4001lem4 - Metamath Proof Explorer

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Theorem 4001lem4 17238

Description: Lemma for 4001prm 17239. Calculate the GCD of 2↑800 − 1≡2310 with 𝑁 = 4001. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.)

**Hypothesis**
Ref	Expression
4001prm.1	⊢ 𝑁 = ;;;4001

**Assertion**
Ref	Expression
4001lem4	⊢ (((2↑;;800) − 1) gcd 𝑁) = 1

**Proof of Theorem 4001lem4**
Step	Hyp	Ref	Expression
1		2nn 12339	. . . 4 ⊢ 2 ∈ ℕ
2		8nn0 12552	. . . . . 6 ⊢ 8 ∈ ℕ₀
3		0nn0 12544	. . . . . 6 ⊢ 0 ∈ ℕ₀
4	2, 3	deccl 12752	. . . . 5 ⊢ ;80 ∈ ℕ₀
5	4, 3	deccl 12752	. . . 4 ⊢ ;;800 ∈ ℕ₀
6		nnexpcl 14138	. . . 4 ⊢ ((2 ∈ ℕ ∧ ;;800 ∈ ℕ₀) → (2↑;;800) ∈ ℕ)
7	1, 5, 6	mp2an 705	. . 3 ⊢ (2↑;;800) ∈ ℕ
8		nnm1nn0 12570	. . 3 ⊢ ((2↑;;800) ∈ ℕ → ((2↑;;800) − 1) ∈ ℕ₀)
9	7, 8	ax-mp 5	. 2 ⊢ ((2↑;;800) − 1) ∈ ℕ₀
10		2nn0 12546	. . . . 5 ⊢ 2 ∈ ℕ₀
11		3nn0 12547	. . . . 5 ⊢ 3 ∈ ℕ₀
12	10, 11	deccl 12752	. . . 4 ⊢ ;23 ∈ ℕ₀
13		1nn0 12545	. . . 4 ⊢ 1 ∈ ℕ₀
14	12, 13	deccl 12752	. . 3 ⊢ ;;231 ∈ ℕ₀
15	14, 3	deccl 12752	. 2 ⊢ ;;;2310 ∈ ℕ₀
16		4001prm.1	. . 3 ⊢ 𝑁 = ;;;4001
17		4nn0 12548	. . . . . 6 ⊢ 4 ∈ ℕ₀
18	17, 3	deccl 12752	. . . . 5 ⊢ ;40 ∈ ℕ₀
19	18, 3	deccl 12752	. . . 4 ⊢ ;;400 ∈ ℕ₀
20		1nn 12269	. . . 4 ⊢ 1 ∈ ℕ
21	19, 20	decnncl 12761	. . 3 ⊢ ;;;4001 ∈ ℕ
22	16, 21	eqeltri 2858	. 2 ⊢ 𝑁 ∈ ℕ
23	16	4001lem2 17236	. . 3 ⊢ ((2↑;;800) mod 𝑁) = (;;;2311 mod 𝑁)
24		0p1e1 12386	. . . 4 ⊢ (0 + 1) = 1
25		eqid 2762	. . . 4 ⊢ ;;;2310 = ;;;2310
26	14, 3, 24, 25	decsuc 12773	. . 3 ⊢ (;;;2310 + 1) = ;;;2311
27	22, 7, 13, 15, 23, 26	modsubi 17166	. 2 ⊢ (((2↑;;800) − 1) mod 𝑁) = (;;;2310 mod 𝑁)
28		6nn0 12550	. . . . . 6 ⊢ 6 ∈ ℕ₀
29	13, 28	deccl 12752	. . . . 5 ⊢ ;16 ∈ ℕ₀
30		9nn0 12553	. . . . 5 ⊢ 9 ∈ ℕ₀
31	29, 30	deccl 12752	. . . 4 ⊢ ;;169 ∈ ℕ₀
32	31, 13	deccl 12752	. . 3 ⊢ ;;;1691 ∈ ℕ₀
33	28, 13	deccl 12752	. . . . 5 ⊢ ;61 ∈ ℕ₀
34	33, 30	deccl 12752	. . . 4 ⊢ ;;619 ∈ ℕ₀
35		5nn0 12549	. . . . . . 7 ⊢ 5 ∈ ℕ₀
36	17, 35	deccl 12752	. . . . . 6 ⊢ ;45 ∈ ℕ₀
37	36, 11	deccl 12752	. . . . 5 ⊢ ;;453 ∈ ℕ₀
38	29, 28	deccl 12752	. . . . . 6 ⊢ ;;166 ∈ ℕ₀
39	13, 10	deccl 12752	. . . . . . . 8 ⊢ ;12 ∈ ℕ₀
40	39, 13	deccl 12752	. . . . . . 7 ⊢ ;;121 ∈ ℕ₀
41	11, 13	deccl 12752	. . . . . . . . 9 ⊢ ;31 ∈ ℕ₀
42	13, 17	deccl 12752	. . . . . . . . . 10 ⊢ ;14 ∈ ℕ₀
43	42	nn0zi 12644	. . . . . . . . . . . . 13 ⊢ ;14 ∈ ℤ
44	11	nn0zi 12644	. . . . . . . . . . . . 13 ⊢ 3 ∈ ℤ
45		gcdcom 16605	. . . . . . . . . . . . 13 ⊢ ((;14 ∈ ℤ ∧ 3 ∈ ℤ) → (;14 gcd 3) = (3 gcd ;14))
46	43, 44, 45	mp2an 705	. . . . . . . . . . . 12 ⊢ (;14 gcd 3) = (3 gcd ;14)
47		3nn 12345	. . . . . . . . . . . . . 14 ⊢ 3 ∈ ℕ
48		4cn 12351	. . . . . . . . . . . . . . . 16 ⊢ 4 ∈ ℂ
49		3cn 12347	. . . . . . . . . . . . . . . 16 ⊢ 3 ∈ ℂ
50		4t3e12 12840	. . . . . . . . . . . . . . . 16 ⊢ (4 · 3) = ;12
51	48, 49, 50	mulcomli 11243	. . . . . . . . . . . . . . 15 ⊢ (3 · 4) = ;12
52		2p2e4 12400	. . . . . . . . . . . . . . 15 ⊢ (2 + 2) = 4
53	13, 10, 10, 51, 52	decaddi 12802	. . . . . . . . . . . . . 14 ⊢ ((3 · 4) + 2) = ;14
54		2lt3 12439	. . . . . . . . . . . . . 14 ⊢ 2 < 3
55	47, 17, 1, 53, 54	ndvdsi 16504	. . . . . . . . . . . . 13 ⊢ ¬ 3 ∥ ;14
56		3prm 16786	. . . . . . . . . . . . . 14 ⊢ 3 ∈ ℙ
57		coprm 16804	. . . . . . . . . . . . . 14 ⊢ ((3 ∈ ℙ ∧ ;14 ∈ ℤ) → (¬ 3 ∥ ;14 ↔ (3 gcd ;14) = 1))
58	56, 43, 57	mp2an 705	. . . . . . . . . . . . 13 ⊢ (¬ 3 ∥ ;14 ↔ (3 gcd ;14) = 1)
59	55, 58	mpbi 233	. . . . . . . . . . . 12 ⊢ (3 gcd ;14) = 1
60	46, 59	eqtri 2785	. . . . . . . . . . 11 ⊢ (;14 gcd 3) = 1
61		eqid 2762	. . . . . . . . . . . 12 ⊢ ;14 = ;14
62	11	dec0h 12764	. . . . . . . . . . . 12 ⊢ 3 = ;03
63		2t1e2 12428	. . . . . . . . . . . . . 14 ⊢ (2 · 1) = 2
64	63, 24	oveq12i 7428	. . . . . . . . . . . . 13 ⊢ ((2 · 1) + (0 + 1)) = (2 + 1)
65		2p1e3 12407	. . . . . . . . . . . . 13 ⊢ (2 + 1) = 3
66	64, 65	eqtri 2785	. . . . . . . . . . . 12 ⊢ ((2 · 1) + (0 + 1)) = 3
67		2t4e8 12435	. . . . . . . . . . . . . 14 ⊢ (2 · 4) = 8
68	67	oveq1i 7426	. . . . . . . . . . . . 13 ⊢ ((2 · 4) + 3) = (8 + 3)
69		8p3e11 12823	. . . . . . . . . . . . 13 ⊢ (8 + 3) = ;11
70	68, 69	eqtri 2785	. . . . . . . . . . . 12 ⊢ ((2 · 4) + 3) = ;11
71	13, 17, 3, 11, 61, 62, 10, 13, 13, 66, 70	decma2c 12795	. . . . . . . . . . 11 ⊢ ((2 · ;14) + 3) = ;31
72	10, 11, 42, 60, 71	gcdi 17167	. . . . . . . . . 10 ⊢ (;31 gcd ;14) = 1
73		eqid 2762	. . . . . . . . . . 11 ⊢ ;31 = ;31
74	49	mullidi 11239	. . . . . . . . . . . . 13 ⊢ (1 · 3) = 3
75		ax-1cn 11183	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℂ
76	75	addridi 11422	. . . . . . . . . . . . 13 ⊢ (1 + 0) = 1
77	74, 76	oveq12i 7428	. . . . . . . . . . . 12 ⊢ ((1 · 3) + (1 + 0)) = (3 + 1)
78		3p1e4 12410	. . . . . . . . . . . 12 ⊢ (3 + 1) = 4
79	77, 78	eqtri 2785	. . . . . . . . . . 11 ⊢ ((1 · 3) + (1 + 0)) = 4
80		1t1e1 12427	. . . . . . . . . . . . 13 ⊢ (1 · 1) = 1
81	80	oveq1i 7426	. . . . . . . . . . . 12 ⊢ ((1 · 1) + 4) = (1 + 4)
82		4p1e5 12411	. . . . . . . . . . . . 13 ⊢ (4 + 1) = 5
83	48, 75, 82	addcomli 11427	. . . . . . . . . . . 12 ⊢ (1 + 4) = 5
84	35	dec0h 12764	. . . . . . . . . . . 12 ⊢ 5 = ;05
85	81, 83, 84	3eqtri 2789	. . . . . . . . . . 11 ⊢ ((1 · 1) + 4) = ;05
86	11, 13, 13, 17, 73, 61, 13, 35, 3, 79, 85	decma2c 12795	. . . . . . . . . 10 ⊢ ((1 · ;31) + ;14) = ;45
87	13, 42, 41, 72, 86	gcdi 17167	. . . . . . . . 9 ⊢ (;45 gcd ;31) = 1
88		eqid 2762	. . . . . . . . . 10 ⊢ ;45 = ;45
89	67, 78	oveq12i 7428	. . . . . . . . . . 11 ⊢ ((2 · 4) + (3 + 1)) = (8 + 4)
90		8p4e12 12824	. . . . . . . . . . 11 ⊢ (8 + 4) = ;12
91	89, 90	eqtri 2785	. . . . . . . . . 10 ⊢ ((2 · 4) + (3 + 1)) = ;12
92		5cn 12354	. . . . . . . . . . . 12 ⊢ 5 ∈ ℂ
93		2cn 12341	. . . . . . . . . . . 12 ⊢ 2 ∈ ℂ
94		5t2e10 12842	. . . . . . . . . . . 12 ⊢ (5 · 2) = ;10
95	92, 93, 94	mulcomli 11243	. . . . . . . . . . 11 ⊢ (2 · 5) = ;10
96	13, 3, 24, 95	decsuc 12773	. . . . . . . . . 10 ⊢ ((2 · 5) + 1) = ;11
97	17, 35, 11, 13, 88, 73, 10, 13, 13, 91, 96	decma2c 12795	. . . . . . . . 9 ⊢ ((2 · ;45) + ;31) = ;;121
98	10, 41, 36, 87, 97	gcdi 17167	. . . . . . . 8 ⊢ (;;121 gcd ;45) = 1
99		eqid 2762	. . . . . . . . 9 ⊢ ;;121 = ;;121
100		eqid 2762	. . . . . . . . . 10 ⊢ ;12 = ;12
101	48	addridi 11422	. . . . . . . . . . 11 ⊢ (4 + 0) = 4
102	17	dec0h 12764	. . . . . . . . . . 11 ⊢ 4 = ;04
103	101, 102	eqtri 2785	. . . . . . . . . 10 ⊢ (4 + 0) = ;04
104		00id 11410	. . . . . . . . . . . 12 ⊢ (0 + 0) = 0
105	80, 104	oveq12i 7428	. . . . . . . . . . 11 ⊢ ((1 · 1) + (0 + 0)) = (1 + 0)
106	105, 76	eqtri 2785	. . . . . . . . . 10 ⊢ ((1 · 1) + (0 + 0)) = 1
107	93	mullidi 11239	. . . . . . . . . . . 12 ⊢ (1 · 2) = 2
108	107	oveq1i 7426	. . . . . . . . . . 11 ⊢ ((1 · 2) + 4) = (2 + 4)
109		4p2e6 12418	. . . . . . . . . . . 12 ⊢ (4 + 2) = 6
110	48, 93, 109	addcomli 11427	. . . . . . . . . . 11 ⊢ (2 + 4) = 6
111	28	dec0h 12764	. . . . . . . . . . 11 ⊢ 6 = ;06
112	108, 110, 111	3eqtri 2789	. . . . . . . . . 10 ⊢ ((1 · 2) + 4) = ;06
113	13, 10, 3, 17, 100, 103, 13, 28, 3, 106, 112	decma2c 12795	. . . . . . . . 9 ⊢ ((1 · ;12) + (4 + 0)) = ;16
114	80	oveq1i 7426	. . . . . . . . . 10 ⊢ ((1 · 1) + 5) = (1 + 5)
115		5p1e6 12412	. . . . . . . . . . 11 ⊢ (5 + 1) = 6
116	92, 75, 115	addcomli 11427	. . . . . . . . . 10 ⊢ (1 + 5) = 6
117	114, 116, 111	3eqtri 2789	. . . . . . . . 9 ⊢ ((1 · 1) + 5) = ;06
118	39, 13, 17, 35, 99, 88, 13, 28, 3, 113, 117	decma2c 12795	. . . . . . . 8 ⊢ ((1 · ;;121) + ;45) = ;;166
119	13, 36, 40, 98, 118	gcdi 17167	. . . . . . 7 ⊢ (;;166 gcd ;;121) = 1
120		eqid 2762	. . . . . . . 8 ⊢ ;;166 = ;;166
121		eqid 2762	. . . . . . . . 9 ⊢ ;16 = ;16
122	13, 10, 65, 100	decsuc 12773	. . . . . . . . 9 ⊢ (;12 + 1) = ;13
123		1p1e2 12389	. . . . . . . . . . 11 ⊢ (1 + 1) = 2
124	63, 123	oveq12i 7428	. . . . . . . . . 10 ⊢ ((2 · 1) + (1 + 1)) = (2 + 2)
125	124, 52	eqtri 2785	. . . . . . . . 9 ⊢ ((2 · 1) + (1 + 1)) = 4
126		6cn 12357	. . . . . . . . . . 11 ⊢ 6 ∈ ℂ
127		6t2e12 12846	. . . . . . . . . . 11 ⊢ (6 · 2) = ;12
128	126, 93, 127	mulcomli 11243	. . . . . . . . . 10 ⊢ (2 · 6) = ;12
129		3p2e5 12416	. . . . . . . . . . 11 ⊢ (3 + 2) = 5
130	49, 93, 129	addcomli 11427	. . . . . . . . . 10 ⊢ (2 + 3) = 5
131	13, 10, 11, 128, 130	decaddi 12802	. . . . . . . . 9 ⊢ ((2 · 6) + 3) = ;15
132	13, 28, 13, 11, 121, 122, 10, 35, 13, 125, 131	decma2c 12795	. . . . . . . 8 ⊢ ((2 · ;16) + (;12 + 1)) = ;45
133	13, 10, 65, 128	decsuc 12773	. . . . . . . 8 ⊢ ((2 · 6) + 1) = ;13
134	29, 28, 39, 13, 120, 99, 10, 11, 13, 132, 133	decma2c 12795	. . . . . . 7 ⊢ ((2 · ;;166) + ;;121) = ;;453
135	10, 40, 38, 119, 134	gcdi 17167	. . . . . 6 ⊢ (;;453 gcd ;;166) = 1
136		eqid 2762	. . . . . . 7 ⊢ ;;453 = ;;453
137	29	nn0cni 12541	. . . . . . . . 9 ⊢ ;16 ∈ ℂ
138	137	addridi 11422	. . . . . . . 8 ⊢ (;16 + 0) = ;16
139	48	mullidi 11239	. . . . . . . . . 10 ⊢ (1 · 4) = 4
140	139, 123	oveq12i 7428	. . . . . . . . 9 ⊢ ((1 · 4) + (1 + 1)) = (4 + 2)
141	140, 109	eqtri 2785	. . . . . . . 8 ⊢ ((1 · 4) + (1 + 1)) = 6
142	92	mullidi 11239	. . . . . . . . . 10 ⊢ (1 · 5) = 5
143	142	oveq1i 7426	. . . . . . . . 9 ⊢ ((1 · 5) + 6) = (5 + 6)
144		6p5e11 12815	. . . . . . . . . 10 ⊢ (6 + 5) = ;11
145	126, 92, 144	addcomli 11427	. . . . . . . . 9 ⊢ (5 + 6) = ;11
146	143, 145	eqtri 2785	. . . . . . . 8 ⊢ ((1 · 5) + 6) = ;11
147	17, 35, 13, 28, 88, 138, 13, 13, 13, 141, 146	decma2c 12795	. . . . . . 7 ⊢ ((1 · ;45) + (;16 + 0)) = ;61
148	74	oveq1i 7426	. . . . . . . 8 ⊢ ((1 · 3) + 6) = (3 + 6)
149		6p3e9 12425	. . . . . . . . 9 ⊢ (6 + 3) = 9
150	126, 49, 149	addcomli 11427	. . . . . . . 8 ⊢ (3 + 6) = 9
151	30	dec0h 12764	. . . . . . . 8 ⊢ 9 = ;09
152	148, 150, 151	3eqtri 2789	. . . . . . 7 ⊢ ((1 · 3) + 6) = ;09
153	36, 11, 29, 28, 136, 120, 13, 30, 3, 147, 152	decma2c 12795	. . . . . 6 ⊢ ((1 · ;;453) + ;;166) = ;;619
154	13, 38, 37, 135, 153	gcdi 17167	. . . . 5 ⊢ (;;619 gcd ;;453) = 1
155		eqid 2762	. . . . . 6 ⊢ ;;619 = ;;619
156		7nn0 12551	. . . . . . 7 ⊢ 7 ∈ ℕ₀
157		eqid 2762	. . . . . . 7 ⊢ ;61 = ;61
158		5p2e7 12421	. . . . . . . 8 ⊢ (5 + 2) = 7
159	17, 35, 10, 88, 158	decaddi 12802	. . . . . . 7 ⊢ (;45 + 2) = ;47
160	101	oveq2i 7427	. . . . . . . 8 ⊢ ((2 · 6) + (4 + 0)) = ((2 · 6) + 4)
161	13, 10, 17, 128, 110	decaddi 12802	. . . . . . . 8 ⊢ ((2 · 6) + 4) = ;16
162	160, 161	eqtri 2785	. . . . . . 7 ⊢ ((2 · 6) + (4 + 0)) = ;16
163	63	oveq1i 7426	. . . . . . . 8 ⊢ ((2 · 1) + 7) = (2 + 7)
164		7cn 12360	. . . . . . . . 9 ⊢ 7 ∈ ℂ
165		7p2e9 12426	. . . . . . . . 9 ⊢ (7 + 2) = 9
166	164, 93, 165	addcomli 11427	. . . . . . . 8 ⊢ (2 + 7) = 9
167	163, 166, 151	3eqtri 2789	. . . . . . 7 ⊢ ((2 · 1) + 7) = ;09
168	28, 13, 17, 156, 157, 159, 10, 30, 3, 162, 167	decma2c 12795	. . . . . 6 ⊢ ((2 · ;61) + (;45 + 2)) = ;;169
169		9cn 12366	. . . . . . . 8 ⊢ 9 ∈ ℂ
170		9t2e18 12864	. . . . . . . 8 ⊢ (9 · 2) = ;18
171	169, 93, 170	mulcomli 11243	. . . . . . 7 ⊢ (2 · 9) = ;18
172	13, 2, 11, 171, 123, 13, 69	decaddci 12803	. . . . . 6 ⊢ ((2 · 9) + 3) = ;21
173	33, 30, 36, 11, 155, 136, 10, 13, 10, 168, 172	decma2c 12795	. . . . 5 ⊢ ((2 · ;;619) + ;;453) = ;;;1691
174	10, 37, 34, 154, 173	gcdi 17167	. . . 4 ⊢ (;;;1691 gcd ;;619) = 1
175		eqid 2762	. . . . 5 ⊢ ;;;1691 = ;;;1691
176		eqid 2762	. . . . . 6 ⊢ ;;169 = ;;169
177	28, 13, 123, 157	decsuc 12773	. . . . . 6 ⊢ (;61 + 1) = ;62
178		6p1e7 12413	. . . . . . . 8 ⊢ (6 + 1) = 7
179	156	dec0h 12764	. . . . . . . 8 ⊢ 7 = ;07
180	178, 179	eqtri 2785	. . . . . . 7 ⊢ (6 + 1) = ;07
181	80, 24	oveq12i 7428	. . . . . . . 8 ⊢ ((1 · 1) + (0 + 1)) = (1 + 1)
182	181, 123	eqtri 2785	. . . . . . 7 ⊢ ((1 · 1) + (0 + 1)) = 2
183	126	mullidi 11239	. . . . . . . . 9 ⊢ (1 · 6) = 6
184	183	oveq1i 7426	. . . . . . . 8 ⊢ ((1 · 6) + 7) = (6 + 7)
185		7p6e13 12820	. . . . . . . . 9 ⊢ (7 + 6) = ;13
186	164, 126, 185	addcomli 11427	. . . . . . . 8 ⊢ (6 + 7) = ;13
187	184, 186	eqtri 2785	. . . . . . 7 ⊢ ((1 · 6) + 7) = ;13
188	13, 28, 3, 156, 121, 180, 13, 11, 13, 182, 187	decma2c 12795	. . . . . 6 ⊢ ((1 · ;16) + (6 + 1)) = ;23
189	169	mullidi 11239	. . . . . . . 8 ⊢ (1 · 9) = 9
190	189	oveq1i 7426	. . . . . . 7 ⊢ ((1 · 9) + 2) = (9 + 2)
191		9p2e11 12829	. . . . . . 7 ⊢ (9 + 2) = ;11
192	190, 191	eqtri 2785	. . . . . 6 ⊢ ((1 · 9) + 2) = ;11
193	29, 30, 28, 10, 176, 177, 13, 13, 13, 188, 192	decma2c 12795	. . . . 5 ⊢ ((1 · ;;169) + (;61 + 1)) = ;;231
194	80	oveq1i 7426	. . . . . 6 ⊢ ((1 · 1) + 9) = (1 + 9)
195		9p1e10 12739	. . . . . . 7 ⊢ (9 + 1) = ;10
196	169, 75, 195	addcomli 11427	. . . . . 6 ⊢ (1 + 9) = ;10
197	194, 196	eqtri 2785	. . . . 5 ⊢ ((1 · 1) + 9) = ;10
198	31, 13, 33, 30, 175, 155, 13, 3, 13, 193, 197	decma2c 12795	. . . 4 ⊢ ((1 · ;;;1691) + ;;619) = ;;;2310
199	13, 34, 32, 174, 198	gcdi 17167	. . 3 ⊢ (;;;2310 gcd ;;;1691) = 1
200		eqid 2762	. . . . . 6 ⊢ ;;231 = ;;231
201	31	nn0cni 12541	. . . . . . 7 ⊢ ;;169 ∈ ℂ
202	201	addridi 11422	. . . . . 6 ⊢ (;;169 + 0) = ;;169
203		eqid 2762	. . . . . . 7 ⊢ ;23 = ;23
204	13, 28, 178, 121	decsuc 12773	. . . . . . 7 ⊢ (;16 + 1) = ;17
205	107, 123	oveq12i 7428	. . . . . . . 8 ⊢ ((1 · 2) + (1 + 1)) = (2 + 2)
206	205, 52	eqtri 2785	. . . . . . 7 ⊢ ((1 · 2) + (1 + 1)) = 4
207	74	oveq1i 7426	. . . . . . . 8 ⊢ ((1 · 3) + 7) = (3 + 7)
208		7p3e10 12817	. . . . . . . . 9 ⊢ (7 + 3) = ;10
209	164, 49, 208	addcomli 11427	. . . . . . . 8 ⊢ (3 + 7) = ;10
210	207, 209	eqtri 2785	. . . . . . 7 ⊢ ((1 · 3) + 7) = ;10
211	10, 11, 13, 156, 203, 204, 13, 3, 13, 206, 210	decma2c 12795	. . . . . 6 ⊢ ((1 · ;23) + (;16 + 1)) = ;40
212	12, 13, 29, 30, 200, 202, 13, 3, 13, 211, 197	decma2c 12795	. . . . 5 ⊢ ((1 · ;;231) + (;;169 + 0)) = ;;400
213	75	mul01i 11425	. . . . . . 7 ⊢ (1 · 0) = 0
214	213	oveq1i 7426	. . . . . 6 ⊢ ((1 · 0) + 1) = (0 + 1)
215	13	dec0h 12764	. . . . . 6 ⊢ 1 = ;01
216	214, 24, 215	3eqtri 2789	. . . . 5 ⊢ ((1 · 0) + 1) = ;01
217	14, 3, 31, 13, 25, 175, 13, 13, 3, 212, 216	decma2c 12795	. . . 4 ⊢ ((1 · ;;;2310) + ;;;1691) = ;;;4001
218	217, 16	eqtr4i 2788	. . 3 ⊢ ((1 · ;;;2310) + ;;;1691) = 𝑁
219	13, 32, 15, 199, 218	gcdi 17167	. 2 ⊢ (𝑁 gcd ;;;2310) = 1
220	9, 15, 22, 27, 219	gcdmodi 17168	1 ⊢ (((2↑;;800) − 1) gcd 𝑁) = 1

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 ↔ wb 209 = wceq 1570 ∈ wcel 2145 class class class wbr 5107 (class class class)co 7416 0cc0 11125 1c1 11126 + caddc 11128 · cmul 11130 − cmin 11466 ℕcn 12258 2c2 12320 3c3 12321 4c4 12322 5c5 12323 6c6 12324 7c7 12325 8c8 12326 9c9 12327 ℕ₀cn0 12529 ℤcz 12616 cdc 12737 ↑cexp 14125 ∥ cdvds 16344 gcd cgcd 16586 ℙcprime 16763

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-cnex 11181 ax-resscn 11182 ax-1cn 11183 ax-icn 11184 ax-addcl 11185 ax-addrcl 11186 ax-mulcl 11187 ax-mulrcl 11188 ax-mulcom 11189 ax-addass 11190 ax-mulass 11191 ax-distr 11192 ax-i2m1 11193 ax-1ne0 11194 ax-1rid 11195 ax-rnegex 11196 ax-rrecex 11197 ax-cnre 11198 ax-pre-lttri 11199 ax-pre-lttrn 11200 ax-pre-ltadd 11201 ax-pre-mulgt0 11202 ax-pre-sup 11203

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 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 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7866 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8458 df-2o 8459 df-er 8699 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-sup 9415 df-inf 9416 df-pnf 11270 df-mnf 11271 df-xr 11272 df-ltxr 11273 df-le 11274 df-sub 11468 df-neg 11469 df-div 11897 df-nn 12259 df-2 12328 df-3 12329 df-4 12330 df-5 12331 df-6 12332 df-7 12333 df-8 12334 df-9 12335 df-n0 12530 df-z 12617 df-dec 12738 df-uz 12889 df-rp 13043 df-fz 13562 df-fl 13853 df-mod 13931 df-seq 14066 df-exp 14126 df-cj 15186 df-re 15187 df-im 15188 df-sqrt 15322 df-abs 15323 df-dvds 16345 df-gcd 16587 df-prm 16764

This theorem is used by: 4001prm 17239

W3C validator