aks4d1p5 - Mathbox for metakunt

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Theorem aks4d1p5 42875

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 489	. . . 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 42874	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑅 ∈ (1...𝐵) ∧ ¬ 𝑅 ∥ 𝐴))
7	6	simpld 499	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ∈ (1...𝐵))
8		elfznn 13588	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ (1...𝐵) → 𝑅 ∈ ℕ)
9	7, 8	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ ℕ)
10	9	nnred 12254	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ ℝ)
11		eluzelz 12878	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ ℤ)
12	2, 11	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℤ)
13		0red 11217	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ∈ ℝ)
14		3re 12327	. . . . . . . . . . . . . . . 16 ⊢ 3 ∈ ℝ
15	14	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 3 ∈ ℝ)
16	12	zred 12706	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ ℝ)
17		3pos 12355	. . . . . . . . . . . . . . . 16 ⊢ 0 < 3
18	17	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 < 3)
19		eluzle 12881	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 3 ≤ 𝑁)
20	2, 19	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 3 ≤ 𝑁)
21	13, 15, 16, 18, 20	ltletrd 11376	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < 𝑁)
22	12, 21	jca 520	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁 ∈ ℤ ∧ 0 < 𝑁))
23		elnnz 12607	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁))
24	22, 23	sylibr 237	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℕ)
25		gcdnncl 16571	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑅 ∈ ℕ) → (𝑁 gcd 𝑅) ∈ ℕ)
26	24, 9, 25	syl2anc 595	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 gcd 𝑅) ∈ ℕ)
27	26	nnred 12254	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 gcd 𝑅) ∈ ℝ)
28	26	nnne0d 12292	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 gcd 𝑅) ≠ 0)
29	10, 27, 28	redivcld 12049	. . . . . . . . 9 ⊢ (𝜑 → (𝑅 / (𝑁 gcd 𝑅)) ∈ ℝ)
30	29	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 / (𝑁 gcd 𝑅)) ∈ ℝ)
31	10	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 𝑅 ∈ ℝ)
32	30, 31	ltnled 11363	. . . . . . 7 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → ((𝑅 / (𝑁 gcd 𝑅)) < 𝑅 ↔ ¬ 𝑅 ≤ (𝑅 / (𝑁 gcd 𝑅))))
33	32	biimprd 251	. . . . . 6 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (¬ 𝑅 ≤ (𝑅 / (𝑁 gcd 𝑅)) → (𝑅 / (𝑁 gcd 𝑅)) < 𝑅))
34	33	imp 411	. . . . 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 4033	. . . . . . . . . . . 12 ⊢ {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴} ⊆ (1...𝐵)
37	36	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴} ⊆ (1...𝐵))
38		elfznn 13588	. . . . . . . . . . . . . . 15 ⊢ (𝑜 ∈ (1...𝐵) → 𝑜 ∈ ℕ)
39	38	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑜 ∈ (1...𝐵)) → 𝑜 ∈ ℕ)
40	39	nnred 12254	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑜 ∈ (1...𝐵)) → 𝑜 ∈ ℝ)
41	40	ex 417	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑜 ∈ (1...𝐵) → 𝑜 ∈ ℝ))
42	41	ssrdv 3942	. . . . . . . . . . 11 ⊢ (𝜑 → (1...𝐵) ⊆ ℝ)
43	37, 42	sstrd 3946	. . . . . . . . . 10 ⊢ (𝜑 → {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴} ⊆ ℝ)
44	43	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴} ⊆ ℝ)
45		fzfid 14016	. . . . . . . . . . . 12 ⊢ (𝜑 → (1...𝐵) ∈ Fin)
46	45, 37	ssfid 9227	. . . . . . . . . . 11 ⊢ (𝜑 → {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴} ∈ Fin)
47	46	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴} ∈ Fin)
48	2, 3, 4	aks4d1p3 42873	. . . . . . . . . . . 12 ⊢ (𝜑 → ∃𝑟 ∈ (1...𝐵) ¬ 𝑟 ∥ 𝐴)
49		rabn0 4345	. . . . . . . . . . . 12 ⊢ ({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴} ≠ ∅ ↔ ∃𝑟 ∈ (1...𝐵) ¬ 𝑟 ∥ 𝐴)
50	48, 49	sylibr 237	. . . . . . . . . . 11 ⊢ (𝜑 → {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴} ≠ ∅)
51	50	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴} ≠ ∅)
52		fiminre 12168	. . . . . . . . . 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 1397	. . . . . . . . 9 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → ∃𝑥 ∈ {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}∀𝑦 ∈ {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}𝑥 ≤ 𝑦)
54		breq1 5111	. . . . . . . . . . 11 ⊢ (𝑟 = (𝑅 / (𝑁 gcd 𝑅)) → (𝑟 ∥ 𝐴 ↔ (𝑅 / (𝑁 gcd 𝑅)) ∥ 𝐴))
55	54	notbid 321	. . . . . . . . . 10 ⊢ (𝑟 = (𝑅 / (𝑁 gcd 𝑅)) → (¬ 𝑟 ∥ 𝐴 ↔ ¬ (𝑅 / (𝑁 gcd 𝑅)) ∥ 𝐴))
56		1zzd 12631	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 1 ∈ ℤ)
57	4	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 = (⌈‘((2 log_b 𝑁)↑5)))
58		2re 12321	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℝ
59	58	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 2 ∈ ℝ)
60		2pos 12351	. . . . . . . . . . . . . . . . 17 ⊢ 0 < 2
61	60	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 < 2)
62		1red 11215	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 1 ∈ ℝ)
63		1lt2 12419	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 < 2
64	63	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 1 < 2)
65	62, 64	ltned 11352	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 1 ≠ 2)
66	65	necomd 3012	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 2 ≠ 1)
67	59, 61, 16, 21, 66	relogbcld 42769	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2 log_b 𝑁) ∈ ℝ)
68		5nn0 12530	. . . . . . . . . . . . . . . 16 ⊢ 5 ∈ ℕ₀
69	68	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 5 ∈ ℕ₀)
70	67, 69	reexpcld 14206	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((2 log_b 𝑁)↑5) ∈ ℝ)
71		ceilcl 13882	. . . . . . . . . . . . . 14 ⊢ (((2 log_b 𝑁)↑5) ∈ ℝ → (⌈‘((2 log_b 𝑁)↑5)) ∈ ℤ)
72	70, 71	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (⌈‘((2 log_b 𝑁)↑5)) ∈ ℤ)
73	57, 72	eqeltrd 2862	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ ℤ)
74	73	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 𝐵 ∈ ℤ)
75	24	nnzd 12623	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℤ)
76		divgcdnnr 16580	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ ℕ ∧ 𝑁 ∈ ℤ) → (𝑅 / (𝑁 gcd 𝑅)) ∈ ℕ)
77	9, 75, 76	syl2anc 595	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑅 / (𝑁 gcd 𝑅)) ∈ ℕ)
78	77	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 / (𝑁 gcd 𝑅)) ∈ ℕ)
79	78	nnzd 12623	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 / (𝑁 gcd 𝑅)) ∈ ℤ)
80	78	nnge1d 12290	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 1 ≤ (𝑅 / (𝑁 gcd 𝑅)))
81	74	zred 12706	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 𝐵 ∈ ℝ)
82	9	nnrpd 13064	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
83	82	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 𝑅 ∈ ℝ⁺)
84	26	nnrpd 13064	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁 gcd 𝑅) ∈ ℝ⁺)
85	84	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑁 gcd 𝑅) ∈ ℝ⁺)
86	31	recnd 11243	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 𝑅 ∈ ℂ)
87	83	rpne0d 13071	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 𝑅 ≠ 0)
88	86, 87	dividd 11995	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 / 𝑅) = 1)
89		simpr 489	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 1 < (𝑁 gcd 𝑅))
90	88, 89	eqbrtrd 5132	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 / 𝑅) < (𝑁 gcd 𝑅))
91	31, 83, 85, 90	ltdiv23d 13133	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 / (𝑁 gcd 𝑅)) < 𝑅)
92	30, 31, 91	ltled 11364	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 / (𝑁 gcd 𝑅)) ≤ 𝑅)
93		elfzle2 13562	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ (1...𝐵) → 𝑅 ≤ 𝐵)
94	7, 93	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ≤ 𝐵)
95	94	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 𝑅 ≤ 𝐵)
96	30, 31, 81, 92, 95	letrd 11373	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 / (𝑁 gcd 𝑅)) ≤ 𝐵)
97	56, 74, 79, 80, 96	elfzd 13549	. . . . . . . . . 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 489	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) ∧ ¬ (𝑅 / (𝑁 gcd 𝑅)) ∥ 𝐴) → ¬ (𝑅 / (𝑁 gcd 𝑅)) ∥ 𝐴)
100		exmidd 908	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → ((𝑅 / (𝑁 gcd 𝑅)) ∥ 𝐴 ∨ ¬ (𝑅 / (𝑁 gcd 𝑅)) ∥ 𝐴))
101	98, 99, 100	mpjaodan 972	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → ¬ (𝑅 / (𝑁 gcd 𝑅)) ∥ 𝐴)
102	55, 97, 101	elrabd 3651	. . . . . . . . 9 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 / (𝑁 gcd 𝑅)) ∈ {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴})
103		lbinfle 12176	. . . . . . . . 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 1397	. . . . . . . 8 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → inf({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}, ℝ, < ) ≤ (𝑅 / (𝑁 gcd 𝑅)))
105	35, 104	eqbrtrd 5132	. . . . . . 7 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 𝑅 ≤ (𝑅 / (𝑁 gcd 𝑅)))
106	31, 30	lenltd 11362	. . . . . . 7 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 ≤ (𝑅 / (𝑁 gcd 𝑅)) ↔ ¬ (𝑅 / (𝑁 gcd 𝑅)) < 𝑅))
107	105, 106	mpbid 235	. . . . . 6 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → ¬ (𝑅 / (𝑁 gcd 𝑅)) < 𝑅)
108	107	adantr 485	. . . . 5 ⊢ (((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) ∧ ¬ 𝑅 ≤ (𝑅 / (𝑁 gcd 𝑅))) → ¬ (𝑅 / (𝑁 gcd 𝑅)) < 𝑅)
109	34, 108	pm2.21dd 198	. . . 4 ⊢ (((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) ∧ ¬ 𝑅 ≤ (𝑅 / (𝑁 gcd 𝑅))) → 𝑅 ≤ (𝑅 / (𝑁 gcd 𝑅)))
110	1, 109	pm2.61dan 824	. . 3 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 𝑅 ≤ (𝑅 / (𝑁 gcd 𝑅)))
111	82	rpred 13066	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ ℝ)
112	111	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 𝑅 ∈ ℝ)
113	91, 107	pm2.21dd 198	. . . . . 6 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑁 gcd 𝑅) ∈ ℕ)
114	113	nnrpd 13064	. . . . 5 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑁 gcd 𝑅) ∈ ℝ⁺)
115	112	recnd 11243	. . . . . . 7 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 𝑅 ∈ ℂ)
116	115, 87	dividd 11995	. . . . . 6 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 / 𝑅) = 1)
117	116, 89	eqbrtrd 5132	. . . . 5 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 / 𝑅) < (𝑁 gcd 𝑅))
118	112, 83, 114, 117	ltdiv23d 13133	. . . 4 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 / (𝑁 gcd 𝑅)) < 𝑅)
119	77	nnred 12254	. . . . . 6 ⊢ (𝜑 → (𝑅 / (𝑁 gcd 𝑅)) ∈ ℝ)
120	119, 111	ltnled 11363	. . . . 5 ⊢ (𝜑 → ((𝑅 / (𝑁 gcd 𝑅)) < 𝑅 ↔ ¬ 𝑅 ≤ (𝑅 / (𝑁 gcd 𝑅))))
121	120	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → ((𝑅 / (𝑁 gcd 𝑅)) < 𝑅 ↔ ¬ 𝑅 ≤ (𝑅 / (𝑁 gcd 𝑅))))
122	118, 121	mpbid 235	. . 3 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → ¬ 𝑅 ≤ (𝑅 / (𝑁 gcd 𝑅)))
123	110, 122	pm2.21dd 198	. 2 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑁 gcd 𝑅) = 1)
124		simpr 489	. . 3 ⊢ (((𝜑 ∧ ¬ 1 < (𝑁 gcd 𝑅)) ∧ (𝑁 gcd 𝑅) = 1) → (𝑁 gcd 𝑅) = 1)
125	26	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 1 < (𝑁 gcd 𝑅)) → (𝑁 gcd 𝑅) ∈ ℕ)
126	125	nnred 12254	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 1 < (𝑁 gcd 𝑅)) → (𝑁 gcd 𝑅) ∈ ℝ)
127	126	adantr 485	. . . . 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 11215	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 1 < (𝑁 gcd 𝑅)) ∧ (𝑁 gcd 𝑅) ∈ (ℤ_≥‘2)) → 1 ∈ ℝ)
130	27, 62	lenltd 11362	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁 gcd 𝑅) ≤ 1 ↔ ¬ 1 < (𝑁 gcd 𝑅)))
131	130	biimprd 251	. . . . . . . 8 ⊢ (𝜑 → (¬ 1 < (𝑁 gcd 𝑅) → (𝑁 gcd 𝑅) ≤ 1))
132	131	imp 411	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 1 < (𝑁 gcd 𝑅)) → (𝑁 gcd 𝑅) ≤ 1)
133	132	adantr 485	. . . . . 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 11374	. . . . 5 ⊢ (((𝜑 ∧ ¬ 1 < (𝑁 gcd 𝑅)) ∧ (𝑁 gcd 𝑅) ∈ (ℤ_≥‘2)) → (𝑁 gcd 𝑅) < 2)
136		eluzle 12881	. . . . . 6 ⊢ ((𝑁 gcd 𝑅) ∈ (ℤ_≥‘2) → 2 ≤ (𝑁 gcd 𝑅))
137	136	adantl 486	. . . . 5 ⊢ (((𝜑 ∧ ¬ 1 < (𝑁 gcd 𝑅)) ∧ (𝑁 gcd 𝑅) ∈ (ℤ_≥‘2)) → 2 ≤ (𝑁 gcd 𝑅))
138	127, 128, 127, 135, 137	ltletrd 11376	. . . 4 ⊢ (((𝜑 ∧ ¬ 1 < (𝑁 gcd 𝑅)) ∧ (𝑁 gcd 𝑅) ∈ (ℤ_≥‘2)) → (𝑁 gcd 𝑅) < (𝑁 gcd 𝑅))
139	127	ltnrd 11350	. . . 4 ⊢ (((𝜑 ∧ ¬ 1 < (𝑁 gcd 𝑅)) ∧ (𝑁 gcd 𝑅) ∈ (ℤ_≥‘2)) → ¬ (𝑁 gcd 𝑅) < (𝑁 gcd 𝑅))
140	138, 139	pm2.21dd 198	. . 3 ⊢ (((𝜑 ∧ ¬ 1 < (𝑁 gcd 𝑅)) ∧ (𝑁 gcd 𝑅) ∈ (ℤ_≥‘2)) → (𝑁 gcd 𝑅) = 1)
141		elnn1uz2 12955	. . . 4 ⊢ ((𝑁 gcd 𝑅) ∈ ℕ ↔ ((𝑁 gcd 𝑅) = 1 ∨ (𝑁 gcd 𝑅) ∈ (ℤ_≥‘2)))
142	125, 141	sylib 221	. . 3 ⊢ ((𝜑 ∧ ¬ 1 < (𝑁 gcd 𝑅)) → ((𝑁 gcd 𝑅) = 1 ∨ (𝑁 gcd 𝑅) ∈ (ℤ_≥‘2)))
143	124, 140, 142	mpjaodan 972	. 2 ⊢ ((𝜑 ∧ ¬ 1 < (𝑁 gcd 𝑅)) → (𝑁 gcd 𝑅) = 1)
144	123, 143	pm2.61dan 824	1 ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 860 = wceq 1569 ∈ wcel 2142 ≠ wne 2957 ∀wral 3078 ∃wrex 3088 {crab 3415 ⊆ wss 3904 ∅c0 4285 class class class wbr 5108 ‘cfv 6536 (class class class)co 7412 Fincfn 8941 infcinf 9399 ℝcr 11105 0cc0 11106 1c1 11107 · cmul 11111 < clt 11249 ≤ cle 11250 − cmin 11447 / cdiv 11877 ℕcn 12239 2c2 12301 3c3 12302 5c5 12304 ℕ₀cn0 12510 ℤcz 12597 ℤ_≥cuz 12868 ℝ⁺crp 13022 ...cfz 13541 ⌊cfl 13830 ⌈cceil 13831 ↑cexp 14104 ∏cprod 15964 ∥ cdvds 16316 gcd cgcd 16558 log_b clogb 26940

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 ax-inf2 9608 ax-cc 10425 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1103 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-symdif 4205 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4595 df-uni 4872 df-int 4912 df-iun 4957 df-iin 4958 df-disj 5076 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5555 df-eprel 5560 df-po 5568 df-so 5569 df-fr 5613 df-se 5614 df-we 5615 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7676 df-ofr 7677 df-om 7861 df-1st 7984 df-2nd 7985 df-supp 8155 df-frecs 8276 df-wrecs 8307 df-recs 8356 df-rdg 8395 df-1o 8451 df-2o 8452 df-oadd 8455 df-omul 8456 df-er 8692 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9320 df-fi 9369 df-sup 9400 df-inf 9401 df-oi 9470 df-dju 9894 df-card 9932 df-acn 9935 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11449 df-neg 11450 df-div 11878 df-nn 12240 df-2 12309 df-3 12310 df-4 12311 df-5 12312 df-6 12313 df-7 12314 df-8 12315 df-9 12316 df-n0 12511 df-z 12598 df-dec 12718 df-uz 12869 df-q 12979 df-rp 13023 df-xneg 13143 df-xadd 13144 df-xmul 13145 df-ioo 13382 df-ioc 13383 df-ico 13384 df-icc 13385 df-fz 13542 df-fzo 13690 df-fl 13832 df-ceil 13833 df-mod 13910 df-seq 14045 df-exp 14105 df-fac 14317 df-bc 14346 df-hash 14374 df-shft 15111 df-cj 15157 df-re 15158 df-im 15159 df-sqrt 15293 df-abs 15294 df-limsup 15529 df-clim 15546 df-rlim 15547 df-sum 15745 df-prod 15965 df-ef 16127 df-e 16128 df-sin 16129 df-cos 16130 df-pi 16132 df-dvds 16317 df-gcd 16559 df-lcm 16654 df-lcmf 16655 df-prm 16736 df-struct 17213 df-sets 17230 df-slot 17248 df-ndx 17260 df-base 17276 df-ress 17297 df-plusg 17329 df-mulr 17330 df-starv 17331 df-sca 17332 df-vsca 17333 df-ip 17334 df-tset 17335 df-ple 17336 df-ds 17338 df-unif 17339 df-hom 17340 df-cco 17341 df-rest 17481 df-topn 17482 df-0g 17500 df-gsum 17501 df-topgen 17502 df-pt 17503 df-prds 17506 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18704 df-sgrp 18783 df-mnd 18799 df-submnd 18848 df-mulg 19140 df-cntz 19393 df-cmn 19858 df-psmet 21525 df-xmet 21526 df-met 21527 df-bl 21528 df-mopn 21529 df-fbas 21530 df-fg 21531 df-cnfld 21534 df-top 23062 df-topon 23079 df-topsp 23101 df-bases 23114 df-cld 23187 df-ntr 23188 df-cls 23189 df-nei 23266 df-lp 23304 df-perf 23305 df-cn 23395 df-cnp 23396 df-haus 23483 df-cmp 23555 df-tx 23730 df-hmeo 23923 df-fil 24014 df-fm 24106 df-flim 24107 df-flf 24108 df-xms 24488 df-ms 24489 df-tms 24490 df-cncf 25048 df-ovol 25634 df-vol 25635 df-mbf 25789 df-itg1 25790 df-itg2 25791 df-ibl 25792 df-itg 25793 df-0p 25840 df-limc 26036 df-dv 26037 df-log 26732 df-cxp 26733 df-logb 26941

This theorem is used by: aks4d1p8 42882

W3C validator