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Theorem fta1glem2 26121

Description: Lemma for fta1g 26122. (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 2725	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑅 ↑_s 𝐾) = (𝑅 ↑_s 𝐾)
3		fta1glem.k	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝐾 = (Base‘𝑅)
4		eqid 2725	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Base‘(𝑅 ↑_s 𝐾)) = (Base‘(𝑅 ↑_s 𝐾))
5		fta1g.1	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑅 ∈ IDomn)
6	3	fvexi 6906	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝐾 ∈ V
7	6	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐾 ∈ V)
8		isidom 21258	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑅 ∈ IDomn ↔ (𝑅 ∈ CRing ∧ 𝑅 ∈ Domn))
9	8	simplbi 496	. . . . . . . . . . . . . . . . . . . . . . . . 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 22260	. . . . . . . . . . . . . . . . . . . . . . . 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 20428	. . . . . . . . . . . . . . . . . . . . . . 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 7090	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑂‘𝐹) ∈ (Base‘(𝑅 ↑_s 𝐾)))
20	2, 3, 4, 5, 7, 19	pwselbas 17470	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑂‘𝐹):𝐾⟶𝐾)
21	20	ffnd 6718	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑂‘𝐹) Fn 𝐾)
22		fniniseg 7064	. . . . . . . . . . . . . . . . . . 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 494	. . . . . . . . . . . . . . . 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 495	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑅 ∈ IDomn → 𝑅 ∈ Domn)
31		domnnzr 21246	. . . . . . . . . . . . . . . . . . 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 493	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑇 ∈ 𝐾)
35		fta1g.w	. . . . . . . . . . . . . . . . 17 ⊢ 𝑊 = (0_g‘𝑅)
36		eqid 2725	. . . . . . . . . . . . . . . . 17 ⊢ (∥_r‘𝑃) = (∥_r‘𝑃)
37	12, 15, 3, 26, 27, 28, 29, 11, 33, 10, 34, 18, 35, 36	facth1 26119	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐺(∥_r‘𝑃)𝐹 ↔ ((𝑂‘𝐹)‘𝑇) = 𝑊))
38	25, 37	mpbird 256	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐺(∥_r‘𝑃)𝐹)
39		nzrring 20459	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring)
40	33, 39	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑅 ∈ Ring)
41		eqid 2725	. . . . . . . . . . . . . . . . . . 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 26117	. . . . . . . . . . . . . . . . . 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 2725	. . . . . . . . . . . . . . . . . 18 ⊢ (Unic_1p‘𝑅) = (Unic_1p‘𝑅)
46	45, 41	mon1puc1p 26104	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ (Monic_1p‘𝑅)) → 𝐺 ∈ (Unic_1p‘𝑅))
47	40, 44, 46	syl2anc 582	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐺 ∈ (Unic_1p‘𝑅))
48		eqid 2725	. . . . . . . . . . . . . . . . 17 ⊢ (._r‘𝑃) = (._r‘𝑃)
49		eqid 2725	. . . . . . . . . . . . . . . . 17 ⊢ (quot_1p‘𝑅) = (quot_1p‘𝑅)
50	12, 36, 15, 45, 48, 49	dvdsq1p 26115	. . . . . . . . . . . . . . . 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 6896	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑂‘𝐹) = (𝑂‘((𝐹(quot_1p‘𝑅)𝐺)(._r‘𝑃)𝐺)))
54	49, 12, 15, 45	q1pcl 26110	. . . . . . . . . . . . . . 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 26098	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ (Monic_1p‘𝑅) → 𝐺 ∈ 𝐵)
57	44, 56	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺 ∈ 𝐵)
58		eqid 2725	. . . . . . . . . . . . . . 15 ⊢ (._r‘(𝑅 ↑_s 𝐾)) = (._r‘(𝑅 ↑_s 𝐾))
59	15, 48, 58	rhmmul 20429	. . . . . . . . . . . . . 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 7090	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) ∈ (Base‘(𝑅 ↑_s 𝐾)))
62	17, 57	ffvelcdmd 7090	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑂‘𝐺) ∈ (Base‘(𝑅 ↑_s 𝐾)))
63		eqid 2725	. . . . . . . . . . . . . 14 ⊢ (._r‘𝑅) = (._r‘𝑅)
64	2, 4, 5, 7, 61, 62, 63, 58	pwsmulrval 17472	. . . . . . . . . . . . 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 2769	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑂‘𝐹) = ((𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) ∘_f (._r‘𝑅)(𝑂‘𝐺)))
66	65	fveq1d 6894	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑂‘𝐹)‘𝑥) = (((𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) ∘_f (._r‘𝑅)(𝑂‘𝐺))‘𝑥))
67	66	adantr 479	. . . . . . . . . 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 17470	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑂‘(𝐹(quot_1p‘𝑅)𝐺)):𝐾⟶𝐾)
69	68	ffnd 6718	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) Fn 𝐾)
70	69	adantr 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → (𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) Fn 𝐾)
71	2, 3, 4, 5, 7, 62	pwselbas 17470	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑂‘𝐺):𝐾⟶𝐾)
72	71	ffnd 6718	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑂‘𝐺) Fn 𝐾)
73	72	adantr 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → (𝑂‘𝐺) Fn 𝐾)
74	6	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → 𝐾 ∈ V)
75		simpr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → 𝑥 ∈ 𝐾)
76		fnfvof 7699	. . . . . . . . . . 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 837	. . . . . . . . . 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 2765	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → ((𝑂‘𝐹)‘𝑥) = (((𝑂‘(𝐹(quot_1p‘𝑅)𝐺))‘𝑥)(._r‘𝑅)((𝑂‘𝐺)‘𝑥)))
79	78	eqeq1d 2727	. . . . . . . 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 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → 𝑅 ∈ Domn)
82	68	ffvelcdmda 7089	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → ((𝑂‘(𝐹(quot_1p‘𝑅)𝐺))‘𝑥) ∈ 𝐾)
83	71	ffvelcdmda 7089	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → ((𝑂‘𝐺)‘𝑥) ∈ 𝐾)
84	3, 63, 35	domneq0 21248	. . . . . . . . 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 278	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → (((𝑂‘𝐹)‘𝑥) = 𝑊 ↔ (((𝑂‘(𝐹(quot_1p‘𝑅)𝐺))‘𝑥) = 𝑊 ∨ ((𝑂‘𝐺)‘𝑥) = 𝑊)))
87	86	pm5.32da 577	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐾 ∧ ((𝑂‘𝐹)‘𝑥) = 𝑊) ↔ (𝑥 ∈ 𝐾 ∧ (((𝑂‘(𝐹(quot_1p‘𝑅)𝐺))‘𝑥) = 𝑊 ∨ ((𝑂‘𝐺)‘𝑥) = 𝑊))))
88		andi 1005	. . . . . 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 286	. . . . 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 7064	. . . . . 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 4141	. . . . . 6 ⊢ (𝑥 ∈ ((^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇}) ↔ (𝑥 ∈ (^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊}) ∨ 𝑥 ∈ {𝑇}))
93		fniniseg 7064	. . . . . . . 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 2811	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (^◡(𝑂‘𝐺) “ {𝑊}) ↔ 𝑥 ∈ {𝑇}))
97		fniniseg 7064	. . . . . . . . 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 280	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ {𝑇} ↔ (𝑥 ∈ 𝐾 ∧ ((𝑂‘𝐺)‘𝑥) = 𝑊)))
100	94, 99	orbi12d 916	. . . . . 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 282	. . . . 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 310	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (^◡(𝑂‘𝐹) “ {𝑊}) ↔ 𝑥 ∈ ((^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇})))
103	102	eqrdv 2723	. . 3 ⊢ (𝜑 → (^◡(𝑂‘𝐹) “ {𝑊}) = ((^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇}))
104	103	fveq2d 6896	. 2 ⊢ (𝜑 → (♯‘(^◡(𝑂‘𝐹) “ {𝑊})) = (♯‘((^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇})))
105		fvex 6905	. . . . . . . . . 10 ⊢ (𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) ∈ V
106	105	cnvex 7931	. . . . . . . . 9 ⊢ ^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) ∈ V
107	106	imaex 7920	. . . . . . . 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 26120	. . . . . . . . 9 ⊢ (𝜑 → (𝐷‘(𝐹(quot_1p‘𝑅)𝐺)) = 𝑁)
113		fveq2 6892	. . . . . . . . . . . 12 ⊢ (𝑔 = (𝐹(quot_1p‘𝑅)𝐺) → (𝐷‘𝑔) = (𝐷‘(𝐹(quot_1p‘𝑅)𝐺)))
114	113	eqeq1d 2727	. . . . . . . . . . 11 ⊢ (𝑔 = (𝐹(quot_1p‘𝑅)𝐺) → ((𝐷‘𝑔) = 𝑁 ↔ (𝐷‘(𝐹(quot_1p‘𝑅)𝐺)) = 𝑁))
115		fveq2 6892	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (𝐹(quot_1p‘𝑅)𝐺) → (𝑂‘𝑔) = (𝑂‘(𝐹(quot_1p‘𝑅)𝐺)))
116	115	cnveqd 5872	. . . . . . . . . . . . . 14 ⊢ (𝑔 = (𝐹(quot_1p‘𝑅)𝐺) → ^◡(𝑂‘𝑔) = ^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)))
117	116	imaeq1d 6057	. . . . . . . . . . . . 13 ⊢ (𝑔 = (𝐹(quot_1p‘𝑅)𝐺) → (^◡(𝑂‘𝑔) “ {𝑊}) = (^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊}))
118	117	fveq2d 6896	. . . . . . . . . . . 12 ⊢ (𝑔 = (𝐹(quot_1p‘𝑅)𝐺) → (♯‘(^◡(𝑂‘𝑔) “ {𝑊})) = (♯‘(^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊})))
119	118, 113	breq12d 5156	. . . . . . . . . . 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 343	. . . . . . . . . 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 3602	. . . . . . . . 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 5169	. . . . . . 7 ⊢ (𝜑 → (♯‘(^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊})) ≤ 𝑁)
125		hashbnd 14327	. . . . . . 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 9067	. . . . . 6 ⊢ {𝑇} ∈ Fin
128		unfi 9195	. . . . . 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 584	. . . . 5 ⊢ (𝜑 → ((^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇}) ∈ Fin)
130		hashcl 14347	. . . . 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 12563	. . 3 ⊢ (𝜑 → (♯‘((^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇})) ∈ ℝ)
133		hashcl 14347	. . . . . 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 12563	. . . 4 ⊢ (𝜑 → (♯‘(^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊})) ∈ ℝ)
136		peano2re 11417	. . . 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 12542	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℕ₀)
139	109, 138	syl 17	. . . . 5 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ₀)
140	111, 139	eqeltrd 2825	. . . 4 ⊢ (𝜑 → (𝐷‘𝐹) ∈ ℕ₀)
141	140	nn0red 12563	. . 3 ⊢ (𝜑 → (𝐷‘𝐹) ∈ ℝ)
142		hashun2 14374	. . . . 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 584	. . . 4 ⊢ (𝜑 → (♯‘((^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇})) ≤ ((♯‘(^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊})) + (♯‘{𝑇})))
144		hashsng 14360	. . . . . 6 ⊢ (𝑇 ∈ (^◡(𝑂‘𝐹) “ {𝑊}) → (♯‘{𝑇}) = 1)
145	1, 144	syl 17	. . . . 5 ⊢ (𝜑 → (♯‘{𝑇}) = 1)
146	145	oveq2d 7432	. . . 4 ⊢ (𝜑 → ((♯‘(^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊})) + (♯‘{𝑇})) = ((♯‘(^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊})) + 1))
147	143, 146	breqtrd 5169	. . 3 ⊢ (𝜑 → (♯‘((^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇})) ≤ ((♯‘(^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊})) + 1))
148	109	nn0red 12563	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℝ)
149		1red 11245	. . . . 5 ⊢ (𝜑 → 1 ∈ ℝ)
150	135, 148, 149, 124	leadd1dd 11858	. . . 4 ⊢ (𝜑 → ((♯‘(^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊})) + 1) ≤ (𝑁 + 1))
151	150, 111	breqtrrd 5171	. . 3 ⊢ (𝜑 → ((♯‘(^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊})) + 1) ≤ (𝐷‘𝐹))
152	132, 137, 141, 147, 151	letrd 11401	. 2 ⊢ (𝜑 → (♯‘((^◡(𝑂‘(𝐹(quot_1p‘𝑅)𝐺)) “ {𝑊}) ∪ {𝑇})) ≤ (𝐷‘𝐹))
153	104, 152	eqbrtrd 5165	1 ⊢ (𝜑 → (♯‘(^◡(𝑂‘𝐹) “ {𝑊})) ≤ (𝐷‘𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∨ wo 845 = wceq 1533 ∈ wcel 2098 ∀wral 3051 Vcvv 3463 ∪ cun 3937 {csn 4624 class class class wbr 5143 ^◡ccnv 5671 “ cima 5675 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 (class class class)co 7416 ∘_f cof 7680 Fincfn 8962 ℝcr 11137 1c1 11139 + caddc 11141 ≤ cle 11279 ℕ₀cn0 12502 ♯chash 14321 Basecbs 17179 ._rcmulr 17233 0_gc0g 17420 ↑_s cpws 17427 -_gcsg 18896 Ringcrg 20177 CRingccrg 20178 ∥_rcdsr 20297 RingHom crh 20412 NzRingcnzr 20455 Domncdomn 21231 IDomncidom 21232 algSccascl 21790 var₁cv1 22103 Poly₁cpl1 22104 eval₁ce1 22242 deg₁ cdg1 26005 Monic_1pcmn1 26079 Unic_1pcuc1p 26080 quot_1pcq1p 26081

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 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 ax-addf 11217

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7682 df-ofr 7683 df-om 7869 df-1st 7991 df-2nd 7992 df-supp 8164 df-tpos 8230 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-oadd 8489 df-er 8723 df-map 8845 df-pm 8846 df-ixp 8915 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-fsupp 9386 df-sup 9465 df-oi 9533 df-dju 9924 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-xnn0 12575 df-z 12589 df-dec 12708 df-uz 12853 df-fz 13517 df-fzo 13660 df-seq 13999 df-hash 14322 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-starv 17247 df-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-ds 17254 df-unif 17255 df-hom 17256 df-cco 17257 df-0g 17422 df-gsum 17423 df-prds 17428 df-pws 17430 df-mre 17565 df-mrc 17566 df-acs 17568 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18739 df-submnd 18740 df-grp 18897 df-minusg 18898 df-sbg 18899 df-mulg 19028 df-subg 19082 df-ghm 19172 df-cntz 19272 df-cmn 19741 df-abl 19742 df-mgp 20079 df-rng 20097 df-ur 20126 df-srg 20131 df-ring 20179 df-cring 20180 df-oppr 20277 df-dvdsr 20300 df-unit 20301 df-invr 20331 df-rhm 20415 df-nzr 20456 df-subrng 20487 df-subrg 20512 df-lmod 20749 df-lss 20820 df-lsp 20860 df-rlreg 21234 df-domn 21235 df-idom 21236 df-cnfld 21284 df-assa 21791 df-asp 21792 df-ascl 21793 df-psr 21846 df-mvr 21847 df-mpl 21848 df-opsr 21850 df-evls 22025 df-evl 22026 df-psr1 22107 df-vr1 22108 df-ply1 22109 df-coe1 22110 df-evl1 22244 df-mdeg 26006 df-deg1 26007 df-mon1 26084 df-uc1p 26085 df-q1p 26086 df-r1p 26087

This theorem is referenced by: fta1g 26122

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