Theorem fta1glem2 24217

Description: Lemma for fta1g 24218. (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 2765	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑅 ↑s 𝐾) = (𝑅 ↑s 𝐾)
3		fta1glem.k	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝐾 = (Base‘𝑅)
4		eqid 2765	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Base‘(𝑅 ↑s 𝐾)) = (Base‘(𝑅 ↑s 𝐾))
5		fta1g.1	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑅 ∈ IDomn)
6	3	fvexi 6389	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝐾 ∈ V
7	6	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐾 ∈ V)
8		isidom 19578	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑅 ∈ IDomn ↔ (𝑅 ∈ CRing ∧ 𝑅 ∈ Domn))
9	8	simplbi 491	. . . . . . . . . . . . . . . . . . . . . . . . 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 19969	. . . . . . . . . . . . . . . . . . . . . . . 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 18995	. . . . . . . . . . . . . . . . . . . . . . 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 6550	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑂‘𝐹) ∈ (Base‘(𝑅 ↑s 𝐾)))
20	2, 3, 4, 5, 7, 19	pwselbas 16415	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑂‘𝐹):𝐾⟶𝐾)
21	20	ffnd 6224	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑂‘𝐹) Fn 𝐾)
22		fniniseg 6528	. . . . . . . . . . . . . . . . . . 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 223	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑇 ∈ 𝐾 ∧ ((𝑂‘𝐹)‘𝑇) = 𝑊))
25	24	simprd 489	. . . . . . . . . . . . . . . 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 490	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑅 ∈ IDomn → 𝑅 ∈ Domn)
31		domnnzr 19569	. . . . . . . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑇 ∈ 𝐾)
35		fta1g.w	. . . . . . . . . . . . . . . . 17 ⊢ 𝑊 = (0g‘𝑅)
36		eqid 2765	. . . . . . . . . . . . . . . . 17 ⊢ (∥r‘𝑃) = (∥r‘𝑃)
37	12, 15, 3, 26, 27, 28, 29, 11, 33, 10, 34, 18, 35, 36	facth1 24215	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐺(∥r‘𝑃)𝐹 ↔ ((𝑂‘𝐹)‘𝑇) = 𝑊))
38	25, 37	mpbird 248	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐺(∥r‘𝑃)𝐹)
39		nzrring 19535	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring)
40	33, 39	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑅 ∈ Ring)
41		eqid 2765	. . . . . . . . . . . . . . . . . . 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 24213	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐺 ∈ (Monic1p‘𝑅) ∧ (𝐷‘𝐺) = 1 ∧ (◡(𝑂‘𝐺) “ {𝑊}) = {𝑇}))
44	43	simp1d 1172	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐺 ∈ (Monic1p‘𝑅))
45		eqid 2765	. . . . . . . . . . . . . . . . . 18 ⊢ (Unic1p‘𝑅) = (Unic1p‘𝑅)
46	45, 41	mon1puc1p 24201	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ (Monic1p‘𝑅)) → 𝐺 ∈ (Unic1p‘𝑅))
47	40, 44, 46	syl2anc 579	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐺 ∈ (Unic1p‘𝑅))
48		eqid 2765	. . . . . . . . . . . . . . . . 17 ⊢ (.r‘𝑃) = (.r‘𝑃)
49		eqid 2765	. . . . . . . . . . . . . . . . 17 ⊢ (quot1p‘𝑅) = (quot1p‘𝑅)
50	12, 36, 15, 45, 48, 49	dvdsq1p 24211	. . . . . . . . . . . . . . . 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 1490	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐺(∥r‘𝑃)𝐹 ↔ 𝐹 = ((𝐹(quot1p‘𝑅)𝐺)(.r‘𝑃)𝐺)))
52	38, 51	mpbid 223	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹 = ((𝐹(quot1p‘𝑅)𝐺)(.r‘𝑃)𝐺))
53	52	fveq2d 6379	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑂‘𝐹) = (𝑂‘((𝐹(quot1p‘𝑅)𝐺)(.r‘𝑃)𝐺)))
54	49, 12, 15, 45	q1pcl 24206	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ (Unic1p‘𝑅)) → (𝐹(quot1p‘𝑅)𝐺) ∈ 𝐵)
55	40, 18, 47, 54	syl3anc 1490	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹(quot1p‘𝑅)𝐺) ∈ 𝐵)
56	12, 15, 41	mon1pcl 24195	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ (Monic1p‘𝑅) → 𝐺 ∈ 𝐵)
57	44, 56	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺 ∈ 𝐵)
58		eqid 2765	. . . . . . . . . . . . . . 15 ⊢ (.r‘(𝑅 ↑s 𝐾)) = (.r‘(𝑅 ↑s 𝐾))
59	15, 48, 58	rhmmul 18996	. . . . . . . . . . . . . 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 1490	. . . . . . . . . . . . 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 6550	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑂‘(𝐹(quot1p‘𝑅)𝐺)) ∈ (Base‘(𝑅 ↑s 𝐾)))
62	17, 57	ffvelrnd 6550	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑂‘𝐺) ∈ (Base‘(𝑅 ↑s 𝐾)))
63		eqid 2765	. . . . . . . . . . . . . 14 ⊢ (.r‘𝑅) = (.r‘𝑅)
64	2, 4, 5, 7, 61, 62, 63, 58	pwsmulrval 16417	. . . . . . . . . . . . 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 2803	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑂‘𝐹) = ((𝑂‘(𝐹(quot1p‘𝑅)𝐺)) ∘𝑓 (.r‘𝑅)(𝑂‘𝐺)))
66	65	fveq1d 6377	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑂‘𝐹)‘𝑥) = (((𝑂‘(𝐹(quot1p‘𝑅)𝐺)) ∘𝑓 (.r‘𝑅)(𝑂‘𝐺))‘𝑥))
67	66	adantr 472	. . . . . . . . . 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 16415	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑂‘(𝐹(quot1p‘𝑅)𝐺)):𝐾⟶𝐾)
69	68	ffnd 6224	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑂‘(𝐹(quot1p‘𝑅)𝐺)) Fn 𝐾)
70	69	adantr 472	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → (𝑂‘(𝐹(quot1p‘𝑅)𝐺)) Fn 𝐾)
71	2, 3, 4, 5, 7, 62	pwselbas 16415	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑂‘𝐺):𝐾⟶𝐾)
72	71	ffnd 6224	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑂‘𝐺) Fn 𝐾)
73	72	adantr 472	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → (𝑂‘𝐺) Fn 𝐾)
74	6	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → 𝐾 ∈ V)
75		simpr 477	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → 𝑥 ∈ 𝐾)
76		fnfvof 7109	. . . . . . . . . . 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 867	. . . . . . . . . 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 2799	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → ((𝑂‘𝐹)‘𝑥) = (((𝑂‘(𝐹(quot1p‘𝑅)𝐺))‘𝑥)(.r‘𝑅)((𝑂‘𝐺)‘𝑥)))
79	78	eqeq1d 2767	. . . . . . . 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 472	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → 𝑅 ∈ Domn)
82	68	ffvelrnda 6549	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → ((𝑂‘(𝐹(quot1p‘𝑅)𝐺))‘𝑥) ∈ 𝐾)
83	71	ffvelrnda 6549	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → ((𝑂‘𝐺)‘𝑥) ∈ 𝐾)
84	3, 63, 35	domneq0 19571	. . . . . . . . 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 1490	. . . . . . . 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 270	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → (((𝑂‘𝐹)‘𝑥) = 𝑊 ↔ (((𝑂‘(𝐹(quot1p‘𝑅)𝐺))‘𝑥) = 𝑊 ∨ ((𝑂‘𝐺)‘𝑥) = 𝑊)))
87	86	pm5.32da 574	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐾 ∧ ((𝑂‘𝐹)‘𝑥) = 𝑊) ↔ (𝑥 ∈ 𝐾 ∧ (((𝑂‘(𝐹(quot1p‘𝑅)𝐺))‘𝑥) = 𝑊 ∨ ((𝑂‘𝐺)‘𝑥) = 𝑊))))
88		andi 1030	. . . . . 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	syl6bb 278	. . . . 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 6528	. . . . . 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 3915	. . . . . 6 ⊢ (𝑥 ∈ ((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇}) ↔ (𝑥 ∈ (◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∨ 𝑥 ∈ {𝑇}))
93		fniniseg 6528	. . . . . . . 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 1174	. . . . . . . . 9 ⊢ (𝜑 → (◡(𝑂‘𝐺) “ {𝑊}) = {𝑇})
96	95	eleq2d 2830	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (◡(𝑂‘𝐺) “ {𝑊}) ↔ 𝑥 ∈ {𝑇}))
97		fniniseg 6528	. . . . . . . . 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 272	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ {𝑇} ↔ (𝑥 ∈ 𝐾 ∧ ((𝑂‘𝐺)‘𝑥) = 𝑊)))
100	94, 99	orbi12d 942	. . . . . 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 274	. . . . 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 302	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (◡(𝑂‘𝐹) “ {𝑊}) ↔ 𝑥 ∈ ((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇})))
103	102	eqrdv 2763	. . 3 ⊢ (𝜑 → (◡(𝑂‘𝐹) “ {𝑊}) = ((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇}))
104	103	fveq2d 6379	. 2 ⊢ (𝜑 → (♯‘(◡(𝑂‘𝐹) “ {𝑊})) = (♯‘((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇})))
105		fvex 6388	. . . . . . . . . 10 ⊢ (𝑂‘(𝐹(quot1p‘𝑅)𝐺)) ∈ V
106	105	cnvex 7311	. . . . . . . . 9 ⊢ ◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) ∈ V
107	106	imaex 7302	. . . . . . . 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 24216	. . . . . . . . 9 ⊢ (𝜑 → (𝐷‘(𝐹(quot1p‘𝑅)𝐺)) = 𝑁)
113		fveq2 6375	. . . . . . . . . . . 12 ⊢ (𝑔 = (𝐹(quot1p‘𝑅)𝐺) → (𝐷‘𝑔) = (𝐷‘(𝐹(quot1p‘𝑅)𝐺)))
114	113	eqeq1d 2767	. . . . . . . . . . 11 ⊢ (𝑔 = (𝐹(quot1p‘𝑅)𝐺) → ((𝐷‘𝑔) = 𝑁 ↔ (𝐷‘(𝐹(quot1p‘𝑅)𝐺)) = 𝑁))
115		fveq2 6375	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (𝐹(quot1p‘𝑅)𝐺) → (𝑂‘𝑔) = (𝑂‘(𝐹(quot1p‘𝑅)𝐺)))
116	115	cnveqd 5466	. . . . . . . . . . . . . 14 ⊢ (𝑔 = (𝐹(quot1p‘𝑅)𝐺) → ◡(𝑂‘𝑔) = ◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)))
117	116	imaeq1d 5647	. . . . . . . . . . . . 13 ⊢ (𝑔 = (𝐹(quot1p‘𝑅)𝐺) → (◡(𝑂‘𝑔) “ {𝑊}) = (◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}))
118	117	fveq2d 6379	. . . . . . . . . . . 12 ⊢ (𝑔 = (𝐹(quot1p‘𝑅)𝐺) → (♯‘(◡(𝑂‘𝑔) “ {𝑊})) = (♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})))
119	118, 113	breq12d 4822	. . . . . . . . . . 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 335	. . . . . . . . . 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 3467	. . . . . . . . 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 4835	. . . . . . 7 ⊢ (𝜑 → (♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})) ≤ 𝑁)
125		hashbnd 13327	. . . . . . 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 1490	. . . . . 6 ⊢ (𝜑 → (◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∈ Fin)
127		snfi 8245	. . . . . 6 ⊢ {𝑇} ∈ Fin
128		unfi 8434	. . . . . 6 ⊢ (((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∈ Fin ∧ {𝑇} ∈ Fin) → ((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇}) ∈ Fin)
129	126, 127, 128	sylancl 580	. . . . 5 ⊢ (𝜑 → ((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇}) ∈ Fin)
130		hashcl 13349	. . . . 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 11599	. . 3 ⊢ (𝜑 → (♯‘((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇})) ∈ ℝ)
133		hashcl 13349	. . . . . 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 11599	. . . 4 ⊢ (𝜑 → (♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})) ∈ ℝ)
136		peano2re 10463	. . . 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 11580	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
139	109, 138	syl 17	. . . . 5 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ0)
140	111, 139	eqeltrd 2844	. . . 4 ⊢ (𝜑 → (𝐷‘𝐹) ∈ ℕ0)
141	140	nn0red 11599	. . 3 ⊢ (𝜑 → (𝐷‘𝐹) ∈ ℝ)
142		hashun2 13374	. . . . 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 580	. . . 4 ⊢ (𝜑 → (♯‘((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇})) ≤ ((♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})) + (♯‘{𝑇})))
144		hashsng 13361	. . . . . 6 ⊢ (𝑇 ∈ (◡(𝑂‘𝐹) “ {𝑊}) → (♯‘{𝑇}) = 1)
145	1, 144	syl 17	. . . . 5 ⊢ (𝜑 → (♯‘{𝑇}) = 1)
146	145	oveq2d 6858	. . . 4 ⊢ (𝜑 → ((♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})) + (♯‘{𝑇})) = ((♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})) + 1))
147	143, 146	breqtrd 4835	. . 3 ⊢ (𝜑 → (♯‘((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇})) ≤ ((♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})) + 1))
148	109	nn0red 11599	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℝ)
149		1red 10294	. . . . 5 ⊢ (𝜑 → 1 ∈ ℝ)
150	135, 148, 149, 124	leadd1dd 10895	. . . 4 ⊢ (𝜑 → ((♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})) + 1) ≤ (𝑁 + 1))
151	150, 111	breqtrrd 4837	. . 3 ⊢ (𝜑 → ((♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})) + 1) ≤ (𝐷‘𝐹))
152	132, 137, 141, 147, 151	letrd 10448	. 2 ⊢ (𝜑 → (♯‘((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇})) ≤ (𝐷‘𝐹))
153	104, 152	eqbrtrd 4831	1 ⊢ (𝜑 → (♯‘(◡(𝑂‘𝐹) “ {𝑊})) ≤ (𝐷‘𝐹))

Syntax hints:   → wi 4   ↔ wb 197   ∧ wa 384   ∨ wo 873   = wceq 1652   ∈ wcel 2155  ∀wral 3055  Vcvv 3350   ∪ cun 3730  {csn 4334   class class class wbr 4809  ^◡ccnv 5276   “ cima 5280   Fn wfn 6063  ⟶wf 6064  ‘cfv 6068  (class class class)co 6842   ∘_𝑓 cof 7093  Fincfn 8160  ℝcr 10188  1c1 10190   + caddc 10192   ≤ cle 10329  ℕ₀cn0 11538  ♯chash 13321  Basecbs 16130  ._rcmulr 16215  0_gc0g 16366   ↑_s cpws 16373  -_gcsg 17691  Ringcrg 18814  CRingccrg 18815  ∥_rcdsr 18905   RingHom crh 18981  NzRingcnzr 19531  Domncdomn 19554  IDomncidom 19555  algSccascl 19585  var₁cv1 19819  Poly₁cpl1 19820  eval₁ce1 19952   deg₁ cdg1 24105  Monic_1pcmn1 24176  Unic_1pcuc1p 24177  quot_1pcq1p 24178

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 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266  ax-pre-sup 10267  ax-addf 10268  ax-mulf 10269

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  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 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-iin 4679  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-of 7095  df-ofr 7096  df-om 7264  df-1st 7366  df-2nd 7367  df-supp 7498  df-tpos 7555  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-2o 7765  df-oadd 7768  df-er 7947  df-map 8062  df-pm 8063  df-ixp 8114  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-fsupp 8483  df-sup 8555  df-oi 8622  df-card 9016  df-cda 9243  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-2 11335  df-3 11336  df-4 11337  df-5 11338  df-6 11339  df-7 11340  df-8 11341  df-9 11342  df-n0 11539  df-xnn0 11611  df-z 11625  df-dec 11741  df-uz 11887  df-fz 12534  df-fzo 12674  df-seq 13009  df-hash 13322  df-struct 16132  df-ndx 16133  df-slot 16134  df-base 16136  df-sets 16137  df-ress 16138  df-plusg 16227  df-mulr 16228  df-starv 16229  df-sca 16230  df-vsca 16231  df-ip 16232  df-tset 16233  df-ple 16234  df-ds 16236  df-unif 16237  df-hom 16238  df-cco 16239  df-0g 16368  df-gsum 16369  df-prds 16374  df-pws 16376  df-mre 16512  df-mrc 16513  df-acs 16515  df-mgm 17508  df-sgrp 17550  df-mnd 17561  df-mhm 17601  df-submnd 17602  df-grp 17692  df-minusg 17693  df-sbg 17694  df-mulg 17808  df-subg 17855  df-ghm 17922  df-cntz 18013  df-cmn 18461  df-abl 18462  df-mgp 18757  df-ur 18769  df-srg 18773  df-ring 18816  df-cring 18817  df-oppr 18890  df-dvdsr 18908  df-unit 18909  df-invr 18939  df-rnghom 18984  df-subrg 19047  df-lmod 19134  df-lss 19202  df-lsp 19244  df-nzr 19532  df-rlreg 19557  df-domn 19558  df-idom 19559  df-assa 19586  df-asp 19587  df-ascl 19588  df-psr 19630  df-mvr 19631  df-mpl 19632  df-opsr 19634  df-evls 19779  df-evl 19780  df-psr1 19823  df-vr1 19824  df-ply1 19825  df-coe1 19826  df-evl1 19954  df-cnfld 20020  df-mdeg 24106  df-deg1 24107  df-mon1 24181  df-uc1p 24182  df-q1p 24183  df-r1p 24184

This theorem is referenced by:  fta1g  24218