4001lem4 - Metamath Proof Explorer

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Theorem 4001lem4 17055

Description: Lemma for 4001prm 17056. 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 12201	. . . 4 ⊢ 2 ∈ ℕ
2		8nn0 12407	. . . . . 6 ⊢ 8 ∈ ℕ₀
3		0nn0 12399	. . . . . 6 ⊢ 0 ∈ ℕ₀
4	2, 3	deccl 12606	. . . . 5 ⊢ ;80 ∈ ℕ₀
5	4, 3	deccl 12606	. . . 4 ⊢ ;;800 ∈ ℕ₀
6		nnexpcl 13981	. . . 4 ⊢ ((2 ∈ ℕ ∧ ;;800 ∈ ℕ₀) → (2↑;;800) ∈ ℕ)
7	1, 5, 6	mp2an 692	. . 3 ⊢ (2↑;;800) ∈ ℕ
8		nnm1nn0 12425	. . 3 ⊢ ((2↑;;800) ∈ ℕ → ((2↑;;800) − 1) ∈ ℕ₀)
9	7, 8	ax-mp 5	. 2 ⊢ ((2↑;;800) − 1) ∈ ℕ₀
10		2nn0 12401	. . . . 5 ⊢ 2 ∈ ℕ₀
11		3nn0 12402	. . . . 5 ⊢ 3 ∈ ℕ₀
12	10, 11	deccl 12606	. . . 4 ⊢ ;23 ∈ ℕ₀
13		1nn0 12400	. . . 4 ⊢ 1 ∈ ℕ₀
14	12, 13	deccl 12606	. . 3 ⊢ ;;231 ∈ ℕ₀
15	14, 3	deccl 12606	. 2 ⊢ ;;;2310 ∈ ℕ₀
16		4001prm.1	. . 3 ⊢ 𝑁 = ;;;4001
17		4nn0 12403	. . . . . 6 ⊢ 4 ∈ ℕ₀
18	17, 3	deccl 12606	. . . . 5 ⊢ ;40 ∈ ℕ₀
19	18, 3	deccl 12606	. . . 4 ⊢ ;;400 ∈ ℕ₀
20		1nn 12139	. . . 4 ⊢ 1 ∈ ℕ
21	19, 20	decnncl 12611	. . 3 ⊢ ;;;4001 ∈ ℕ
22	16, 21	eqeltri 2824	. 2 ⊢ 𝑁 ∈ ℕ
23	16	4001lem2 17053	. . 3 ⊢ ((2↑;;800) mod 𝑁) = (;;;2311 mod 𝑁)
24		0p1e1 12245	. . . 4 ⊢ (0 + 1) = 1
25		eqid 2729	. . . 4 ⊢ ;;;2310 = ;;;2310
26	14, 3, 24, 25	decsuc 12622	. . 3 ⊢ (;;;2310 + 1) = ;;;2311
27	22, 7, 13, 15, 23, 26	modsubi 16984	. 2 ⊢ (((2↑;;800) − 1) mod 𝑁) = (;;;2310 mod 𝑁)
28		6nn0 12405	. . . . . 6 ⊢ 6 ∈ ℕ₀
29	13, 28	deccl 12606	. . . . 5 ⊢ ;16 ∈ ℕ₀
30		9nn0 12408	. . . . 5 ⊢ 9 ∈ ℕ₀
31	29, 30	deccl 12606	. . . 4 ⊢ ;;169 ∈ ℕ₀
32	31, 13	deccl 12606	. . 3 ⊢ ;;;1691 ∈ ℕ₀
33	28, 13	deccl 12606	. . . . 5 ⊢ ;61 ∈ ℕ₀
34	33, 30	deccl 12606	. . . 4 ⊢ ;;619 ∈ ℕ₀
35		5nn0 12404	. . . . . . 7 ⊢ 5 ∈ ℕ₀
36	17, 35	deccl 12606	. . . . . 6 ⊢ ;45 ∈ ℕ₀
37	36, 11	deccl 12606	. . . . 5 ⊢ ;;453 ∈ ℕ₀
38	29, 28	deccl 12606	. . . . . 6 ⊢ ;;166 ∈ ℕ₀
39	13, 10	deccl 12606	. . . . . . . 8 ⊢ ;12 ∈ ℕ₀
40	39, 13	deccl 12606	. . . . . . 7 ⊢ ;;121 ∈ ℕ₀
41	11, 13	deccl 12606	. . . . . . . . 9 ⊢ ;31 ∈ ℕ₀
42	13, 17	deccl 12606	. . . . . . . . . 10 ⊢ ;14 ∈ ℕ₀
43	42	nn0zi 12500	. . . . . . . . . . . . 13 ⊢ ;14 ∈ ℤ
44	11	nn0zi 12500	. . . . . . . . . . . . 13 ⊢ 3 ∈ ℤ
45		gcdcom 16424	. . . . . . . . . . . . 13 ⊢ ((;14 ∈ ℤ ∧ 3 ∈ ℤ) → (;14 gcd 3) = (3 gcd ;14))
46	43, 44, 45	mp2an 692	. . . . . . . . . . . 12 ⊢ (;14 gcd 3) = (3 gcd ;14)
47		3nn 12207	. . . . . . . . . . . . . 14 ⊢ 3 ∈ ℕ
48		4cn 12213	. . . . . . . . . . . . . . . 16 ⊢ 4 ∈ ℂ
49		3cn 12209	. . . . . . . . . . . . . . . 16 ⊢ 3 ∈ ℂ
50		4t3e12 12689	. . . . . . . . . . . . . . . 16 ⊢ (4 · 3) = ;12
51	48, 49, 50	mulcomli 11124	. . . . . . . . . . . . . . 15 ⊢ (3 · 4) = ;12
52		2p2e4 12258	. . . . . . . . . . . . . . 15 ⊢ (2 + 2) = 4
53	13, 10, 10, 51, 52	decaddi 12651	. . . . . . . . . . . . . 14 ⊢ ((3 · 4) + 2) = ;14
54		2lt3 12295	. . . . . . . . . . . . . 14 ⊢ 2 < 3
55	47, 17, 1, 53, 54	ndvdsi 16323	. . . . . . . . . . . . 13 ⊢ ¬ 3 ∥ ;14
56		3prm 16605	. . . . . . . . . . . . . 14 ⊢ 3 ∈ ℙ
57		coprm 16622	. . . . . . . . . . . . . 14 ⊢ ((3 ∈ ℙ ∧ ;14 ∈ ℤ) → (¬ 3 ∥ ;14 ↔ (3 gcd ;14) = 1))
58	56, 43, 57	mp2an 692	. . . . . . . . . . . . 13 ⊢ (¬ 3 ∥ ;14 ↔ (3 gcd ;14) = 1)
59	55, 58	mpbi 230	. . . . . . . . . . . 12 ⊢ (3 gcd ;14) = 1
60	46, 59	eqtri 2752	. . . . . . . . . . 11 ⊢ (;14 gcd 3) = 1
61		eqid 2729	. . . . . . . . . . . 12 ⊢ ;14 = ;14
62	11	dec0h 12613	. . . . . . . . . . . 12 ⊢ 3 = ;03
63		2t1e2 12286	. . . . . . . . . . . . . 14 ⊢ (2 · 1) = 2
64	63, 24	oveq12i 7361	. . . . . . . . . . . . 13 ⊢ ((2 · 1) + (0 + 1)) = (2 + 1)
65		2p1e3 12265	. . . . . . . . . . . . 13 ⊢ (2 + 1) = 3
66	64, 65	eqtri 2752	. . . . . . . . . . . 12 ⊢ ((2 · 1) + (0 + 1)) = 3
67		2cn 12203	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℂ
68		4t2e8 12291	. . . . . . . . . . . . . . 15 ⊢ (4 · 2) = 8
69	48, 67, 68	mulcomli 11124	. . . . . . . . . . . . . 14 ⊢ (2 · 4) = 8
70	69	oveq1i 7359	. . . . . . . . . . . . 13 ⊢ ((2 · 4) + 3) = (8 + 3)
71		8p3e11 12672	. . . . . . . . . . . . 13 ⊢ (8 + 3) = ;11
72	70, 71	eqtri 2752	. . . . . . . . . . . 12 ⊢ ((2 · 4) + 3) = ;11
73	13, 17, 3, 11, 61, 62, 10, 13, 13, 66, 72	decma2c 12644	. . . . . . . . . . 11 ⊢ ((2 · ;14) + 3) = ;31
74	10, 11, 42, 60, 73	gcdi 16985	. . . . . . . . . 10 ⊢ (;31 gcd ;14) = 1
75		eqid 2729	. . . . . . . . . . 11 ⊢ ;31 = ;31
76	49	mullidi 11120	. . . . . . . . . . . . 13 ⊢ (1 · 3) = 3
77		ax-1cn 11067	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℂ
78	77	addridi 11303	. . . . . . . . . . . . 13 ⊢ (1 + 0) = 1
79	76, 78	oveq12i 7361	. . . . . . . . . . . 12 ⊢ ((1 · 3) + (1 + 0)) = (3 + 1)
80		3p1e4 12268	. . . . . . . . . . . 12 ⊢ (3 + 1) = 4
81	79, 80	eqtri 2752	. . . . . . . . . . 11 ⊢ ((1 · 3) + (1 + 0)) = 4
82		1t1e1 12285	. . . . . . . . . . . . 13 ⊢ (1 · 1) = 1
83	82	oveq1i 7359	. . . . . . . . . . . 12 ⊢ ((1 · 1) + 4) = (1 + 4)
84		4p1e5 12269	. . . . . . . . . . . . 13 ⊢ (4 + 1) = 5
85	48, 77, 84	addcomli 11308	. . . . . . . . . . . 12 ⊢ (1 + 4) = 5
86	35	dec0h 12613	. . . . . . . . . . . 12 ⊢ 5 = ;05
87	83, 85, 86	3eqtri 2756	. . . . . . . . . . 11 ⊢ ((1 · 1) + 4) = ;05
88	11, 13, 13, 17, 75, 61, 13, 35, 3, 81, 87	decma2c 12644	. . . . . . . . . 10 ⊢ ((1 · ;31) + ;14) = ;45
89	13, 42, 41, 74, 88	gcdi 16985	. . . . . . . . 9 ⊢ (;45 gcd ;31) = 1
90		eqid 2729	. . . . . . . . . 10 ⊢ ;45 = ;45
91	69, 80	oveq12i 7361	. . . . . . . . . . 11 ⊢ ((2 · 4) + (3 + 1)) = (8 + 4)
92		8p4e12 12673	. . . . . . . . . . 11 ⊢ (8 + 4) = ;12
93	91, 92	eqtri 2752	. . . . . . . . . 10 ⊢ ((2 · 4) + (3 + 1)) = ;12
94		5cn 12216	. . . . . . . . . . . 12 ⊢ 5 ∈ ℂ
95		5t2e10 12691	. . . . . . . . . . . 12 ⊢ (5 · 2) = ;10
96	94, 67, 95	mulcomli 11124	. . . . . . . . . . 11 ⊢ (2 · 5) = ;10
97	13, 3, 24, 96	decsuc 12622	. . . . . . . . . 10 ⊢ ((2 · 5) + 1) = ;11
98	17, 35, 11, 13, 90, 75, 10, 13, 13, 93, 97	decma2c 12644	. . . . . . . . 9 ⊢ ((2 · ;45) + ;31) = ;;121
99	10, 41, 36, 89, 98	gcdi 16985	. . . . . . . 8 ⊢ (;;121 gcd ;45) = 1
100		eqid 2729	. . . . . . . . 9 ⊢ ;;121 = ;;121
101		eqid 2729	. . . . . . . . . 10 ⊢ ;12 = ;12
102	48	addridi 11303	. . . . . . . . . . 11 ⊢ (4 + 0) = 4
103	17	dec0h 12613	. . . . . . . . . . 11 ⊢ 4 = ;04
104	102, 103	eqtri 2752	. . . . . . . . . 10 ⊢ (4 + 0) = ;04
105		00id 11291	. . . . . . . . . . . 12 ⊢ (0 + 0) = 0
106	82, 105	oveq12i 7361	. . . . . . . . . . 11 ⊢ ((1 · 1) + (0 + 0)) = (1 + 0)
107	106, 78	eqtri 2752	. . . . . . . . . 10 ⊢ ((1 · 1) + (0 + 0)) = 1
108	67	mullidi 11120	. . . . . . . . . . . 12 ⊢ (1 · 2) = 2
109	108	oveq1i 7359	. . . . . . . . . . 11 ⊢ ((1 · 2) + 4) = (2 + 4)
110		4p2e6 12276	. . . . . . . . . . . 12 ⊢ (4 + 2) = 6
111	48, 67, 110	addcomli 11308	. . . . . . . . . . 11 ⊢ (2 + 4) = 6
112	28	dec0h 12613	. . . . . . . . . . 11 ⊢ 6 = ;06
113	109, 111, 112	3eqtri 2756	. . . . . . . . . 10 ⊢ ((1 · 2) + 4) = ;06
114	13, 10, 3, 17, 101, 104, 13, 28, 3, 107, 113	decma2c 12644	. . . . . . . . 9 ⊢ ((1 · ;12) + (4 + 0)) = ;16
115	82	oveq1i 7359	. . . . . . . . . 10 ⊢ ((1 · 1) + 5) = (1 + 5)
116		5p1e6 12270	. . . . . . . . . . 11 ⊢ (5 + 1) = 6
117	94, 77, 116	addcomli 11308	. . . . . . . . . 10 ⊢ (1 + 5) = 6
118	115, 117, 112	3eqtri 2756	. . . . . . . . 9 ⊢ ((1 · 1) + 5) = ;06
119	39, 13, 17, 35, 100, 90, 13, 28, 3, 114, 118	decma2c 12644	. . . . . . . 8 ⊢ ((1 · ;;121) + ;45) = ;;166
120	13, 36, 40, 99, 119	gcdi 16985	. . . . . . 7 ⊢ (;;166 gcd ;;121) = 1
121		eqid 2729	. . . . . . . 8 ⊢ ;;166 = ;;166
122		eqid 2729	. . . . . . . . 9 ⊢ ;16 = ;16
123	13, 10, 65, 101	decsuc 12622	. . . . . . . . 9 ⊢ (;12 + 1) = ;13
124		1p1e2 12248	. . . . . . . . . . 11 ⊢ (1 + 1) = 2
125	63, 124	oveq12i 7361	. . . . . . . . . 10 ⊢ ((2 · 1) + (1 + 1)) = (2 + 2)
126	125, 52	eqtri 2752	. . . . . . . . 9 ⊢ ((2 · 1) + (1 + 1)) = 4
127		6cn 12219	. . . . . . . . . . 11 ⊢ 6 ∈ ℂ
128		6t2e12 12695	. . . . . . . . . . 11 ⊢ (6 · 2) = ;12
129	127, 67, 128	mulcomli 11124	. . . . . . . . . 10 ⊢ (2 · 6) = ;12
130		3p2e5 12274	. . . . . . . . . . 11 ⊢ (3 + 2) = 5
131	49, 67, 130	addcomli 11308	. . . . . . . . . 10 ⊢ (2 + 3) = 5
132	13, 10, 11, 129, 131	decaddi 12651	. . . . . . . . 9 ⊢ ((2 · 6) + 3) = ;15
133	13, 28, 13, 11, 122, 123, 10, 35, 13, 126, 132	decma2c 12644	. . . . . . . 8 ⊢ ((2 · ;16) + (;12 + 1)) = ;45
134	13, 10, 65, 129	decsuc 12622	. . . . . . . 8 ⊢ ((2 · 6) + 1) = ;13
135	29, 28, 39, 13, 121, 100, 10, 11, 13, 133, 134	decma2c 12644	. . . . . . 7 ⊢ ((2 · ;;166) + ;;121) = ;;453
136	10, 40, 38, 120, 135	gcdi 16985	. . . . . 6 ⊢ (;;453 gcd ;;166) = 1
137		eqid 2729	. . . . . . 7 ⊢ ;;453 = ;;453
138	29	nn0cni 12396	. . . . . . . . 9 ⊢ ;16 ∈ ℂ
139	138	addridi 11303	. . . . . . . 8 ⊢ (;16 + 0) = ;16
140	48	mullidi 11120	. . . . . . . . . 10 ⊢ (1 · 4) = 4
141	140, 124	oveq12i 7361	. . . . . . . . 9 ⊢ ((1 · 4) + (1 + 1)) = (4 + 2)
142	141, 110	eqtri 2752	. . . . . . . 8 ⊢ ((1 · 4) + (1 + 1)) = 6
143	94	mullidi 11120	. . . . . . . . . 10 ⊢ (1 · 5) = 5
144	143	oveq1i 7359	. . . . . . . . 9 ⊢ ((1 · 5) + 6) = (5 + 6)
145		6p5e11 12664	. . . . . . . . . 10 ⊢ (6 + 5) = ;11
146	127, 94, 145	addcomli 11308	. . . . . . . . 9 ⊢ (5 + 6) = ;11
147	144, 146	eqtri 2752	. . . . . . . 8 ⊢ ((1 · 5) + 6) = ;11
148	17, 35, 13, 28, 90, 139, 13, 13, 13, 142, 147	decma2c 12644	. . . . . . 7 ⊢ ((1 · ;45) + (;16 + 0)) = ;61
149	76	oveq1i 7359	. . . . . . . 8 ⊢ ((1 · 3) + 6) = (3 + 6)
150		6p3e9 12283	. . . . . . . . 9 ⊢ (6 + 3) = 9
151	127, 49, 150	addcomli 11308	. . . . . . . 8 ⊢ (3 + 6) = 9
152	30	dec0h 12613	. . . . . . . 8 ⊢ 9 = ;09
153	149, 151, 152	3eqtri 2756	. . . . . . 7 ⊢ ((1 · 3) + 6) = ;09
154	36, 11, 29, 28, 137, 121, 13, 30, 3, 148, 153	decma2c 12644	. . . . . 6 ⊢ ((1 · ;;453) + ;;166) = ;;619
155	13, 38, 37, 136, 154	gcdi 16985	. . . . 5 ⊢ (;;619 gcd ;;453) = 1
156		eqid 2729	. . . . . 6 ⊢ ;;619 = ;;619
157		7nn0 12406	. . . . . . 7 ⊢ 7 ∈ ℕ₀
158		eqid 2729	. . . . . . 7 ⊢ ;61 = ;61
159		5p2e7 12279	. . . . . . . 8 ⊢ (5 + 2) = 7
160	17, 35, 10, 90, 159	decaddi 12651	. . . . . . 7 ⊢ (;45 + 2) = ;47
161	102	oveq2i 7360	. . . . . . . 8 ⊢ ((2 · 6) + (4 + 0)) = ((2 · 6) + 4)
162	13, 10, 17, 129, 111	decaddi 12651	. . . . . . . 8 ⊢ ((2 · 6) + 4) = ;16
163	161, 162	eqtri 2752	. . . . . . 7 ⊢ ((2 · 6) + (4 + 0)) = ;16
164	63	oveq1i 7359	. . . . . . . 8 ⊢ ((2 · 1) + 7) = (2 + 7)
165		7cn 12222	. . . . . . . . 9 ⊢ 7 ∈ ℂ
166		7p2e9 12284	. . . . . . . . 9 ⊢ (7 + 2) = 9
167	165, 67, 166	addcomli 11308	. . . . . . . 8 ⊢ (2 + 7) = 9
168	164, 167, 152	3eqtri 2756	. . . . . . 7 ⊢ ((2 · 1) + 7) = ;09
169	28, 13, 17, 157, 158, 160, 10, 30, 3, 163, 168	decma2c 12644	. . . . . 6 ⊢ ((2 · ;61) + (;45 + 2)) = ;;169
170		9cn 12228	. . . . . . . 8 ⊢ 9 ∈ ℂ
171		9t2e18 12713	. . . . . . . 8 ⊢ (9 · 2) = ;18
172	170, 67, 171	mulcomli 11124	. . . . . . 7 ⊢ (2 · 9) = ;18
173	13, 2, 11, 172, 124, 13, 71	decaddci 12652	. . . . . 6 ⊢ ((2 · 9) + 3) = ;21
174	33, 30, 36, 11, 156, 137, 10, 13, 10, 169, 173	decma2c 12644	. . . . 5 ⊢ ((2 · ;;619) + ;;453) = ;;;1691
175	10, 37, 34, 155, 174	gcdi 16985	. . . 4 ⊢ (;;;1691 gcd ;;619) = 1
176		eqid 2729	. . . . 5 ⊢ ;;;1691 = ;;;1691
177		eqid 2729	. . . . . 6 ⊢ ;;169 = ;;169
178	28, 13, 124, 158	decsuc 12622	. . . . . 6 ⊢ (;61 + 1) = ;62
179		6p1e7 12271	. . . . . . . 8 ⊢ (6 + 1) = 7
180	157	dec0h 12613	. . . . . . . 8 ⊢ 7 = ;07
181	179, 180	eqtri 2752	. . . . . . 7 ⊢ (6 + 1) = ;07
182	82, 24	oveq12i 7361	. . . . . . . 8 ⊢ ((1 · 1) + (0 + 1)) = (1 + 1)
183	182, 124	eqtri 2752	. . . . . . 7 ⊢ ((1 · 1) + (0 + 1)) = 2
184	127	mullidi 11120	. . . . . . . . 9 ⊢ (1 · 6) = 6
185	184	oveq1i 7359	. . . . . . . 8 ⊢ ((1 · 6) + 7) = (6 + 7)
186		7p6e13 12669	. . . . . . . . 9 ⊢ (7 + 6) = ;13
187	165, 127, 186	addcomli 11308	. . . . . . . 8 ⊢ (6 + 7) = ;13
188	185, 187	eqtri 2752	. . . . . . 7 ⊢ ((1 · 6) + 7) = ;13
189	13, 28, 3, 157, 122, 181, 13, 11, 13, 183, 188	decma2c 12644	. . . . . 6 ⊢ ((1 · ;16) + (6 + 1)) = ;23
190	170	mullidi 11120	. . . . . . . 8 ⊢ (1 · 9) = 9
191	190	oveq1i 7359	. . . . . . 7 ⊢ ((1 · 9) + 2) = (9 + 2)
192		9p2e11 12678	. . . . . . 7 ⊢ (9 + 2) = ;11
193	191, 192	eqtri 2752	. . . . . 6 ⊢ ((1 · 9) + 2) = ;11
194	29, 30, 28, 10, 177, 178, 13, 13, 13, 189, 193	decma2c 12644	. . . . 5 ⊢ ((1 · ;;169) + (;61 + 1)) = ;;231
195	82	oveq1i 7359	. . . . . 6 ⊢ ((1 · 1) + 9) = (1 + 9)
196		9p1e10 12593	. . . . . . 7 ⊢ (9 + 1) = ;10
197	170, 77, 196	addcomli 11308	. . . . . 6 ⊢ (1 + 9) = ;10
198	195, 197	eqtri 2752	. . . . 5 ⊢ ((1 · 1) + 9) = ;10
199	31, 13, 33, 30, 176, 156, 13, 3, 13, 194, 198	decma2c 12644	. . . 4 ⊢ ((1 · ;;;1691) + ;;619) = ;;;2310
200	13, 34, 32, 175, 199	gcdi 16985	. . 3 ⊢ (;;;2310 gcd ;;;1691) = 1
201		eqid 2729	. . . . . 6 ⊢ ;;231 = ;;231
202	31	nn0cni 12396	. . . . . . 7 ⊢ ;;169 ∈ ℂ
203	202	addridi 11303	. . . . . 6 ⊢ (;;169 + 0) = ;;169
204		eqid 2729	. . . . . . 7 ⊢ ;23 = ;23
205	13, 28, 179, 122	decsuc 12622	. . . . . . 7 ⊢ (;16 + 1) = ;17
206	108, 124	oveq12i 7361	. . . . . . . 8 ⊢ ((1 · 2) + (1 + 1)) = (2 + 2)
207	206, 52	eqtri 2752	. . . . . . 7 ⊢ ((1 · 2) + (1 + 1)) = 4
208	76	oveq1i 7359	. . . . . . . 8 ⊢ ((1 · 3) + 7) = (3 + 7)
209		7p3e10 12666	. . . . . . . . 9 ⊢ (7 + 3) = ;10
210	165, 49, 209	addcomli 11308	. . . . . . . 8 ⊢ (3 + 7) = ;10
211	208, 210	eqtri 2752	. . . . . . 7 ⊢ ((1 · 3) + 7) = ;10
212	10, 11, 13, 157, 204, 205, 13, 3, 13, 207, 211	decma2c 12644	. . . . . 6 ⊢ ((1 · ;23) + (;16 + 1)) = ;40
213	12, 13, 29, 30, 201, 203, 13, 3, 13, 212, 198	decma2c 12644	. . . . 5 ⊢ ((1 · ;;231) + (;;169 + 0)) = ;;400
214	77	mul01i 11306	. . . . . . 7 ⊢ (1 · 0) = 0
215	214	oveq1i 7359	. . . . . 6 ⊢ ((1 · 0) + 1) = (0 + 1)
216	13	dec0h 12613	. . . . . 6 ⊢ 1 = ;01
217	215, 24, 216	3eqtri 2756	. . . . 5 ⊢ ((1 · 0) + 1) = ;01
218	14, 3, 31, 13, 25, 176, 13, 13, 3, 213, 217	decma2c 12644	. . . 4 ⊢ ((1 · ;;;2310) + ;;;1691) = ;;;4001
219	218, 16	eqtr4i 2755	. . 3 ⊢ ((1 · ;;;2310) + ;;;1691) = 𝑁
220	13, 32, 15, 200, 219	gcdi 16985	. 2 ⊢ (𝑁 gcd ;;;2310) = 1
221	9, 15, 22, 27, 220	gcdmodi 16986	1 ⊢ (((2↑;;800) − 1) gcd 𝑁) = 1

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 206 = wceq 1540 ∈ wcel 2109 class class class wbr 5092 (class class class)co 7349 0cc0 11009 1c1 11010 + caddc 11012 · cmul 11014 − cmin 11347 ℕcn 12128 2c2 12183 3c3 12184 4c4 12185 5c5 12186 6c6 12187 7c7 12188 8c8 12189 9c9 12190 ℕ₀cn0 12384 ℤcz 12471 cdc 12591 ↑cexp 13968 ∥ cdvds 16163 gcd cgcd 16405 ℙcprime 16582

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-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087

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 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7800 df-1st 7924 df-2nd 7925 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-2o 8389 df-er 8625 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-sup 9332 df-inf 9333 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-z 12472 df-dec 12592 df-uz 12736 df-rp 12894 df-fz 13411 df-fl 13696 df-mod 13774 df-seq 13909 df-exp 13969 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-dvds 16164 df-gcd 16406 df-prm 16583

This theorem is referenced by: 4001prm 17056

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