aks4d1p5 - Mathbox for metakunt

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Theorem aks4d1p5 42530

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 42529	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑅 ∈ (1...𝐵) ∧ ¬ 𝑅 ∥ 𝐴))
7	6	simpld 494	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ∈ (1...𝐵))
8		elfznn 13496	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ (1...𝐵) → 𝑅 ∈ ℕ)
9	7, 8	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ ℕ)
10	9	nnred 12178	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ ℝ)
11		eluzelz 12787	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ ℤ)
12	2, 11	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℤ)
13		0red 11136	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ∈ ℝ)
14		3re 12250	. . . . . . . . . . . . . . . 16 ⊢ 3 ∈ ℝ
15	14	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 3 ∈ ℝ)
16	12	zred 12622	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ ℝ)
17		3pos 12275	. . . . . . . . . . . . . . . 16 ⊢ 0 < 3
18	17	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 < 3)
19		eluzle 12790	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 3 ≤ 𝑁)
20	2, 19	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 3 ≤ 𝑁)
21	13, 15, 16, 18, 20	ltletrd 11295	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < 𝑁)
22	12, 21	jca 511	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁 ∈ ℤ ∧ 0 < 𝑁))
23		elnnz 12523	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁))
24	22, 23	sylibr 234	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℕ)
25		gcdnncl 16465	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑅 ∈ ℕ) → (𝑁 gcd 𝑅) ∈ ℕ)
26	24, 9, 25	syl2anc 585	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 gcd 𝑅) ∈ ℕ)
27	26	nnred 12178	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 gcd 𝑅) ∈ ℝ)
28	26	nnne0d 12216	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 gcd 𝑅) ≠ 0)
29	10, 27, 28	redivcld 11972	. . . . . . . . 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 11282	. . . . . . 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 4021	. . . . . . . . . . . 12 ⊢ {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴} ⊆ (1...𝐵)
37	36	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴} ⊆ (1...𝐵))
38		elfznn 13496	. . . . . . . . . . . . . . 15 ⊢ (𝑜 ∈ (1...𝐵) → 𝑜 ∈ ℕ)
39	38	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑜 ∈ (1...𝐵)) → 𝑜 ∈ ℕ)
40	39	nnred 12178	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑜 ∈ (1...𝐵)) → 𝑜 ∈ ℝ)
41	40	ex 412	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑜 ∈ (1...𝐵) → 𝑜 ∈ ℝ))
42	41	ssrdv 3928	. . . . . . . . . . 11 ⊢ (𝜑 → (1...𝐵) ⊆ ℝ)
43	37, 42	sstrd 3933	. . . . . . . . . 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 13924	. . . . . . . . . . . 12 ⊢ (𝜑 → (1...𝐵) ∈ Fin)
46	45, 37	ssfid 9170	. . . . . . . . . . 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 42528	. . . . . . . . . . . 12 ⊢ (𝜑 → ∃𝑟 ∈ (1...𝐵) ¬ 𝑟 ∥ 𝐴)
49		rabn0 4330	. . . . . . . . . . . 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 12092	. . . . . . . . . 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 1374	. . . . . . . . 9 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → ∃𝑥 ∈ {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}∀𝑦 ∈ {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}𝑥 ≤ 𝑦)
54		breq1 5089	. . . . . . . . . . 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 12547	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 1 ∈ ℤ)
57	4	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 = (⌈‘((2 log_b 𝑁)↑5)))
58		2re 12244	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℝ
59	58	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 2 ∈ ℝ)
60		2pos 12273	. . . . . . . . . . . . . . . . 17 ⊢ 0 < 2
61	60	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 < 2)
62		1red 11134	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 1 ∈ ℝ)
63		1lt2 12336	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 < 2
64	63	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 1 < 2)
65	62, 64	ltned 11271	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 1 ≠ 2)
66	65	necomd 2988	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 2 ≠ 1)
67	59, 61, 16, 21, 66	relogbcld 42424	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2 log_b 𝑁) ∈ ℝ)
68		5nn0 12446	. . . . . . . . . . . . . . . 16 ⊢ 5 ∈ ℕ₀
69	68	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 5 ∈ ℕ₀)
70	67, 69	reexpcld 14114	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((2 log_b 𝑁)↑5) ∈ ℝ)
71		ceilcl 13790	. . . . . . . . . . . . . 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 2837	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ ℤ)
74	73	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 𝐵 ∈ ℤ)
75	24	nnzd 12539	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℤ)
76		divgcdnnr 16474	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ ℕ ∧ 𝑁 ∈ ℤ) → (𝑅 / (𝑁 gcd 𝑅)) ∈ ℕ)
77	9, 75, 76	syl2anc 585	. . . . . . . . . . . . 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 12539	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 / (𝑁 gcd 𝑅)) ∈ ℤ)
80	78	nnge1d 12214	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 1 ≤ (𝑅 / (𝑁 gcd 𝑅)))
81	74	zred 12622	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 𝐵 ∈ ℝ)
82	9	nnrpd 12973	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
83	82	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 𝑅 ∈ ℝ⁺)
84	26	nnrpd 12973	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁 gcd 𝑅) ∈ ℝ⁺)
85	84	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑁 gcd 𝑅) ∈ ℝ⁺)
86	31	recnd 11162	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 𝑅 ∈ ℂ)
87	83	rpne0d 12980	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 𝑅 ≠ 0)
88	86, 87	dividd 11918	. . . . . . . . . . . . . . 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 5108	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 / 𝑅) < (𝑁 gcd 𝑅))
91	31, 83, 85, 90	ltdiv23d 13042	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 / (𝑁 gcd 𝑅)) < 𝑅)
92	30, 31, 91	ltled 11283	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 / (𝑁 gcd 𝑅)) ≤ 𝑅)
93		elfzle2 13471	. . . . . . . . . . . . . 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 11292	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 / (𝑁 gcd 𝑅)) ≤ 𝐵)
97	56, 74, 79, 80, 96	elfzd 13458	. . . . . . . . . 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 896	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → ((𝑅 / (𝑁 gcd 𝑅)) ∥ 𝐴 ∨ ¬ (𝑅 / (𝑁 gcd 𝑅)) ∥ 𝐴))
101	98, 99, 100	mpjaodan 961	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → ¬ (𝑅 / (𝑁 gcd 𝑅)) ∥ 𝐴)
102	55, 97, 101	elrabd 3637	. . . . . . . . 9 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 / (𝑁 gcd 𝑅)) ∈ {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴})
103		lbinfle 12100	. . . . . . . . 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 1374	. . . . . . . 8 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → inf({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}, ℝ, < ) ≤ (𝑅 / (𝑁 gcd 𝑅)))
105	35, 104	eqbrtrd 5108	. . . . . . 7 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 𝑅 ≤ (𝑅 / (𝑁 gcd 𝑅)))
106	31, 30	lenltd 11281	. . . . . . 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 813	. . 3 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 𝑅 ≤ (𝑅 / (𝑁 gcd 𝑅)))
111	82	rpred 12975	. . . . . 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 12973	. . . . 5 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑁 gcd 𝑅) ∈ ℝ⁺)
115	112	recnd 11162	. . . . . . 7 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 𝑅 ∈ ℂ)
116	115, 87	dividd 11918	. . . . . 6 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 / 𝑅) = 1)
117	116, 89	eqbrtrd 5108	. . . . 5 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 / 𝑅) < (𝑁 gcd 𝑅))
118	112, 83, 114, 117	ltdiv23d 13042	. . . 4 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 / (𝑁 gcd 𝑅)) < 𝑅)
119	77	nnred 12178	. . . . . 6 ⊢ (𝜑 → (𝑅 / (𝑁 gcd 𝑅)) ∈ ℝ)
120	119, 111	ltnled 11282	. . . . 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 12178	. . . . . 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 11134	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 1 < (𝑁 gcd 𝑅)) ∧ (𝑁 gcd 𝑅) ∈ (ℤ_≥‘2)) → 1 ∈ ℝ)
130	27, 62	lenltd 11281	. . . . . . . . 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 11293	. . . . 5 ⊢ (((𝜑 ∧ ¬ 1 < (𝑁 gcd 𝑅)) ∧ (𝑁 gcd 𝑅) ∈ (ℤ_≥‘2)) → (𝑁 gcd 𝑅) < 2)
136		eluzle 12790	. . . . . 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 11295	. . . 4 ⊢ (((𝜑 ∧ ¬ 1 < (𝑁 gcd 𝑅)) ∧ (𝑁 gcd 𝑅) ∈ (ℤ_≥‘2)) → (𝑁 gcd 𝑅) < (𝑁 gcd 𝑅))
139	127	ltnrd 11269	. . . 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 12864	. . . 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 961	. 2 ⊢ ((𝜑 ∧ ¬ 1 < (𝑁 gcd 𝑅)) → (𝑁 gcd 𝑅) = 1)
144	123, 143	pm2.61dan 813	1 ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 ∃wrex 3062 {crab 3390 ⊆ wss 3890 ∅c0 4274 class class class wbr 5086 ‘cfv 6490 (class class class)co 7358 Fincfn 8884 infcinf 9345 ℝcr 11026 0cc0 11027 1c1 11028 · cmul 11032 < clt 11168 ≤ cle 11169 − cmin 11366 / cdiv 11796 ℕcn 12163 2c2 12225 3c3 12226 5c5 12228 ℕ₀cn0 12426 ℤcz 12513 ℤ_≥cuz 12777 ℝ⁺crp 12931 ...cfz 13450 ⌊cfl 13738 ⌈cceil 13739 ↑cexp 14012 ∏cprod 15857 ∥ cdvds 16210 gcd cgcd 16452 log_b clogb 26745

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5368 ax-un 7680 ax-inf2 9551 ax-cc 10346 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104 ax-pre-sup 11105 ax-addf 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-symdif 4194 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-disj 5054 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-ofr 7623 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8102 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-2o 8397 df-oadd 8400 df-omul 8401 df-er 8634 df-map 8766 df-pm 8767 df-ixp 8837 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fsupp 9266 df-fi 9315 df-sup 9346 df-inf 9347 df-oi 9416 df-dju 9814 df-card 9852 df-acn 9855 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12164 df-2 12233 df-3 12234 df-4 12235 df-5 12236 df-6 12237 df-7 12238 df-8 12239 df-9 12240 df-n0 12427 df-z 12514 df-dec 12634 df-uz 12778 df-q 12888 df-rp 12932 df-xneg 13052 df-xadd 13053 df-xmul 13054 df-ioo 13291 df-ioc 13292 df-ico 13293 df-icc 13294 df-fz 13451 df-fzo 13598 df-fl 13740 df-ceil 13741 df-mod 13818 df-seq 13953 df-exp 14013 df-fac 14225 df-bc 14254 df-hash 14282 df-shft 15018 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-limsup 15422 df-clim 15439 df-rlim 15440 df-sum 15638 df-prod 15858 df-ef 16021 df-e 16022 df-sin 16023 df-cos 16024 df-pi 16026 df-dvds 16211 df-gcd 16453 df-lcm 16548 df-lcmf 16549 df-prm 16630 df-struct 17106 df-sets 17123 df-slot 17141 df-ndx 17153 df-base 17169 df-ress 17190 df-plusg 17222 df-mulr 17223 df-starv 17224 df-sca 17225 df-vsca 17226 df-ip 17227 df-tset 17228 df-ple 17229 df-ds 17231 df-unif 17232 df-hom 17233 df-cco 17234 df-rest 17374 df-topn 17375 df-0g 17393 df-gsum 17394 df-topgen 17395 df-pt 17396 df-prds 17399 df-xrs 17455 df-qtop 17460 df-imas 17461 df-xps 17463 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-submnd 18741 df-mulg 19033 df-cntz 19281 df-cmn 19746 df-psmet 21334 df-xmet 21335 df-met 21336 df-bl 21337 df-mopn 21338 df-fbas 21339 df-fg 21340 df-cnfld 21343 df-top 22868 df-topon 22885 df-topsp 22907 df-bases 22920 df-cld 22993 df-ntr 22994 df-cls 22995 df-nei 23072 df-lp 23110 df-perf 23111 df-cn 23201 df-cnp 23202 df-haus 23289 df-cmp 23361 df-tx 23536 df-hmeo 23729 df-fil 23820 df-fm 23912 df-flim 23913 df-flf 23914 df-xms 24294 df-ms 24295 df-tms 24296 df-cncf 24854 df-ovol 25440 df-vol 25441 df-mbf 25595 df-itg1 25596 df-itg2 25597 df-ibl 25598 df-itg 25599 df-0p 25646 df-limc 25842 df-dv 25843 df-log 26536 df-cxp 26537 df-logb 26746

This theorem is referenced by: aks4d1p8 42537

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