Theorem fta1lem 24356

Description: Lemma for fta1 24357. (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 4474	. . . . . . . . . 10 ⊢ (𝐹 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ↔ (𝐹 ∈ (Poly‘ℂ) ∧ 𝐹 ≠ 0𝑝))
3	1, 2	sylib 209	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 ∈ (Poly‘ℂ) ∧ 𝐹 ≠ 0𝑝))
4	3	simpld 488	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (Poly‘ℂ))
5		fta1.5	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ (◡𝐹 “ {0}))
6		plyf 24248	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Poly‘ℂ) → 𝐹:ℂ⟶ℂ)
7		ffn 6225	. . . . . . . . . . 11 ⊢ (𝐹:ℂ⟶ℂ → 𝐹 Fn ℂ)
8		fniniseg 6530	. . . . . . . . . . 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 223	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ∈ ℂ ∧ (𝐹‘𝐴) = 0))
11	10	simpld 488	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℂ)
12	10	simprd 489	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝐴) = 0)
13		eqid 2765	. . . . . . . . 9 ⊢ (Xp ∘𝑓 − (ℂ × {𝐴})) = (Xp ∘𝑓 − (ℂ × {𝐴}))
14	13	facth 24355	. . . . . . . 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 1490	. . . . . . 7 ⊢ (𝜑 → 𝐹 = ((Xp ∘𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴})))))
16	15	cnveqd 5468	. . . . . 6 ⊢ (𝜑 → ◡𝐹 = ◡((Xp ∘𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴})))))
17	16	imaeq1d 5649	. . . . 5 ⊢ (𝜑 → (◡𝐹 “ {0}) = (◡((Xp ∘𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴})))) “ {0}))
18		cnex 10272	. . . . . . 7 ⊢ ℂ ∈ V
19	18	a1i 11	. . . . . 6 ⊢ (𝜑 → ℂ ∈ V)
20		ssid 3785	. . . . . . . . 9 ⊢ ℂ ⊆ ℂ
21		ax-1cn 10249	. . . . . . . . 9 ⊢ 1 ∈ ℂ
22		plyid 24259	. . . . . . . . 9 ⊢ ((ℂ ⊆ ℂ ∧ 1 ∈ ℂ) → Xp ∈ (Poly‘ℂ))
23	20, 21, 22	mp2an 683	. . . . . . . 8 ⊢ Xp ∈ (Poly‘ℂ)
24		plyconst 24256	. . . . . . . . 9 ⊢ ((ℂ ⊆ ℂ ∧ 𝐴 ∈ ℂ) → (ℂ × {𝐴}) ∈ (Poly‘ℂ))
25	20, 11, 24	sylancr 581	. . . . . . . 8 ⊢ (𝜑 → (ℂ × {𝐴}) ∈ (Poly‘ℂ))
26		plysubcl 24272	. . . . . . . 8 ⊢ ((Xp ∈ (Poly‘ℂ) ∧ (ℂ × {𝐴}) ∈ (Poly‘ℂ)) → (Xp ∘𝑓 − (ℂ × {𝐴})) ∈ (Poly‘ℂ))
27	23, 25, 26	sylancr 581	. . . . . . 7 ⊢ (𝜑 → (Xp ∘𝑓 − (ℂ × {𝐴})) ∈ (Poly‘ℂ))
28		plyf 24248	. . . . . . 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 24353	. . . . . . . . . . . 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 1173	. . . . . . . . . 10 ⊢ (𝜑 → (deg‘(Xp ∘𝑓 − (ℂ × {𝐴}))) = 1)
33		ax-1ne0 10260	. . . . . . . . . . 11 ⊢ 1 ≠ 0
34	33	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 1 ≠ 0)
35	32, 34	eqnetrd 3004	. . . . . . . . 9 ⊢ (𝜑 → (deg‘(Xp ∘𝑓 − (ℂ × {𝐴}))) ≠ 0)
36		fveq2 6377	. . . . . . . . . . 11 ⊢ ((Xp ∘𝑓 − (ℂ × {𝐴})) = 0𝑝 → (deg‘(Xp ∘𝑓 − (ℂ × {𝐴}))) = (deg‘0𝑝))
37		dgr0 24312	. . . . . . . . . . 11 ⊢ (deg‘0𝑝) = 0
38	36, 37	syl6eq 2815	. . . . . . . . . 10 ⊢ ((Xp ∘𝑓 − (ℂ × {𝐴})) = 0𝑝 → (deg‘(Xp ∘𝑓 − (ℂ × {𝐴}))) = 0)
39	38	necon3i 2969	. . . . . . . . 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 24351	. . . . . . . 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 1490	. . . . . . 7 ⊢ (𝜑 → (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) ∈ (Poly‘ℂ))
43		plyf 24248	. . . . . . 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 24331	. . . . . 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 1490	. . . . 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 1174	. . . . . 6 ⊢ (𝜑 → (◡(Xp ∘𝑓 − (ℂ × {𝐴})) “ {0}) = {𝐴})
48	47	uneq1d 3930	. . . . 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 2803	. . . 4 ⊢ (𝜑 → (◡𝐹 “ {0}) = ({𝐴} ∪ (◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0})))
50		fta1.1	. . . 4 ⊢ 𝑅 = (◡𝐹 “ {0})
51		uncom 3921	. . . 4 ⊢ ((◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴}) = ({𝐴} ∪ (◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0}))
52	49, 50, 51	3eqtr4g 2824	. . 3 ⊢ (𝜑 → 𝑅 = ((◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴}))
53	21	a1i 11	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℂ)
54		dgrcl 24283	. . . . . . . . 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 11602	. . . . . . 7 ⊢ (𝜑 → (deg‘(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴})))) ∈ ℂ)
57		fta1.2	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ ℕ0)
58	57	nn0cnd 11602	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ ℂ)
59		addcom 10478	. . . . . . . . 9 ⊢ ((1 ∈ ℂ ∧ 𝐷 ∈ ℂ) → (1 + 𝐷) = (𝐷 + 1))
60	21, 58, 59	sylancr 581	. . . . . . . 8 ⊢ (𝜑 → (1 + 𝐷) = (𝐷 + 1))
61	15	fveq2d 6381	. . . . . . . . 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 489	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ≠ 0𝑝)
64	15	eqcomd 2771	. . . . . . . . . . . 12 ⊢ (𝜑 → ((Xp ∘𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴})))) = 𝐹)
65		0cnd 10288	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ∈ ℂ)
66		mul01 10471	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → (𝑥 · 0) = 0)
67	66	adantl 473	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝑥 · 0) = 0)
68	19, 29, 65, 65, 67	caofid1 7127	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((Xp ∘𝑓 − (ℂ × {𝐴})) ∘𝑓 · (ℂ × {0})) = (ℂ × {0}))
69		df-0p 23731	. . . . . . . . . . . . . 14 ⊢ 0𝑝 = (ℂ × {0})
70	69	oveq2i 6855	. . . . . . . . . . . . 13 ⊢ ((Xp ∘𝑓 − (ℂ × {𝐴})) ∘𝑓 · 0𝑝) = ((Xp ∘𝑓 − (ℂ × {𝐴})) ∘𝑓 · (ℂ × {0}))
71	68, 70, 69	3eqtr4g 2824	. . . . . . . . . . . 12 ⊢ (𝜑 → ((Xp ∘𝑓 − (ℂ × {𝐴})) ∘𝑓 · 0𝑝) = 0𝑝)
72	63, 64, 71	3netr4d 3014	. . . . . . . . . . 11 ⊢ (𝜑 → ((Xp ∘𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴})))) ≠ ((Xp ∘𝑓 − (ℂ × {𝐴})) ∘𝑓 · 0𝑝))
73		oveq2 6852	. . . . . . . . . . . 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 2969	. . . . . . . . . . 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 2765	. . . . . . . . . . 11 ⊢ (deg‘(Xp ∘𝑓 − (ℂ × {𝐴}))) = (deg‘(Xp ∘𝑓 − (ℂ × {𝐴})))
77		eqid 2765	. . . . . . . . . . 11 ⊢ (deg‘(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴})))) = (deg‘(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))))
78	76, 77	dgrmul 24320	. . . . . . . . . 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 867	. . . . . . . . 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 2807	. . . . . . . 8 ⊢ (𝜑 → (𝐷 + 1) = ((deg‘(Xp ∘𝑓 − (ℂ × {𝐴}))) + (deg‘(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))))))
81	32	oveq1d 6859	. . . . . . . 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 2804	. . . . . . 7 ⊢ (𝜑 → (1 + (deg‘(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))))) = (1 + 𝐷))
83	53, 56, 58, 82	addcanad 10497	. . . . . 6 ⊢ (𝜑 → (deg‘(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴})))) = 𝐷)
84		fveqeq2 6386	. . . . . . . 8 ⊢ (𝑔 = (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) → ((deg‘𝑔) = 𝐷 ↔ (deg‘(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴})))) = 𝐷))
85		cnveq 5466	. . . . . . . . . . 11 ⊢ (𝑔 = (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) → ◡𝑔 = ◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))))
86	85	imaeq1d 5649	. . . . . . . . . 10 ⊢ (𝑔 = (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) → (◡𝑔 “ {0}) = (◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0}))
87	86	eleq1d 2829	. . . . . . . . 9 ⊢ (𝑔 = (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) → ((◡𝑔 “ {0}) ∈ Fin ↔ (◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0}) ∈ Fin))
88	86	fveq2d 6381	. . . . . . . . . 10 ⊢ (𝑔 = (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) → (♯‘(◡𝑔 “ {0})) = (♯‘(◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0})))
89		fveq2 6377	. . . . . . . . . 10 ⊢ (𝑔 = (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) → (deg‘𝑔) = (deg‘(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴})))))
90	88, 89	breq12d 4824	. . . . . . . . 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 624	. . . . . . . 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 335	. . . . . . 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 4474	. . . . . . . 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 578	. . . . . . 7 ⊢ (𝜑 → (𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) ∈ ((Poly‘ℂ) ∖ {0𝑝}))
96	92, 93, 95	rspcdva 3468	. . . . . 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 488	. . . 4 ⊢ (𝜑 → (◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0}) ∈ Fin)
99		snfi 8247	. . . 4 ⊢ {𝐴} ∈ Fin
100		unfi 8436	. . . 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 580	. . 3 ⊢ (𝜑 → ((◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴}) ∈ Fin)
102	52, 101	eqeltrd 2844	. 2 ⊢ (𝜑 → 𝑅 ∈ Fin)
103	52	fveq2d 6381	. . 3 ⊢ (𝜑 → (♯‘𝑅) = (♯‘((◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})))
104		hashcl 13352	. . . . . 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 11601	. . . 4 ⊢ (𝜑 → (♯‘((◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ∈ ℝ)
107		hashcl 13352	. . . . . . 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 11601	. . . . 5 ⊢ (𝜑 → (♯‘(◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0})) ∈ ℝ)
110		peano2re 10465	. . . . 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 24283	. . . . . 6 ⊢ (𝐹 ∈ (Poly‘ℂ) → (deg‘𝐹) ∈ ℕ0)
113	4, 112	syl 17	. . . . 5 ⊢ (𝜑 → (deg‘𝐹) ∈ ℕ0)
114	113	nn0red 11601	. . . 4 ⊢ (𝜑 → (deg‘𝐹) ∈ ℝ)
115		hashun2 13377	. . . . . 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 580	. . . . 5 ⊢ (𝜑 → (♯‘((◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ≤ ((♯‘(◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0})) + (♯‘{𝐴})))
117		hashsng 13364	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (♯‘{𝐴}) = 1)
118	11, 117	syl 17	. . . . . 6 ⊢ (𝜑 → (♯‘{𝐴}) = 1)
119	118	oveq2d 6860	. . . . 5 ⊢ (𝜑 → ((♯‘(◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0})) + (♯‘{𝐴})) = ((♯‘(◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0})) + 1))
120	116, 119	breqtrd 4837	. . . 4 ⊢ (𝜑 → (♯‘((◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ≤ ((♯‘(◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0})) + 1))
121	57	nn0red 11601	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ ℝ)
122		1red 10296	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℝ)
123	97	simprd 489	. . . . . . 7 ⊢ (𝜑 → (♯‘(◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0})) ≤ (deg‘(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴})))))
124	123, 83	breqtrd 4837	. . . . . 6 ⊢ (𝜑 → (♯‘(◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0})) ≤ 𝐷)
125	109, 121, 122, 124	leadd1dd 10897	. . . . 5 ⊢ (𝜑 → ((♯‘(◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0})) + 1) ≤ (𝐷 + 1))
126	125, 62	breqtrrd 4839	. . . 4 ⊢ (𝜑 → ((♯‘(◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0})) + 1) ≤ (deg‘𝐹))
127	106, 111, 114, 120, 126	letrd 10450	. . 3 ⊢ (𝜑 → (♯‘((◡(𝐹 quot (Xp ∘𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ≤ (deg‘𝐹))
128	103, 127	eqbrtrd 4833	. 2 ⊢ (𝜑 → (♯‘𝑅) ≤ (deg‘𝐹))
129	102, 128	jca 507	1 ⊢ (𝜑 → (𝑅 ∈ Fin ∧ (♯‘𝑅) ≤ (deg‘𝐹)))

Syntax hints:   → wi 4   ↔ wb 197   ∧ wa 384   ∧ w3a 1107   = wceq 1652   ∈ wcel 2155   ≠ wne 2937  ∀wral 3055  Vcvv 3350   ∖ cdif 3731   ∪ cun 3732   ⊆ wss 3734  {csn 4336   class class class wbr 4811   × cxp 5277  ^◡ccnv 5278   “ cima 5282   Fn wfn 6065  ⟶wf 6066  ‘cfv 6070  (class class class)co 6844   ∘_𝑓 cof 7095  Fincfn 8162  ℂcc 10189  ℝcr 10190  0cc0 10191  1c1 10192   + caddc 10194   · cmul 10196   ≤ cle 10331   − cmin 10522  ℕ₀cn0 11540  ♯chash 13324  0_𝑝c0p 23730  Polycply 24234  X_pcidp 24235  degcdgr 24237   quot cquot 24339

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7149  ax-inf2 8755  ax-cnex 10247  ax-resscn 10248  ax-1cn 10249  ax-icn 10250  ax-addcl 10251  ax-addrcl 10252  ax-mulcl 10253  ax-mulrcl 10254  ax-mulcom 10255  ax-addass 10256  ax-mulass 10257  ax-distr 10258  ax-i2m1 10259  ax-1ne0 10260  ax-1rid 10261  ax-rnegex 10262  ax-rrecex 10263  ax-cnre 10264  ax-pre-lttri 10265  ax-pre-lttrn 10266  ax-pre-ltadd 10267  ax-pre-mulgt0 10268  ax-pre-sup 10269  ax-addf 10270

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-se 5239  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-isom 6079  df-riota 6805  df-ov 6847  df-oprab 6848  df-mpt2 6849  df-of 7097  df-om 7266  df-1st 7368  df-2nd 7369  df-wrecs 7612  df-recs 7674  df-rdg 7712  df-1o 7766  df-oadd 7770  df-er 7949  df-map 8064  df-pm 8065  df-en 8163  df-dom 8164  df-sdom 8165  df-fin 8166  df-sup 8557  df-inf 8558  df-oi 8624  df-card 9018  df-cda 9245  df-pnf 10332  df-mnf 10333  df-xr 10334  df-ltxr 10335  df-le 10336  df-sub 10524  df-neg 10525  df-div 10941  df-nn 11277  df-2 11337  df-3 11338  df-n0 11541  df-xnn0 11613  df-z 11627  df-uz 11890  df-rp 12032  df-fz 12537  df-fzo 12677  df-fl 12804  df-seq 13012  df-exp 13071  df-hash 13325  df-cj 14127  df-re 14128  df-im 14129  df-sqrt 14263  df-abs 14264  df-clim 14507  df-rlim 14508  df-sum 14705  df-0p 23731  df-ply 24238  df-idp 24239  df-coe 24240  df-dgr 24241  df-quot 24340

This theorem is referenced by:  fta1  24357