aks4d1p5 - Mathbox for metakunt

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Theorem aks4d1p5 42061

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 42060	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑅 ∈ (1...𝐵) ∧ ¬ 𝑅 ∥ 𝐴))
7	6	simpld 494	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ∈ (1...𝐵))
8		elfznn 13490	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ (1...𝐵) → 𝑅 ∈ ℕ)
9	7, 8	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ ℕ)
10	9	nnred 12177	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ ℝ)
11		eluzelz 12779	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ ℤ)
12	2, 11	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℤ)
13		0red 11153	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ∈ ℝ)
14		3re 12242	. . . . . . . . . . . . . . . 16 ⊢ 3 ∈ ℝ
15	14	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 3 ∈ ℝ)
16	12	zred 12614	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ ℝ)
17		3pos 12267	. . . . . . . . . . . . . . . 16 ⊢ 0 < 3
18	17	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 < 3)
19		eluzle 12782	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 3 ≤ 𝑁)
20	2, 19	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 3 ≤ 𝑁)
21	13, 15, 16, 18, 20	ltletrd 11310	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < 𝑁)
22	12, 21	jca 511	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁 ∈ ℤ ∧ 0 < 𝑁))
23		elnnz 12515	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁))
24	22, 23	sylibr 234	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℕ)
25		gcdnncl 16453	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑅 ∈ ℕ) → (𝑁 gcd 𝑅) ∈ ℕ)
26	24, 9, 25	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 gcd 𝑅) ∈ ℕ)
27	26	nnred 12177	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 gcd 𝑅) ∈ ℝ)
28	26	nnne0d 12212	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 gcd 𝑅) ≠ 0)
29	10, 27, 28	redivcld 11986	. . . . . . . . 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 11297	. . . . . . 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 4039	. . . . . . . . . . . 12 ⊢ {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴} ⊆ (1...𝐵)
37	36	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴} ⊆ (1...𝐵))
38		elfznn 13490	. . . . . . . . . . . . . . 15 ⊢ (𝑜 ∈ (1...𝐵) → 𝑜 ∈ ℕ)
39	38	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑜 ∈ (1...𝐵)) → 𝑜 ∈ ℕ)
40	39	nnred 12177	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑜 ∈ (1...𝐵)) → 𝑜 ∈ ℝ)
41	40	ex 412	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑜 ∈ (1...𝐵) → 𝑜 ∈ ℝ))
42	41	ssrdv 3949	. . . . . . . . . . 11 ⊢ (𝜑 → (1...𝐵) ⊆ ℝ)
43	37, 42	sstrd 3954	. . . . . . . . . 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 13914	. . . . . . . . . . . 12 ⊢ (𝜑 → (1...𝐵) ∈ Fin)
46	45, 37	ssfid 9188	. . . . . . . . . . 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 42059	. . . . . . . . . . . 12 ⊢ (𝜑 → ∃𝑟 ∈ (1...𝐵) ¬ 𝑟 ∥ 𝐴)
49		rabn0 4348	. . . . . . . . . . . 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 12106	. . . . . . . . . 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 1373	. . . . . . . . 9 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → ∃𝑥 ∈ {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}∀𝑦 ∈ {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}𝑥 ≤ 𝑦)
54		breq1 5105	. . . . . . . . . . 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 12540	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 1 ∈ ℤ)
57	4	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 = (⌈‘((2 log_b 𝑁)↑5)))
58		2re 12236	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℝ
59	58	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 2 ∈ ℝ)
60		2pos 12265	. . . . . . . . . . . . . . . . 17 ⊢ 0 < 2
61	60	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 < 2)
62		1red 11151	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 1 ∈ ℝ)
63		1lt2 12328	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 < 2
64	63	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 1 < 2)
65	62, 64	ltned 11286	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 1 ≠ 2)
66	65	necomd 2980	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 2 ≠ 1)
67	59, 61, 16, 21, 66	relogbcld 41954	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2 log_b 𝑁) ∈ ℝ)
68		5nn0 12438	. . . . . . . . . . . . . . . 16 ⊢ 5 ∈ ℕ₀
69	68	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 5 ∈ ℕ₀)
70	67, 69	reexpcld 14104	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((2 log_b 𝑁)↑5) ∈ ℝ)
71		ceilcl 13780	. . . . . . . . . . . . . 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 2828	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ ℤ)
74	73	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 𝐵 ∈ ℤ)
75	24	nnzd 12532	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℤ)
76		divgcdnnr 16462	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 / (𝑁 gcd 𝑅)) ∈ ℕ)
79	78	nnzd 12532	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 / (𝑁 gcd 𝑅)) ∈ ℤ)
80	78	nnge1d 12210	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 1 ≤ (𝑅 / (𝑁 gcd 𝑅)))
81	74	zred 12614	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 𝐵 ∈ ℝ)
82	9	nnrpd 12969	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
83	82	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 𝑅 ∈ ℝ⁺)
84	26	nnrpd 12969	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁 gcd 𝑅) ∈ ℝ⁺)
85	84	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑁 gcd 𝑅) ∈ ℝ⁺)
86	31	recnd 11178	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 𝑅 ∈ ℂ)
87	83	rpne0d 12976	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 𝑅 ≠ 0)
88	86, 87	dividd 11932	. . . . . . . . . . . . . . 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 5124	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 / 𝑅) < (𝑁 gcd 𝑅))
91	31, 83, 85, 90	ltdiv23d 13038	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 / (𝑁 gcd 𝑅)) < 𝑅)
92	30, 31, 91	ltled 11298	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 / (𝑁 gcd 𝑅)) ≤ 𝑅)
93		elfzle2 13465	. . . . . . . . . . . . . 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 11307	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 / (𝑁 gcd 𝑅)) ≤ 𝐵)
97	56, 74, 79, 80, 96	elfzd 13452	. . . . . . . . . 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 895	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → ((𝑅 / (𝑁 gcd 𝑅)) ∥ 𝐴 ∨ ¬ (𝑅 / (𝑁 gcd 𝑅)) ∥ 𝐴))
101	98, 99, 100	mpjaodan 960	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → ¬ (𝑅 / (𝑁 gcd 𝑅)) ∥ 𝐴)
102	55, 97, 101	elrabd 3658	. . . . . . . . 9 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 / (𝑁 gcd 𝑅)) ∈ {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴})
103		lbinfle 12114	. . . . . . . . 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 1373	. . . . . . . 8 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → inf({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}, ℝ, < ) ≤ (𝑅 / (𝑁 gcd 𝑅)))
105	35, 104	eqbrtrd 5124	. . . . . . 7 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 𝑅 ≤ (𝑅 / (𝑁 gcd 𝑅)))
106	31, 30	lenltd 11296	. . . . . . 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 12971	. . . . . 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 12969	. . . . 5 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑁 gcd 𝑅) ∈ ℝ⁺)
115	112	recnd 11178	. . . . . . 7 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → 𝑅 ∈ ℂ)
116	115, 87	dividd 11932	. . . . . 6 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 / 𝑅) = 1)
117	116, 89	eqbrtrd 5124	. . . . 5 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 / 𝑅) < (𝑁 gcd 𝑅))
118	112, 83, 114, 117	ltdiv23d 13038	. . . 4 ⊢ ((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) → (𝑅 / (𝑁 gcd 𝑅)) < 𝑅)
119	77	nnred 12177	. . . . . 6 ⊢ (𝜑 → (𝑅 / (𝑁 gcd 𝑅)) ∈ ℝ)
120	119, 111	ltnled 11297	. . . . 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 12177	. . . . . 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 11151	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 1 < (𝑁 gcd 𝑅)) ∧ (𝑁 gcd 𝑅) ∈ (ℤ_≥‘2)) → 1 ∈ ℝ)
130	27, 62	lenltd 11296	. . . . . . . . 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 11308	. . . . 5 ⊢ (((𝜑 ∧ ¬ 1 < (𝑁 gcd 𝑅)) ∧ (𝑁 gcd 𝑅) ∈ (ℤ_≥‘2)) → (𝑁 gcd 𝑅) < 2)
136		eluzle 12782	. . . . . 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 11310	. . . 4 ⊢ (((𝜑 ∧ ¬ 1 < (𝑁 gcd 𝑅)) ∧ (𝑁 gcd 𝑅) ∈ (ℤ_≥‘2)) → (𝑁 gcd 𝑅) < (𝑁 gcd 𝑅))
139	127	ltnrd 11284	. . . 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 12860	. . . 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 960	. 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 847 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 ∃wrex 3053 {crab 3402 ⊆ wss 3911 ∅c0 4292 class class class wbr 5102 ‘cfv 6499 (class class class)co 7369 Fincfn 8895 infcinf 9368 ℝcr 11043 0cc0 11044 1c1 11045 · cmul 11049 < clt 11184 ≤ cle 11185 − cmin 11381 / cdiv 11811 ℕcn 12162 2c2 12217 3c3 12218 5c5 12220 ℕ₀cn0 12418 ℤcz 12505 ℤ_≥cuz 12769 ℝ⁺crp 12927 ...cfz 13444 ⌊cfl 13728 ⌈cceil 13729 ↑cexp 14002 ∏cprod 15845 ∥ cdvds 16198 gcd cgcd 16440 log_b clogb 26707

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-inf2 9570 ax-cc 10364 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122 ax-addf 11123

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-symdif 4212 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-disj 5070 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-ofr 7634 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-oadd 8415 df-omul 8416 df-er 8648 df-map 8778 df-pm 8779 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-fi 9338 df-sup 9369 df-inf 9370 df-oi 9439 df-dju 9830 df-card 9868 df-acn 9871 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-dec 12626 df-uz 12770 df-q 12884 df-rp 12928 df-xneg 13048 df-xadd 13049 df-xmul 13050 df-ioo 13286 df-ioc 13287 df-ico 13288 df-icc 13289 df-fz 13445 df-fzo 13592 df-fl 13730 df-ceil 13731 df-mod 13808 df-seq 13943 df-exp 14003 df-fac 14215 df-bc 14244 df-hash 14272 df-shft 15009 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-limsup 15413 df-clim 15430 df-rlim 15431 df-sum 15629 df-prod 15846 df-ef 16009 df-e 16010 df-sin 16011 df-cos 16012 df-pi 16014 df-dvds 16199 df-gcd 16441 df-lcm 16536 df-lcmf 16537 df-prm 16618 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17361 df-topn 17362 df-0g 17380 df-gsum 17381 df-topgen 17382 df-pt 17383 df-prds 17386 df-xrs 17441 df-qtop 17446 df-imas 17447 df-xps 17449 df-mre 17523 df-mrc 17524 df-acs 17526 df-mgm 18549 df-sgrp 18628 df-mnd 18644 df-submnd 18693 df-mulg 18982 df-cntz 19231 df-cmn 19696 df-psmet 21288 df-xmet 21289 df-met 21290 df-bl 21291 df-mopn 21292 df-fbas 21293 df-fg 21294 df-cnfld 21297 df-top 22814 df-topon 22831 df-topsp 22853 df-bases 22866 df-cld 22939 df-ntr 22940 df-cls 22941 df-nei 23018 df-lp 23056 df-perf 23057 df-cn 23147 df-cnp 23148 df-haus 23235 df-cmp 23307 df-tx 23482 df-hmeo 23675 df-fil 23766 df-fm 23858 df-flim 23859 df-flf 23860 df-xms 24241 df-ms 24242 df-tms 24243 df-cncf 24804 df-ovol 25398 df-vol 25399 df-mbf 25553 df-itg1 25554 df-itg2 25555 df-ibl 25556 df-itg 25557 df-0p 25604 df-limc 25800 df-dv 25801 df-log 26498 df-cxp 26499 df-logb 26708

This theorem is referenced by: aks4d1p8 42068

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