Theorem fta1glem2 25064

Description: Lemma for fta1g 25065. (Contributed by Mario Carneiro, 12-Jun-2015.)

Hypotheses

Ref	Expression
fta1g.p	⊢ 𝑃 = (Poly1‘𝑅)
fta1g.b	⊢ 𝐵 = (Base‘𝑃)
fta1g.d	⊢ 𝐷 = ( deg1 ‘𝑅)
fta1g.o	⊢ 𝑂 = (eval1‘𝑅)
fta1g.w	⊢ 𝑊 = (0g‘𝑅)
fta1g.z	⊢ 0 = (0g‘𝑃)
fta1g.1	⊢ (𝜑 → 𝑅 ∈ IDomn)
fta1g.2	⊢ (𝜑 → 𝐹 ∈ 𝐵)
fta1glem.k	⊢ 𝐾 = (Base‘𝑅)
fta1glem.x	⊢ 𝑋 = (var1‘𝑅)
fta1glem.m	⊢ − = (-g‘𝑃)
fta1glem.a	⊢ 𝐴 = (algSc‘𝑃)
fta1glem.g	⊢ 𝐺 = (𝑋 − (𝐴‘𝑇))
fta1glem.3	⊢ (𝜑 → 𝑁 ∈ ℕ0)
fta1glem.4	⊢ (𝜑 → (𝐷‘𝐹) = (𝑁 + 1))
fta1glem.5	⊢ (𝜑 → 𝑇 ∈ (◡(𝑂‘𝐹) “ {𝑊}))
fta1glem.6	⊢ (𝜑 → ∀𝑔 ∈ 𝐵 ((𝐷‘𝑔) = 𝑁 → (♯‘(◡(𝑂‘𝑔) “ {𝑊})) ≤ (𝐷‘𝑔)))

Assertion

Ref	Expression
fta1glem2	⊢ (𝜑 → (♯‘(◡(𝑂‘𝐹) “ {𝑊})) ≤ (𝐷‘𝐹))

Distinct variable groups:   𝐵,𝑔   𝐷,𝑔   𝑔,𝐹   𝑔,𝑁   𝑔,𝑂   𝑔,𝐺   𝑃,𝑔   𝑅,𝑔   𝑔,𝑊

Allowed substitution hints:   𝜑(𝑔)   𝐴(𝑔)   𝑇(𝑔)   𝐾(𝑔)   − (𝑔)   𝑋(𝑔)   0 (𝑔)

Proof of Theorem fta1glem2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fta1glem.5	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑇 ∈ (◡(𝑂‘𝐹) “ {𝑊}))
2		eqid 2737	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑅 ↑s 𝐾) = (𝑅 ↑s 𝐾)
3		fta1glem.k	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝐾 = (Base‘𝑅)
4		eqid 2737	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Base‘(𝑅 ↑s 𝐾)) = (Base‘(𝑅 ↑s 𝐾))
5		fta1g.1	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑅 ∈ IDomn)
6	3	fvexi 6731	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝐾 ∈ V
7	6	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐾 ∈ V)
8		isidom 20342	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑅 ∈ IDomn ↔ (𝑅 ∈ CRing ∧ 𝑅 ∈ Domn))
9	8	simplbi 501	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑅 ∈ IDomn → 𝑅 ∈ CRing)
10	5, 9	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝑅 ∈ CRing)
11		fta1g.o	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑂 = (eval1‘𝑅)
12		fta1g.p	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑃 = (Poly1‘𝑅)
13	11, 12, 2, 3	evl1rhm 21248	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑅 ∈ CRing → 𝑂 ∈ (𝑃 RingHom (𝑅 ↑s 𝐾)))
14	10, 13	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑂 ∈ (𝑃 RingHom (𝑅 ↑s 𝐾)))
15		fta1g.b	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝐵 = (Base‘𝑃)
16	15, 4	rhmf 19746	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑂 ∈ (𝑃 RingHom (𝑅 ↑s 𝐾)) → 𝑂:𝐵⟶(Base‘(𝑅 ↑s 𝐾)))
17	14, 16	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑂:𝐵⟶(Base‘(𝑅 ↑s 𝐾)))
18		fta1g.2	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
19	17, 18	ffvelrnd 6905	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑂‘𝐹) ∈ (Base‘(𝑅 ↑s 𝐾)))
20	2, 3, 4, 5, 7, 19	pwselbas 16994	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑂‘𝐹):𝐾⟶𝐾)
21	20	ffnd 6546	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑂‘𝐹) Fn 𝐾)
22		fniniseg 6880	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑂‘𝐹) Fn 𝐾 → (𝑇 ∈ (◡(𝑂‘𝐹) “ {𝑊}) ↔ (𝑇 ∈ 𝐾 ∧ ((𝑂‘𝐹)‘𝑇) = 𝑊)))
23	21, 22	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑇 ∈ (◡(𝑂‘𝐹) “ {𝑊}) ↔ (𝑇 ∈ 𝐾 ∧ ((𝑂‘𝐹)‘𝑇) = 𝑊)))
24	1, 23	mpbid 235	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑇 ∈ 𝐾 ∧ ((𝑂‘𝐹)‘𝑇) = 𝑊))
25	24	simprd 499	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑂‘𝐹)‘𝑇) = 𝑊)
26		fta1glem.x	. . . . . . . . . . . . . . . . 17 ⊢ 𝑋 = (var1‘𝑅)
27		fta1glem.m	. . . . . . . . . . . . . . . . 17 ⊢ − = (-g‘𝑃)
28		fta1glem.a	. . . . . . . . . . . . . . . . 17 ⊢ 𝐴 = (algSc‘𝑃)
29		fta1glem.g	. . . . . . . . . . . . . . . . 17 ⊢ 𝐺 = (𝑋 − (𝐴‘𝑇))
30	8	simprbi 500	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑅 ∈ IDomn → 𝑅 ∈ Domn)
31		domnnzr 20333	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ NzRing)
32	30, 31	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑅 ∈ IDomn → 𝑅 ∈ NzRing)
33	5, 32	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑅 ∈ NzRing)
34	24	simpld 498	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑇 ∈ 𝐾)
35		fta1g.w	. . . . . . . . . . . . . . . . 17 ⊢ 𝑊 = (0g‘𝑅)
36		eqid 2737	. . . . . . . . . . . . . . . . 17 ⊢ (∥r‘𝑃) = (∥r‘𝑃)
37	12, 15, 3, 26, 27, 28, 29, 11, 33, 10, 34, 18, 35, 36	facth1 25062	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐺(∥r‘𝑃)𝐹 ↔ ((𝑂‘𝐹)‘𝑇) = 𝑊))
38	25, 37	mpbird 260	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐺(∥r‘𝑃)𝐹)
39		nzrring 20299	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring)
40	33, 39	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑅 ∈ Ring)
41		eqid 2737	. . . . . . . . . . . . . . . . . . 19 ⊢ (Monic1p‘𝑅) = (Monic1p‘𝑅)
42		fta1g.d	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐷 = ( deg1 ‘𝑅)
43	12, 15, 3, 26, 27, 28, 29, 11, 33, 10, 34, 41, 42, 35	ply1remlem 25060	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐺 ∈ (Monic1p‘𝑅) ∧ (𝐷‘𝐺) = 1 ∧ (◡(𝑂‘𝐺) “ {𝑊}) = {𝑇}))
44	43	simp1d 1144	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐺 ∈ (Monic1p‘𝑅))
45		eqid 2737	. . . . . . . . . . . . . . . . . 18 ⊢ (Unic1p‘𝑅) = (Unic1p‘𝑅)
46	45, 41	mon1puc1p 25048	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ (Monic1p‘𝑅)) → 𝐺 ∈ (Unic1p‘𝑅))
47	40, 44, 46	syl2anc 587	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐺 ∈ (Unic1p‘𝑅))
48		eqid 2737	. . . . . . . . . . . . . . . . 17 ⊢ (.r‘𝑃) = (.r‘𝑃)
49		eqid 2737	. . . . . . . . . . . . . . . . 17 ⊢ (quot1p‘𝑅) = (quot1p‘𝑅)
50	12, 36, 15, 45, 48, 49	dvdsq1p 25058	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ (Unic1p‘𝑅)) → (𝐺(∥r‘𝑃)𝐹 ↔ 𝐹 = ((𝐹(quot1p‘𝑅)𝐺)(.r‘𝑃)𝐺)))
51	40, 18, 47, 50	syl3anc 1373	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐺(∥r‘𝑃)𝐹 ↔ 𝐹 = ((𝐹(quot1p‘𝑅)𝐺)(.r‘𝑃)𝐺)))
52	38, 51	mpbid 235	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹 = ((𝐹(quot1p‘𝑅)𝐺)(.r‘𝑃)𝐺))
53	52	fveq2d 6721	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑂‘𝐹) = (𝑂‘((𝐹(quot1p‘𝑅)𝐺)(.r‘𝑃)𝐺)))
54	49, 12, 15, 45	q1pcl 25053	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ (Unic1p‘𝑅)) → (𝐹(quot1p‘𝑅)𝐺) ∈ 𝐵)
55	40, 18, 47, 54	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹(quot1p‘𝑅)𝐺) ∈ 𝐵)
56	12, 15, 41	mon1pcl 25042	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ (Monic1p‘𝑅) → 𝐺 ∈ 𝐵)
57	44, 56	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺 ∈ 𝐵)
58		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (.r‘(𝑅 ↑s 𝐾)) = (.r‘(𝑅 ↑s 𝐾))
59	15, 48, 58	rhmmul 19747	. . . . . . . . . . . . . 14 ⊢ ((𝑂 ∈ (𝑃 RingHom (𝑅 ↑s 𝐾)) ∧ (𝐹(quot1p‘𝑅)𝐺) ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝑂‘((𝐹(quot1p‘𝑅)𝐺)(.r‘𝑃)𝐺)) = ((𝑂‘(𝐹(quot1p‘𝑅)𝐺))(.r‘(𝑅 ↑s 𝐾))(𝑂‘𝐺)))
60	14, 55, 57, 59	syl3anc 1373	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑂‘((𝐹(quot1p‘𝑅)𝐺)(.r‘𝑃)𝐺)) = ((𝑂‘(𝐹(quot1p‘𝑅)𝐺))(.r‘(𝑅 ↑s 𝐾))(𝑂‘𝐺)))
61	17, 55	ffvelrnd 6905	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑂‘(𝐹(quot1p‘𝑅)𝐺)) ∈ (Base‘(𝑅 ↑s 𝐾)))
62	17, 57	ffvelrnd 6905	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑂‘𝐺) ∈ (Base‘(𝑅 ↑s 𝐾)))
63		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (.r‘𝑅) = (.r‘𝑅)
64	2, 4, 5, 7, 61, 62, 63, 58	pwsmulrval 16996	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑂‘(𝐹(quot1p‘𝑅)𝐺))(.r‘(𝑅 ↑s 𝐾))(𝑂‘𝐺)) = ((𝑂‘(𝐹(quot1p‘𝑅)𝐺)) ∘f (.r‘𝑅)(𝑂‘𝐺)))
65	53, 60, 64	3eqtrd 2781	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑂‘𝐹) = ((𝑂‘(𝐹(quot1p‘𝑅)𝐺)) ∘f (.r‘𝑅)(𝑂‘𝐺)))
66	65	fveq1d 6719	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑂‘𝐹)‘𝑥) = (((𝑂‘(𝐹(quot1p‘𝑅)𝐺)) ∘f (.r‘𝑅)(𝑂‘𝐺))‘𝑥))
67	66	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → ((𝑂‘𝐹)‘𝑥) = (((𝑂‘(𝐹(quot1p‘𝑅)𝐺)) ∘f (.r‘𝑅)(𝑂‘𝐺))‘𝑥))
68	2, 3, 4, 5, 7, 61	pwselbas 16994	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑂‘(𝐹(quot1p‘𝑅)𝐺)):𝐾⟶𝐾)
69	68	ffnd 6546	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑂‘(𝐹(quot1p‘𝑅)𝐺)) Fn 𝐾)
70	69	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → (𝑂‘(𝐹(quot1p‘𝑅)𝐺)) Fn 𝐾)
71	2, 3, 4, 5, 7, 62	pwselbas 16994	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑂‘𝐺):𝐾⟶𝐾)
72	71	ffnd 6546	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑂‘𝐺) Fn 𝐾)
73	72	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → (𝑂‘𝐺) Fn 𝐾)
74	6	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → 𝐾 ∈ V)
75		simpr 488	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → 𝑥 ∈ 𝐾)
76		fnfvof 7485	. . . . . . . . . . 11 ⊢ ((((𝑂‘(𝐹(quot1p‘𝑅)𝐺)) Fn 𝐾 ∧ (𝑂‘𝐺) Fn 𝐾) ∧ (𝐾 ∈ V ∧ 𝑥 ∈ 𝐾)) → (((𝑂‘(𝐹(quot1p‘𝑅)𝐺)) ∘f (.r‘𝑅)(𝑂‘𝐺))‘𝑥) = (((𝑂‘(𝐹(quot1p‘𝑅)𝐺))‘𝑥)(.r‘𝑅)((𝑂‘𝐺)‘𝑥)))
77	70, 73, 74, 75, 76	syl22anc 839	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → (((𝑂‘(𝐹(quot1p‘𝑅)𝐺)) ∘f (.r‘𝑅)(𝑂‘𝐺))‘𝑥) = (((𝑂‘(𝐹(quot1p‘𝑅)𝐺))‘𝑥)(.r‘𝑅)((𝑂‘𝐺)‘𝑥)))
78	67, 77	eqtrd 2777	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → ((𝑂‘𝐹)‘𝑥) = (((𝑂‘(𝐹(quot1p‘𝑅)𝐺))‘𝑥)(.r‘𝑅)((𝑂‘𝐺)‘𝑥)))
79	78	eqeq1d 2739	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → (((𝑂‘𝐹)‘𝑥) = 𝑊 ↔ (((𝑂‘(𝐹(quot1p‘𝑅)𝐺))‘𝑥)(.r‘𝑅)((𝑂‘𝐺)‘𝑥)) = 𝑊))
80	5, 30	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ Domn)
81	80	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → 𝑅 ∈ Domn)
82	68	ffvelrnda 6904	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → ((𝑂‘(𝐹(quot1p‘𝑅)𝐺))‘𝑥) ∈ 𝐾)
83	71	ffvelrnda 6904	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → ((𝑂‘𝐺)‘𝑥) ∈ 𝐾)
84	3, 63, 35	domneq0 20335	. . . . . . . . 9 ⊢ ((𝑅 ∈ Domn ∧ ((𝑂‘(𝐹(quot1p‘𝑅)𝐺))‘𝑥) ∈ 𝐾 ∧ ((𝑂‘𝐺)‘𝑥) ∈ 𝐾) → ((((𝑂‘(𝐹(quot1p‘𝑅)𝐺))‘𝑥)(.r‘𝑅)((𝑂‘𝐺)‘𝑥)) = 𝑊 ↔ (((𝑂‘(𝐹(quot1p‘𝑅)𝐺))‘𝑥) = 𝑊 ∨ ((𝑂‘𝐺)‘𝑥) = 𝑊)))
85	81, 82, 83, 84	syl3anc 1373	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → ((((𝑂‘(𝐹(quot1p‘𝑅)𝐺))‘𝑥)(.r‘𝑅)((𝑂‘𝐺)‘𝑥)) = 𝑊 ↔ (((𝑂‘(𝐹(quot1p‘𝑅)𝐺))‘𝑥) = 𝑊 ∨ ((𝑂‘𝐺)‘𝑥) = 𝑊)))
86	79, 85	bitrd 282	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → (((𝑂‘𝐹)‘𝑥) = 𝑊 ↔ (((𝑂‘(𝐹(quot1p‘𝑅)𝐺))‘𝑥) = 𝑊 ∨ ((𝑂‘𝐺)‘𝑥) = 𝑊)))
87	86	pm5.32da 582	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐾 ∧ ((𝑂‘𝐹)‘𝑥) = 𝑊) ↔ (𝑥 ∈ 𝐾 ∧ (((𝑂‘(𝐹(quot1p‘𝑅)𝐺))‘𝑥) = 𝑊 ∨ ((𝑂‘𝐺)‘𝑥) = 𝑊))))
88		andi 1008	. . . . . 6 ⊢ ((𝑥 ∈ 𝐾 ∧ (((𝑂‘(𝐹(quot1p‘𝑅)𝐺))‘𝑥) = 𝑊 ∨ ((𝑂‘𝐺)‘𝑥) = 𝑊)) ↔ ((𝑥 ∈ 𝐾 ∧ ((𝑂‘(𝐹(quot1p‘𝑅)𝐺))‘𝑥) = 𝑊) ∨ (𝑥 ∈ 𝐾 ∧ ((𝑂‘𝐺)‘𝑥) = 𝑊)))
89	87, 88	bitrdi 290	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐾 ∧ ((𝑂‘𝐹)‘𝑥) = 𝑊) ↔ ((𝑥 ∈ 𝐾 ∧ ((𝑂‘(𝐹(quot1p‘𝑅)𝐺))‘𝑥) = 𝑊) ∨ (𝑥 ∈ 𝐾 ∧ ((𝑂‘𝐺)‘𝑥) = 𝑊))))
90		fniniseg 6880	. . . . . 6 ⊢ ((𝑂‘𝐹) Fn 𝐾 → (𝑥 ∈ (◡(𝑂‘𝐹) “ {𝑊}) ↔ (𝑥 ∈ 𝐾 ∧ ((𝑂‘𝐹)‘𝑥) = 𝑊)))
91	21, 90	syl 17	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (◡(𝑂‘𝐹) “ {𝑊}) ↔ (𝑥 ∈ 𝐾 ∧ ((𝑂‘𝐹)‘𝑥) = 𝑊)))
92		elun 4063	. . . . . 6 ⊢ (𝑥 ∈ ((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇}) ↔ (𝑥 ∈ (◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∨ 𝑥 ∈ {𝑇}))
93		fniniseg 6880	. . . . . . . 8 ⊢ ((𝑂‘(𝐹(quot1p‘𝑅)𝐺)) Fn 𝐾 → (𝑥 ∈ (◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ↔ (𝑥 ∈ 𝐾 ∧ ((𝑂‘(𝐹(quot1p‘𝑅)𝐺))‘𝑥) = 𝑊)))
94	69, 93	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ↔ (𝑥 ∈ 𝐾 ∧ ((𝑂‘(𝐹(quot1p‘𝑅)𝐺))‘𝑥) = 𝑊)))
95	43	simp3d 1146	. . . . . . . . 9 ⊢ (𝜑 → (◡(𝑂‘𝐺) “ {𝑊}) = {𝑇})
96	95	eleq2d 2823	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (◡(𝑂‘𝐺) “ {𝑊}) ↔ 𝑥 ∈ {𝑇}))
97		fniniseg 6880	. . . . . . . . 9 ⊢ ((𝑂‘𝐺) Fn 𝐾 → (𝑥 ∈ (◡(𝑂‘𝐺) “ {𝑊}) ↔ (𝑥 ∈ 𝐾 ∧ ((𝑂‘𝐺)‘𝑥) = 𝑊)))
98	72, 97	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (◡(𝑂‘𝐺) “ {𝑊}) ↔ (𝑥 ∈ 𝐾 ∧ ((𝑂‘𝐺)‘𝑥) = 𝑊)))
99	96, 98	bitr3d 284	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ {𝑇} ↔ (𝑥 ∈ 𝐾 ∧ ((𝑂‘𝐺)‘𝑥) = 𝑊)))
100	94, 99	orbi12d 919	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ (◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∨ 𝑥 ∈ {𝑇}) ↔ ((𝑥 ∈ 𝐾 ∧ ((𝑂‘(𝐹(quot1p‘𝑅)𝐺))‘𝑥) = 𝑊) ∨ (𝑥 ∈ 𝐾 ∧ ((𝑂‘𝐺)‘𝑥) = 𝑊))))
101	92, 100	syl5bb 286	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇}) ↔ ((𝑥 ∈ 𝐾 ∧ ((𝑂‘(𝐹(quot1p‘𝑅)𝐺))‘𝑥) = 𝑊) ∨ (𝑥 ∈ 𝐾 ∧ ((𝑂‘𝐺)‘𝑥) = 𝑊))))
102	89, 91, 101	3bitr4d 314	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (◡(𝑂‘𝐹) “ {𝑊}) ↔ 𝑥 ∈ ((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇})))
103	102	eqrdv 2735	. . 3 ⊢ (𝜑 → (◡(𝑂‘𝐹) “ {𝑊}) = ((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇}))
104	103	fveq2d 6721	. 2 ⊢ (𝜑 → (♯‘(◡(𝑂‘𝐹) “ {𝑊})) = (♯‘((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇})))
105		fvex 6730	. . . . . . . . . 10 ⊢ (𝑂‘(𝐹(quot1p‘𝑅)𝐺)) ∈ V
106	105	cnvex 7703	. . . . . . . . 9 ⊢ ◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) ∈ V
107	106	imaex 7694	. . . . . . . 8 ⊢ (◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∈ V
108	107	a1i 11	. . . . . . 7 ⊢ (𝜑 → (◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∈ V)
109		fta1glem.3	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
110		fta1g.z	. . . . . . . . . 10 ⊢ 0 = (0g‘𝑃)
111		fta1glem.4	. . . . . . . . . 10 ⊢ (𝜑 → (𝐷‘𝐹) = (𝑁 + 1))
112	12, 15, 42, 11, 35, 110, 5, 18, 3, 26, 27, 28, 29, 109, 111, 1	fta1glem1 25063	. . . . . . . . 9 ⊢ (𝜑 → (𝐷‘(𝐹(quot1p‘𝑅)𝐺)) = 𝑁)
113		fveq2 6717	. . . . . . . . . . . 12 ⊢ (𝑔 = (𝐹(quot1p‘𝑅)𝐺) → (𝐷‘𝑔) = (𝐷‘(𝐹(quot1p‘𝑅)𝐺)))
114	113	eqeq1d 2739	. . . . . . . . . . 11 ⊢ (𝑔 = (𝐹(quot1p‘𝑅)𝐺) → ((𝐷‘𝑔) = 𝑁 ↔ (𝐷‘(𝐹(quot1p‘𝑅)𝐺)) = 𝑁))
115		fveq2 6717	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (𝐹(quot1p‘𝑅)𝐺) → (𝑂‘𝑔) = (𝑂‘(𝐹(quot1p‘𝑅)𝐺)))
116	115	cnveqd 5744	. . . . . . . . . . . . . 14 ⊢ (𝑔 = (𝐹(quot1p‘𝑅)𝐺) → ◡(𝑂‘𝑔) = ◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)))
117	116	imaeq1d 5928	. . . . . . . . . . . . 13 ⊢ (𝑔 = (𝐹(quot1p‘𝑅)𝐺) → (◡(𝑂‘𝑔) “ {𝑊}) = (◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}))
118	117	fveq2d 6721	. . . . . . . . . . . 12 ⊢ (𝑔 = (𝐹(quot1p‘𝑅)𝐺) → (♯‘(◡(𝑂‘𝑔) “ {𝑊})) = (♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})))
119	118, 113	breq12d 5066	. . . . . . . . . . 11 ⊢ (𝑔 = (𝐹(quot1p‘𝑅)𝐺) → ((♯‘(◡(𝑂‘𝑔) “ {𝑊})) ≤ (𝐷‘𝑔) ↔ (♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})) ≤ (𝐷‘(𝐹(quot1p‘𝑅)𝐺))))
120	114, 119	imbi12d 348	. . . . . . . . . 10 ⊢ (𝑔 = (𝐹(quot1p‘𝑅)𝐺) → (((𝐷‘𝑔) = 𝑁 → (♯‘(◡(𝑂‘𝑔) “ {𝑊})) ≤ (𝐷‘𝑔)) ↔ ((𝐷‘(𝐹(quot1p‘𝑅)𝐺)) = 𝑁 → (♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})) ≤ (𝐷‘(𝐹(quot1p‘𝑅)𝐺)))))
121		fta1glem.6	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑔 ∈ 𝐵 ((𝐷‘𝑔) = 𝑁 → (♯‘(◡(𝑂‘𝑔) “ {𝑊})) ≤ (𝐷‘𝑔)))
122	120, 121, 55	rspcdva 3539	. . . . . . . . 9 ⊢ (𝜑 → ((𝐷‘(𝐹(quot1p‘𝑅)𝐺)) = 𝑁 → (♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})) ≤ (𝐷‘(𝐹(quot1p‘𝑅)𝐺))))
123	112, 122	mpd 15	. . . . . . . 8 ⊢ (𝜑 → (♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})) ≤ (𝐷‘(𝐹(quot1p‘𝑅)𝐺)))
124	123, 112	breqtrd 5079	. . . . . . 7 ⊢ (𝜑 → (♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})) ≤ 𝑁)
125		hashbnd 13902	. . . . . . 7 ⊢ (((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∈ V ∧ 𝑁 ∈ ℕ0 ∧ (♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})) ≤ 𝑁) → (◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∈ Fin)
126	108, 109, 124, 125	syl3anc 1373	. . . . . 6 ⊢ (𝜑 → (◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∈ Fin)
127		snfi 8721	. . . . . 6 ⊢ {𝑇} ∈ Fin
128		unfi 8850	. . . . . 6 ⊢ (((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∈ Fin ∧ {𝑇} ∈ Fin) → ((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇}) ∈ Fin)
129	126, 127, 128	sylancl 589	. . . . 5 ⊢ (𝜑 → ((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇}) ∈ Fin)
130		hashcl 13923	. . . . 5 ⊢ (((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇}) ∈ Fin → (♯‘((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇})) ∈ ℕ0)
131	129, 130	syl 17	. . . 4 ⊢ (𝜑 → (♯‘((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇})) ∈ ℕ0)
132	131	nn0red 12151	. . 3 ⊢ (𝜑 → (♯‘((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇})) ∈ ℝ)
133		hashcl 13923	. . . . . 6 ⊢ ((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∈ Fin → (♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})) ∈ ℕ0)
134	126, 133	syl 17	. . . . 5 ⊢ (𝜑 → (♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})) ∈ ℕ0)
135	134	nn0red 12151	. . . 4 ⊢ (𝜑 → (♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})) ∈ ℝ)
136		peano2re 11005	. . . 4 ⊢ ((♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})) ∈ ℝ → ((♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})) + 1) ∈ ℝ)
137	135, 136	syl 17	. . 3 ⊢ (𝜑 → ((♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})) + 1) ∈ ℝ)
138		peano2nn0 12130	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
139	109, 138	syl 17	. . . . 5 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ0)
140	111, 139	eqeltrd 2838	. . . 4 ⊢ (𝜑 → (𝐷‘𝐹) ∈ ℕ0)
141	140	nn0red 12151	. . 3 ⊢ (𝜑 → (𝐷‘𝐹) ∈ ℝ)
142		hashun2 13950	. . . . 5 ⊢ (((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∈ Fin ∧ {𝑇} ∈ Fin) → (♯‘((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇})) ≤ ((♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})) + (♯‘{𝑇})))
143	126, 127, 142	sylancl 589	. . . 4 ⊢ (𝜑 → (♯‘((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇})) ≤ ((♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})) + (♯‘{𝑇})))
144		hashsng 13936	. . . . . 6 ⊢ (𝑇 ∈ (◡(𝑂‘𝐹) “ {𝑊}) → (♯‘{𝑇}) = 1)
145	1, 144	syl 17	. . . . 5 ⊢ (𝜑 → (♯‘{𝑇}) = 1)
146	145	oveq2d 7229	. . . 4 ⊢ (𝜑 → ((♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})) + (♯‘{𝑇})) = ((♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})) + 1))
147	143, 146	breqtrd 5079	. . 3 ⊢ (𝜑 → (♯‘((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇})) ≤ ((♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})) + 1))
148	109	nn0red 12151	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℝ)
149		1red 10834	. . . . 5 ⊢ (𝜑 → 1 ∈ ℝ)
150	135, 148, 149, 124	leadd1dd 11446	. . . 4 ⊢ (𝜑 → ((♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})) + 1) ≤ (𝑁 + 1))
151	150, 111	breqtrrd 5081	. . 3 ⊢ (𝜑 → ((♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})) + 1) ≤ (𝐷‘𝐹))
152	132, 137, 141, 147, 151	letrd 10989	. 2 ⊢ (𝜑 → (♯‘((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇})) ≤ (𝐷‘𝐹))
153	104, 152	eqbrtrd 5075	1 ⊢ (𝜑 → (♯‘(◡(𝑂‘𝐹) “ {𝑊})) ≤ (𝐷‘𝐹))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 847   = wceq 1543   ∈ wcel 2110  ∀wral 3061  Vcvv 3408   ∪ cun 3864  {csn 4541   class class class wbr 5053  ^◡ccnv 5550   “ cima 5554   Fn wfn 6375  ⟶wf 6376  ‘cfv 6380  (class class class)co 7213   ∘_f cof 7467  Fincfn 8626  ℝcr 10728  1c1 10730   + caddc 10732   ≤ cle 10868  ℕ₀cn0 12090  ♯chash 13896  Basecbs 16760  ._rcmulr 16803  0_gc0g 16944   ↑_s cpws 16951  -_gcsg 18367  Ringcrg 19562  CRingccrg 19563  ∥_rcdsr 19656   RingHom crh 19732  NzRingcnzr 20295  Domncdomn 20318  IDomncidom 20319  algSccascl 20814  var₁cv1 21097  Poly₁cpl1 21098  eval₁ce1 21230   deg₁ cdg1 24949  Monic_1pcmn1 25023  Unic_1pcuc1p 25024  quot_1pcq1p 25025

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806  ax-pre-sup 10807  ax-addf 10808  ax-mulf 10809

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-iin 4907  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-se 5510  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-isom 6389  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-of 7469  df-ofr 7470  df-om 7645  df-1st 7761  df-2nd 7762  df-supp 7904  df-tpos 7968  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-oadd 8206  df-er 8391  df-map 8510  df-pm 8511  df-ixp 8579  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-fsupp 8986  df-sup 9058  df-oi 9126  df-dju 9517  df-card 9555  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-nn 11831  df-2 11893  df-3 11894  df-4 11895  df-5 11896  df-6 11897  df-7 11898  df-8 11899  df-9 11900  df-n0 12091  df-xnn0 12163  df-z 12177  df-dec 12294  df-uz 12439  df-fz 13096  df-fzo 13239  df-seq 13575  df-hash 13897  df-struct 16700  df-sets 16717  df-slot 16735  df-ndx 16745  df-base 16761  df-ress 16785  df-plusg 16815  df-mulr 16816  df-starv 16817  df-sca 16818  df-vsca 16819  df-ip 16820  df-tset 16821  df-ple 16822  df-ds 16824  df-unif 16825  df-hom 16826  df-cco 16827  df-0g 16946  df-gsum 16947  df-prds 16952  df-pws 16954  df-mre 17089  df-mrc 17090  df-acs 17092  df-mgm 18114  df-sgrp 18163  df-mnd 18174  df-mhm 18218  df-submnd 18219  df-grp 18368  df-minusg 18369  df-sbg 18370  df-mulg 18489  df-subg 18540  df-ghm 18620  df-cntz 18711  df-cmn 19172  df-abl 19173  df-mgp 19505  df-ur 19517  df-srg 19521  df-ring 19564  df-cring 19565  df-oppr 19641  df-dvdsr 19659  df-unit 19660  df-invr 19690  df-rnghom 19735  df-subrg 19798  df-lmod 19901  df-lss 19969  df-lsp 20009  df-nzr 20296  df-rlreg 20321  df-domn 20322  df-idom 20323  df-cnfld 20364  df-assa 20815  df-asp 20816  df-ascl 20817  df-psr 20868  df-mvr 20869  df-mpl 20870  df-opsr 20872  df-evls 21032  df-evl 21033  df-psr1 21101  df-vr1 21102  df-ply1 21103  df-coe1 21104  df-evl1 21232  df-mdeg 24950  df-deg1 24951  df-mon1 25028  df-uc1p 25029  df-q1p 25030  df-r1p 25031

This theorem is referenced by:  fta1g  25065