Theorem fta1lem 25467

Description: Lemma for fta1 25468. (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 4720	. . . . . . . . . 10 ⊢ (𝐹 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ↔ (𝐹 ∈ (Poly‘ℂ) ∧ 𝐹 ≠ 0𝑝))
3	1, 2	sylib 217	. . . . . . . . 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 25359	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Poly‘ℂ) → 𝐹:ℂ⟶ℂ)
7		ffn 6600	. . . . . . . . . . 11 ⊢ (𝐹:ℂ⟶ℂ → 𝐹 Fn ℂ)
8		fniniseg 6937	. . . . . . . . . . 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 231	. . . . . . . . 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 2738	. . . . . . . . 9 ⊢ (Xp ∘f − (ℂ × {𝐴})) = (Xp ∘f − (ℂ × {𝐴}))
14	13	facth 25466	. . . . . . . 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 1370	. . . . . . 7 ⊢ (𝜑 → 𝐹 = ((Xp ∘f − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xp ∘f − (ℂ × {𝐴})))))
16	15	cnveqd 5784	. . . . . 6 ⊢ (𝜑 → ◡𝐹 = ◡((Xp ∘f − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xp ∘f − (ℂ × {𝐴})))))
17	16	imaeq1d 5968	. . . . 5 ⊢ (𝜑 → (◡𝐹 “ {0}) = (◡((Xp ∘f − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xp ∘f − (ℂ × {𝐴})))) “ {0}))
18		cnex 10952	. . . . . . 7 ⊢ ℂ ∈ V
19	18	a1i 11	. . . . . 6 ⊢ (𝜑 → ℂ ∈ V)
20		ssid 3943	. . . . . . . . 9 ⊢ ℂ ⊆ ℂ
21		ax-1cn 10929	. . . . . . . . 9 ⊢ 1 ∈ ℂ
22		plyid 25370	. . . . . . . . 9 ⊢ ((ℂ ⊆ ℂ ∧ 1 ∈ ℂ) → Xp ∈ (Poly‘ℂ))
23	20, 21, 22	mp2an 689	. . . . . . . 8 ⊢ Xp ∈ (Poly‘ℂ)
24		plyconst 25367	. . . . . . . . 9 ⊢ ((ℂ ⊆ ℂ ∧ 𝐴 ∈ ℂ) → (ℂ × {𝐴}) ∈ (Poly‘ℂ))
25	20, 11, 24	sylancr 587	. . . . . . . 8 ⊢ (𝜑 → (ℂ × {𝐴}) ∈ (Poly‘ℂ))
26		plysubcl 25383	. . . . . . . 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 25359	. . . . . . 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 25464	. . . . . . . . . . . 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 1142	. . . . . . . . . 10 ⊢ (𝜑 → (deg‘(Xp ∘f − (ℂ × {𝐴}))) = 1)
33		ax-1ne0 10940	. . . . . . . . . . 11 ⊢ 1 ≠ 0
34	33	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 1 ≠ 0)
35	32, 34	eqnetrd 3011	. . . . . . . . 9 ⊢ (𝜑 → (deg‘(Xp ∘f − (ℂ × {𝐴}))) ≠ 0)
36		fveq2 6774	. . . . . . . . . . 11 ⊢ ((Xp ∘f − (ℂ × {𝐴})) = 0𝑝 → (deg‘(Xp ∘f − (ℂ × {𝐴}))) = (deg‘0𝑝))
37		dgr0 25423	. . . . . . . . . . 11 ⊢ (deg‘0𝑝) = 0
38	36, 37	eqtrdi 2794	. . . . . . . . . 10 ⊢ ((Xp ∘f − (ℂ × {𝐴})) = 0𝑝 → (deg‘(Xp ∘f − (ℂ × {𝐴}))) = 0)
39	38	necon3i 2976	. . . . . . . . 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 25462	. . . . . . . 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 1370	. . . . . . 7 ⊢ (𝜑 → (𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) ∈ (Poly‘ℂ))
43		plyf 25359	. . . . . . 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 25442	. . . . . 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 1370	. . . . 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 1143	. . . . . 6 ⊢ (𝜑 → (◡(Xp ∘f − (ℂ × {𝐴})) “ {0}) = {𝐴})
48	47	uneq1d 4096	. . . . 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 2782	. . . 4 ⊢ (𝜑 → (◡𝐹 “ {0}) = ({𝐴} ∪ (◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})))
50		fta1.1	. . . 4 ⊢ 𝑅 = (◡𝐹 “ {0})
51		uncom 4087	. . . 4 ⊢ ((◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴}) = ({𝐴} ∪ (◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}))
52	49, 50, 51	3eqtr4g 2803	. . 3 ⊢ (𝜑 → 𝑅 = ((◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴}))
53	21	a1i 11	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℂ)
54		dgrcl 25394	. . . . . . . . 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 12295	. . . . . . 7 ⊢ (𝜑 → (deg‘(𝐹 quot (Xp ∘f − (ℂ × {𝐴})))) ∈ ℂ)
57		fta1.2	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ ℕ0)
58	57	nn0cnd 12295	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ ℂ)
59		addcom 11161	. . . . . . . . 9 ⊢ ((1 ∈ ℂ ∧ 𝐷 ∈ ℂ) → (1 + 𝐷) = (𝐷 + 1))
60	21, 58, 59	sylancr 587	. . . . . . . 8 ⊢ (𝜑 → (1 + 𝐷) = (𝐷 + 1))
61	15	fveq2d 6778	. . . . . . . . 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 2744	. . . . . . . . . . . 12 ⊢ (𝜑 → ((Xp ∘f − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xp ∘f − (ℂ × {𝐴})))) = 𝐹)
65		0cnd 10968	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ∈ ℂ)
66		mul01 11154	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → (𝑥 · 0) = 0)
67	66	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝑥 · 0) = 0)
68	19, 29, 65, 65, 67	caofid1 7566	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((Xp ∘f − (ℂ × {𝐴})) ∘f · (ℂ × {0})) = (ℂ × {0}))
69		df-0p 24834	. . . . . . . . . . . . . 14 ⊢ 0𝑝 = (ℂ × {0})
70	69	oveq2i 7286	. . . . . . . . . . . . 13 ⊢ ((Xp ∘f − (ℂ × {𝐴})) ∘f · 0𝑝) = ((Xp ∘f − (ℂ × {𝐴})) ∘f · (ℂ × {0}))
71	68, 70, 69	3eqtr4g 2803	. . . . . . . . . . . 12 ⊢ (𝜑 → ((Xp ∘f − (ℂ × {𝐴})) ∘f · 0𝑝) = 0𝑝)
72	63, 64, 71	3netr4d 3021	. . . . . . . . . . 11 ⊢ (𝜑 → ((Xp ∘f − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xp ∘f − (ℂ × {𝐴})))) ≠ ((Xp ∘f − (ℂ × {𝐴})) ∘f · 0𝑝))
73		oveq2 7283	. . . . . . . . . . . 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 2976	. . . . . . . . . . 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 2738	. . . . . . . . . . 11 ⊢ (deg‘(Xp ∘f − (ℂ × {𝐴}))) = (deg‘(Xp ∘f − (ℂ × {𝐴})))
77		eqid 2738	. . . . . . . . . . 11 ⊢ (deg‘(𝐹 quot (Xp ∘f − (ℂ × {𝐴})))) = (deg‘(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))))
78	76, 77	dgrmul 25431	. . . . . . . . . 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 836	. . . . . . . . 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 2786	. . . . . . . 8 ⊢ (𝜑 → (𝐷 + 1) = ((deg‘(Xp ∘f − (ℂ × {𝐴}))) + (deg‘(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))))))
81	32	oveq1d 7290	. . . . . . . 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 2783	. . . . . . 7 ⊢ (𝜑 → (1 + (deg‘(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))))) = (1 + 𝐷))
83	53, 56, 58, 82	addcanad 11180	. . . . . 6 ⊢ (𝜑 → (deg‘(𝐹 quot (Xp ∘f − (ℂ × {𝐴})))) = 𝐷)
84		fveqeq2 6783	. . . . . . . 8 ⊢ (𝑔 = (𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) → ((deg‘𝑔) = 𝐷 ↔ (deg‘(𝐹 quot (Xp ∘f − (ℂ × {𝐴})))) = 𝐷))
85		cnveq 5782	. . . . . . . . . . 11 ⊢ (𝑔 = (𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) → ◡𝑔 = ◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))))
86	85	imaeq1d 5968	. . . . . . . . . 10 ⊢ (𝑔 = (𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) → (◡𝑔 “ {0}) = (◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}))
87	86	eleq1d 2823	. . . . . . . . 9 ⊢ (𝑔 = (𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) → ((◡𝑔 “ {0}) ∈ Fin ↔ (◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}) ∈ Fin))
88	86	fveq2d 6778	. . . . . . . . . 10 ⊢ (𝑔 = (𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) → (♯‘(◡𝑔 “ {0})) = (♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})))
89		fveq2 6774	. . . . . . . . . 10 ⊢ (𝑔 = (𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) → (deg‘𝑔) = (deg‘(𝐹 quot (Xp ∘f − (ℂ × {𝐴})))))
90	88, 89	breq12d 5087	. . . . . . . . 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 631	. . . . . . . 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 345	. . . . . . 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 4720	. . . . . . . 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 3562	. . . . . 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 8834	. . . 4 ⊢ {𝐴} ∈ Fin
100		unfi 8955	. . . 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 2839	. 2 ⊢ (𝜑 → 𝑅 ∈ Fin)
103	52	fveq2d 6778	. . 3 ⊢ (𝜑 → (♯‘𝑅) = (♯‘((◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})))
104		hashcl 14071	. . . . . 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 12294	. . . 4 ⊢ (𝜑 → (♯‘((◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ∈ ℝ)
107		hashcl 14071	. . . . . . 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 12294	. . . . 5 ⊢ (𝜑 → (♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})) ∈ ℝ)
110		peano2re 11148	. . . . 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 25394	. . . . . 6 ⊢ (𝐹 ∈ (Poly‘ℂ) → (deg‘𝐹) ∈ ℕ0)
113	4, 112	syl 17	. . . . 5 ⊢ (𝜑 → (deg‘𝐹) ∈ ℕ0)
114	113	nn0red 12294	. . . 4 ⊢ (𝜑 → (deg‘𝐹) ∈ ℝ)
115		hashun2 14098	. . . . . 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 14084	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (♯‘{𝐴}) = 1)
118	11, 117	syl 17	. . . . . 6 ⊢ (𝜑 → (♯‘{𝐴}) = 1)
119	118	oveq2d 7291	. . . . 5 ⊢ (𝜑 → ((♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})) + (♯‘{𝐴})) = ((♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})) + 1))
120	116, 119	breqtrd 5100	. . . 4 ⊢ (𝜑 → (♯‘((◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ≤ ((♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})) + 1))
121	57	nn0red 12294	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ ℝ)
122		1red 10976	. . . . . 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 5100	. . . . . 6 ⊢ (𝜑 → (♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})) ≤ 𝐷)
125	109, 121, 122, 124	leadd1dd 11589	. . . . 5 ⊢ (𝜑 → ((♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})) + 1) ≤ (𝐷 + 1))
126	125, 62	breqtrrd 5102	. . . 4 ⊢ (𝜑 → ((♯‘(◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0})) + 1) ≤ (deg‘𝐹))
127	106, 111, 114, 120, 126	letrd 11132	. . 3 ⊢ (𝜑 → (♯‘((◡(𝐹 quot (Xp ∘f − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ≤ (deg‘𝐹))
128	103, 127	eqbrtrd 5096	. 2 ⊢ (𝜑 → (♯‘𝑅) ≤ (deg‘𝐹))
129	102, 128	jca 512	1 ⊢ (𝜑 → (𝑅 ∈ Fin ∧ (♯‘𝑅) ≤ (deg‘𝐹)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  Vcvv 3432   ∖ cdif 3884   ∪ cun 3885   ⊆ wss 3887  {csn 4561   class class class wbr 5074   × cxp 5587  ^◡ccnv 5588   “ cima 5592   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   ∘_f cof 7531  Fincfn 8733  ℂcc 10869  ℝcr 10870  0cc0 10871  1c1 10872   + caddc 10874   · cmul 10876   ≤ cle 11010   − cmin 11205  ℕ₀cn0 12233  ♯chash 14044  0_𝑝c0p 24833  Polycply 25345  X_pcidp 25346  degcdgr 25348   quot cquot 25450

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-of 7533  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-oadd 8301  df-er 8498  df-map 8617  df-pm 8618  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-sup 9201  df-inf 9202  df-oi 9269  df-dju 9659  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-n0 12234  df-xnn0 12306  df-z 12320  df-uz 12583  df-rp 12731  df-fz 13240  df-fzo 13383  df-fl 13512  df-seq 13722  df-exp 13783  df-hash 14045  df-cj 14810  df-re 14811  df-im 14812  df-sqrt 14946  df-abs 14947  df-clim 15197  df-rlim 15198  df-sum 15398  df-0p 24834  df-ply 25349  df-idp 25350  df-coe 25351  df-dgr 25352  df-quot 25451

This theorem is referenced by:  fta1  25468