aks4d1p5 - Mathbox for metakunt

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Theorem aks4d1p5 42037

Description: Show that 𝑁 and 𝑅 are coprime for AKS existence theorem. Precondition will be eliminated in further theorem. (Contributed by metakunt, 30-Oct-2024.)

**Hypotheses**
Ref	Expression
aks4d1p5.1	⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘3))
aks4d1p5.2	⊢ 𝐴 = ((𝑁↑(⌊‘(2 log_b 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))((𝑁↑𝑘) − 1))
aks4d1p5.3	⊢ 𝐵 = (⌈‘((2 log_b 𝑁)↑5))
aks4d1p5.4	⊢ 𝑅 = inf({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}, ℝ, < )
aks4d1p5.5	⊢ (((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) ∧ (𝑅 / (𝑁 gcd 𝑅)) ∥ 𝐴) → ¬ (𝑅 / (𝑁 gcd 𝑅)) ∥ 𝐴)

**Assertion**
Ref	Expression
aks4d1p5	⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)

Distinct variable groups: 𝐴,𝑟 𝐵,𝑟 𝑘,𝑁 𝑁,𝑟 𝑅,𝑟 𝜑,𝑘 Allowed substitution hints: 𝜑(𝑟) 𝐴(𝑘) 𝐵(𝑘) 𝑅(𝑘)

Proof of Theorem aks4d1p5 Dummy variables 𝑥 𝑦 𝑜 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 484	. . . 4 ⊢ (((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) ∧ 𝑅 ≤ (𝑅 / (𝑁 gcd 𝑅))) → 𝑅 ≤ (𝑅 / (𝑁 gcd 𝑅)))
2		aks4d1p5.1	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘3))
3		aks4d1p5.2	. . . . . . . . . . . . . 14 ⊢ 𝐴 = ((𝑁↑(⌊‘(2 log_b 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))((𝑁↑𝑘) − 1))
4		aks4d1p5.3	. . . . . . . . . . . . . 14 ⊢ 𝐵 = (⌈‘((2 log_b 𝑁)↑5))
5		aks4d1p5.4	. . . . . . . . . . . . . 14 ⊢ 𝑅 = inf({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}, ℝ, < )
6	2, 3, 4, 5	aks4d1p4 42036	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑅 ∈ (1...𝐵) ∧ ¬ 𝑅 ∥ 𝐴))
7	6	simpld 494	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ∈ (1...𝐵))
8		elfznn 13613	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ (1...𝐵) → 𝑅 ∈ ℕ)
9	7, 8	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ ℕ)
10	9	nnred 12308	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ ℝ)
11		eluzelz 12913	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ ℤ)
12	2, 11	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℤ)
13		0red 11293	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ∈ ℝ)
14		3re 12373	. . . . . . . . . . . . . . . 16 ⊢ 3 ∈ ℝ
15	14	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 3 ∈ ℝ)
16	12	zred 12747	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ ℝ)
17		3pos 12398	. . . . . . . . . . . . . . . 16 ⊢ 0 < 3
18	17	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 < 3)
19		eluzle 12916	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 3 ≤ 𝑁)
20	2, 19	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 3 ≤ 𝑁)
21	13, 15, 16, 18, 20	ltletrd 11450	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < 𝑁)
22	12, 21	jca 511	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁 ∈ ℤ ∧ 0 < 𝑁))
23		elnnz 12649	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁))
24	22, 23	sylibr 234	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℕ)
25		gcdnncl 16553	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑅 ∈ ℕ) → (𝑁 gcd 𝑅) ∈ ℕ)
26	24, 9, 25	syl2anc 583	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 gcd 𝑅) ∈ ℕ)
27	26	nnred 12308	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 gcd 𝑅) ∈ ℝ)
28	26	nnne0d 12343	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 gcd 𝑅) ≠ 0)
29	10, 27, 28	redivcld 12122	. . . . . . . . 9 ⊢ (𝜑 → (𝑅 / (𝑁 gcd 𝑅)) ∈ ℝ)
30	29	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 / (𝑁 gcd 𝑅)) ∈ ℝ)
31	10	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 𝑅 ∈ ℝ)
32	30, 31	ltnled 11437	. . . . . . 7 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → ((𝑅 / (𝑁 gcd 𝑅)) < 𝑅 ↔ ¬ 𝑅 ≤ (𝑅 / (𝑁 gcd 𝑅))))
33	32	biimprd 248	. . . . . 6 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (¬ 𝑅 ≤ (𝑅 / (𝑁 gcd 𝑅)) → (𝑅 / (𝑁 gcd 𝑅)) < 𝑅))
34	33	imp 406	. . . . 5 ⊢ (((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) ∧ ¬ 𝑅 ≤ (𝑅 / (𝑁 gcd 𝑅))) → (𝑅 / (𝑁 gcd 𝑅)) < 𝑅)
35	5	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 𝑅 = inf({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}, ℝ, < ))
36		ssrab2 4103	. . . . . . . . . . . 12 ⊢ {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴} ⊆ (1...𝐵)
37	36	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴} ⊆ (1...𝐵))
38		elfznn 13613	. . . . . . . . . . . . . . 15 ⊢ (𝑜 ∈ (1...𝐵) → 𝑜 ∈ ℕ)
39	38	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑜 ∈ (1...𝐵)) → 𝑜 ∈ ℕ)
40	39	nnred 12308	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑜 ∈ (1...𝐵)) → 𝑜 ∈ ℝ)
41	40	ex 412	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑜 ∈ (1...𝐵) → 𝑜 ∈ ℝ))
42	41	ssrdv 4014	. . . . . . . . . . 11 ⊢ (𝜑 → (1...𝐵) ⊆ ℝ)
43	37, 42	sstrd 4019	. . . . . . . . . 10 ⊢ (𝜑 → {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴} ⊆ ℝ)
44	43	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴} ⊆ ℝ)
45		fzfid 14024	. . . . . . . . . . . 12 ⊢ (𝜑 → (1...𝐵) ∈ Fin)
46	45, 37	ssfid 9329	. . . . . . . . . . 11 ⊢ (𝜑 → {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴} ∈ Fin)
47	46	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴} ∈ Fin)
48	2, 3, 4	aks4d1p3 42035	. . . . . . . . . . . 12 ⊢ (𝜑 → ∃𝑟 ∈ (1...𝐵) ¬ 𝑟 ∥ 𝐴)
49		rabn0 4412	. . . . . . . . . . . 12 ⊢ ({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴} ≠ ∅ ↔ ∃𝑟 ∈ (1...𝐵) ¬ 𝑟 ∥ 𝐴)
50	48, 49	sylibr 234	. . . . . . . . . . 11 ⊢ (𝜑 → {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴} ≠ ∅)
51	50	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴} ≠ ∅)
52		fiminre 12242	. . . . . . . . . 10 ⊢ (({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴} ⊆ ℝ ∧ {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴} ∈ Fin ∧ {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴} ≠ ∅) → ∃𝑥 ∈ {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}∀𝑦 ∈ {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}𝑥 ≤ 𝑦)
53	44, 47, 51, 52	syl3anc 1371	. . . . . . . . 9 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → ∃𝑥 ∈ {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}∀𝑦 ∈ {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}𝑥 ≤ 𝑦)
54		breq1 5169	. . . . . . . . . . 11 ⊢ (𝑟 = (𝑅 / (𝑁 gcd 𝑅)) → (𝑟 ∥ 𝐴 ↔ (𝑅 / (𝑁 gcd 𝑅)) ∥ 𝐴))
55	54	notbid 318	. . . . . . . . . 10 ⊢ (𝑟 = (𝑅 / (𝑁 gcd 𝑅)) → (¬ 𝑟 ∥ 𝐴 ↔ ¬ (𝑅 / (𝑁 gcd 𝑅)) ∥ 𝐴))
56		1zzd 12674	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 1 ∈ ℤ)
57	4	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 = (⌈‘((2 log_b 𝑁)↑5)))
58		2re 12367	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℝ
59	58	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 2 ∈ ℝ)
60		2pos 12396	. . . . . . . . . . . . . . . . 17 ⊢ 0 < 2
61	60	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 < 2)
62		1red 11291	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 1 ∈ ℝ)
63		1lt2 12464	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 < 2
64	63	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 1 < 2)
65	62, 64	ltned 11426	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 1 ≠ 2)
66	65	necomd 3002	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 2 ≠ 1)
67	59, 61, 16, 21, 66	relogbcld 41929	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2 log_b 𝑁) ∈ ℝ)
68		5nn0 12573	. . . . . . . . . . . . . . . 16 ⊢ 5 ∈ ℕ₀
69	68	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 5 ∈ ℕ₀)
70	67, 69	reexpcld 14213	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((2 log_b 𝑁)↑5) ∈ ℝ)
71		ceilcl 13893	. . . . . . . . . . . . . 14 ⊢ (((2 log_b 𝑁)↑5) ∈ ℝ → (⌈‘((2 log_b 𝑁)↑5)) ∈ ℤ)
72	70, 71	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (⌈‘((2 log_b 𝑁)↑5)) ∈ ℤ)
73	57, 72	eqeltrd 2844	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ ℤ)
74	73	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 𝐵 ∈ ℤ)
75	24	nnzd 12666	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℤ)
76		divgcdnnr 16562	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ ℕ ∧ 𝑁 ∈ ℤ) → (𝑅 / (𝑁 gcd 𝑅)) ∈ ℕ)
77	9, 75, 76	syl2anc 583	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑅 / (𝑁 gcd 𝑅)) ∈ ℕ)
78	77	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 / (𝑁 gcd 𝑅)) ∈ ℕ)
79	78	nnzd 12666	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 / (𝑁 gcd 𝑅)) ∈ ℤ)
80	78	nnge1d 12341	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 1 ≤ (𝑅 / (𝑁 gcd 𝑅)))
81	74	zred 12747	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 𝐵 ∈ ℝ)
82	9	nnrpd 13097	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
83	82	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 𝑅 ∈ ℝ⁺)
84	26	nnrpd 13097	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁 gcd 𝑅) ∈ ℝ⁺)
85	84	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑁 gcd 𝑅) ∈ ℝ⁺)
86	31	recnd 11318	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 𝑅 ∈ ℂ)
87	83	rpne0d 13104	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 𝑅 ≠ 0)
88	86, 87	dividd 12068	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 / 𝑅) = 1)
89		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 1 < (𝑁 gcd 𝑅))
90	88, 89	eqbrtrd 5188	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 / 𝑅) < (𝑁 gcd 𝑅))
91	31, 83, 85, 90	ltdiv23d 13166	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 / (𝑁 gcd 𝑅)) < 𝑅)
92	30, 31, 91	ltled 11438	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 / (𝑁 gcd 𝑅)) ≤ 𝑅)
93		elfzle2 13588	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ (1...𝐵) → 𝑅 ≤ 𝐵)
94	7, 93	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ≤ 𝐵)
95	94	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 𝑅 ≤ 𝐵)
96	30, 31, 81, 92, 95	letrd 11447	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 / (𝑁 gcd 𝑅)) ≤ 𝐵)
97	56, 74, 79, 80, 96	elfzd 13575	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 / (𝑁 gcd 𝑅)) ∈ (1...𝐵))
98		aks4d1p5.5	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) ∧ (𝑅 / (𝑁 gcd 𝑅)) ∥ 𝐴) → ¬ (𝑅 / (𝑁 gcd 𝑅)) ∥ 𝐴)
99		simpr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) ∧ ¬ (𝑅 / (𝑁 gcd 𝑅)) ∥ 𝐴) → ¬ (𝑅 / (𝑁 gcd 𝑅)) ∥ 𝐴)
100		exmidd 894	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → ((𝑅 / (𝑁 gcd 𝑅)) ∥ 𝐴 ∨ ¬ (𝑅 / (𝑁 gcd 𝑅)) ∥ 𝐴))
101	98, 99, 100	mpjaodan 959	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → ¬ (𝑅 / (𝑁 gcd 𝑅)) ∥ 𝐴)
102	55, 97, 101	elrabd 3710	. . . . . . . . 9 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 / (𝑁 gcd 𝑅)) ∈ {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴})
103		lbinfle 12250	. . . . . . . . 9 ⊢ (({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴} ⊆ ℝ ∧ ∃𝑥 ∈ {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}∀𝑦 ∈ {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}𝑥 ≤ 𝑦 ∧ (𝑅 / (𝑁 gcd 𝑅)) ∈ {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}) → inf({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}, ℝ, < ) ≤ (𝑅 / (𝑁 gcd 𝑅)))
104	44, 53, 102, 103	syl3anc 1371	. . . . . . . 8 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → inf({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}, ℝ, < ) ≤ (𝑅 / (𝑁 gcd 𝑅)))
105	35, 104	eqbrtrd 5188	. . . . . . 7 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 𝑅 ≤ (𝑅 / (𝑁 gcd 𝑅)))
106	31, 30	lenltd 11436	. . . . . . 7 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 ≤ (𝑅 / (𝑁 gcd 𝑅)) ↔ ¬ (𝑅 / (𝑁 gcd 𝑅)) < 𝑅))
107	105, 106	mpbid 232	. . . . . 6 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → ¬ (𝑅 / (𝑁 gcd 𝑅)) < 𝑅)
108	107	adantr 480	. . . . 5 ⊢ (((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) ∧ ¬ 𝑅 ≤ (𝑅 / (𝑁 gcd 𝑅))) → ¬ (𝑅 / (𝑁 gcd 𝑅)) < 𝑅)
109	34, 108	pm2.21dd 195	. . . 4 ⊢ (((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) ∧ ¬ 𝑅 ≤ (𝑅 / (𝑁 gcd 𝑅))) → 𝑅 ≤ (𝑅 / (𝑁 gcd 𝑅)))
110	1, 109	pm2.61dan 812	. . 3 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 𝑅 ≤ (𝑅 / (𝑁 gcd 𝑅)))
111	82	rpred 13099	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ ℝ)
112	111	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 𝑅 ∈ ℝ)
113	91, 107	pm2.21dd 195	. . . . . 6 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑁 gcd 𝑅) ∈ ℕ)
114	113	nnrpd 13097	. . . . 5 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑁 gcd 𝑅) ∈ ℝ⁺)
115	112	recnd 11318	. . . . . . 7 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 𝑅 ∈ ℂ)
116	115, 87	dividd 12068	. . . . . 6 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 / 𝑅) = 1)
117	116, 89	eqbrtrd 5188	. . . . 5 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 / 𝑅) < (𝑁 gcd 𝑅))
118	112, 83, 114, 117	ltdiv23d 13166	. . . 4 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 / (𝑁 gcd 𝑅)) < 𝑅)
119	77	nnred 12308	. . . . . 6 ⊢ (𝜑 → (𝑅 / (𝑁 gcd 𝑅)) ∈ ℝ)
120	119, 111	ltnled 11437	. . . . 5 ⊢ (𝜑 → ((𝑅 / (𝑁 gcd 𝑅)) < 𝑅 ↔ ¬ 𝑅 ≤ (𝑅 / (𝑁 gcd 𝑅))))
121	120	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → ((𝑅 / (𝑁 gcd 𝑅)) < 𝑅 ↔ ¬ 𝑅 ≤ (𝑅 / (𝑁 gcd 𝑅))))
122	118, 121	mpbid 232	. . 3 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → ¬ 𝑅 ≤ (𝑅 / (𝑁 gcd 𝑅)))
123	110, 122	pm2.21dd 195	. 2 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑁 gcd 𝑅) = 1)
124		simpr 484	. . 3 ⊢ (((𝜑 ∧ ¬ 1 < (𝑁 gcd 𝑅)) ∧ (𝑁 gcd 𝑅) = 1) → (𝑁 gcd 𝑅) = 1)
125	26	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 1 < (𝑁 gcd 𝑅)) → (𝑁 gcd 𝑅) ∈ ℕ)
126	125	nnred 12308	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 1 < (𝑁 gcd 𝑅)) → (𝑁 gcd 𝑅) ∈ ℝ)
127	126	adantr 480	. . . . 5 ⊢ (((𝜑 ∧ ¬ 1 < (𝑁 gcd 𝑅)) ∧ (𝑁 gcd 𝑅) ∈ (ℤ_≥‘2)) → (𝑁 gcd 𝑅) ∈ ℝ)
128	58	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ ¬ 1 < (𝑁 gcd 𝑅)) ∧ (𝑁 gcd 𝑅) ∈ (ℤ_≥‘2)) → 2 ∈ ℝ)
129		1red 11291	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 1 < (𝑁 gcd 𝑅)) ∧ (𝑁 gcd 𝑅) ∈ (ℤ_≥‘2)) → 1 ∈ ℝ)
130	27, 62	lenltd 11436	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁 gcd 𝑅) ≤ 1 ↔ ¬ 1 < (𝑁 gcd 𝑅)))
131	130	biimprd 248	. . . . . . . 8 ⊢ (𝜑 → (¬ 1 < (𝑁 gcd 𝑅) → (𝑁 gcd 𝑅) ≤ 1))
132	131	imp 406	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 1 < (𝑁 gcd 𝑅)) → (𝑁 gcd 𝑅) ≤ 1)
133	132	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 1 < (𝑁 gcd 𝑅)) ∧ (𝑁 gcd 𝑅) ∈ (ℤ_≥‘2)) → (𝑁 gcd 𝑅) ≤ 1)
134	63	a1i 11	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 1 < (𝑁 gcd 𝑅)) ∧ (𝑁 gcd 𝑅) ∈ (ℤ_≥‘2)) → 1 < 2)
135	127, 129, 128, 133, 134	lelttrd 11448	. . . . 5 ⊢ (((𝜑 ∧ ¬ 1 < (𝑁 gcd 𝑅)) ∧ (𝑁 gcd 𝑅) ∈ (ℤ_≥‘2)) → (𝑁 gcd 𝑅) < 2)
136		eluzle 12916	. . . . . 6 ⊢ ((𝑁 gcd 𝑅) ∈ (ℤ_≥‘2) → 2 ≤ (𝑁 gcd 𝑅))
137	136	adantl 481	. . . . 5 ⊢ (((𝜑 ∧ ¬ 1 < (𝑁 gcd 𝑅)) ∧ (𝑁 gcd 𝑅) ∈ (ℤ_≥‘2)) → 2 ≤ (𝑁 gcd 𝑅))
138	127, 128, 127, 135, 137	ltletrd 11450	. . . 4 ⊢ (((𝜑 ∧ ¬ 1 < (𝑁 gcd 𝑅)) ∧ (𝑁 gcd 𝑅) ∈ (ℤ_≥‘2)) → (𝑁 gcd 𝑅) < (𝑁 gcd 𝑅))
139	127	ltnrd 11424	. . . 4 ⊢ (((𝜑 ∧ ¬ 1 < (𝑁 gcd 𝑅)) ∧ (𝑁 gcd 𝑅) ∈ (ℤ_≥‘2)) → ¬ (𝑁 gcd 𝑅) < (𝑁 gcd 𝑅))
140	138, 139	pm2.21dd 195	. . 3 ⊢ (((𝜑 ∧ ¬ 1 < (𝑁 gcd 𝑅)) ∧ (𝑁 gcd 𝑅) ∈ (ℤ_≥‘2)) → (𝑁 gcd 𝑅) = 1)
141		elnn1uz2 12990	. . . 4 ⊢ ((𝑁 gcd 𝑅) ∈ ℕ ↔ ((𝑁 gcd 𝑅) = 1 ∨ (𝑁 gcd 𝑅) ∈ (ℤ_≥‘2)))
142	125, 141	sylib 218	. . 3 ⊢ ((𝜑 ∧ ¬ 1 < (𝑁 gcd 𝑅)) → ((𝑁 gcd 𝑅) = 1 ∨ (𝑁 gcd 𝑅) ∈ (ℤ_≥‘2)))
143	124, 140, 142	mpjaodan 959	. 2 ⊢ ((𝜑 ∧ ¬ 1 < (𝑁 gcd 𝑅)) → (𝑁 gcd 𝑅) = 1)
144	123, 143	pm2.61dan 812	1 ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 846 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ∀wral 3067 ∃wrex 3076 {crab 3443 ⊆ wss 3976 ∅c0 4352 class class class wbr 5166 ‘cfv 6573 (class class class)co 7448 Fincfn 9003 infcinf 9510 ℝcr 11183 0cc0 11184 1c1 11185 · cmul 11189 < clt 11324 ≤ cle 11325 − cmin 11520 / cdiv 11947 ℕcn 12293 2c2 12348 3c3 12349 5c5 12351 ℕ₀cn0 12553 ℤcz 12639 ℤ_≥cuz 12903 ℝ⁺crp 13057 ...cfz 13567 ⌊cfl 13841 ⌈cceil 13842 ↑cexp 14112 ∏cprod 15951 ∥ cdvds 16302 gcd cgcd 16540 log_b clogb 26825

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cc 10504 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-symdif 4272 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-disj 5134 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-ofr 7715 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-oadd 8526 df-omul 8527 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-dju 9970 df-card 10008 df-acn 10011 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ioo 13411 df-ioc 13412 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-fl 13843 df-ceil 13844 df-mod 13921 df-seq 14053 df-exp 14113 df-fac 14323 df-bc 14352 df-hash 14380 df-shft 15116 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-limsup 15517 df-clim 15534 df-rlim 15535 df-sum 15735 df-prod 15952 df-ef 16115 df-e 16116 df-sin 16117 df-cos 16118 df-pi 16120 df-dvds 16303 df-gcd 16541 df-lcm 16637 df-lcmf 16638 df-prm 16719 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-pt 17504 df-prds 17507 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-submnd 18819 df-mulg 19108 df-cntz 19357 df-cmn 19824 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-fbas 21384 df-fg 21385 df-cnfld 21388 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-lp 23165 df-perf 23166 df-cn 23256 df-cnp 23257 df-haus 23344 df-cmp 23416 df-tx 23591 df-hmeo 23784 df-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-xms 24351 df-ms 24352 df-tms 24353 df-cncf 24923 df-ovol 25518 df-vol 25519 df-mbf 25673 df-itg1 25674 df-itg2 25675 df-ibl 25676 df-itg 25677 df-0p 25724 df-limc 25921 df-dv 25922 df-log 26616 df-cxp 26617 df-logb 26826

This theorem is referenced by: aks4d1p8 42044

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