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Theorem fta1glem2 26058

Description: Lemma for fta1g 26059. (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 = (0_g‘𝑅)
fta1g.z	⊢ 0 = (0_g‘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 2726	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑅 ↑_s 𝐾) = (𝑅 ↑_s 𝐾)
3		fta1glem.k	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝐾 = (Base‘𝑅)
4		eqid 2726	. . . . . . . . . . . . . . . . . . . . 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 21216	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑅 ∈ IDomn ↔ (𝑅 ∈ CRing ∧ 𝑅 ∈ Domn))
9	8	simplbi 497	. . . . . . . . . . . . . . . . . . . . . . . . 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 22206	. . . . . . . . . . . . . . . . . . . . . . . 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 20387	. . . . . . . . . . . . . . . . . . . . . . 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	ffvelcdmd 7081	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑂‘𝐹) ∈ (Base‘(𝑅 ↑_s 𝐾)))
20	2, 3, 4, 5, 7, 19	pwselbas 17444	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑂‘𝐹):𝐾⟶𝐾)
21	20	ffnd 6712	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑂‘𝐹) Fn 𝐾)
22		fniniseg 7055	. . . . . . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑇 ∈ 𝐾 ∧ ((𝑂‘𝐹)‘𝑇) = 𝑊))
25	24	simprd 495	. . . . . . . . . . . . . . . 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 496	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑅 ∈ IDomn → 𝑅 ∈ Domn)
31		domnnzr 21205	. . . . . . . . . . . . . . . . . . 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 494	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑇 ∈ 𝐾)
35		fta1g.w	. . . . . . . . . . . . . . . . 17 ⊢ 𝑊 = (0_g‘𝑅)
36		eqid 2726	. . . . . . . . . . . . . . . . 17 ⊢ (∥_r‘𝑃) = (∥_r‘𝑃)
37	12, 15, 3, 26, 27, 28, 29, 11, 33, 10, 34, 18, 35, 36	facth1 26056	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐺(∥_r‘𝑃)𝐹 ↔ ((𝑂‘𝐹)‘𝑇) = 𝑊))
38	25, 37	mpbird 257	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐺(∥_r‘𝑃)𝐹)
39		nzrring 20418	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring)
40	33, 39	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑅 ∈ Ring)
41		eqid 2726	. . . . . . . . . . . . . . . . . . 19 ⊢ (Monic_1p‘𝑅) = (Monic_1p‘𝑅)
42		fta1g.d	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐷 = ( deg₁ ‘𝑅)
43	12, 15, 3, 26, 27, 28, 29, 11, 33, 10, 34, 41, 42, 35	ply1remlem 26054	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐺 ∈ (Monic_1p‘𝑅) ∧ (𝐷‘𝐺) = 1 ∧ (^◡(𝑂‘𝐺) “ {𝑊}) = {𝑇}))
44	43	simp1d 1139	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐺 ∈ (Monic_1p‘𝑅))
45		eqid 2726	. . . . . . . . . . . . . . . . . 18 ⊢ (Unic_1p‘𝑅) = (Unic_1p‘𝑅)
46	45, 41	mon1puc1p 26041	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ (Monic_1p‘𝑅)) → 𝐺 ∈ (Unic_1p‘𝑅))
47	40, 44, 46	syl2anc 583	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐺 ∈ (Unic_1p‘𝑅))
48		eqid 2726	. . . . . . . . . . . . . . . . 17 ⊢ (._r‘𝑃) = (._r‘𝑃)
49		eqid 2726	. . . . . . . . . . . . . . . . 17 ⊢ (quot_1p‘𝑅) = (quot_1p‘𝑅)
50	12, 36, 15, 45, 48, 49	dvdsq1p 26052	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ (Unic_1p‘𝑅)) → (𝐺(∥_r‘𝑃)𝐹 ↔ 𝐹 = ((𝐹(quot_1p‘𝑅)𝐺)(._r‘𝑃)𝐺)))
51	40, 18, 47, 50	syl3anc 1368	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐺(∥_r‘𝑃)𝐹 ↔ 𝐹 = ((𝐹(quot_1p‘𝑅)𝐺)(._r‘𝑃)𝐺)))
52	38, 51	mpbid 231	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹 = ((𝐹(quot_1p‘𝑅)𝐺)(._r‘𝑃)𝐺))
53	52	fveq2d 6889	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑂‘𝐹) = (𝑂‘((𝐹(quot_1p‘𝑅)𝐺)(._r‘𝑃)𝐺)))
54	49, 12, 15, 45	q1pcl 26047	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ (Unic_1p‘𝑅)) → (𝐹(quot_1p‘𝑅)𝐺) ∈ 𝐵)
55	40, 18, 47, 54	syl3anc 1368	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹(quot_1p‘𝑅)𝐺) ∈ 𝐵)
56	12, 15, 41	mon1pcl 26035	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ (Monic_1p‘𝑅) → 𝐺 ∈ 𝐵)
57	44, 56	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺 ∈ 𝐵)
58		eqid 2726	. . . . . . . . . . . . . . 15 ⊢ (._r‘(𝑅 ↑_s 𝐾)) = (._r‘(𝑅 ↑_s 𝐾))
59	15, 48, 58	rhmmul 20388	. . . . . . . . . . . . . 14 ⊢ ((𝑂 ∈ (𝑃 RingHom (𝑅 ↑_s 𝐾)) ∧ (𝐹(quot_1p‘𝑅)𝐺) ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝑂‘((𝐹(quot_1p‘𝑅)𝐺)(._r‘𝑃)𝐺)) = ((𝑂‘(𝐹(quot_1p‘𝑅)𝐺))(._r‘(𝑅 ↑_s 𝐾))(𝑂‘𝐺)))
60	14, 55, 57, 59	syl3anc 1368	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑂‘((𝐹(quot_1p‘𝑅)𝐺)(._r‘𝑃)𝐺)) = ((𝑂‘(𝐹(quot_1p‘𝑅)𝐺))(._r‘(𝑅 ↑_s 𝐾))(𝑂‘𝐺)))
61	17, 55	ffvelcdmd 7081	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) ∈ (Base‘(𝑅 ↑_s 𝐾)))
62	17, 57	ffvelcdmd 7081	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑂‘𝐺) ∈ (Base‘(𝑅 ↑_s 𝐾)))
63		eqid 2726	. . . . . . . . . . . . . 14 ⊢ (._r‘𝑅) = (._r‘𝑅)
64	2, 4, 5, 7, 61, 62, 63, 58	pwsmulrval 17446	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑂‘(𝐹(quot_1p‘𝑅)𝐺))(._r‘(𝑅 ↑_s 𝐾))(𝑂‘𝐺)) = ((𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) ∘_f (._r‘𝑅)(𝑂‘𝐺)))
65	53, 60, 64	3eqtrd 2770	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑂‘𝐹) = ((𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) ∘_f (._r‘𝑅)(𝑂‘𝐺)))
66	65	fveq1d 6887	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑂‘𝐹)‘𝑥) = (((𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) ∘_f (._r‘𝑅)(𝑂‘𝐺))‘𝑥))
67	66	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → ((𝑂‘𝐹)‘𝑥) = (((𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) ∘_f (._r‘𝑅)(𝑂‘𝐺))‘𝑥))
68	2, 3, 4, 5, 7, 61	pwselbas 17444	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑂‘(𝐹(quot_1p‘𝑅)𝐺)):𝐾⟶𝐾)
69	68	ffnd 6712	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) Fn 𝐾)
70	69	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → (𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) Fn 𝐾)
71	2, 3, 4, 5, 7, 62	pwselbas 17444	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑂‘𝐺):𝐾⟶𝐾)
72	71	ffnd 6712	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑂‘𝐺) Fn 𝐾)
73	72	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → (𝑂‘𝐺) Fn 𝐾)
74	6	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → 𝐾 ∈ V)
75		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → 𝑥 ∈ 𝐾)
76		fnfvof 7684	. . . . . . . . . . 11 ⊢ ((((𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) Fn 𝐾 ∧ (𝑂‘𝐺) Fn 𝐾) ∧ (𝐾 ∈ V ∧ 𝑥 ∈ 𝐾)) → (((𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) ∘_f (._r‘𝑅)(𝑂‘𝐺))‘𝑥) = (((𝑂‘(𝐹(quot_1p‘𝑅)𝐺))‘𝑥)(._r‘𝑅)((𝑂‘𝐺)‘𝑥)))
77	70, 73, 74, 75, 76	syl22anc 836	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → (((𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) ∘_f (._r‘𝑅)(𝑂‘𝐺))‘𝑥) = (((𝑂‘(𝐹(quot_1p‘𝑅)𝐺))‘𝑥)(._r‘𝑅)((𝑂‘𝐺)‘𝑥)))
78	67, 77	eqtrd 2766	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → ((𝑂‘𝐹)‘𝑥) = (((𝑂‘(𝐹(quot_1p‘𝑅)𝐺))‘𝑥)(._r‘𝑅)((𝑂‘𝐺)‘𝑥)))
79	78	eqeq1d 2728	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → (((𝑂‘𝐹)‘𝑥) = 𝑊 ↔ (((𝑂‘(𝐹(quot_1p‘𝑅)𝐺))‘𝑥)(._r‘𝑅)((𝑂‘𝐺)‘𝑥)) = 𝑊))
80	5, 30	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ Domn)
81	80	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → 𝑅 ∈ Domn)
82	68	ffvelcdmda 7080	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → ((𝑂‘(𝐹(quot_1p‘𝑅)𝐺))‘𝑥) ∈ 𝐾)
83	71	ffvelcdmda 7080	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → ((𝑂‘𝐺)‘𝑥) ∈ 𝐾)
84	3, 63, 35	domneq0 21207	. . . . . . . . 9 ⊢ ((𝑅 ∈ Domn ∧ ((𝑂‘(𝐹(quot_1p‘𝑅)𝐺))‘𝑥) ∈ 𝐾 ∧ ((𝑂‘𝐺)‘𝑥) ∈ 𝐾) → ((((𝑂‘(𝐹(quot_1p‘𝑅)𝐺))‘𝑥)(._r‘𝑅)((𝑂‘𝐺)‘𝑥)) = 𝑊 ↔ (((𝑂‘(𝐹(quot_1p‘𝑅)𝐺))‘𝑥) = 𝑊 ∨ ((𝑂‘𝐺)‘𝑥) = 𝑊)))
85	81, 82, 83, 84	syl3anc 1368	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → ((((𝑂‘(𝐹(quot_1p‘𝑅)𝐺))‘𝑥)(._r‘𝑅)((𝑂‘𝐺)‘𝑥)) = 𝑊 ↔ (((𝑂‘(𝐹(quot_1p‘𝑅)𝐺))‘𝑥) = 𝑊 ∨ ((𝑂‘𝐺)‘𝑥) = 𝑊)))
86	79, 85	bitrd 279	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → (((𝑂‘𝐹)‘𝑥) = 𝑊 ↔ (((𝑂‘(𝐹(quot_1p‘𝑅)𝐺))‘𝑥) = 𝑊 ∨ ((𝑂‘𝐺)‘𝑥) = 𝑊)))
87	86	pm5.32da 578	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐾 ∧ ((𝑂‘𝐹)‘𝑥) = 𝑊) ↔ (𝑥 ∈ 𝐾 ∧ (((𝑂‘(𝐹(quot_1p‘𝑅)𝐺))‘𝑥) = 𝑊 ∨ ((𝑂‘𝐺)‘𝑥) = 𝑊))))
88		andi 1004	. . . . . 6 ⊢ ((𝑥 ∈ 𝐾 ∧ (((𝑂‘(𝐹(quot_1p‘𝑅)𝐺))‘𝑥) = 𝑊 ∨ ((𝑂‘𝐺)‘𝑥) = 𝑊)) ↔ ((𝑥 ∈ 𝐾 ∧ ((𝑂‘(𝐹(quot_1p‘𝑅)𝐺))‘𝑥) = 𝑊) ∨ (𝑥 ∈ 𝐾 ∧ ((𝑂‘𝐺)‘𝑥) = 𝑊)))
89	87, 88	bitrdi 287	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐾 ∧ ((𝑂‘𝐹)‘𝑥) = 𝑊) ↔ ((𝑥 ∈ 𝐾 ∧ ((𝑂‘(𝐹(quot_1p‘𝑅)𝐺))‘𝑥) = 𝑊) ∨ (𝑥 ∈ 𝐾 ∧ ((𝑂‘𝐺)‘𝑥) = 𝑊))))
90		fniniseg 7055	. . . . . 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 4143	. . . . . 6 ⊢ (𝑥 ∈ ((^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇}) ↔ (𝑥 ∈ (^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊}) ∨ 𝑥 ∈ {𝑇}))
93		fniniseg 7055	. . . . . . . 8 ⊢ ((𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) Fn 𝐾 → (𝑥 ∈ (^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊}) ↔ (𝑥 ∈ 𝐾 ∧ ((𝑂‘(𝐹(quot_1p‘𝑅)𝐺))‘𝑥) = 𝑊)))
94	69, 93	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊}) ↔ (𝑥 ∈ 𝐾 ∧ ((𝑂‘(𝐹(quot_1p‘𝑅)𝐺))‘𝑥) = 𝑊)))
95	43	simp3d 1141	. . . . . . . . 9 ⊢ (𝜑 → (^◡(𝑂‘𝐺) “ {𝑊}) = {𝑇})
96	95	eleq2d 2813	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (^◡(𝑂‘𝐺) “ {𝑊}) ↔ 𝑥 ∈ {𝑇}))
97		fniniseg 7055	. . . . . . . . 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 281	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ {𝑇} ↔ (𝑥 ∈ 𝐾 ∧ ((𝑂‘𝐺)‘𝑥) = 𝑊)))
100	94, 99	orbi12d 915	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ (^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊}) ∨ 𝑥 ∈ {𝑇}) ↔ ((𝑥 ∈ 𝐾 ∧ ((𝑂‘(𝐹(quot_1p‘𝑅)𝐺))‘𝑥) = 𝑊) ∨ (𝑥 ∈ 𝐾 ∧ ((𝑂‘𝐺)‘𝑥) = 𝑊))))
101	92, 100	bitrid 283	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ((^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇}) ↔ ((𝑥 ∈ 𝐾 ∧ ((𝑂‘(𝐹(quot_1p‘𝑅)𝐺))‘𝑥) = 𝑊) ∨ (𝑥 ∈ 𝐾 ∧ ((𝑂‘𝐺)‘𝑥) = 𝑊))))
102	89, 91, 101	3bitr4d 311	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (^◡(𝑂‘𝐹) “ {𝑊}) ↔ 𝑥 ∈ ((^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇})))
103	102	eqrdv 2724	. . 3 ⊢ (𝜑 → (^◡(𝑂‘𝐹) “ {𝑊}) = ((^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇}))
104	103	fveq2d 6889	. 2 ⊢ (𝜑 → (♯‘(^◡(𝑂‘𝐹) “ {𝑊})) = (♯‘((^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇})))
105		fvex 6898	. . . . . . . . . 10 ⊢ (𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) ∈ V
106	105	cnvex 7915	. . . . . . . . 9 ⊢ ^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) ∈ V
107	106	imaex 7904	. . . . . . . 8 ⊢ (^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊}) ∈ V
108	107	a1i 11	. . . . . . 7 ⊢ (𝜑 → (^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊}) ∈ V)
109		fta1glem.3	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
110		fta1g.z	. . . . . . . . . 10 ⊢ 0 = (0_g‘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 26057	. . . . . . . . 9 ⊢ (𝜑 → (𝐷‘(𝐹(quot_1p‘𝑅)𝐺)) = 𝑁)
113		fveq2 6885	. . . . . . . . . . . 12 ⊢ (𝑔 = (𝐹(quot_1p‘𝑅)𝐺) → (𝐷‘𝑔) = (𝐷‘(𝐹(quot_1p‘𝑅)𝐺)))
114	113	eqeq1d 2728	. . . . . . . . . . 11 ⊢ (𝑔 = (𝐹(quot_1p‘𝑅)𝐺) → ((𝐷‘𝑔) = 𝑁 ↔ (𝐷‘(𝐹(quot_1p‘𝑅)𝐺)) = 𝑁))
115		fveq2 6885	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (𝐹(quot_1p‘𝑅)𝐺) → (𝑂‘𝑔) = (𝑂‘(𝐹(quot_1p‘𝑅)𝐺)))
116	115	cnveqd 5869	. . . . . . . . . . . . . 14 ⊢ (𝑔 = (𝐹(quot_1p‘𝑅)𝐺) → ^◡(𝑂‘𝑔) = ^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)))
117	116	imaeq1d 6052	. . . . . . . . . . . . 13 ⊢ (𝑔 = (𝐹(quot_1p‘𝑅)𝐺) → (^◡(𝑂‘𝑔) “ {𝑊}) = (^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊}))
118	117	fveq2d 6889	. . . . . . . . . . . 12 ⊢ (𝑔 = (𝐹(quot_1p‘𝑅)𝐺) → (♯‘(^◡(𝑂‘𝑔) “ {𝑊})) = (♯‘(^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊})))
119	118, 113	breq12d 5154	. . . . . . . . . . 11 ⊢ (𝑔 = (𝐹(quot_1p‘𝑅)𝐺) → ((♯‘(^◡(𝑂‘𝑔) “ {𝑊})) ≤ (𝐷‘𝑔) ↔ (♯‘(^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊})) ≤ (𝐷‘(𝐹(quot_1p‘𝑅)𝐺))))
120	114, 119	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑔 = (𝐹(quot_1p‘𝑅)𝐺) → (((𝐷‘𝑔) = 𝑁 → (♯‘(^◡(𝑂‘𝑔) “ {𝑊})) ≤ (𝐷‘𝑔)) ↔ ((𝐷‘(𝐹(quot_1p‘𝑅)𝐺)) = 𝑁 → (♯‘(^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊})) ≤ (𝐷‘(𝐹(quot_1p‘𝑅)𝐺)))))
121		fta1glem.6	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑔 ∈ 𝐵 ((𝐷‘𝑔) = 𝑁 → (♯‘(^◡(𝑂‘𝑔) “ {𝑊})) ≤ (𝐷‘𝑔)))
122	120, 121, 55	rspcdva 3607	. . . . . . . . 9 ⊢ (𝜑 → ((𝐷‘(𝐹(quot_1p‘𝑅)𝐺)) = 𝑁 → (♯‘(^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊})) ≤ (𝐷‘(𝐹(quot_1p‘𝑅)𝐺))))
123	112, 122	mpd 15	. . . . . . . 8 ⊢ (𝜑 → (♯‘(^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊})) ≤ (𝐷‘(𝐹(quot_1p‘𝑅)𝐺)))
124	123, 112	breqtrd 5167	. . . . . . 7 ⊢ (𝜑 → (♯‘(^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊})) ≤ 𝑁)
125		hashbnd 14301	. . . . . . 7 ⊢ (((^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊}) ∈ V ∧ 𝑁 ∈ ℕ₀ ∧ (♯‘(^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊})) ≤ 𝑁) → (^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊}) ∈ Fin)
126	108, 109, 124, 125	syl3anc 1368	. . . . . 6 ⊢ (𝜑 → (^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊}) ∈ Fin)
127		snfi 9046	. . . . . 6 ⊢ {𝑇} ∈ Fin
128		unfi 9174	. . . . . 6 ⊢ (((^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊}) ∈ Fin ∧ {𝑇} ∈ Fin) → ((^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇}) ∈ Fin)
129	126, 127, 128	sylancl 585	. . . . 5 ⊢ (𝜑 → ((^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇}) ∈ Fin)
130		hashcl 14321	. . . . 5 ⊢ (((^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇}) ∈ Fin → (♯‘((^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇})) ∈ ℕ₀)
131	129, 130	syl 17	. . . 4 ⊢ (𝜑 → (♯‘((^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇})) ∈ ℕ₀)
132	131	nn0red 12537	. . 3 ⊢ (𝜑 → (♯‘((^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇})) ∈ ℝ)
133		hashcl 14321	. . . . . 6 ⊢ ((^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊}) ∈ Fin → (♯‘(^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊})) ∈ ℕ₀)
134	126, 133	syl 17	. . . . 5 ⊢ (𝜑 → (♯‘(^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊})) ∈ ℕ₀)
135	134	nn0red 12537	. . . 4 ⊢ (𝜑 → (♯‘(^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊})) ∈ ℝ)
136		peano2re 11391	. . . 4 ⊢ ((♯‘(^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊})) ∈ ℝ → ((♯‘(^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊})) + 1) ∈ ℝ)
137	135, 136	syl 17	. . 3 ⊢ (𝜑 → ((♯‘(^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊})) + 1) ∈ ℝ)
138		peano2nn0 12516	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℕ₀)
139	109, 138	syl 17	. . . . 5 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ₀)
140	111, 139	eqeltrd 2827	. . . 4 ⊢ (𝜑 → (𝐷‘𝐹) ∈ ℕ₀)
141	140	nn0red 12537	. . 3 ⊢ (𝜑 → (𝐷‘𝐹) ∈ ℝ)
142		hashun2 14348	. . . . 5 ⊢ (((^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊}) ∈ Fin ∧ {𝑇} ∈ Fin) → (♯‘((^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇})) ≤ ((♯‘(^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊})) + (♯‘{𝑇})))
143	126, 127, 142	sylancl 585	. . . 4 ⊢ (𝜑 → (♯‘((^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇})) ≤ ((♯‘(^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊})) + (♯‘{𝑇})))
144		hashsng 14334	. . . . . 6 ⊢ (𝑇 ∈ (^◡(𝑂‘𝐹) “ {𝑊}) → (♯‘{𝑇}) = 1)
145	1, 144	syl 17	. . . . 5 ⊢ (𝜑 → (♯‘{𝑇}) = 1)
146	145	oveq2d 7421	. . . 4 ⊢ (𝜑 → ((♯‘(^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊})) + (♯‘{𝑇})) = ((♯‘(^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊})) + 1))
147	143, 146	breqtrd 5167	. . 3 ⊢ (𝜑 → (♯‘((^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇})) ≤ ((♯‘(^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊})) + 1))
148	109	nn0red 12537	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℝ)
149		1red 11219	. . . . 5 ⊢ (𝜑 → 1 ∈ ℝ)
150	135, 148, 149, 124	leadd1dd 11832	. . . 4 ⊢ (𝜑 → ((♯‘(^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊})) + 1) ≤ (𝑁 + 1))
151	150, 111	breqtrrd 5169	. . 3 ⊢ (𝜑 → ((♯‘(^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊})) + 1) ≤ (𝐷‘𝐹))
152	132, 137, 141, 147, 151	letrd 11375	. 2 ⊢ (𝜑 → (♯‘((^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇})) ≤ (𝐷‘𝐹))
153	104, 152	eqbrtrd 5163	1 ⊢ (𝜑 → (♯‘(^◡(𝑂‘𝐹) “ {𝑊})) ≤ (𝐷‘𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 844 = wceq 1533 ∈ wcel 2098 ∀wral 3055 Vcvv 3468 ∪ cun 3941 {csn 4623 class class class wbr 5141 ^◡ccnv 5668 “ cima 5672 Fn wfn 6532 ⟶wf 6533 ‘cfv 6537 (class class class)co 7405 ∘_f cof 7665 Fincfn 8941 ℝcr 11111 1c1 11113 + caddc 11115 ≤ cle 11253 ℕ₀cn0 12476 ♯chash 14295 Basecbs 17153 ._rcmulr 17207 0_gc0g 17394 ↑_s cpws 17401 -_gcsg 18865 Ringcrg 20138 CRingccrg 20139 ∥_rcdsr 20256 RingHom crh 20371 NzRingcnzr 20414 Domncdomn 21190 IDomncidom 21191 algSccascl 21747 var₁cv1 22050 Poly₁cpl1 22051 eval₁ce1 22188 deg₁ cdg1 25942 Monic_1pcmn1 26016 Unic_1pcuc1p 26017 quot_1pcq1p 26018

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-ofr 7668 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-tpos 8212 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-oadd 8471 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-sup 9439 df-oi 9507 df-dju 9898 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-xnn0 12549 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13491 df-fzo 13634 df-seq 13973 df-hash 14296 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-starv 17221 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-unif 17229 df-hom 17230 df-cco 17231 df-0g 17396 df-gsum 17397 df-prds 17402 df-pws 17404 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-mhm 18713 df-submnd 18714 df-grp 18866 df-minusg 18867 df-sbg 18868 df-mulg 18996 df-subg 19050 df-ghm 19139 df-cntz 19233 df-cmn 19702 df-abl 19703 df-mgp 20040 df-rng 20058 df-ur 20087 df-srg 20092 df-ring 20140 df-cring 20141 df-oppr 20236 df-dvdsr 20259 df-unit 20260 df-invr 20290 df-rhm 20374 df-nzr 20415 df-subrng 20446 df-subrg 20471 df-lmod 20708 df-lss 20779 df-lsp 20819 df-rlreg 21193 df-domn 21194 df-idom 21195 df-cnfld 21241 df-assa 21748 df-asp 21749 df-ascl 21750 df-psr 21803 df-mvr 21804 df-mpl 21805 df-opsr 21807 df-evls 21977 df-evl 21978 df-psr1 22054 df-vr1 22055 df-ply1 22056 df-coe1 22057 df-evl1 22190 df-mdeg 25943 df-deg1 25944 df-mon1 26021 df-uc1p 26022 df-q1p 26023 df-r1p 26024

This theorem is referenced by: fta1g 26059

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