Theorem fta1glem2 24763

Description: Lemma for fta1g 24764. (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 2824	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑅 ↑s 𝐾) = (𝑅 ↑s 𝐾)
3		fta1glem.k	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝐾 = (Base‘𝑅)
4		eqid 2824	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Base‘(𝑅 ↑s 𝐾)) = (Base‘(𝑅 ↑s 𝐾))
5		fta1g.1	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑅 ∈ IDomn)
6	3	fvexi 6687	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝐾 ∈ V
7	6	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐾 ∈ V)
8		isidom 20080	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑅 ∈ IDomn ↔ (𝑅 ∈ CRing ∧ 𝑅 ∈ Domn))
9	8	simplbi 500	. . . . . . . . . . . . . . . . . . . . . . . . 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 20498	. . . . . . . . . . . . . . . . . . . . . . . 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 19481	. . . . . . . . . . . . . . . . . . . . . . 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 6855	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑂‘𝐹) ∈ (Base‘(𝑅 ↑s 𝐾)))
20	2, 3, 4, 5, 7, 19	pwselbas 16765	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑂‘𝐹):𝐾⟶𝐾)
21	20	ffnd 6518	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑂‘𝐹) Fn 𝐾)
22		fniniseg 6833	. . . . . . . . . . . . . . . . . . 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 234	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑇 ∈ 𝐾 ∧ ((𝑂‘𝐹)‘𝑇) = 𝑊))
25	24	simprd 498	. . . . . . . . . . . . . . . 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 499	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑅 ∈ IDomn → 𝑅 ∈ Domn)
31		domnnzr 20071	. . . . . . . . . . . . . . . . . . 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 497	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑇 ∈ 𝐾)
35		fta1g.w	. . . . . . . . . . . . . . . . 17 ⊢ 𝑊 = (0g‘𝑅)
36		eqid 2824	. . . . . . . . . . . . . . . . 17 ⊢ (∥r‘𝑃) = (∥r‘𝑃)
37	12, 15, 3, 26, 27, 28, 29, 11, 33, 10, 34, 18, 35, 36	facth1 24761	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐺(∥r‘𝑃)𝐹 ↔ ((𝑂‘𝐹)‘𝑇) = 𝑊))
38	25, 37	mpbird 259	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐺(∥r‘𝑃)𝐹)
39		nzrring 20037	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring)
40	33, 39	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑅 ∈ Ring)
41		eqid 2824	. . . . . . . . . . . . . . . . . . 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 24759	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐺 ∈ (Monic1p‘𝑅) ∧ (𝐷‘𝐺) = 1 ∧ (◡(𝑂‘𝐺) “ {𝑊}) = {𝑇}))
44	43	simp1d 1138	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐺 ∈ (Monic1p‘𝑅))
45		eqid 2824	. . . . . . . . . . . . . . . . . 18 ⊢ (Unic1p‘𝑅) = (Unic1p‘𝑅)
46	45, 41	mon1puc1p 24747	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ (Monic1p‘𝑅)) → 𝐺 ∈ (Unic1p‘𝑅))
47	40, 44, 46	syl2anc 586	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐺 ∈ (Unic1p‘𝑅))
48		eqid 2824	. . . . . . . . . . . . . . . . 17 ⊢ (.r‘𝑃) = (.r‘𝑃)
49		eqid 2824	. . . . . . . . . . . . . . . . 17 ⊢ (quot1p‘𝑅) = (quot1p‘𝑅)
50	12, 36, 15, 45, 48, 49	dvdsq1p 24757	. . . . . . . . . . . . . . . 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 1367	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐺(∥r‘𝑃)𝐹 ↔ 𝐹 = ((𝐹(quot1p‘𝑅)𝐺)(.r‘𝑃)𝐺)))
52	38, 51	mpbid 234	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹 = ((𝐹(quot1p‘𝑅)𝐺)(.r‘𝑃)𝐺))
53	52	fveq2d 6677	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑂‘𝐹) = (𝑂‘((𝐹(quot1p‘𝑅)𝐺)(.r‘𝑃)𝐺)))
54	49, 12, 15, 45	q1pcl 24752	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ (Unic1p‘𝑅)) → (𝐹(quot1p‘𝑅)𝐺) ∈ 𝐵)
55	40, 18, 47, 54	syl3anc 1367	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹(quot1p‘𝑅)𝐺) ∈ 𝐵)
56	12, 15, 41	mon1pcl 24741	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ (Monic1p‘𝑅) → 𝐺 ∈ 𝐵)
57	44, 56	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺 ∈ 𝐵)
58		eqid 2824	. . . . . . . . . . . . . . 15 ⊢ (.r‘(𝑅 ↑s 𝐾)) = (.r‘(𝑅 ↑s 𝐾))
59	15, 48, 58	rhmmul 19482	. . . . . . . . . . . . . 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 1367	. . . . . . . . . . . . 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 6855	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑂‘(𝐹(quot1p‘𝑅)𝐺)) ∈ (Base‘(𝑅 ↑s 𝐾)))
62	17, 57	ffvelrnd 6855	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑂‘𝐺) ∈ (Base‘(𝑅 ↑s 𝐾)))
63		eqid 2824	. . . . . . . . . . . . . 14 ⊢ (.r‘𝑅) = (.r‘𝑅)
64	2, 4, 5, 7, 61, 62, 63, 58	pwsmulrval 16767	. . . . . . . . . . . . 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 2863	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑂‘𝐹) = ((𝑂‘(𝐹(quot1p‘𝑅)𝐺)) ∘f (.r‘𝑅)(𝑂‘𝐺)))
66	65	fveq1d 6675	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑂‘𝐹)‘𝑥) = (((𝑂‘(𝐹(quot1p‘𝑅)𝐺)) ∘f (.r‘𝑅)(𝑂‘𝐺))‘𝑥))
67	66	adantr 483	. . . . . . . . . 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 16765	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑂‘(𝐹(quot1p‘𝑅)𝐺)):𝐾⟶𝐾)
69	68	ffnd 6518	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑂‘(𝐹(quot1p‘𝑅)𝐺)) Fn 𝐾)
70	69	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → (𝑂‘(𝐹(quot1p‘𝑅)𝐺)) Fn 𝐾)
71	2, 3, 4, 5, 7, 62	pwselbas 16765	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑂‘𝐺):𝐾⟶𝐾)
72	71	ffnd 6518	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑂‘𝐺) Fn 𝐾)
73	72	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → (𝑂‘𝐺) Fn 𝐾)
74	6	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → 𝐾 ∈ V)
75		simpr 487	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → 𝑥 ∈ 𝐾)
76		fnfvof 7426	. . . . . . . . . . 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 836	. . . . . . . . . 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 2859	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → ((𝑂‘𝐹)‘𝑥) = (((𝑂‘(𝐹(quot1p‘𝑅)𝐺))‘𝑥)(.r‘𝑅)((𝑂‘𝐺)‘𝑥)))
79	78	eqeq1d 2826	. . . . . . . 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 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → 𝑅 ∈ Domn)
82	68	ffvelrnda 6854	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → ((𝑂‘(𝐹(quot1p‘𝑅)𝐺))‘𝑥) ∈ 𝐾)
83	71	ffvelrnda 6854	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → ((𝑂‘𝐺)‘𝑥) ∈ 𝐾)
84	3, 63, 35	domneq0 20073	. . . . . . . . 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 1367	. . . . . . . 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 281	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → (((𝑂‘𝐹)‘𝑥) = 𝑊 ↔ (((𝑂‘(𝐹(quot1p‘𝑅)𝐺))‘𝑥) = 𝑊 ∨ ((𝑂‘𝐺)‘𝑥) = 𝑊)))
87	86	pm5.32da 581	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐾 ∧ ((𝑂‘𝐹)‘𝑥) = 𝑊) ↔ (𝑥 ∈ 𝐾 ∧ (((𝑂‘(𝐹(quot1p‘𝑅)𝐺))‘𝑥) = 𝑊 ∨ ((𝑂‘𝐺)‘𝑥) = 𝑊))))
88		andi 1004	. . . . . 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 289	. . . . 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 6833	. . . . . 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 4128	. . . . . 6 ⊢ (𝑥 ∈ ((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇}) ↔ (𝑥 ∈ (◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∨ 𝑥 ∈ {𝑇}))
93		fniniseg 6833	. . . . . . . 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 1140	. . . . . . . . 9 ⊢ (𝜑 → (◡(𝑂‘𝐺) “ {𝑊}) = {𝑇})
96	95	eleq2d 2901	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (◡(𝑂‘𝐺) “ {𝑊}) ↔ 𝑥 ∈ {𝑇}))
97		fniniseg 6833	. . . . . . . . 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 283	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ {𝑇} ↔ (𝑥 ∈ 𝐾 ∧ ((𝑂‘𝐺)‘𝑥) = 𝑊)))
100	94, 99	orbi12d 915	. . . . . 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 285	. . . . 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 313	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (◡(𝑂‘𝐹) “ {𝑊}) ↔ 𝑥 ∈ ((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇})))
103	102	eqrdv 2822	. . 3 ⊢ (𝜑 → (◡(𝑂‘𝐹) “ {𝑊}) = ((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇}))
104	103	fveq2d 6677	. 2 ⊢ (𝜑 → (♯‘(◡(𝑂‘𝐹) “ {𝑊})) = (♯‘((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇})))
105		fvex 6686	. . . . . . . . . 10 ⊢ (𝑂‘(𝐹(quot1p‘𝑅)𝐺)) ∈ V
106	105	cnvex 7633	. . . . . . . . 9 ⊢ ◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) ∈ V
107	106	imaex 7624	. . . . . . . 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 24762	. . . . . . . . 9 ⊢ (𝜑 → (𝐷‘(𝐹(quot1p‘𝑅)𝐺)) = 𝑁)
113		fveq2 6673	. . . . . . . . . . . 12 ⊢ (𝑔 = (𝐹(quot1p‘𝑅)𝐺) → (𝐷‘𝑔) = (𝐷‘(𝐹(quot1p‘𝑅)𝐺)))
114	113	eqeq1d 2826	. . . . . . . . . . 11 ⊢ (𝑔 = (𝐹(quot1p‘𝑅)𝐺) → ((𝐷‘𝑔) = 𝑁 ↔ (𝐷‘(𝐹(quot1p‘𝑅)𝐺)) = 𝑁))
115		fveq2 6673	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (𝐹(quot1p‘𝑅)𝐺) → (𝑂‘𝑔) = (𝑂‘(𝐹(quot1p‘𝑅)𝐺)))
116	115	cnveqd 5749	. . . . . . . . . . . . . 14 ⊢ (𝑔 = (𝐹(quot1p‘𝑅)𝐺) → ◡(𝑂‘𝑔) = ◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)))
117	116	imaeq1d 5931	. . . . . . . . . . . . 13 ⊢ (𝑔 = (𝐹(quot1p‘𝑅)𝐺) → (◡(𝑂‘𝑔) “ {𝑊}) = (◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}))
118	117	fveq2d 6677	. . . . . . . . . . . 12 ⊢ (𝑔 = (𝐹(quot1p‘𝑅)𝐺) → (♯‘(◡(𝑂‘𝑔) “ {𝑊})) = (♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})))
119	118, 113	breq12d 5082	. . . . . . . . . . 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 3628	. . . . . . . . 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 5095	. . . . . . 7 ⊢ (𝜑 → (♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})) ≤ 𝑁)
125		hashbnd 13699	. . . . . . 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 1367	. . . . . 6 ⊢ (𝜑 → (◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∈ Fin)
127		snfi 8597	. . . . . 6 ⊢ {𝑇} ∈ Fin
128		unfi 8788	. . . . . 6 ⊢ (((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∈ Fin ∧ {𝑇} ∈ Fin) → ((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇}) ∈ Fin)
129	126, 127, 128	sylancl 588	. . . . 5 ⊢ (𝜑 → ((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇}) ∈ Fin)
130		hashcl 13720	. . . . 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 11959	. . 3 ⊢ (𝜑 → (♯‘((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇})) ∈ ℝ)
133		hashcl 13720	. . . . . 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 11959	. . . 4 ⊢ (𝜑 → (♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})) ∈ ℝ)
136		peano2re 10816	. . . 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 11940	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
139	109, 138	syl 17	. . . . 5 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ0)
140	111, 139	eqeltrd 2916	. . . 4 ⊢ (𝜑 → (𝐷‘𝐹) ∈ ℕ0)
141	140	nn0red 11959	. . 3 ⊢ (𝜑 → (𝐷‘𝐹) ∈ ℝ)
142		hashun2 13747	. . . . 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 588	. . . 4 ⊢ (𝜑 → (♯‘((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇})) ≤ ((♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})) + (♯‘{𝑇})))
144		hashsng 13733	. . . . . 6 ⊢ (𝑇 ∈ (◡(𝑂‘𝐹) “ {𝑊}) → (♯‘{𝑇}) = 1)
145	1, 144	syl 17	. . . . 5 ⊢ (𝜑 → (♯‘{𝑇}) = 1)
146	145	oveq2d 7175	. . . 4 ⊢ (𝜑 → ((♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})) + (♯‘{𝑇})) = ((♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})) + 1))
147	143, 146	breqtrd 5095	. . 3 ⊢ (𝜑 → (♯‘((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇})) ≤ ((♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})) + 1))
148	109	nn0red 11959	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℝ)
149		1red 10645	. . . . 5 ⊢ (𝜑 → 1 ∈ ℝ)
150	135, 148, 149, 124	leadd1dd 11257	. . . 4 ⊢ (𝜑 → ((♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})) + 1) ≤ (𝑁 + 1))
151	150, 111	breqtrrd 5097	. . 3 ⊢ (𝜑 → ((♯‘(◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊})) + 1) ≤ (𝐷‘𝐹))
152	132, 137, 141, 147, 151	letrd 10800	. 2 ⊢ (𝜑 → (♯‘((◡(𝑂‘(𝐹(quot1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇})) ≤ (𝐷‘𝐹))
153	104, 152	eqbrtrd 5091	1 ⊢ (𝜑 → (♯‘(◡(𝑂‘𝐹) “ {𝑊})) ≤ (𝐷‘𝐹))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   = wceq 1536   ∈ wcel 2113  ∀wral 3141  Vcvv 3497   ∪ cun 3937  {csn 4570   class class class wbr 5069  ^◡ccnv 5557   “ cima 5561   Fn wfn 6353  ⟶wf 6354  ‘cfv 6358  (class class class)co 7159   ∘_f cof 7410  Fincfn 8512  ℝcr 10539  1c1 10541   + caddc 10543   ≤ cle 10679  ℕ₀cn0 11900  ♯chash 13693  Basecbs 16486  ._rcmulr 16569  0_gc0g 16716   ↑_s cpws 16723  -_gcsg 18108  Ringcrg 19300  CRingccrg 19301  ∥_rcdsr 19391   RingHom crh 19467  NzRingcnzr 20033  Domncdomn 20056  IDomncidom 20057  algSccascl 20087  var₁cv1 20347  Poly₁cpl1 20348  eval₁ce1 20480   deg₁ cdg1 24651  Monic_1pcmn1 24722  Unic_1pcuc1p 24723  quot_1pcq1p 24724

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464  ax-cnex 10596  ax-resscn 10597  ax-1cn 10598  ax-icn 10599  ax-addcl 10600  ax-addrcl 10601  ax-mulcl 10602  ax-mulrcl 10603  ax-mulcom 10604  ax-addass 10605  ax-mulass 10606  ax-distr 10607  ax-i2m1 10608  ax-1ne0 10609  ax-1rid 10610  ax-rnegex 10611  ax-rrecex 10612  ax-cnre 10613  ax-pre-lttri 10614  ax-pre-lttrn 10615  ax-pre-ltadd 10616  ax-pre-mulgt0 10617  ax-pre-sup 10618  ax-addf 10619  ax-mulf 10620

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-nel 3127  df-ral 3146  df-rex 3147  df-reu 3148  df-rmo 3149  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-int 4880  df-iun 4924  df-iin 4925  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-se 5518  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-isom 6367  df-riota 7117  df-ov 7162  df-oprab 7163  df-mpo 7164  df-of 7412  df-ofr 7413  df-om 7584  df-1st 7692  df-2nd 7693  df-supp 7834  df-tpos 7895  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-1o 8105  df-2o 8106  df-oadd 8109  df-er 8292  df-map 8411  df-pm 8412  df-ixp 8465  df-en 8513  df-dom 8514  df-sdom 8515  df-fin 8516  df-fsupp 8837  df-sup 8909  df-oi 8977  df-dju 9333  df-card 9371  df-pnf 10680  df-mnf 10681  df-xr 10682  df-ltxr 10683  df-le 10684  df-sub 10875  df-neg 10876  df-nn 11642  df-2 11703  df-3 11704  df-4 11705  df-5 11706  df-6 11707  df-7 11708  df-8 11709  df-9 11710  df-n0 11901  df-xnn0 11971  df-z 11985  df-dec 12102  df-uz 12247  df-fz 12896  df-fzo 13037  df-seq 13373  df-hash 13694  df-struct 16488  df-ndx 16489  df-slot 16490  df-base 16492  df-sets 16493  df-ress 16494  df-plusg 16581  df-mulr 16582  df-starv 16583  df-sca 16584  df-vsca 16585  df-ip 16586  df-tset 16587  df-ple 16588  df-ds 16590  df-unif 16591  df-hom 16592  df-cco 16593  df-0g 16718  df-gsum 16719  df-prds 16724  df-pws 16726  df-mre 16860  df-mrc 16861  df-acs 16863  df-mgm 17855  df-sgrp 17904  df-mnd 17915  df-mhm 17959  df-submnd 17960  df-grp 18109  df-minusg 18110  df-sbg 18111  df-mulg 18228  df-subg 18279  df-ghm 18359  df-cntz 18450  df-cmn 18911  df-abl 18912  df-mgp 19243  df-ur 19255  df-srg 19259  df-ring 19302  df-cring 19303  df-oppr 19376  df-dvdsr 19394  df-unit 19395  df-invr 19425  df-rnghom 19470  df-subrg 19536  df-lmod 19639  df-lss 19707  df-lsp 19747  df-nzr 20034  df-rlreg 20059  df-domn 20060  df-idom 20061  df-assa 20088  df-asp 20089  df-ascl 20090  df-psr 20139  df-mvr 20140  df-mpl 20141  df-opsr 20143  df-evls 20289  df-evl 20290  df-psr1 20351  df-vr1 20352  df-ply1 20353  df-coe1 20354  df-evl1 20482  df-cnfld 20549  df-mdeg 24652  df-deg1 24653  df-mon1 24727  df-uc1p 24728  df-q1p 24729  df-r1p 24730

This theorem is referenced by:  fta1g  24764