Theorem fta1glem2 26309

Description: Lemma for fta1g 26310. (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 2770	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑅 ↑s 𝐾) = (𝑅 ↑s 𝐾)
3		fta1glem.k	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝐾 = (Base‘𝑅)
4		eqid 2770	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Base‘(𝑅 ↑s 𝐾)) = (Base‘(𝑅 ↑s 𝐾))
5		fta1g.1	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑅 ∈ IDomn)
6	3	fvexi 6899	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝐾 ∈ V
7	6	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐾 ∈ V)
8		isidom 20812	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑅 ∈ IDomn ↔ (𝑅 ∈ CRing ∧ 𝑅 ∈ Domn))
9	8	simplbi 501	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑅 ∈ IDomn → 𝑅 ∈ CRing)
10	5, 9	syl 18	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝑅 ∈ CRing)
11		fta1g.o	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑂 = (eval1‘𝑅)
12		fta1g.p	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑃 = (Poly1‘𝑅)
13	11, 12, 2, 3	evl1rhm 22475	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑅 ∈ CRing → 𝑂 ∈ (𝑃 RingHom (𝑅 ↑s 𝐾)))
14	10, 13	syl 18	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑂 ∈ (𝑃 RingHom (𝑅 ↑s 𝐾)))
15		fta1g.b	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝐵 = (Base‘𝑃)
16	15, 4	rhmf 20569	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑂 ∈ (𝑃 RingHom (𝑅 ↑s 𝐾)) → 𝑂:𝐵⟶(Base‘(𝑅 ↑s 𝐾)))
17	14, 16	syl 18	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑂:𝐵⟶(Base‘(𝑅 ↑s 𝐾)))
18		fta1g.2	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
19	17, 18	ffvelcdmd 7084	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑂‘𝐹) ∈ (Base‘(𝑅 ↑s 𝐾)))
20	2, 3, 4, 5, 7, 19	pwselbas 17545	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑂‘𝐹):𝐾⟶𝐾)
21	20	ffnd 6710	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑂‘𝐹) Fn 𝐾)
22		fniniseg 7059	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑂‘𝐹) Fn 𝐾 → (𝑇 ∈ (◡(𝑂‘𝐹) “ {𝑊}) ↔ (𝑇 ∈ 𝐾 ∧ ((𝑂‘𝐹)‘𝑇) = 𝑊)))
23	21, 22	syl 18	. . . . . . . . . . . . . . . . . 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 500	. . . . . . . . . . . . . . . 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 502	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑅 ∈ IDomn → 𝑅 ∈ Domn)
31		domnnzr 20794	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ NzRing)
32	30, 31	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑅 ∈ IDomn → 𝑅 ∈ NzRing)
33	5, 32	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑅 ∈ NzRing)
34	24	simpld 499	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑇 ∈ 𝐾)
35		fta1g.w	. . . . . . . . . . . . . . . . 17 ⊢ 𝑊 = (0g‘𝑅)
36		eqid 2770	. . . . . . . . . . . . . . . . 17 ⊢ (∥r‘𝑃) = (∥r‘𝑃)
37	12, 15, 3, 26, 27, 28, 29, 11, 33, 10, 34, 18, 35, 36	facth1 26307	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐺(∥r‘𝑃)𝐹 ↔ ((𝑂‘𝐹)‘𝑇) = 𝑊))
38	25, 37	mpbird 260	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐺(∥r‘𝑃)𝐹)
39		nzrring 20602	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring)
40	33, 39	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑅 ∈ Ring)
41		eqid 2770	. . . . . . . . . . . . . . . . . . 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 26305	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐺 ∈ (Monic1p‘𝑅) ∧ (𝐷‘𝐺) = 1 ∧ (◡(𝑂‘𝐺) “ {𝑊}) = {𝑇}))
44	43	simp1d 1158	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐺 ∈ (Monic1p‘𝑅))
45		eqid 2770	. . . . . . . . . . . . . . . . . 18 ⊢ (Unic1p‘𝑅) = (Unic1p‘𝑅)
46	45, 41	mon1puc1p 26291	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ (Monic1p‘𝑅)) → 𝐺 ∈ (Unic1p‘𝑅))
47	40, 44, 46	syl2anc 595	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐺 ∈ (Unic1p‘𝑅))
48		eqid 2770	. . . . . . . . . . . . . . . . 17 ⊢ (.r‘𝑃) = (.r‘𝑃)
49		eqid 2770	. . . . . . . . . . . . . . . . 17 ⊢ (quot1p‘𝑅) = (quot1p‘𝑅)
50	12, 36, 15, 45, 48, 49	dvdsq1p 26303	. . . . . . . . . . . . . . . 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 1396	. . . . . . . . . . . . . . 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 6889	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑂‘𝐹) = (𝑂‘((𝐹(quot1p‘𝑅)𝐺)(.r‘𝑃)𝐺)))
54	49, 12, 15, 45	q1pcl 26297	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ (Unic1p‘𝑅)) → (𝐹(quot1p‘𝑅)𝐺) ∈ 𝐵)
55	40, 18, 47, 54	syl3anc 1396	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹(quot1p‘𝑅)𝐺) ∈ 𝐵)
56	12, 15, 41	mon1pcl 26285	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ (Monic1p‘𝑅) → 𝐺 ∈ 𝐵)
57	44, 56	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺 ∈ 𝐵)
58		eqid 2770	. . . . . . . . . . . . . . 15 ⊢ (.r‘(𝑅 ↑s 𝐾)) = (.r‘(𝑅 ↑s 𝐾))
59	15, 48, 58	rhmmul 20571	. . . . . . . . . . . . . 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 1396	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑂‘((𝐹(quot1p‘𝑅)𝐺)(.r‘𝑃)𝐺)) = ((𝑂‘(𝐹(quot1p‘𝑅)𝐺))(.r‘(𝑅 ↑s 𝐾))(𝑂‘𝐺)))
61	17, 55	ffvelcdmd 7084	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑂‘(𝐹(quot1p‘𝑅)𝐺)) ∈ (Base‘(𝑅 ↑s 𝐾)))
62	17, 57	ffvelcdmd 7084	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑂‘𝐺) ∈ (Base‘(𝑅 ↑s 𝐾)))
63		eqid 2770	. . . . . . . . . . . . . 14 ⊢ (.r‘𝑅) = (.r‘𝑅)
64	2, 4, 5, 7, 61, 62, 63, 58	pwsmulrval 17548	. . . . . . . . . . . . 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 2809	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑂‘𝐹) = ((𝑂‘(𝐹(quot1p‘𝑅)𝐺)) ∘f (.r‘𝑅)(𝑂‘𝐺)))
66	65	fveq1d 6887	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑂‘𝐹)‘𝑥) = (((𝑂‘(𝐹(quot1p‘𝑅)𝐺)) ∘f (.r‘𝑅)(𝑂‘𝐺))‘𝑥))
67	66	adantr 485	. . . . . . . . . 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 17545	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑂‘(𝐹(quot1p‘𝑅)𝐺)):𝐾⟶𝐾)
69	68	ffnd 6710	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑂‘(𝐹(quot1p‘𝑅)𝐺)) Fn 𝐾)
70	69	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → (𝑂‘(𝐹(quot1p‘𝑅)𝐺)) Fn 𝐾)
71	2, 3, 4, 5, 7, 62	pwselbas 17545	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑂‘𝐺):𝐾⟶𝐾)
72	71	ffnd 6710	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑂‘𝐺) Fn 𝐾)
73	72	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → (𝑂‘𝐺) Fn 𝐾)
74	6	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → 𝐾 ∈ V)
75		simpr 489	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → 𝑥 ∈ 𝐾)
76		fnfvof 7695	. . . . . . . . . . 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 851	. . . . . . . . . 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 2805	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → ((𝑂‘𝐹)‘𝑥) = (((𝑂‘(𝐹(quot1p‘𝑅)𝐺))‘𝑥)(.r‘𝑅)((𝑂‘𝐺)‘𝑥)))
79	78	eqeq1d 2772	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → (((𝑂‘𝐹)‘𝑥) = 𝑊 ↔ (((𝑂‘(𝐹(quot1p‘𝑅)𝐺))‘𝑥)(.r‘𝑅)((𝑂‘𝐺)‘𝑥)) = 𝑊))
80	5, 30	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ Domn)
81	80	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → 𝑅 ∈ Domn)
82	68	ffvelcdmda 7083	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → ((𝑂‘(𝐹(quot1p‘𝑅)𝐺))‘𝑥) ∈ 𝐾)
83	71	ffvelcdmda 7083	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → ((𝑂‘𝐺)‘𝑥) ∈ 𝐾)
84	3, 63, 35	domneq0 20796	. . . . . . . . 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 1396	. . . . . . . 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 589	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐾 ∧ ((𝑂‘𝐹)‘𝑥) = 𝑊) ↔ (𝑥 ∈ 𝐾 ∧ (((𝑂‘(𝐹(quot1p‘𝑅)𝐺))‘𝑥) = 𝑊 ∨ ((𝑂‘𝐺)‘𝑥) = 𝑊))))
88		andi 1023	. . . . . 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 7059	. . . . . 6 ⊢ ((𝑂‘𝐹) Fn 𝐾 → (𝑥 ∈ (◡(𝑂‘𝐹) “ {𝑊}) ↔ (𝑥 ∈ 𝐾 ∧ ((𝑂‘𝐹)‘𝑥) = 𝑊)))
91	21, 90	syl 18	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (◡(𝑂‘𝐹) “ {𝑊}) ↔ (𝑥 ∈ 𝐾 ∧ ((𝑂‘𝐹)‘𝑥) = 𝑊)))
92		elun 4115	. . . . . 6 ⊢ (𝑥 ∈ ((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇}) ↔ (𝑥 ∈ (◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∨ 𝑥 ∈ {𝑇}))
93		fniniseg 7059	. . . . . . . 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 18	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ↔ (𝑥 ∈ 𝐾 ∧ ((𝑂‘(𝐹(quot1p‘𝑅)𝐺))‘𝑥) = 𝑊)))
95	43	simp3d 1160	. . . . . . . . 9 ⊢ (𝜑 → (◡(𝑂‘𝐺) “ {𝑊}) = {𝑇})
96	95	eleq2d 2856	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (◡(𝑂‘𝐺) “ {𝑊}) ↔ 𝑥 ∈ {𝑇}))
97		fniniseg 7059	. . . . . . . . 9 ⊢ ((𝑂‘𝐺) Fn 𝐾 → (𝑥 ∈ (◡(𝑂‘𝐺) “ {𝑊}) ↔ (𝑥 ∈ 𝐾 ∧ ((𝑂‘𝐺)‘𝑥) = 𝑊)))
98	72, 97	syl 18	. . . . . . . 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 931	. . . . . 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	bitrid 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 2768	. . 3 ⊢ (𝜑 → (◡(𝑂‘𝐹) “ {𝑊}) = ((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇}))
104	103	fveq2d 6889	. 2 ⊢ (𝜑 → (♯‘(◡(𝑂‘𝐹) “ {𝑊})) = (♯‘((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇})))
105		fvex 6898	. . . . . . . . . 10 ⊢ (𝑂‘(𝐹(quot1p‘𝑅)𝐺)) ∈ V
106	105	cnvex 7925	. . . . . . . . 9 ⊢ ◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) ∈ V
107	106	imaex 7914	. . . . . . . 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 26308	. . . . . . . . 9 ⊢ (𝜑 → (𝐷‘(𝐹(quot1p‘𝑅)𝐺)) = 𝑁)
113		fveq2 6885	. . . . . . . . . . . 12 ⊢ (𝑔 = (𝐹(quot1p‘𝑅)𝐺) → (𝐷‘𝑔) = (𝐷‘(𝐹(quot1p‘𝑅)𝐺)))
114	113	eqeq1d 2772	. . . . . . . . . . 11 ⊢ (𝑔 = (𝐹(quot1p‘𝑅)𝐺) → ((𝐷‘𝑔) = 𝑁 ↔ (𝐷‘(𝐹(quot1p‘𝑅)𝐺)) = 𝑁))
115		fveq2 6885	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (𝐹(quot1p‘𝑅)𝐺) → (𝑂‘𝑔) = (𝑂‘(𝐹(quot1p‘𝑅)𝐺)))
116	115	cnveqd 5865	. . . . . . . . . . . . . 14 ⊢ (𝑔 = (𝐹(quot1p‘𝑅)𝐺) → ◡(𝑂‘𝑔) = ◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)))
117	116	imaeq1d 6065	. . . . . . . . . . . . 13 ⊢ (𝑔 = (𝐹(quot1p‘𝑅)𝐺) → (◡(𝑂‘𝑔) “ {𝑊}) = (◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}))
118	117	fveq2d 6889	. . . . . . . . . . . 12 ⊢ (𝑔 = (𝐹(quot1p‘𝑅)𝐺) → (♯‘(◡(𝑂‘𝑔) “ {𝑊})) = (♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})))
119	118, 113	breq12d 5127	. . . . . . . . . . 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 347	. . . . . . . . . 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 3590	. . . . . . . . 9 ⊢ (𝜑 → ((𝐷‘(𝐹(quot1p‘𝑅)𝐺)) = 𝑁 → (♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})) ≤ (𝐷‘(𝐹(quot1p‘𝑅)𝐺))))
123	112, 122	mpd 16	. . . . . . . 8 ⊢ (𝜑 → (♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})) ≤ (𝐷‘(𝐹(quot1p‘𝑅)𝐺)))
124	123, 112	breqtrd 5142	. . . . . . 7 ⊢ (𝜑 → (♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})) ≤ 𝑁)
125		hashbnd 14375	. . . . . . 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 1396	. . . . . 6 ⊢ (𝜑 → (◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∈ Fin)
127		snfi 9043	. . . . . 6 ⊢ {𝑇} ∈ Fin
128		unfi 9158	. . . . . 6 ⊢ (((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∈ Fin ∧ {𝑇} ∈ Fin) → ((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇}) ∈ Fin)
129	126, 127, 128	sylancl 597	. . . . 5 ⊢ (𝜑 → ((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇}) ∈ Fin)
130		hashcl 14395	. . . . 5 ⊢ (((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇}) ∈ Fin → (♯‘((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇})) ∈ ℕ0)
131	129, 130	syl 18	. . . 4 ⊢ (𝜑 → (♯‘((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇})) ∈ ℕ0)
132	131	nn0red 12569	. . 3 ⊢ (𝜑 → (♯‘((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇})) ∈ ℝ)
133		hashcl 14395	. . . . . 6 ⊢ ((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∈ Fin → (♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})) ∈ ℕ0)
134	126, 133	syl 18	. . . . 5 ⊢ (𝜑 → (♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})) ∈ ℕ0)
135	134	nn0red 12569	. . . 4 ⊢ (𝜑 → (♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})) ∈ ℝ)
136		peano2re 11386	. . . 4 ⊢ ((♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})) ∈ ℝ → ((♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})) + 1) ∈ ℝ)
137	135, 136	syl 18	. . 3 ⊢ (𝜑 → ((♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})) + 1) ∈ ℝ)
138		peano2nn0 12547	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
139	109, 138	syl 18	. . . . 5 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ0)
140	111, 139	eqeltrd 2870	. . . 4 ⊢ (𝜑 → (𝐷‘𝐹) ∈ ℕ0)
141	140	nn0red 12569	. . 3 ⊢ (𝜑 → (𝐷‘𝐹) ∈ ℝ)
142		hashun2 14422	. . . . 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 597	. . . 4 ⊢ (𝜑 → (♯‘((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇})) ≤ ((♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})) + (♯‘{𝑇})))
144		hashsng 14408	. . . . . 6 ⊢ (𝑇 ∈ (◡(𝑂‘𝐹) “ {𝑊}) → (♯‘{𝑇}) = 1)
145	1, 144	syl 18	. . . . 5 ⊢ (𝜑 → (♯‘{𝑇}) = 1)
146	145	oveq2d 7430	. . . 4 ⊢ (𝜑 → ((♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})) + (♯‘{𝑇})) = ((♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})) + 1))
147	143, 146	breqtrd 5142	. . 3 ⊢ (𝜑 → (♯‘((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇})) ≤ ((♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})) + 1))
148	109	nn0red 12569	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℝ)
149		1red 11212	. . . . 5 ⊢ (𝜑 → 1 ∈ ℝ)
150	135, 148, 149, 124	leadd1dd 11831	. . . 4 ⊢ (𝜑 → ((♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})) + 1) ≤ (𝑁 + 1))
151	150, 111	breqtrrd 5144	. . 3 ⊢ (𝜑 → ((♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})) + 1) ≤ (𝐷‘𝐹))
152	132, 137, 141, 147, 151	letrd 11370	. 2 ⊢ (𝜑 → (♯‘((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇})) ≤ (𝐷‘𝐹))
153	104, 152	eqbrtrd 5138	1 ⊢ (𝜑 → (♯‘(◡(𝑂‘𝐹) “ {𝑊})) ≤ (𝐷‘𝐹))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1568   ∈ wcel 2150  ∀wral 3086  Vcvv 3462   ∪ cun 3911  {csn 4594   class class class wbr 5114  ^◡ccnv 5664   “ cima 5668   Fn wfn 6535  ⟶wf 6536  ‘cfv 6540  (class class class)co 7414   ∘_f cof 7676  Fincfn 8946  ℝcr 11102  1c1 11104   + caddc 11106   ≤ cle 11247  ℕ₀cn0 12507  ♯chash 14369  Basecbs 17272  ._rcmulr 17314  0_gc0g 17495   ↑_s cpws 17502  -_gcsg 19005  Ringcrg 20318  CRingccrg 20319  ∥_rcdsr 20439   RingHom crh 20554  NzRingcnzr 20598  Domncdomn 20780  IDomncidom 20781  algSccascl 21985  var₁cv1 22319  Poly₁cpl1 22320  eval₁ce1 22457  deg₁cdg1 26194  Monic_1pcmn1 26266  Unic_1pcuc1p 26267  quot_1pcq1p 26268

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5340  ax-pr 5408  ax-un 7736  ax-cnex 11159  ax-resscn 11160  ax-1cn 11161  ax-icn 11162  ax-addcl 11163  ax-addrcl 11164  ax-mulcl 11165  ax-mulrcl 11166  ax-mulcom 11167  ax-addass 11168  ax-mulass 11169  ax-distr 11170  ax-i2m1 11171  ax-1ne0 11172  ax-1rid 11173  ax-rnegex 11174  ax-rrecex 11175  ax-cnre 11176  ax-pre-lttri 11177  ax-pre-lttrn 11178  ax-pre-ltadd 11179  ax-pre-mulgt0 11180  ax-pre-sup 11181  ax-addf 11182

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2099  df-mo 2574  df-eu 2604  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-ne 2966  df-nel 3072  df-ral 3087  df-rex 3097  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3464  df-sbc 3753  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-uni 4878  df-int 4918  df-iun 4963  df-iin 4964  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5560  df-eprel 5565  df-po 5573  df-so 5574  df-fr 5618  df-se 5619  df-we 5620  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-of 7678  df-ofr 7679  df-om 7866  df-1st 7989  df-2nd 7990  df-supp 8160  df-tpos 8225  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-1o 8456  df-2o 8457  df-oadd 8460  df-er 8697  df-map 8829  df-pm 8830  df-ixp 8899  df-en 8947  df-dom 8948  df-sdom 8949  df-fin 8950  df-fsupp 9325  df-sup 9405  df-oi 9475  df-dju 9890  df-card 9928  df-pnf 11248  df-mnf 11249  df-xr 11250  df-ltxr 11251  df-le 11252  df-sub 11446  df-neg 11447  df-nn 12237  df-2 12306  df-3 12307  df-4 12308  df-5 12309  df-6 12310  df-7 12311  df-8 12312  df-9 12313  df-n0 12508  df-xnn0 12581  df-z 12595  df-dec 12715  df-uz 12866  df-fz 13539  df-fzo 13686  df-seq 14041  df-hash 14370  df-struct 17210  df-sets 17227  df-slot 17245  df-ndx 17257  df-base 17273  df-ress 17294  df-plusg 17326  df-mulr 17327  df-starv 17328  df-sca 17329  df-vsca 17330  df-ip 17331  df-tset 17332  df-ple 17333  df-ds 17335  df-unif 17336  df-hom 17337  df-cco 17338  df-0g 17497  df-gsum 17498  df-prds 17503  df-pws 17505  df-mre 17641  df-mrc 17642  df-acs 17644  df-mgm 18701  df-sgrp 18780  df-mnd 18796  df-mhm 18844  df-submnd 18845  df-grp 19006  df-minusg 19007  df-sbg 19008  df-mulg 19137  df-subg 19192  df-ghm 19287  df-cntz 19390  df-cmn 19855  df-abl 19856  df-mgp 20220  df-rng 20234  df-ur 20267  df-srg 20272  df-ring 20320  df-cring 20321  df-oppr 20422  df-dvdsr 20442  df-unit 20443  df-invr 20473  df-rhm 20557  df-nzr 20599  df-subrng 20634  df-subrg 20658  df-rlreg 20782  df-domn 20783  df-idom 20784  df-lmod 20966  df-lss 21036  df-lsp 21076  df-cnfld 21506  df-assa 21986  df-asp 21987  df-ascl 21988  df-psr 22042  df-mvr 22043  df-mpl 22044  df-opsr 22046  df-evls 22208  df-evl 22209  df-psr1 22323  df-vr1 22324  df-ply1 22325  df-coe1 22326  df-evl1 22459  df-mdeg 26195  df-deg1 26196  df-mon1 26271  df-uc1p 26272  df-q1p 26273  df-r1p 26274

This theorem is referenced by:  fta1g  26310