Theorem fta1lem 23866

Description: Lemma for fta1 23867. (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 4260	. . . . . . . . . 10 ⊢ (𝐹 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ↔ (𝐹 ∈ (Poly‘ℂ) ∧ 𝐹 ≠ 0𝑝))
3	1, 2	sylib 207	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 ∈ (Poly‘ℂ) ∧ 𝐹 ≠ 0𝑝))
4	3	simpld 474	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (Poly‘ℂ))
5		fta1.5	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ (◡𝐹 “ {0}))
6		plyf 23758	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Poly‘ℂ) → 𝐹:ℂ⟶ℂ)
7		ffn 5958	. . . . . . . . . . 11 ⊢ (𝐹:ℂ⟶ℂ → 𝐹 Fn ℂ)
8		fniniseg 6246	. . . . . . . . . . 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 221	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ∈ ℂ ∧ (𝐹‘𝐴) = 0))
11	10	simpld 474	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℂ)
12	10	simprd 478	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝐴) = 0)
13		eqid 2610	. . . . . . . . 9 ⊢ (Xp ∘𝑓 − (ℂ × {𝐴})) = (Xp ∘𝑓 − (ℂ × {𝐴}))
14	13	facth 23865	. . . . . . . 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 1318	. . . . . . 7 ⊢ (𝜑 → 𝐹 = ((Xp ∘𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴})))))
16	15	cnveqd 5220	. . . . . 6 ⊢ (𝜑 → ◡𝐹 = ◡((Xp ∘𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴})))))
17	16	imaeq1d 5384	. . . . 5 ⊢ (𝜑 → (◡𝐹 “ {0}) = (◡((Xp ∘𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴})))) “ {0}))
18		cnex 9896	. . . . . . 7 ⊢ ℂ ∈ V
19	18	a1i 11	. . . . . 6 ⊢ (𝜑 → ℂ ∈ V)
20		ssid 3587	. . . . . . . . 9 ⊢ ℂ ⊆ ℂ
21		ax-1cn 9873	. . . . . . . . 9 ⊢ 1 ∈ ℂ
22		plyid 23769	. . . . . . . . 9 ⊢ ((ℂ ⊆ ℂ ∧ 1 ∈ ℂ) → Xp ∈ (Poly‘ℂ))
23	20, 21, 22	mp2an 704	. . . . . . . 8 ⊢ Xp ∈ (Poly‘ℂ)
24		plyconst 23766	. . . . . . . . 9 ⊢ ((ℂ ⊆ ℂ ∧ 𝐴 ∈ ℂ) → (ℂ × {𝐴}) ∈ (Poly‘ℂ))
25	20, 11, 24	sylancr 694	. . . . . . . 8 ⊢ (𝜑 → (ℂ × {𝐴}) ∈ (Poly‘ℂ))
26		plysubcl 23782	. . . . . . . 8 ⊢ ((Xp ∈ (Poly‘ℂ) ∧ (ℂ × {𝐴}) ∈ (Poly‘ℂ)) → (Xp ∘𝑓 − (ℂ × {𝐴})) ∈ (Poly‘ℂ))
27	23, 25, 26	sylancr 694	. . . . . . 7 ⊢ (𝜑 → (Xp ∘𝑓 − (ℂ × {𝐴})) ∈ (Poly‘ℂ))
28		plyf 23758	. . . . . . 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 23863	. . . . . . . . . . . 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 1067	. . . . . . . . . 10 ⊢ (𝜑 → (deg‘(Xp ∘𝑓 − (ℂ × {𝐴}))) = 1)
33		ax-1ne0 9884	. . . . . . . . . . 11 ⊢ 1 ≠ 0
34	33	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 1 ≠ 0)
35	32, 34	eqnetrd 2849	. . . . . . . . 9 ⊢ (𝜑 → (deg‘(Xp ∘𝑓 − (ℂ × {𝐴}))) ≠ 0)
36		fveq2 6103	. . . . . . . . . . 11 ⊢ ((Xp ∘𝑓 − (ℂ × {𝐴})) = 0𝑝 → (deg‘(Xp ∘𝑓 − (ℂ × {𝐴}))) = (deg‘0𝑝))
37		dgr0 23822	. . . . . . . . . . 11 ⊢ (deg‘0𝑝) = 0
38	36, 37	syl6eq 2660	. . . . . . . . . 10 ⊢ ((Xp ∘𝑓 − (ℂ × {𝐴})) = 0𝑝 → (deg‘(Xp ∘𝑓 − (ℂ × {𝐴}))) = 0)
39	38	necon3i 2814	. . . . . . . . 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 23861	. . . . . . . 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 1318	. . . . . . 7 ⊢ (𝜑 → (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) ∈ (Poly‘ℂ))
43		plyf 23758	. . . . . . 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 23841	. . . . . 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 1318	. . . . 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 1068	. . . . . 6 ⊢ (𝜑 → (◡(Xp ∘𝑓 − (ℂ × {𝐴})) “ {0}) = {𝐴})
48	47	uneq1d 3728	. . . . 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 2648	. . . 4 ⊢ (𝜑 → (◡𝐹 “ {0}) = ({𝐴} ∪ (◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0})))
50		fta1.1	. . . 4 ⊢ 𝑅 = (◡𝐹 “ {0})
51		uncom 3719	. . . 4 ⊢ ((◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴}) = ({𝐴} ∪ (◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0}))
52	49, 50, 51	3eqtr4g 2669	. . 3 ⊢ (𝜑 → 𝑅 = ((◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴}))
53	3	simprd 478	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ≠ 0𝑝)
54	15	eqcomd 2616	. . . . . . . . 9 ⊢ (𝜑 → ((Xp ∘𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴})))) = 𝐹)
55		0cnd 9912	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ∈ ℂ)
56		mul01 10094	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ → (𝑥 · 0) = 0)
57	56	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝑥 · 0) = 0)
58	19, 29, 55, 55, 57	caofid1 6825	. . . . . . . . . 10 ⊢ (𝜑 → ((Xp ∘𝑓 − (ℂ × {𝐴})) ∘𝑓 · (ℂ × {0})) = (ℂ × {0}))
59		df-0p 23243	. . . . . . . . . . 11 ⊢ 0𝑝 = (ℂ × {0})
60	59	oveq2i 6560	. . . . . . . . . 10 ⊢ ((Xp ∘𝑓 − (ℂ × {𝐴})) ∘𝑓 · 0𝑝) = ((Xp ∘𝑓 − (ℂ × {𝐴})) ∘𝑓 · (ℂ × {0}))
61	58, 60, 59	3eqtr4g 2669	. . . . . . . . 9 ⊢ (𝜑 → ((Xp ∘𝑓 − (ℂ × {𝐴})) ∘𝑓 · 0𝑝) = 0𝑝)
62	53, 54, 61	3netr4d 2859	. . . . . . . 8 ⊢ (𝜑 → ((Xp ∘𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴})))) ≠ ((Xp ∘𝑓 − (ℂ × {𝐴})) ∘𝑓 · 0𝑝))
63		oveq2 6557	. . . . . . . . 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 2814	. . . . . . . 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 4260	. . . . . . 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 695	. . . . . 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 23793	. . . . . . . . 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 11230	. . . . . . 7 ⊢ (𝜑 → (deg‘(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴})))) ∈ ℂ)
73		fta1.2	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ ℕ0)
74	73	nn0cnd 11230	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ ℂ)
75		addcom 10101	. . . . . . . . 9 ⊢ ((1 ∈ ℂ ∧ 𝐷 ∈ ℂ) → (1 + 𝐷) = (𝐷 + 1))
76	21, 74, 75	sylancr 694	. . . . . . . 8 ⊢ (𝜑 → (1 + 𝐷) = (𝐷 + 1))
77	15	fveq2d 6107	. . . . . . . . 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 2610	. . . . . . . . . . 11 ⊢ (deg‘(Xp ∘𝑓 − (ℂ × {𝐴}))) = (deg‘(Xp ∘𝑓 − (ℂ × {𝐴})))
80		eqid 2610	. . . . . . . . . . 11 ⊢ (deg‘(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴})))) = (deg‘(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))))
81	79, 80	dgrmul 23830	. . . . . . . . . 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 1319	. . . . . . . . 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 2652	. . . . . . . 8 ⊢ (𝜑 → (𝐷 + 1) = ((deg‘(Xp ∘𝑓 − (ℂ × {𝐴}))) + (deg‘(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))))))
84	32	oveq1d 6564	. . . . . . . 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 2649	. . . . . . 7 ⊢ (𝜑 → (1 + (deg‘(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))))) = (1 + 𝐷))
86	69, 72, 74, 85	addcanad 10120	. . . . . 6 ⊢ (𝜑 → (deg‘(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴})))) = 𝐷)
87		fveq2 6103	. . . . . . . . 9 ⊢ (𝑔 = (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) → (deg‘𝑔) = (deg‘(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴})))))
88	87	eqeq1d 2612	. . . . . . . 8 ⊢ (𝑔 = (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) → ((deg‘𝑔) = 𝐷 ↔ (deg‘(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴})))) = 𝐷))
89		cnveq 5218	. . . . . . . . . . 11 ⊢ (𝑔 = (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) → ◡𝑔 = ◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))))
90	89	imaeq1d 5384	. . . . . . . . . 10 ⊢ (𝑔 = (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) → (◡𝑔 “ {0}) = (◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0}))
91	90	eleq1d 2672	. . . . . . . . 9 ⊢ (𝑔 = (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) → ((◡𝑔 “ {0}) ∈ Fin ↔ (◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0}) ∈ Fin))
92	90	fveq2d 6107	. . . . . . . . . 10 ⊢ (𝑔 = (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) → (#‘(◡𝑔 “ {0})) = (#‘(◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0})))
93	92, 87	breq12d 4596	. . . . . . . . 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 743	. . . . . . . 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 333	. . . . . . 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 3278	. . . . . 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 64	. . . . 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 474	. . . 4 ⊢ (𝜑 → (◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0}) ∈ Fin)
99		snfi 7923	. . . 4 ⊢ {𝐴} ∈ Fin
100		unfi 8112	. . . 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 693	. . 3 ⊢ (𝜑 → ((◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴}) ∈ Fin)
102	52, 101	eqeltrd 2688	. 2 ⊢ (𝜑 → 𝑅 ∈ Fin)
103	52	fveq2d 6107	. . 3 ⊢ (𝜑 → (#‘𝑅) = (#‘((◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})))
104		hashcl 13009	. . . . . 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 11229	. . . 4 ⊢ (𝜑 → (#‘((◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ∈ ℝ)
107		hashcl 13009	. . . . . . 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 11229	. . . . 5 ⊢ (𝜑 → (#‘(◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0})) ∈ ℝ)
110		peano2re 10088	. . . . 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 23793	. . . . . 6 ⊢ (𝐹 ∈ (Poly‘ℂ) → (deg‘𝐹) ∈ ℕ0)
113	4, 112	syl 17	. . . . 5 ⊢ (𝜑 → (deg‘𝐹) ∈ ℕ0)
114	113	nn0red 11229	. . . 4 ⊢ (𝜑 → (deg‘𝐹) ∈ ℝ)
115		hashun2 13033	. . . . . 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 693	. . . . 5 ⊢ (𝜑 → (#‘((◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ≤ ((#‘(◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0})) + (#‘{𝐴})))
117		hashsng 13020	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (#‘{𝐴}) = 1)
118	11, 117	syl 17	. . . . . 6 ⊢ (𝜑 → (#‘{𝐴}) = 1)
119	118	oveq2d 6565	. . . . 5 ⊢ (𝜑 → ((#‘(◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0})) + (#‘{𝐴})) = ((#‘(◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0})) + 1))
120	116, 119	breqtrd 4609	. . . 4 ⊢ (𝜑 → (#‘((◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ≤ ((#‘(◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0})) + 1))
121	73	nn0red 11229	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ ℝ)
122		1red 9934	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℝ)
123	97	simprd 478	. . . . . . 7 ⊢ (𝜑 → (#‘(◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0})) ≤ (deg‘(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴})))))
124	123, 86	breqtrd 4609	. . . . . 6 ⊢ (𝜑 → (#‘(◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0})) ≤ 𝐷)
125	109, 121, 122, 124	leadd1dd 10520	. . . . 5 ⊢ (𝜑 → ((#‘(◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0})) + 1) ≤ (𝐷 + 1))
126	125, 78	breqtrrd 4611	. . . 4 ⊢ (𝜑 → ((#‘(◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0})) + 1) ≤ (deg‘𝐹))
127	106, 111, 114, 120, 126	letrd 10073	. . 3 ⊢ (𝜑 → (#‘((◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ≤ (deg‘𝐹))
128	103, 127	eqbrtrd 4605	. 2 ⊢ (𝜑 → (#‘𝑅) ≤ (deg‘𝐹))
129	102, 128	jca 553	1 ⊢ (𝜑 → (𝑅 ∈ Fin ∧ (#‘𝑅) ≤ (deg‘𝐹)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538   ⊆ wss 3540  {csn 4125   class class class wbr 4583   × cxp 5036  ^◡ccnv 5037   “ cima 5041   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ∘_𝑓 cof 6793  Fincfn 7841  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   · cmul 9820   ≤ cle 9954   − cmin 10145  ℕ₀cn0 11169  #chash 12979  0_𝑝c0p 23242  Polycply 23744  X_pcidp 23745  degcdgr 23747   quot cquot 23849

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-rlim 14068  df-sum 14265  df-0p 23243  df-ply 23748  df-idp 23749  df-coe 23750  df-dgr 23751  df-quot 23850

This theorem is referenced by:  fta1  23867