ ∧ (𝐹‘𝐴) = 0))
11	10	simpld 475	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℂ)
12	10	simprd 479	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝐴) = 0)
13		eqid 2622	. . . . . . . . 9 ⊢ (Xp ∘𝑓 − (ℂ × {𝐴})) = (Xp ∘𝑓 − (ℂ × {𝐴}))
14	13	facth 24061	. . . . . . . 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 1326	. . . . . . 7 ⊢ (𝜑 → 𝐹 = ((Xp ∘𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴})))))
16	15	cnveqd 5298	. . . . . 6 ⊢ (𝜑 → ◡𝐹 = ◡((Xp ∘𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴})))))
17	16	imaeq1d 5465	. . . . 5 ⊢ (𝜑 → (◡𝐹 “ {0}) = (◡((Xp ∘𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴})))) “ {0}))
18		cnex 10017	. . . . . . 7 ⊢ ℂ ∈ V
19	18	a1i 11	. . . . . 6 ⊢ (𝜑 → ℂ ∈ V)
20		ssid 3624	. . . . . . . . 9 ⊢ ℂ ⊆ ℂ
21		ax-1cn 9994	. . . . . . . . 9 ⊢ 1 ∈ ℂ
22		plyid 23965	. . . . . . . . 9 ⊢ ((ℂ ⊆ ℂ ∧ 1 ∈ ℂ) → Xp ∈ (Poly‘ℂ))
23	20, 21, 22	mp2an 708	. . . . . . . 8 ⊢ Xp ∈ (Poly‘ℂ)
24		plyconst 23962	. . . . . . . . 9 ⊢ ((ℂ ⊆ ℂ ∧ 𝐴 ∈ ℂ) → (ℂ × {𝐴}) ∈ (Poly‘ℂ))
25	20, 11, 24	sylancr 695	. . . . . . . 8 ⊢ (𝜑 → (ℂ × {𝐴}) ∈ (Poly‘ℂ))
26		plysubcl 23978	. . . . . . . 8 ⊢ ((Xp ∈ (Poly‘ℂ) ∧ (ℂ × {𝐴}) ∈ (Poly‘ℂ)) → (Xp ∘𝑓 − (ℂ × {𝐴})) ∈ (Poly‘ℂ))
27	23, 25, 26	sylancr 695	. . . . . . 7 ⊢ (𝜑 → (Xp ∘𝑓 − (ℂ × {𝐴})) ∈ (Poly‘ℂ))
28		plyf 23954	. . . . . . 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 24059	. . . . . . . . . . . 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 1074	. . . . . . . . . 10 ⊢ (𝜑 → (deg‘(Xp ∘𝑓 − (ℂ × {𝐴}))) = 1)
33		ax-1ne0 10005	. . . . . . . . . . 11 ⊢ 1 ≠ 0
34	33	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 1 ≠ 0)
35	32, 34	eqnetrd 2861	. . . . . . . . 9 ⊢ (𝜑 → (deg‘(Xp ∘𝑓 − (ℂ × {𝐴}))) ≠ 0)
36		fveq2 6191	. . . . . . . . . . 11 ⊢ ((Xp ∘𝑓 − (ℂ × {𝐴})) = 0𝑝 → (deg‘(Xp ∘𝑓 − (ℂ × {𝐴}))) = (deg‘0𝑝))
37		dgr0 24018	. . . . . . . . . . 11 ⊢ (deg‘0𝑝) = 0
38	36, 37	syl6eq 2672	. . . . . . . . . 10 ⊢ ((Xp ∘𝑓 − (ℂ × {𝐴})) = 0𝑝 → (deg‘(Xp ∘𝑓 − (ℂ × {𝐴}))) = 0)
39	38	necon3i 2826	. . . . . . . . 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 24057	. . . . . . . 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 1326	. . . . . . 7 ⊢ (𝜑 → (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) ∈ (Poly‘ℂ))
43		plyf 23954	. . . . . . 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 24037	. . . . . 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 1326	. . . . 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 1075	. . . . . 6 ⊢ (𝜑 → (◡(Xp ∘𝑓 − (ℂ × {𝐴})) “ {0}) = {𝐴})
48	47	uneq1d 3766	. . . . 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 2660	. . . 4 ⊢ (𝜑 → (◡𝐹 “ {0}) = ({𝐴} ∪ (◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0})))
50		fta1.1	. . . 4 ⊢ 𝑅 = (◡𝐹 “ {0})
51		uncom 3757	. . . 4 ⊢ ((◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴}) = ({𝐴} ∪ (◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0}))
52	49, 50, 51	3eqtr4g 2681	. . 3 ⊢ (𝜑 → 𝑅 = ((◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴}))
53	3	simprd 479	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ≠ 0𝑝)
54	15	eqcomd 2628	. . . . . . . . 9 ⊢ (𝜑 → ((Xp ∘𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴})))) = 𝐹)
55		0cnd 10033	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ∈ ℂ)
56		mul01 10215	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ → (𝑥 · 0) = 0)
57	56	adantl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝑥 · 0) = 0)
58	19, 29, 55, 55, 57	caofid1 6927	. . . . . . . . . 10 ⊢ (𝜑 → ((Xp ∘𝑓 − (ℂ × {𝐴})) ∘𝑓 · (ℂ × {0})) = (ℂ × {0}))
59		df-0p 23437	. . . . . . . . . . 11 ⊢ 0𝑝 = (ℂ × {0})
60	59	oveq2i 6661	. . . . . . . . . 10 ⊢ ((Xp ∘𝑓 − (ℂ × {𝐴})) ∘𝑓 · 0𝑝) = ((Xp ∘𝑓 − (ℂ × {𝐴})) ∘𝑓 · (ℂ × {0}))
61	58, 60, 59	3eqtr4g 2681	. . . . . . . . 9 ⊢ (𝜑 → ((Xp ∘𝑓 − (ℂ × {𝐴})) ∘𝑓 · 0𝑝) = 0𝑝)
62	53, 54, 61	3netr4d 2871	. . . . . . . 8 ⊢ (𝜑 → ((Xp ∘𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴})))) ≠ ((Xp ∘𝑓 − (ℂ × {𝐴})) ∘𝑓 · 0𝑝))
63		oveq2 6658	. . . . . . . . 9 ⊢ ((𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) = 0𝑝 → ((Xp ∘𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴})))) = ((Xp ∘𝑓 − (ℂ × {𝐴})) ∘𝑓 · 0𝑝))
64	63	necon3i 2826	. . . . . . . 8 ⊢ (((Xp ∘𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴})))) ≠ ((Xp ∘𝑓 − (ℂ × {𝐴})) ∘𝑓 · 0𝑝) → (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) ≠ 0𝑝)
65	62, 64	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) ≠ 0𝑝)
66		eldifsn 4317	. . . . . . 7 ⊢ ((𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) ∈ ((Poly‘ℂ) ∖ {0𝑝}) ↔ ((𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) ∈ (Poly‘ℂ) ∧ (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) ≠ 0𝑝))
67	42, 65, 66	sylanbrc 698	. . . . . 6 ⊢ (𝜑 → (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) ∈ ((Poly‘ℂ) ∖ {0𝑝}))
68		fta1.6	. . . . . 6 ⊢ (𝜑 → ∀𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑔) = 𝐷 → ((◡𝑔 “ {0}) ∈ Fin ∧ (#‘(◡𝑔 “ {0})) ≤ (deg‘𝑔))))
69	21	a1i 11	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℂ)
70		dgrcl 23989	. . . . . . . . 9 ⊢ ((𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) ∈ (Poly‘ℂ) → (deg‘(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴})))) ∈ ℕ0)
71	42, 70	syl 17	. . . . . . . 8 ⊢ (𝜑 → (deg‘(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴})))) ∈ ℕ0)
72	71	nn0cnd 11353	. . . . . . 7 ⊢ (𝜑 → (deg‘(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴})))) ∈ ℂ)
73		fta1.2	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ ℕ0)
74	73	nn0cnd 11353	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ ℂ)
75		addcom 10222	. . . . . . . . 9 ⊢ ((1 ∈ ℂ ∧ 𝐷 ∈ ℂ) → (1 + 𝐷) = (𝐷 + 1))
76	21, 74, 75	sylancr 695	. . . . . . . 8 ⊢ (𝜑 → (1 + 𝐷) = (𝐷 + 1))
77	15	fveq2d 6195	. . . . . . . . 9 ⊢ (𝜑 → (deg‘𝐹) = (deg‘((Xp ∘𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))))))
78		fta1.4	. . . . . . . . 9 ⊢ (𝜑 → (deg‘𝐹) = (𝐷 + 1))
79		eqid 2622	. . . . . . . . . . 11 ⊢ (deg‘(Xp ∘𝑓 − (ℂ × {𝐴}))) = (deg‘(Xp ∘𝑓 − (ℂ × {𝐴})))
80		eqid 2622	. . . . . . . . . . 11 ⊢ (deg‘(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴})))) = (deg‘(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))))
81	79, 80	dgrmul 24026	. . . . . . . . . 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}))))))
82	27, 40, 42, 65, 81	syl22anc 1327	. . . . . . . . 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}))))))
83	77, 78, 82	3eqtr3d 2664	. . . . . . . 8 ⊢ (𝜑 → (𝐷 + 1) = ((deg‘(Xp ∘𝑓 − (ℂ × {𝐴}))) + (deg‘(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))))))
84	32	oveq1d 6665	. . . . . . . 8 ⊢ (𝜑 → ((deg‘(Xp ∘𝑓 − (ℂ × {𝐴}))) + (deg‘(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))))) = (1 + (deg‘(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))))))
85	76, 83, 84	3eqtrrd 2661	. . . . . . 7 ⊢ (𝜑 → (1 + (deg‘(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))))) = (1 + 𝐷))
86	69, 72, 74, 85	addcanad 10241	. . . . . 6 ⊢ (𝜑 → (deg‘(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴})))) = 𝐷)
87		fveq2 6191	. . . . . . . . 9 ⊢ (𝑔 = (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) → (deg‘𝑔) = (deg‘(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴})))))
88	87	eqeq1d 2624	. . . . . . . 8 ⊢ (𝑔 = (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) → ((deg‘𝑔) = 𝐷 ↔ (deg‘(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴})))) = 𝐷))
89		cnveq 5296	. . . . . . . . . . 11 ⊢ (𝑔 = (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) → ◡𝑔 = ◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))))
90	89	imaeq1d 5465	. . . . . . . . . 10 ⊢ (𝑔 = (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) → (◡𝑔 “ {0}) = (◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0}))
91	90	eleq1d 2686	. . . . . . . . 9 ⊢ (𝑔 = (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) → ((◡𝑔 “ {0}) ∈ Fin ↔ (◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0}) ∈ Fin))
92	90	fveq2d 6195	. . . . . . . . . 10 ⊢ (𝑔 = (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) → (#‘(◡𝑔 “ {0})) = (#‘(◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0})))
93	92, 87	breq12d 4666	. . . . . . . . 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}))))))
94	91, 93	anbi12d 747	. . . . . . . 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})))))))
95	88, 94	imbi12d 334	. . . . . . 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}))))))))
96	95	rspcv 3305	. . . . . 6 ⊢ ((𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) ∈ ((Poly‘ℂ) ∖ {0𝑝}) → (∀𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝})((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}))))))))
97	67, 68, 86, 96	syl3c 66	. . . . 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 475	. . . 4 ⊢ (𝜑 → (◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0}) ∈ Fin)
99		snfi 8038	. . . 4 ⊢ {𝐴} ∈ Fin
100		unfi 8227	. . . 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 694	. . 3 ⊢ (𝜑 → ((◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴}) ∈ Fin)
102	52, 101	eqeltrd 2701	. 2 ⊢ (𝜑 → 𝑅 ∈ Fin)
103	52	fveq2d 6195	. . 3 ⊢ (𝜑 → (#‘𝑅) = (#‘((◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})))
104		hashcl 13147	. . . . . 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 11352	. . . 4 ⊢ (𝜑 → (#‘((◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ∈ ℝ)
107		hashcl 13147	. . . . . . 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 11352	. . . . 5 ⊢ (𝜑 → (#‘(◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0})) ∈ ℝ)
110		peano2re 10209	. . . . 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 23989	. . . . . 6 ⊢ (𝐹 ∈ (Poly‘ℂ) → (deg‘𝐹) ∈ ℕ0)
113	4, 112	syl 17	. . . . 5 ⊢ (𝜑 → (deg‘𝐹) ∈ ℕ0)
114	113	nn0red 11352	. . . 4 ⊢ (𝜑 → (deg‘𝐹) ∈ ℝ)
115		hashun2 13172	. . . . . 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 694	. . . . 5 ⊢ (𝜑 → (#‘((◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ≤ ((#‘(◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0})) + (#‘{𝐴})))
117		hashsng 13159	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (#‘{𝐴}) = 1)
118	11, 117	syl 17	. . . . . 6 ⊢ (𝜑 → (#‘{𝐴}) = 1)
119	118	oveq2d 6666	. . . . 5 ⊢ (𝜑 → ((#‘(◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0})) + (#‘{𝐴})) = ((#‘(◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0})) + 1))
120	116, 119	breqtrd 4679	. . . 4 ⊢ (𝜑 → (#‘((◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ≤ ((#‘(◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0})) + 1))
121	73	nn0red 11352	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ ℝ)
122		1red 10055	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℝ)
123	97	simprd 479	. . . . . . 7 ⊢ (𝜑 → (#‘(◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0})) ≤ (deg‘(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴})))))
124	123, 86	breqtrd 4679	. . . . . 6 ⊢ (𝜑 → (#‘(◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0})) ≤ 𝐷)
125	109, 121, 122, 124	leadd1dd 10641	. . . . 5 ⊢ (𝜑 → ((#‘(◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0})) + 1) ≤ (𝐷 + 1))
126	125, 78	breqtrrd 4681	. . . 4 ⊢ (𝜑 → ((#‘(◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0})) + 1) ≤ (deg‘𝐹))
127	106, 111, 114, 120, 126	letrd 10194	. . 3 ⊢ (𝜑 → (#‘((◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ≤ (deg‘𝐹))
128	103, 127	eqbrtrd 4675	. 2 ⊢ (𝜑 → (#‘𝑅) ≤ (deg‘𝐹))
129	102, 128	jca 554	1 ⊢ (𝜑 → (𝑅 ∈ Fin ∧ (#‘𝑅) ≤ (deg‘𝐹)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  Vcvv 3200   ∖ cdif 3571   ∪ cun 3572   ⊆ wss 3574  {csn 4177   class class class wbr 4653   × cxp 5112  ^◡ccnv 5113   “ cima 5117   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ∘_𝑓 cof 6895  Fincfn 7955  ℂcc 9934  ℝcr 9935  0cc0 9936  1c1 9937   + caddc 9939   · cmul 9941   ≤ cle 10075   − cmin 10266  ℕ₀cn0 11292  #chash 13117  0_𝑝c0p 23436  Polycply 23940  X_pcidp 23941  degcdgr 23943   quot cquot 24045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417  df-0p 23437  df-ply 23944  df-idp 23945  df-coe 23946  df-dgr 23947  df-quot 24046

This theorem is referenced by:  fta1  24063