Theorem fta1lem 24583

Description: Lemma for fta1 24584. (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 4632	. . . . . . . . . 10 ⊢ (𝐹 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ↔ (𝐹 ∈ (Poly‘ℂ) ∧ 𝐹 ≠ 0𝑝))
3	1, 2	sylib 219	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 ∈ (Poly‘ℂ) ∧ 𝐹 ≠ 0𝑝))
4	3	simpld 495	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (Poly‘ℂ))
5		fta1.5	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ (◡𝐹 “ {0}))
6		plyf 24475	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Poly‘ℂ) → 𝐹:ℂ⟶ℂ)
7		ffn 6389	. . . . . . . . . . 11 ⊢ (𝐹:ℂ⟶ℂ → 𝐹 Fn ℂ)
8		fniniseg 6702	. . . . . . . . . . 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 233	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ∈ ℂ ∧ (𝐹‘𝐴) = 0))
11	10	simpld 495	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℂ)
12	10	simprd 496	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝐴) = 0)
13		eqid 2797	. . . . . . . . 9 ⊢ (Xp ∘𝑓 − (ℂ × {𝐴})) = (Xp ∘𝑓 − (ℂ × {𝐴}))
14	13	facth 24582	. . . . . . . 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 1364	. . . . . . 7 ⊢ (𝜑 → 𝐹 = ((Xp ∘𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴})))))
16	15	cnveqd 5639	. . . . . 6 ⊢ (𝜑 → ◡𝐹 = ◡((Xp ∘𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴})))))
17	16	imaeq1d 5812	. . . . 5 ⊢ (𝜑 → (◡𝐹 “ {0}) = (◡((Xp ∘𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴})))) “ {0}))
18		cnex 10471	. . . . . . 7 ⊢ ℂ ∈ V
19	18	a1i 11	. . . . . 6 ⊢ (𝜑 → ℂ ∈ V)
20		ssid 3916	. . . . . . . . 9 ⊢ ℂ ⊆ ℂ
21		ax-1cn 10448	. . . . . . . . 9 ⊢ 1 ∈ ℂ
22		plyid 24486	. . . . . . . . 9 ⊢ ((ℂ ⊆ ℂ ∧ 1 ∈ ℂ) → Xp ∈ (Poly‘ℂ))
23	20, 21, 22	mp2an 688	. . . . . . . 8 ⊢ Xp ∈ (Poly‘ℂ)
24		plyconst 24483	. . . . . . . . 9 ⊢ ((ℂ ⊆ ℂ ∧ 𝐴 ∈ ℂ) → (ℂ × {𝐴}) ∈ (Poly‘ℂ))
25	20, 11, 24	sylancr 587	. . . . . . . 8 ⊢ (𝜑 → (ℂ × {𝐴}) ∈ (Poly‘ℂ))
26		plysubcl 24499	. . . . . . . 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 24475	. . . . . . 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 24580	. . . . . . . . . . . 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 1136	. . . . . . . . . 10 ⊢ (𝜑 → (deg‘(Xp ∘𝑓 − (ℂ × {𝐴}))) = 1)
33		ax-1ne0 10459	. . . . . . . . . . 11 ⊢ 1 ≠ 0
34	33	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 1 ≠ 0)
35	32, 34	eqnetrd 3053	. . . . . . . . 9 ⊢ (𝜑 → (deg‘(Xp ∘𝑓 − (ℂ × {𝐴}))) ≠ 0)
36		fveq2 6545	. . . . . . . . . . 11 ⊢ ((Xp ∘𝑓 − (ℂ × {𝐴})) = 0𝑝 → (deg‘(Xp ∘𝑓 − (ℂ × {𝐴}))) = (deg‘0𝑝))
37		dgr0 24539	. . . . . . . . . . 11 ⊢ (deg‘0𝑝) = 0
38	36, 37	syl6eq 2849	. . . . . . . . . 10 ⊢ ((Xp ∘𝑓 − (ℂ × {𝐴})) = 0𝑝 → (deg‘(Xp ∘𝑓 − (ℂ × {𝐴}))) = 0)
39	38	necon3i 3018	. . . . . . . . 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 24578	. . . . . . . 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 1364	. . . . . . 7 ⊢ (𝜑 → (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) ∈ (Poly‘ℂ))
43		plyf 24475	. . . . . . 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 24558	. . . . . 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 1364	. . . . 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 1137	. . . . . 6 ⊢ (𝜑 → (◡(Xp ∘𝑓 − (ℂ × {𝐴})) “ {0}) = {𝐴})
48	47	uneq1d 4065	. . . . 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 2837	. . . 4 ⊢ (𝜑 → (◡𝐹 “ {0}) = ({𝐴} ∪ (◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0})))
50		fta1.1	. . . 4 ⊢ 𝑅 = (◡𝐹 “ {0})
51		uncom 4056	. . . 4 ⊢ ((◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴}) = ({𝐴} ∪ (◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0}))
52	49, 50, 51	3eqtr4g 2858	. . 3 ⊢ (𝜑 → 𝑅 = ((◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴}))
53	21	a1i 11	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℂ)
54		dgrcl 24510	. . . . . . . . 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 11811	. . . . . . 7 ⊢ (𝜑 → (deg‘(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴})))) ∈ ℂ)
57		fta1.2	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ ℕ0)
58	57	nn0cnd 11811	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ ℂ)
59		addcom 10679	. . . . . . . . 9 ⊢ ((1 ∈ ℂ ∧ 𝐷 ∈ ℂ) → (1 + 𝐷) = (𝐷 + 1))
60	21, 58, 59	sylancr 587	. . . . . . . 8 ⊢ (𝜑 → (1 + 𝐷) = (𝐷 + 1))
61	15	fveq2d 6549	. . . . . . . . 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 496	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ≠ 0𝑝)
64	15	eqcomd 2803	. . . . . . . . . . . 12 ⊢ (𝜑 → ((Xp ∘𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴})))) = 𝐹)
65		0cnd 10487	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ∈ ℂ)
66		mul01 10672	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → (𝑥 · 0) = 0)
67	66	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝑥 · 0) = 0)
68	19, 29, 65, 65, 67	caofid1 7304	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((Xp ∘𝑓 − (ℂ × {𝐴})) ∘𝑓 · (ℂ × {0})) = (ℂ × {0}))
69		df-0p 23958	. . . . . . . . . . . . . 14 ⊢ 0𝑝 = (ℂ × {0})
70	69	oveq2i 7034	. . . . . . . . . . . . 13 ⊢ ((Xp ∘𝑓 − (ℂ × {𝐴})) ∘𝑓 · 0𝑝) = ((Xp ∘𝑓 − (ℂ × {𝐴})) ∘𝑓 · (ℂ × {0}))
71	68, 70, 69	3eqtr4g 2858	. . . . . . . . . . . 12 ⊢ (𝜑 → ((Xp ∘𝑓 − (ℂ × {𝐴})) ∘𝑓 · 0𝑝) = 0𝑝)
72	63, 64, 71	3netr4d 3063	. . . . . . . . . . 11 ⊢ (𝜑 → ((Xp ∘𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴})))) ≠ ((Xp ∘𝑓 − (ℂ × {𝐴})) ∘𝑓 · 0𝑝))
73		oveq2 7031	. . . . . . . . . . . 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 3018	. . . . . . . . . . 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 2797	. . . . . . . . . . 11 ⊢ (deg‘(Xp ∘𝑓 − (ℂ × {𝐴}))) = (deg‘(Xp ∘𝑓 − (ℂ × {𝐴})))
77		eqid 2797	. . . . . . . . . . 11 ⊢ (deg‘(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴})))) = (deg‘(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))))
78	76, 77	dgrmul 24547	. . . . . . . . . 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 835	. . . . . . . . 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 2841	. . . . . . . 8 ⊢ (𝜑 → (𝐷 + 1) = ((deg‘(Xp ∘𝑓 − (ℂ × {𝐴}))) + (deg‘(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))))))
81	32	oveq1d 7038	. . . . . . . 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 2838	. . . . . . 7 ⊢ (𝜑 → (1 + (deg‘(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))))) = (1 + 𝐷))
83	53, 56, 58, 82	addcanad 10698	. . . . . 6 ⊢ (𝜑 → (deg‘(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴})))) = 𝐷)
84		fveqeq2 6554	. . . . . . . 8 ⊢ (𝑔 = (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) → ((deg‘𝑔) = 𝐷 ↔ (deg‘(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴})))) = 𝐷))
85		cnveq 5637	. . . . . . . . . . 11 ⊢ (𝑔 = (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) → ◡𝑔 = ◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))))
86	85	imaeq1d 5812	. . . . . . . . . 10 ⊢ (𝑔 = (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) → (◡𝑔 “ {0}) = (◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0}))
87	86	eleq1d 2869	. . . . . . . . 9 ⊢ (𝑔 = (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) → ((◡𝑔 “ {0}) ∈ Fin ↔ (◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0}) ∈ Fin))
88	86	fveq2d 6549	. . . . . . . . . 10 ⊢ (𝑔 = (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) → (♯‘(◡𝑔 “ {0})) = (♯‘(◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0})))
89		fveq2 6545	. . . . . . . . . 10 ⊢ (𝑔 = (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) → (deg‘𝑔) = (deg‘(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴})))))
90	88, 89	breq12d 4981	. . . . . . . . 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 630	. . . . . . . 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 346	. . . . . . 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 4632	. . . . . . . 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 3567	. . . . . 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 495	. . . 4 ⊢ (𝜑 → (◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0}) ∈ Fin)
99		snfi 8449	. . . 4 ⊢ {𝐴} ∈ Fin
100		unfi 8638	. . . 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 2885	. 2 ⊢ (𝜑 → 𝑅 ∈ Fin)
103	52	fveq2d 6549	. . 3 ⊢ (𝜑 → (♯‘𝑅) = (♯‘((◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})))
104		hashcl 13571	. . . . . 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 11810	. . . 4 ⊢ (𝜑 → (♯‘((◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ∈ ℝ)
107		hashcl 13571	. . . . . . 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 11810	. . . . 5 ⊢ (𝜑 → (♯‘(◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0})) ∈ ℝ)
110		peano2re 10666	. . . . 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 24510	. . . . . 6 ⊢ (𝐹 ∈ (Poly‘ℂ) → (deg‘𝐹) ∈ ℕ0)
113	4, 112	syl 17	. . . . 5 ⊢ (𝜑 → (deg‘𝐹) ∈ ℕ0)
114	113	nn0red 11810	. . . 4 ⊢ (𝜑 → (deg‘𝐹) ∈ ℝ)
115		hashun2 13596	. . . . . 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 13583	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (♯‘{𝐴}) = 1)
118	11, 117	syl 17	. . . . . 6 ⊢ (𝜑 → (♯‘{𝐴}) = 1)
119	118	oveq2d 7039	. . . . 5 ⊢ (𝜑 → ((♯‘(◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0})) + (♯‘{𝐴})) = ((♯‘(◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0})) + 1))
120	116, 119	breqtrd 4994	. . . 4 ⊢ (𝜑 → (♯‘((◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ≤ ((♯‘(◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0})) + 1))
121	57	nn0red 11810	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ ℝ)
122		1red 10495	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℝ)
123	97	simprd 496	. . . . . . 7 ⊢ (𝜑 → (♯‘(◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0})) ≤ (deg‘(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴})))))
124	123, 83	breqtrd 4994	. . . . . 6 ⊢ (𝜑 → (♯‘(◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0})) ≤ 𝐷)
125	109, 121, 122, 124	leadd1dd 11108	. . . . 5 ⊢ (𝜑 → ((♯‘(◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0})) + 1) ≤ (𝐷 + 1))
126	125, 62	breqtrrd 4996	. . . 4 ⊢ (𝜑 → ((♯‘(◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0})) + 1) ≤ (deg‘𝐹))
127	106, 111, 114, 120, 126	letrd 10650	. . 3 ⊢ (𝜑 → (♯‘((◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ≤ (deg‘𝐹))
128	103, 127	eqbrtrd 4990	. 2 ⊢ (𝜑 → (♯‘𝑅) ≤ (deg‘𝐹))
129	102, 128	jca 512	1 ⊢ (𝜑 → (𝑅 ∈ Fin ∧ (♯‘𝑅) ≤ (deg‘𝐹)))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1080   = wceq 1525   ∈ wcel 2083   ≠ wne 2986  ∀wral 3107  Vcvv 3440   ∖ cdif 3862   ∪ cun 3863   ⊆ wss 3865  {csn 4478   class class class wbr 4968   × cxp 5448  ^◡ccnv 5449   “ cima 5453   Fn wfn 6227  ⟶wf 6228  ‘cfv 6232  (class class class)co 7023   ∘_𝑓 cof 7272  Fincfn 8364  ℂcc 10388  ℝcr 10389  0cc0 10390  1c1 10391   + caddc 10393   · cmul 10395   ≤ cle 10529   − cmin 10723  ℕ₀cn0 11751  ♯chash 13544  0_𝑝c0p 23957  Polycply 24461  X_pcidp 24462  degcdgr 24464   quot cquot 24566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326  ax-inf2 8957  ax-cnex 10446  ax-resscn 10447  ax-1cn 10448  ax-icn 10449  ax-addcl 10450  ax-addrcl 10451  ax-mulcl 10452  ax-mulrcl 10453  ax-mulcom 10454  ax-addass 10455  ax-mulass 10456  ax-distr 10457  ax-i2m1 10458  ax-1ne0 10459  ax-1rid 10460  ax-rnegex 10461  ax-rrecex 10462  ax-cnre 10463  ax-pre-lttri 10464  ax-pre-lttrn 10465  ax-pre-ltadd 10466  ax-pre-mulgt0 10467  ax-pre-sup 10468  ax-addf 10469

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-fal 1538  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-int 4789  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-se 5410  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-isom 6241  df-riota 6984  df-ov 7026  df-oprab 7027  df-mpo 7028  df-of 7274  df-om 7444  df-1st 7552  df-2nd 7553  df-wrecs 7805  df-recs 7867  df-rdg 7905  df-1o 7960  df-oadd 7964  df-er 8146  df-map 8265  df-pm 8266  df-en 8365  df-dom 8366  df-sdom 8367  df-fin 8368  df-sup 8759  df-inf 8760  df-oi 8827  df-dju 9183  df-card 9221  df-pnf 10530  df-mnf 10531  df-xr 10532  df-ltxr 10533  df-le 10534  df-sub 10725  df-neg 10726  df-div 11152  df-nn 11493  df-2 11554  df-3 11555  df-n0 11752  df-xnn0 11822  df-z 11836  df-uz 12098  df-rp 12244  df-fz 12747  df-fzo 12888  df-fl 13016  df-seq 13224  df-exp 13284  df-hash 13545  df-cj 14296  df-re 14297  df-im 14298  df-sqrt 14432  df-abs 14433  df-clim 14683  df-rlim 14684  df-sum 14881  df-0p 23958  df-ply 24465  df-idp 24466  df-coe 24467  df-dgr 24468  df-quot 24567

This theorem is referenced by:  fta1  24584