4001lem4 - Metamath Proof Explorer

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Theorem 4001lem4 17077

Description: Lemma for 4001prm 17078. 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 12285	. . . 4 ⊢ 2 ∈ ℕ
2		8nn0 12495	. . . . . 6 ⊢ 8 ∈ ℕ₀
3		0nn0 12487	. . . . . 6 ⊢ 0 ∈ ℕ₀
4	2, 3	deccl 12692	. . . . 5 ⊢ ;80 ∈ ℕ₀
5	4, 3	deccl 12692	. . . 4 ⊢ ;;800 ∈ ℕ₀
6		nnexpcl 14040	. . . 4 ⊢ ((2 ∈ ℕ ∧ ;;800 ∈ ℕ₀) → (2↑;;800) ∈ ℕ)
7	1, 5, 6	mp2an 691	. . 3 ⊢ (2↑;;800) ∈ ℕ
8		nnm1nn0 12513	. . 3 ⊢ ((2↑;;800) ∈ ℕ → ((2↑;;800) − 1) ∈ ℕ₀)
9	7, 8	ax-mp 5	. 2 ⊢ ((2↑;;800) − 1) ∈ ℕ₀
10		2nn0 12489	. . . . 5 ⊢ 2 ∈ ℕ₀
11		3nn0 12490	. . . . 5 ⊢ 3 ∈ ℕ₀
12	10, 11	deccl 12692	. . . 4 ⊢ ;23 ∈ ℕ₀
13		1nn0 12488	. . . 4 ⊢ 1 ∈ ℕ₀
14	12, 13	deccl 12692	. . 3 ⊢ ;;231 ∈ ℕ₀
15	14, 3	deccl 12692	. 2 ⊢ ;;;2310 ∈ ℕ₀
16		4001prm.1	. . 3 ⊢ 𝑁 = ;;;4001
17		4nn0 12491	. . . . . 6 ⊢ 4 ∈ ℕ₀
18	17, 3	deccl 12692	. . . . 5 ⊢ ;40 ∈ ℕ₀
19	18, 3	deccl 12692	. . . 4 ⊢ ;;400 ∈ ℕ₀
20		1nn 12223	. . . 4 ⊢ 1 ∈ ℕ
21	19, 20	decnncl 12697	. . 3 ⊢ ;;;4001 ∈ ℕ
22	16, 21	eqeltri 2830	. 2 ⊢ 𝑁 ∈ ℕ
23	16	4001lem2 17075	. . 3 ⊢ ((2↑;;800) mod 𝑁) = (;;;2311 mod 𝑁)
24		0p1e1 12334	. . . 4 ⊢ (0 + 1) = 1
25		eqid 2733	. . . 4 ⊢ ;;;2310 = ;;;2310
26	14, 3, 24, 25	decsuc 12708	. . 3 ⊢ (;;;2310 + 1) = ;;;2311
27	22, 7, 13, 15, 23, 26	modsubi 17005	. 2 ⊢ (((2↑;;800) − 1) mod 𝑁) = (;;;2310 mod 𝑁)
28		6nn0 12493	. . . . . 6 ⊢ 6 ∈ ℕ₀
29	13, 28	deccl 12692	. . . . 5 ⊢ ;16 ∈ ℕ₀
30		9nn0 12496	. . . . 5 ⊢ 9 ∈ ℕ₀
31	29, 30	deccl 12692	. . . 4 ⊢ ;;169 ∈ ℕ₀
32	31, 13	deccl 12692	. . 3 ⊢ ;;;1691 ∈ ℕ₀
33	28, 13	deccl 12692	. . . . 5 ⊢ ;61 ∈ ℕ₀
34	33, 30	deccl 12692	. . . 4 ⊢ ;;619 ∈ ℕ₀
35		5nn0 12492	. . . . . . 7 ⊢ 5 ∈ ℕ₀
36	17, 35	deccl 12692	. . . . . 6 ⊢ ;45 ∈ ℕ₀
37	36, 11	deccl 12692	. . . . 5 ⊢ ;;453 ∈ ℕ₀
38	29, 28	deccl 12692	. . . . . 6 ⊢ ;;166 ∈ ℕ₀
39	13, 10	deccl 12692	. . . . . . . 8 ⊢ ;12 ∈ ℕ₀
40	39, 13	deccl 12692	. . . . . . 7 ⊢ ;;121 ∈ ℕ₀
41	11, 13	deccl 12692	. . . . . . . . 9 ⊢ ;31 ∈ ℕ₀
42	13, 17	deccl 12692	. . . . . . . . . 10 ⊢ ;14 ∈ ℕ₀
43	42	nn0zi 12587	. . . . . . . . . . . . 13 ⊢ ;14 ∈ ℤ
44	11	nn0zi 12587	. . . . . . . . . . . . 13 ⊢ 3 ∈ ℤ
45		gcdcom 16454	. . . . . . . . . . . . 13 ⊢ ((;14 ∈ ℤ ∧ 3 ∈ ℤ) → (;14 gcd 3) = (3 gcd ;14))
46	43, 44, 45	mp2an 691	. . . . . . . . . . . 12 ⊢ (;14 gcd 3) = (3 gcd ;14)
47		3nn 12291	. . . . . . . . . . . . . 14 ⊢ 3 ∈ ℕ
48		4cn 12297	. . . . . . . . . . . . . . . 16 ⊢ 4 ∈ ℂ
49		3cn 12293	. . . . . . . . . . . . . . . 16 ⊢ 3 ∈ ℂ
50		4t3e12 12775	. . . . . . . . . . . . . . . 16 ⊢ (4 · 3) = ;12
51	48, 49, 50	mulcomli 11223	. . . . . . . . . . . . . . 15 ⊢ (3 · 4) = ;12
52		2p2e4 12347	. . . . . . . . . . . . . . 15 ⊢ (2 + 2) = 4
53	13, 10, 10, 51, 52	decaddi 12737	. . . . . . . . . . . . . 14 ⊢ ((3 · 4) + 2) = ;14
54		2lt3 12384	. . . . . . . . . . . . . 14 ⊢ 2 < 3
55	47, 17, 1, 53, 54	ndvdsi 16355	. . . . . . . . . . . . 13 ⊢ ¬ 3 ∥ ;14
56		3prm 16631	. . . . . . . . . . . . . 14 ⊢ 3 ∈ ℙ
57		coprm 16648	. . . . . . . . . . . . . 14 ⊢ ((3 ∈ ℙ ∧ ;14 ∈ ℤ) → (¬ 3 ∥ ;14 ↔ (3 gcd ;14) = 1))
58	56, 43, 57	mp2an 691	. . . . . . . . . . . . 13 ⊢ (¬ 3 ∥ ;14 ↔ (3 gcd ;14) = 1)
59	55, 58	mpbi 229	. . . . . . . . . . . 12 ⊢ (3 gcd ;14) = 1
60	46, 59	eqtri 2761	. . . . . . . . . . 11 ⊢ (;14 gcd 3) = 1
61		eqid 2733	. . . . . . . . . . . 12 ⊢ ;14 = ;14
62	11	dec0h 12699	. . . . . . . . . . . 12 ⊢ 3 = ;03
63		2t1e2 12375	. . . . . . . . . . . . . 14 ⊢ (2 · 1) = 2
64	63, 24	oveq12i 7421	. . . . . . . . . . . . 13 ⊢ ((2 · 1) + (0 + 1)) = (2 + 1)
65		2p1e3 12354	. . . . . . . . . . . . 13 ⊢ (2 + 1) = 3
66	64, 65	eqtri 2761	. . . . . . . . . . . 12 ⊢ ((2 · 1) + (0 + 1)) = 3
67		2cn 12287	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℂ
68		4t2e8 12380	. . . . . . . . . . . . . . 15 ⊢ (4 · 2) = 8
69	48, 67, 68	mulcomli 11223	. . . . . . . . . . . . . 14 ⊢ (2 · 4) = 8
70	69	oveq1i 7419	. . . . . . . . . . . . 13 ⊢ ((2 · 4) + 3) = (8 + 3)
71		8p3e11 12758	. . . . . . . . . . . . 13 ⊢ (8 + 3) = ;11
72	70, 71	eqtri 2761	. . . . . . . . . . . 12 ⊢ ((2 · 4) + 3) = ;11
73	13, 17, 3, 11, 61, 62, 10, 13, 13, 66, 72	decma2c 12730	. . . . . . . . . . 11 ⊢ ((2 · ;14) + 3) = ;31
74	10, 11, 42, 60, 73	gcdi 17006	. . . . . . . . . 10 ⊢ (;31 gcd ;14) = 1
75		eqid 2733	. . . . . . . . . . 11 ⊢ ;31 = ;31
76	49	mullidi 11219	. . . . . . . . . . . . 13 ⊢ (1 · 3) = 3
77		ax-1cn 11168	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℂ
78	77	addridi 11401	. . . . . . . . . . . . 13 ⊢ (1 + 0) = 1
79	76, 78	oveq12i 7421	. . . . . . . . . . . 12 ⊢ ((1 · 3) + (1 + 0)) = (3 + 1)
80		3p1e4 12357	. . . . . . . . . . . 12 ⊢ (3 + 1) = 4
81	79, 80	eqtri 2761	. . . . . . . . . . 11 ⊢ ((1 · 3) + (1 + 0)) = 4
82		1t1e1 12374	. . . . . . . . . . . . 13 ⊢ (1 · 1) = 1
83	82	oveq1i 7419	. . . . . . . . . . . 12 ⊢ ((1 · 1) + 4) = (1 + 4)
84		4p1e5 12358	. . . . . . . . . . . . 13 ⊢ (4 + 1) = 5
85	48, 77, 84	addcomli 11406	. . . . . . . . . . . 12 ⊢ (1 + 4) = 5
86	35	dec0h 12699	. . . . . . . . . . . 12 ⊢ 5 = ;05
87	83, 85, 86	3eqtri 2765	. . . . . . . . . . 11 ⊢ ((1 · 1) + 4) = ;05
88	11, 13, 13, 17, 75, 61, 13, 35, 3, 81, 87	decma2c 12730	. . . . . . . . . 10 ⊢ ((1 · ;31) + ;14) = ;45
89	13, 42, 41, 74, 88	gcdi 17006	. . . . . . . . 9 ⊢ (;45 gcd ;31) = 1
90		eqid 2733	. . . . . . . . . 10 ⊢ ;45 = ;45
91	69, 80	oveq12i 7421	. . . . . . . . . . 11 ⊢ ((2 · 4) + (3 + 1)) = (8 + 4)
92		8p4e12 12759	. . . . . . . . . . 11 ⊢ (8 + 4) = ;12
93	91, 92	eqtri 2761	. . . . . . . . . 10 ⊢ ((2 · 4) + (3 + 1)) = ;12
94		5cn 12300	. . . . . . . . . . . 12 ⊢ 5 ∈ ℂ
95		5t2e10 12777	. . . . . . . . . . . 12 ⊢ (5 · 2) = ;10
96	94, 67, 95	mulcomli 11223	. . . . . . . . . . 11 ⊢ (2 · 5) = ;10
97	13, 3, 24, 96	decsuc 12708	. . . . . . . . . 10 ⊢ ((2 · 5) + 1) = ;11
98	17, 35, 11, 13, 90, 75, 10, 13, 13, 93, 97	decma2c 12730	. . . . . . . . 9 ⊢ ((2 · ;45) + ;31) = ;;121
99	10, 41, 36, 89, 98	gcdi 17006	. . . . . . . 8 ⊢ (;;121 gcd ;45) = 1
100		eqid 2733	. . . . . . . . 9 ⊢ ;;121 = ;;121
101		eqid 2733	. . . . . . . . . 10 ⊢ ;12 = ;12
102	48	addridi 11401	. . . . . . . . . . 11 ⊢ (4 + 0) = 4
103	17	dec0h 12699	. . . . . . . . . . 11 ⊢ 4 = ;04
104	102, 103	eqtri 2761	. . . . . . . . . 10 ⊢ (4 + 0) = ;04
105		00id 11389	. . . . . . . . . . . 12 ⊢ (0 + 0) = 0
106	82, 105	oveq12i 7421	. . . . . . . . . . 11 ⊢ ((1 · 1) + (0 + 0)) = (1 + 0)
107	106, 78	eqtri 2761	. . . . . . . . . 10 ⊢ ((1 · 1) + (0 + 0)) = 1
108	67	mullidi 11219	. . . . . . . . . . . 12 ⊢ (1 · 2) = 2
109	108	oveq1i 7419	. . . . . . . . . . 11 ⊢ ((1 · 2) + 4) = (2 + 4)
110		4p2e6 12365	. . . . . . . . . . . 12 ⊢ (4 + 2) = 6
111	48, 67, 110	addcomli 11406	. . . . . . . . . . 11 ⊢ (2 + 4) = 6
112	28	dec0h 12699	. . . . . . . . . . 11 ⊢ 6 = ;06
113	109, 111, 112	3eqtri 2765	. . . . . . . . . 10 ⊢ ((1 · 2) + 4) = ;06
114	13, 10, 3, 17, 101, 104, 13, 28, 3, 107, 113	decma2c 12730	. . . . . . . . 9 ⊢ ((1 · ;12) + (4 + 0)) = ;16
115	82	oveq1i 7419	. . . . . . . . . 10 ⊢ ((1 · 1) + 5) = (1 + 5)
116		5p1e6 12359	. . . . . . . . . . 11 ⊢ (5 + 1) = 6
117	94, 77, 116	addcomli 11406	. . . . . . . . . 10 ⊢ (1 + 5) = 6
118	115, 117, 112	3eqtri 2765	. . . . . . . . 9 ⊢ ((1 · 1) + 5) = ;06
119	39, 13, 17, 35, 100, 90, 13, 28, 3, 114, 118	decma2c 12730	. . . . . . . 8 ⊢ ((1 · ;;121) + ;45) = ;;166
120	13, 36, 40, 99, 119	gcdi 17006	. . . . . . 7 ⊢ (;;166 gcd ;;121) = 1
121		eqid 2733	. . . . . . . 8 ⊢ ;;166 = ;;166
122		eqid 2733	. . . . . . . . 9 ⊢ ;16 = ;16
123	13, 10, 65, 101	decsuc 12708	. . . . . . . . 9 ⊢ (;12 + 1) = ;13
124		1p1e2 12337	. . . . . . . . . . 11 ⊢ (1 + 1) = 2
125	63, 124	oveq12i 7421	. . . . . . . . . 10 ⊢ ((2 · 1) + (1 + 1)) = (2 + 2)
126	125, 52	eqtri 2761	. . . . . . . . 9 ⊢ ((2 · 1) + (1 + 1)) = 4
127		6cn 12303	. . . . . . . . . . 11 ⊢ 6 ∈ ℂ
128		6t2e12 12781	. . . . . . . . . . 11 ⊢ (6 · 2) = ;12
129	127, 67, 128	mulcomli 11223	. . . . . . . . . 10 ⊢ (2 · 6) = ;12
130		3p2e5 12363	. . . . . . . . . . 11 ⊢ (3 + 2) = 5
131	49, 67, 130	addcomli 11406	. . . . . . . . . 10 ⊢ (2 + 3) = 5
132	13, 10, 11, 129, 131	decaddi 12737	. . . . . . . . 9 ⊢ ((2 · 6) + 3) = ;15
133	13, 28, 13, 11, 122, 123, 10, 35, 13, 126, 132	decma2c 12730	. . . . . . . 8 ⊢ ((2 · ;16) + (;12 + 1)) = ;45
134	13, 10, 65, 129	decsuc 12708	. . . . . . . 8 ⊢ ((2 · 6) + 1) = ;13
135	29, 28, 39, 13, 121, 100, 10, 11, 13, 133, 134	decma2c 12730	. . . . . . 7 ⊢ ((2 · ;;166) + ;;121) = ;;453
136	10, 40, 38, 120, 135	gcdi 17006	. . . . . 6 ⊢ (;;453 gcd ;;166) = 1
137		eqid 2733	. . . . . . 7 ⊢ ;;453 = ;;453
138	29	nn0cni 12484	. . . . . . . . 9 ⊢ ;16 ∈ ℂ
139	138	addridi 11401	. . . . . . . 8 ⊢ (;16 + 0) = ;16
140	48	mullidi 11219	. . . . . . . . . 10 ⊢ (1 · 4) = 4
141	140, 124	oveq12i 7421	. . . . . . . . 9 ⊢ ((1 · 4) + (1 + 1)) = (4 + 2)
142	141, 110	eqtri 2761	. . . . . . . 8 ⊢ ((1 · 4) + (1 + 1)) = 6
143	94	mullidi 11219	. . . . . . . . . 10 ⊢ (1 · 5) = 5
144	143	oveq1i 7419	. . . . . . . . 9 ⊢ ((1 · 5) + 6) = (5 + 6)
145		6p5e11 12750	. . . . . . . . . 10 ⊢ (6 + 5) = ;11
146	127, 94, 145	addcomli 11406	. . . . . . . . 9 ⊢ (5 + 6) = ;11
147	144, 146	eqtri 2761	. . . . . . . 8 ⊢ ((1 · 5) + 6) = ;11
148	17, 35, 13, 28, 90, 139, 13, 13, 13, 142, 147	decma2c 12730	. . . . . . 7 ⊢ ((1 · ;45) + (;16 + 0)) = ;61
149	76	oveq1i 7419	. . . . . . . 8 ⊢ ((1 · 3) + 6) = (3 + 6)
150		6p3e9 12372	. . . . . . . . 9 ⊢ (6 + 3) = 9
151	127, 49, 150	addcomli 11406	. . . . . . . 8 ⊢ (3 + 6) = 9
152	30	dec0h 12699	. . . . . . . 8 ⊢ 9 = ;09
153	149, 151, 152	3eqtri 2765	. . . . . . 7 ⊢ ((1 · 3) + 6) = ;09
154	36, 11, 29, 28, 137, 121, 13, 30, 3, 148, 153	decma2c 12730	. . . . . 6 ⊢ ((1 · ;;453) + ;;166) = ;;619
155	13, 38, 37, 136, 154	gcdi 17006	. . . . 5 ⊢ (;;619 gcd ;;453) = 1
156		eqid 2733	. . . . . 6 ⊢ ;;619 = ;;619
157		7nn0 12494	. . . . . . 7 ⊢ 7 ∈ ℕ₀
158		eqid 2733	. . . . . . 7 ⊢ ;61 = ;61
159		5p2e7 12368	. . . . . . . 8 ⊢ (5 + 2) = 7
160	17, 35, 10, 90, 159	decaddi 12737	. . . . . . 7 ⊢ (;45 + 2) = ;47
161	102	oveq2i 7420	. . . . . . . 8 ⊢ ((2 · 6) + (4 + 0)) = ((2 · 6) + 4)
162	13, 10, 17, 129, 111	decaddi 12737	. . . . . . . 8 ⊢ ((2 · 6) + 4) = ;16
163	161, 162	eqtri 2761	. . . . . . 7 ⊢ ((2 · 6) + (4 + 0)) = ;16
164	63	oveq1i 7419	. . . . . . . 8 ⊢ ((2 · 1) + 7) = (2 + 7)
165		7cn 12306	. . . . . . . . 9 ⊢ 7 ∈ ℂ
166		7p2e9 12373	. . . . . . . . 9 ⊢ (7 + 2) = 9
167	165, 67, 166	addcomli 11406	. . . . . . . 8 ⊢ (2 + 7) = 9
168	164, 167, 152	3eqtri 2765	. . . . . . 7 ⊢ ((2 · 1) + 7) = ;09
169	28, 13, 17, 157, 158, 160, 10, 30, 3, 163, 168	decma2c 12730	. . . . . 6 ⊢ ((2 · ;61) + (;45 + 2)) = ;;169
170		9cn 12312	. . . . . . . 8 ⊢ 9 ∈ ℂ
171		9t2e18 12799	. . . . . . . 8 ⊢ (9 · 2) = ;18
172	170, 67, 171	mulcomli 11223	. . . . . . 7 ⊢ (2 · 9) = ;18
173	13, 2, 11, 172, 124, 13, 71	decaddci 12738	. . . . . 6 ⊢ ((2 · 9) + 3) = ;21
174	33, 30, 36, 11, 156, 137, 10, 13, 10, 169, 173	decma2c 12730	. . . . 5 ⊢ ((2 · ;;619) + ;;453) = ;;;1691
175	10, 37, 34, 155, 174	gcdi 17006	. . . 4 ⊢ (;;;1691 gcd ;;619) = 1
176		eqid 2733	. . . . 5 ⊢ ;;;1691 = ;;;1691
177		eqid 2733	. . . . . 6 ⊢ ;;169 = ;;169
178	28, 13, 124, 158	decsuc 12708	. . . . . 6 ⊢ (;61 + 1) = ;62
179		6p1e7 12360	. . . . . . . 8 ⊢ (6 + 1) = 7
180	157	dec0h 12699	. . . . . . . 8 ⊢ 7 = ;07
181	179, 180	eqtri 2761	. . . . . . 7 ⊢ (6 + 1) = ;07
182	82, 24	oveq12i 7421	. . . . . . . 8 ⊢ ((1 · 1) + (0 + 1)) = (1 + 1)
183	182, 124	eqtri 2761	. . . . . . 7 ⊢ ((1 · 1) + (0 + 1)) = 2
184	127	mullidi 11219	. . . . . . . . 9 ⊢ (1 · 6) = 6
185	184	oveq1i 7419	. . . . . . . 8 ⊢ ((1 · 6) + 7) = (6 + 7)
186		7p6e13 12755	. . . . . . . . 9 ⊢ (7 + 6) = ;13
187	165, 127, 186	addcomli 11406	. . . . . . . 8 ⊢ (6 + 7) = ;13
188	185, 187	eqtri 2761	. . . . . . 7 ⊢ ((1 · 6) + 7) = ;13
189	13, 28, 3, 157, 122, 181, 13, 11, 13, 183, 188	decma2c 12730	. . . . . 6 ⊢ ((1 · ;16) + (6 + 1)) = ;23
190	170	mullidi 11219	. . . . . . . 8 ⊢ (1 · 9) = 9
191	190	oveq1i 7419	. . . . . . 7 ⊢ ((1 · 9) + 2) = (9 + 2)
192		9p2e11 12764	. . . . . . 7 ⊢ (9 + 2) = ;11
193	191, 192	eqtri 2761	. . . . . 6 ⊢ ((1 · 9) + 2) = ;11
194	29, 30, 28, 10, 177, 178, 13, 13, 13, 189, 193	decma2c 12730	. . . . 5 ⊢ ((1 · ;;169) + (;61 + 1)) = ;;231
195	82	oveq1i 7419	. . . . . 6 ⊢ ((1 · 1) + 9) = (1 + 9)
196		9p1e10 12679	. . . . . . 7 ⊢ (9 + 1) = ;10
197	170, 77, 196	addcomli 11406	. . . . . 6 ⊢ (1 + 9) = ;10
198	195, 197	eqtri 2761	. . . . 5 ⊢ ((1 · 1) + 9) = ;10
199	31, 13, 33, 30, 176, 156, 13, 3, 13, 194, 198	decma2c 12730	. . . 4 ⊢ ((1 · ;;;1691) + ;;619) = ;;;2310
200	13, 34, 32, 175, 199	gcdi 17006	. . 3 ⊢ (;;;2310 gcd ;;;1691) = 1
201		eqid 2733	. . . . . 6 ⊢ ;;231 = ;;231
202	31	nn0cni 12484	. . . . . . 7 ⊢ ;;169 ∈ ℂ
203	202	addridi 11401	. . . . . 6 ⊢ (;;169 + 0) = ;;169
204		eqid 2733	. . . . . . 7 ⊢ ;23 = ;23
205	13, 28, 179, 122	decsuc 12708	. . . . . . 7 ⊢ (;16 + 1) = ;17
206	108, 124	oveq12i 7421	. . . . . . . 8 ⊢ ((1 · 2) + (1 + 1)) = (2 + 2)
207	206, 52	eqtri 2761	. . . . . . 7 ⊢ ((1 · 2) + (1 + 1)) = 4
208	76	oveq1i 7419	. . . . . . . 8 ⊢ ((1 · 3) + 7) = (3 + 7)
209		7p3e10 12752	. . . . . . . . 9 ⊢ (7 + 3) = ;10
210	165, 49, 209	addcomli 11406	. . . . . . . 8 ⊢ (3 + 7) = ;10
211	208, 210	eqtri 2761	. . . . . . 7 ⊢ ((1 · 3) + 7) = ;10
212	10, 11, 13, 157, 204, 205, 13, 3, 13, 207, 211	decma2c 12730	. . . . . 6 ⊢ ((1 · ;23) + (;16 + 1)) = ;40
213	12, 13, 29, 30, 201, 203, 13, 3, 13, 212, 198	decma2c 12730	. . . . 5 ⊢ ((1 · ;;231) + (;;169 + 0)) = ;;400
214	77	mul01i 11404	. . . . . . 7 ⊢ (1 · 0) = 0
215	214	oveq1i 7419	. . . . . 6 ⊢ ((1 · 0) + 1) = (0 + 1)
216	13	dec0h 12699	. . . . . 6 ⊢ 1 = ;01
217	215, 24, 216	3eqtri 2765	. . . . 5 ⊢ ((1 · 0) + 1) = ;01
218	14, 3, 31, 13, 25, 176, 13, 13, 3, 213, 217	decma2c 12730	. . . 4 ⊢ ((1 · ;;;2310) + ;;;1691) = ;;;4001
219	218, 16	eqtr4i 2764	. . 3 ⊢ ((1 · ;;;2310) + ;;;1691) = 𝑁
220	13, 32, 15, 200, 219	gcdi 17006	. 2 ⊢ (𝑁 gcd ;;;2310) = 1
221	9, 15, 22, 27, 220	gcdmodi 17007	1 ⊢ (((2↑;;800) − 1) gcd 𝑁) = 1

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 205 = wceq 1542 ∈ wcel 2107 class class class wbr 5149 (class class class)co 7409 0cc0 11110 1c1 11111 + caddc 11113 · cmul 11115 − cmin 11444 ℕcn 12212 2c2 12267 3c3 12268 4c4 12269 5c5 12270 6c6 12271 7c7 12272 8c8 12273 9c9 12274 ℕ₀cn0 12472 ℤcz 12558 cdc 12677 ↑cexp 14027 ∥ cdvds 16197 gcd cgcd 16435 ℙcprime 16608

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9437 df-inf 9438 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-rp 12975 df-fz 13485 df-fl 13757 df-mod 13835 df-seq 13967 df-exp 14028 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-dvds 16198 df-gcd 16436 df-prm 16609

This theorem is referenced by: 4001prm 17078

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