aks4d1p5 - Mathbox for metakunt

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Theorem aks4d1p5 41031

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 485	. . . 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 41030	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑅 ∈ (1...𝐵) ∧ ¬ 𝑅 ∥ 𝐴))
7	6	simpld 495	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ∈ (1...𝐵))
8		elfznn 13532	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ (1...𝐵) → 𝑅 ∈ ℕ)
9	7, 8	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ ℕ)
10	9	nnred 12229	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ ℝ)
11		eluzelz 12834	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ ℤ)
12	2, 11	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℤ)
13		0red 11219	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ∈ ℝ)
14		3re 12294	. . . . . . . . . . . . . . . 16 ⊢ 3 ∈ ℝ
15	14	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 3 ∈ ℝ)
16	12	zred 12668	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ ℝ)
17		3pos 12319	. . . . . . . . . . . . . . . 16 ⊢ 0 < 3
18	17	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 < 3)
19		eluzle 12837	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 3 ≤ 𝑁)
20	2, 19	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 3 ≤ 𝑁)
21	13, 15, 16, 18, 20	ltletrd 11376	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < 𝑁)
22	12, 21	jca 512	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁 ∈ ℤ ∧ 0 < 𝑁))
23		elnnz 12570	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁))
24	22, 23	sylibr 233	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℕ)
25		gcdnncl 16450	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑅 ∈ ℕ) → (𝑁 gcd 𝑅) ∈ ℕ)
26	24, 9, 25	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 gcd 𝑅) ∈ ℕ)
27	26	nnred 12229	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 gcd 𝑅) ∈ ℝ)
28	26	nnne0d 12264	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 gcd 𝑅) ≠ 0)
29	10, 27, 28	redivcld 12044	. . . . . . . . 9 ⊢ (𝜑 → (𝑅 / (𝑁 gcd 𝑅)) ∈ ℝ)
30	29	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 / (𝑁 gcd 𝑅)) ∈ ℝ)
31	10	adantr 481	. . . . . . . 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 247	. . . . . 6 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (¬ 𝑅 ≤ (𝑅 / (𝑁 gcd 𝑅)) → (𝑅 / (𝑁 gcd 𝑅)) < 𝑅))
34	33	imp 407	. . . . 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 4077	. . . . . . . . . . . 12 ⊢ {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴} ⊆ (1...𝐵)
37	36	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴} ⊆ (1...𝐵))
38		elfznn 13532	. . . . . . . . . . . . . . 15 ⊢ (𝑜 ∈ (1...𝐵) → 𝑜 ∈ ℕ)
39	38	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑜 ∈ (1...𝐵)) → 𝑜 ∈ ℕ)
40	39	nnred 12229	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑜 ∈ (1...𝐵)) → 𝑜 ∈ ℝ)
41	40	ex 413	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑜 ∈ (1...𝐵) → 𝑜 ∈ ℝ))
42	41	ssrdv 3988	. . . . . . . . . . 11 ⊢ (𝜑 → (1...𝐵) ⊆ ℝ)
43	37, 42	sstrd 3992	. . . . . . . . . 10 ⊢ (𝜑 → {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴} ⊆ ℝ)
44	43	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴} ⊆ ℝ)
45		fzfid 13940	. . . . . . . . . . . 12 ⊢ (𝜑 → (1...𝐵) ∈ Fin)
46	45, 37	ssfid 9269	. . . . . . . . . . 11 ⊢ (𝜑 → {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴} ∈ Fin)
47	46	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴} ∈ Fin)
48	2, 3, 4	aks4d1p3 41029	. . . . . . . . . . . 12 ⊢ (𝜑 → ∃𝑟 ∈ (1...𝐵) ¬ 𝑟 ∥ 𝐴)
49		rabn0 4385	. . . . . . . . . . . 12 ⊢ ({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴} ≠ ∅ ↔ ∃𝑟 ∈ (1...𝐵) ¬ 𝑟 ∥ 𝐴)
50	48, 49	sylibr 233	. . . . . . . . . . 11 ⊢ (𝜑 → {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴} ≠ ∅)
51	50	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴} ≠ ∅)
52		fiminre 12163	. . . . . . . . . 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 5151	. . . . . . . . . . 11 ⊢ (𝑟 = (𝑅 / (𝑁 gcd 𝑅)) → (𝑟 ∥ 𝐴 ↔ (𝑅 / (𝑁 gcd 𝑅)) ∥ 𝐴))
55	54	notbid 317	. . . . . . . . . 10 ⊢ (𝑟 = (𝑅 / (𝑁 gcd 𝑅)) → (¬ 𝑟 ∥ 𝐴 ↔ ¬ (𝑅 / (𝑁 gcd 𝑅)) ∥ 𝐴))
56		1zzd 12595	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 1 ∈ ℤ)
57	4	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 = (⌈‘((2 log_b 𝑁)↑5)))
58		2re 12288	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℝ
59	58	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 2 ∈ ℝ)
60		2pos 12317	. . . . . . . . . . . . . . . . 17 ⊢ 0 < 2
61	60	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 < 2)
62		1red 11217	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 1 ∈ ℝ)
63		1lt2 12385	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 < 2
64	63	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 1 < 2)
65	62, 64	ltned 11352	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 1 ≠ 2)
66	65	necomd 2996	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 2 ≠ 1)
67	59, 61, 16, 21, 66	relogbcld 40924	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2 log_b 𝑁) ∈ ℝ)
68		5nn0 12494	. . . . . . . . . . . . . . . 16 ⊢ 5 ∈ ℕ₀
69	68	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 5 ∈ ℕ₀)
70	67, 69	reexpcld 14130	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((2 log_b 𝑁)↑5) ∈ ℝ)
71		ceilcl 13809	. . . . . . . . . . . . . 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 2833	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ ℤ)
74	73	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 𝐵 ∈ ℤ)
75	24	nnzd 12587	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℤ)
76		divgcdnnr 16459	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ ℕ ∧ 𝑁 ∈ ℤ) → (𝑅 / (𝑁 gcd 𝑅)) ∈ ℕ)
77	9, 75, 76	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑅 / (𝑁 gcd 𝑅)) ∈ ℕ)
78	77	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 / (𝑁 gcd 𝑅)) ∈ ℕ)
79	78	nnzd 12587	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 / (𝑁 gcd 𝑅)) ∈ ℤ)
80	78	nnge1d 12262	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 1 ≤ (𝑅 / (𝑁 gcd 𝑅)))
81	74	zred 12668	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 𝐵 ∈ ℝ)
82	9	nnrpd 13016	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
83	82	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 𝑅 ∈ ℝ⁺)
84	26	nnrpd 13016	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁 gcd 𝑅) ∈ ℝ⁺)
85	84	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑁 gcd 𝑅) ∈ ℝ⁺)
86	31	recnd 11244	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 𝑅 ∈ ℂ)
87	83	rpne0d 13023	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 𝑅 ≠ 0)
88	86, 87	dividd 11990	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 / 𝑅) = 1)
89		simpr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 1 < (𝑁 gcd 𝑅))
90	88, 89	eqbrtrd 5170	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 / 𝑅) < (𝑁 gcd 𝑅))
91	31, 83, 85, 90	ltdiv23d 13085	. . . . . . . . . . . . 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 13507	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ (1...𝐵) → 𝑅 ≤ 𝐵)
94	7, 93	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ≤ 𝐵)
95	94	adantr 481	. . . . . . . . . . . 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 13494	. . . . . . . . . 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 485	. . . . . . . . . . 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 957	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → ¬ (𝑅 / (𝑁 gcd 𝑅)) ∥ 𝐴)
102	55, 97, 101	elrabd 3685	. . . . . . . . 9 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 / (𝑁 gcd 𝑅)) ∈ {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴})
103		lbinfle 12171	. . . . . . . . 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 5170	. . . . . . 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 231	. . . . . 6 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → ¬ (𝑅 / (𝑁 gcd 𝑅)) < 𝑅)
108	107	adantr 481	. . . . 5 ⊢ (((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) ∧ ¬ 𝑅 ≤ (𝑅 / (𝑁 gcd 𝑅))) → ¬ (𝑅 / (𝑁 gcd 𝑅)) < 𝑅)
109	34, 108	pm2.21dd 194	. . . 4 ⊢ (((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) ∧ ¬ 𝑅 ≤ (𝑅 / (𝑁 gcd 𝑅))) → 𝑅 ≤ (𝑅 / (𝑁 gcd 𝑅)))
110	1, 109	pm2.61dan 811	. . 3 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 𝑅 ≤ (𝑅 / (𝑁 gcd 𝑅)))
111	82	rpred 13018	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ ℝ)
112	111	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 𝑅 ∈ ℝ)
113	91, 107	pm2.21dd 194	. . . . . 6 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑁 gcd 𝑅) ∈ ℕ)
114	113	nnrpd 13016	. . . . 5 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑁 gcd 𝑅) ∈ ℝ⁺)
115	112	recnd 11244	. . . . . . 7 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 𝑅 ∈ ℂ)
116	115, 87	dividd 11990	. . . . . 6 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 / 𝑅) = 1)
117	116, 89	eqbrtrd 5170	. . . . 5 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 / 𝑅) < (𝑁 gcd 𝑅))
118	112, 83, 114, 117	ltdiv23d 13085	. . . 4 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 / (𝑁 gcd 𝑅)) < 𝑅)
119	77	nnred 12229	. . . . . 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 481	. . . 4 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → ((𝑅 / (𝑁 gcd 𝑅)) < 𝑅 ↔ ¬ 𝑅 ≤ (𝑅 / (𝑁 gcd 𝑅))))
122	118, 121	mpbid 231	. . 3 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → ¬ 𝑅 ≤ (𝑅 / (𝑁 gcd 𝑅)))
123	110, 122	pm2.21dd 194	. 2 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑁 gcd 𝑅) = 1)
124		simpr 485	. . 3 ⊢ (((𝜑 ∧ ¬ 1 < (𝑁 gcd 𝑅)) ∧ (𝑁 gcd 𝑅) = 1) → (𝑁 gcd 𝑅) = 1)
125	26	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 1 < (𝑁 gcd 𝑅)) → (𝑁 gcd 𝑅) ∈ ℕ)
126	125	nnred 12229	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 1 < (𝑁 gcd 𝑅)) → (𝑁 gcd 𝑅) ∈ ℝ)
127	126	adantr 481	. . . . 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 11217	. . . . . 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 247	. . . . . . . 8 ⊢ (𝜑 → (¬ 1 < (𝑁 gcd 𝑅) → (𝑁 gcd 𝑅) ≤ 1))
132	131	imp 407	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 1 < (𝑁 gcd 𝑅)) → (𝑁 gcd 𝑅) ≤ 1)
133	132	adantr 481	. . . . . 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 12837	. . . . . 6 ⊢ ((𝑁 gcd 𝑅) ∈ (ℤ_≥‘2) → 2 ≤ (𝑁 gcd 𝑅))
137	136	adantl 482	. . . . 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 194	. . 3 ⊢ (((𝜑 ∧ ¬ 1 < (𝑁 gcd 𝑅)) ∧ (𝑁 gcd 𝑅) ∈ (ℤ_≥‘2)) → (𝑁 gcd 𝑅) = 1)
141		elnn1uz2 12911	. . . 4 ⊢ ((𝑁 gcd 𝑅) ∈ ℕ ↔ ((𝑁 gcd 𝑅) = 1 ∨ (𝑁 gcd 𝑅) ∈ (ℤ_≥‘2)))
142	125, 141	sylib 217	. . 3 ⊢ ((𝜑 ∧ ¬ 1 < (𝑁 gcd 𝑅)) → ((𝑁 gcd 𝑅) = 1 ∨ (𝑁 gcd 𝑅) ∈ (ℤ_≥‘2)))
143	124, 140, 142	mpjaodan 957	. 2 ⊢ ((𝜑 ∧ ¬ 1 < (𝑁 gcd 𝑅)) → (𝑁 gcd 𝑅) = 1)
144	123, 143	pm2.61dan 811	1 ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 {crab 3432 ⊆ wss 3948 ∅c0 4322 class class class wbr 5148 ‘cfv 6543 (class class class)co 7411 Fincfn 8941 infcinf 9438 ℝcr 11111 0cc0 11112 1c1 11113 · cmul 11117 < clt 11250 ≤ cle 11251 − cmin 11446 / cdiv 11873 ℕcn 12214 2c2 12269 3c3 12270 5c5 12272 ℕ₀cn0 12474 ℤcz 12560 ℤ_≥cuz 12824 ℝ⁺crp 12976 ...cfz 13486 ⌊cfl 13757 ⌈cceil 13758 ↑cexp 14029 ∏cprod 15851 ∥ cdvds 16199 gcd cgcd 16437 log_b clogb 26276

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-inf2 9638 ax-cc 10432 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 ax-mulf 11192

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-symdif 4242 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-disj 5114 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 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 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-ofr 7673 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-oadd 8472 df-omul 8473 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-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-dju 9898 df-card 9936 df-acn 9939 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-div 11874 df-nn 12215 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12475 df-z 12561 df-dec 12680 df-uz 12825 df-q 12935 df-rp 12977 df-xneg 13094 df-xadd 13095 df-xmul 13096 df-ioo 13330 df-ioc 13331 df-ico 13332 df-icc 13333 df-fz 13487 df-fzo 13630 df-fl 13759 df-ceil 13760 df-mod 13837 df-seq 13969 df-exp 14030 df-fac 14236 df-bc 14265 df-hash 14293 df-shft 15016 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-limsup 15417 df-clim 15434 df-rlim 15435 df-sum 15635 df-prod 15852 df-ef 16013 df-e 16014 df-sin 16015 df-cos 16016 df-pi 16018 df-dvds 16200 df-gcd 16438 df-lcm 16529 df-lcmf 16530 df-prm 16611 df-struct 17082 df-sets 17099 df-slot 17117 df-ndx 17129 df-base 17147 df-ress 17176 df-plusg 17212 df-mulr 17213 df-starv 17214 df-sca 17215 df-vsca 17216 df-ip 17217 df-tset 17218 df-ple 17219 df-ds 17221 df-unif 17222 df-hom 17223 df-cco 17224 df-rest 17370 df-topn 17371 df-0g 17389 df-gsum 17390 df-topgen 17391 df-pt 17392 df-prds 17395 df-xrs 17450 df-qtop 17455 df-imas 17456 df-xps 17458 df-mre 17532 df-mrc 17533 df-acs 17535 df-mgm 18563 df-sgrp 18612 df-mnd 18628 df-submnd 18674 df-mulg 18953 df-cntz 19183 df-cmn 19652 df-psmet 20942 df-xmet 20943 df-met 20944 df-bl 20945 df-mopn 20946 df-fbas 20947 df-fg 20948 df-cnfld 20951 df-top 22403 df-topon 22420 df-topsp 22442 df-bases 22456 df-cld 22530 df-ntr 22531 df-cls 22532 df-nei 22609 df-lp 22647 df-perf 22648 df-cn 22738 df-cnp 22739 df-haus 22826 df-cmp 22898 df-tx 23073 df-hmeo 23266 df-fil 23357 df-fm 23449 df-flim 23450 df-flf 23451 df-xms 23833 df-ms 23834 df-tms 23835 df-cncf 24401 df-ovol 24988 df-vol 24989 df-mbf 25143 df-itg1 25144 df-itg2 25145 df-ibl 25146 df-itg 25147 df-0p 25194 df-limc 25390 df-dv 25391 df-log 26072 df-cxp 26073 df-logb 26277

This theorem is referenced by: aks4d1p8 41038

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