4001lem4 - Metamath Proof Explorer

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Theorem 4001lem4 16476

Description: Lemma for 4001prm 16477. 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 11709	. . . 4 ⊢ 2 ∈ ℕ
2		8nn0 11919	. . . . . 6 ⊢ 8 ∈ ℕ₀
3		0nn0 11911	. . . . . 6 ⊢ 0 ∈ ℕ₀
4	2, 3	deccl 12112	. . . . 5 ⊢ ;80 ∈ ℕ₀
5	4, 3	deccl 12112	. . . 4 ⊢ ;;800 ∈ ℕ₀
6		nnexpcl 13441	. . . 4 ⊢ ((2 ∈ ℕ ∧ ;;800 ∈ ℕ₀) → (2↑;;800) ∈ ℕ)
7	1, 5, 6	mp2an 690	. . 3 ⊢ (2↑;;800) ∈ ℕ
8		nnm1nn0 11937	. . 3 ⊢ ((2↑;;800) ∈ ℕ → ((2↑;;800) − 1) ∈ ℕ₀)
9	7, 8	ax-mp 5	. 2 ⊢ ((2↑;;800) − 1) ∈ ℕ₀
10		2nn0 11913	. . . . 5 ⊢ 2 ∈ ℕ₀
11		3nn0 11914	. . . . 5 ⊢ 3 ∈ ℕ₀
12	10, 11	deccl 12112	. . . 4 ⊢ ;23 ∈ ℕ₀
13		1nn0 11912	. . . 4 ⊢ 1 ∈ ℕ₀
14	12, 13	deccl 12112	. . 3 ⊢ ;;231 ∈ ℕ₀
15	14, 3	deccl 12112	. 2 ⊢ ;;;2310 ∈ ℕ₀
16		4001prm.1	. . 3 ⊢ 𝑁 = ;;;4001
17		4nn0 11915	. . . . . 6 ⊢ 4 ∈ ℕ₀
18	17, 3	deccl 12112	. . . . 5 ⊢ ;40 ∈ ℕ₀
19	18, 3	deccl 12112	. . . 4 ⊢ ;;400 ∈ ℕ₀
20		1nn 11648	. . . 4 ⊢ 1 ∈ ℕ
21	19, 20	decnncl 12117	. . 3 ⊢ ;;;4001 ∈ ℕ
22	16, 21	eqeltri 2909	. 2 ⊢ 𝑁 ∈ ℕ
23	16	4001lem2 16474	. . 3 ⊢ ((2↑;;800) mod 𝑁) = (;;;2311 mod 𝑁)
24		0p1e1 11758	. . . 4 ⊢ (0 + 1) = 1
25		eqid 2821	. . . 4 ⊢ ;;;2310 = ;;;2310
26	14, 3, 24, 25	decsuc 12128	. . 3 ⊢ (;;;2310 + 1) = ;;;2311
27	22, 7, 13, 15, 23, 26	modsubi 16407	. 2 ⊢ (((2↑;;800) − 1) mod 𝑁) = (;;;2310 mod 𝑁)
28		6nn0 11917	. . . . . 6 ⊢ 6 ∈ ℕ₀
29	13, 28	deccl 12112	. . . . 5 ⊢ ;16 ∈ ℕ₀
30		9nn0 11920	. . . . 5 ⊢ 9 ∈ ℕ₀
31	29, 30	deccl 12112	. . . 4 ⊢ ;;169 ∈ ℕ₀
32	31, 13	deccl 12112	. . 3 ⊢ ;;;1691 ∈ ℕ₀
33	28, 13	deccl 12112	. . . . 5 ⊢ ;61 ∈ ℕ₀
34	33, 30	deccl 12112	. . . 4 ⊢ ;;619 ∈ ℕ₀
35		5nn0 11916	. . . . . . 7 ⊢ 5 ∈ ℕ₀
36	17, 35	deccl 12112	. . . . . 6 ⊢ ;45 ∈ ℕ₀
37	36, 11	deccl 12112	. . . . 5 ⊢ ;;453 ∈ ℕ₀
38	29, 28	deccl 12112	. . . . . 6 ⊢ ;;166 ∈ ℕ₀
39	13, 10	deccl 12112	. . . . . . . 8 ⊢ ;12 ∈ ℕ₀
40	39, 13	deccl 12112	. . . . . . 7 ⊢ ;;121 ∈ ℕ₀
41	11, 13	deccl 12112	. . . . . . . . 9 ⊢ ;31 ∈ ℕ₀
42	13, 17	deccl 12112	. . . . . . . . . 10 ⊢ ;14 ∈ ℕ₀
43	42	nn0zi 12006	. . . . . . . . . . . . 13 ⊢ ;14 ∈ ℤ
44	11	nn0zi 12006	. . . . . . . . . . . . 13 ⊢ 3 ∈ ℤ
45		gcdcom 15861	. . . . . . . . . . . . 13 ⊢ ((;14 ∈ ℤ ∧ 3 ∈ ℤ) → (;14 gcd 3) = (3 gcd ;14))
46	43, 44, 45	mp2an 690	. . . . . . . . . . . 12 ⊢ (;14 gcd 3) = (3 gcd ;14)
47		3nn 11715	. . . . . . . . . . . . . 14 ⊢ 3 ∈ ℕ
48		4cn 11721	. . . . . . . . . . . . . . . 16 ⊢ 4 ∈ ℂ
49		3cn 11717	. . . . . . . . . . . . . . . 16 ⊢ 3 ∈ ℂ
50		4t3e12 12195	. . . . . . . . . . . . . . . 16 ⊢ (4 · 3) = ;12
51	48, 49, 50	mulcomli 10649	. . . . . . . . . . . . . . 15 ⊢ (3 · 4) = ;12
52		2p2e4 11771	. . . . . . . . . . . . . . 15 ⊢ (2 + 2) = 4
53	13, 10, 10, 51, 52	decaddi 12157	. . . . . . . . . . . . . 14 ⊢ ((3 · 4) + 2) = ;14
54		2lt3 11808	. . . . . . . . . . . . . 14 ⊢ 2 < 3
55	47, 17, 1, 53, 54	ndvdsi 15762	. . . . . . . . . . . . 13 ⊢ ¬ 3 ∥ ;14
56		3prm 16037	. . . . . . . . . . . . . 14 ⊢ 3 ∈ ℙ
57		coprm 16054	. . . . . . . . . . . . . 14 ⊢ ((3 ∈ ℙ ∧ ;14 ∈ ℤ) → (¬ 3 ∥ ;14 ↔ (3 gcd ;14) = 1))
58	56, 43, 57	mp2an 690	. . . . . . . . . . . . 13 ⊢ (¬ 3 ∥ ;14 ↔ (3 gcd ;14) = 1)
59	55, 58	mpbi 232	. . . . . . . . . . . 12 ⊢ (3 gcd ;14) = 1
60	46, 59	eqtri 2844	. . . . . . . . . . 11 ⊢ (;14 gcd 3) = 1
61		eqid 2821	. . . . . . . . . . . 12 ⊢ ;14 = ;14
62	11	dec0h 12119	. . . . . . . . . . . 12 ⊢ 3 = ;03
63		2t1e2 11799	. . . . . . . . . . . . . 14 ⊢ (2 · 1) = 2
64	63, 24	oveq12i 7167	. . . . . . . . . . . . 13 ⊢ ((2 · 1) + (0 + 1)) = (2 + 1)
65		2p1e3 11778	. . . . . . . . . . . . 13 ⊢ (2 + 1) = 3
66	64, 65	eqtri 2844	. . . . . . . . . . . 12 ⊢ ((2 · 1) + (0 + 1)) = 3
67		2cn 11711	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℂ
68		4t2e8 11804	. . . . . . . . . . . . . . 15 ⊢ (4 · 2) = 8
69	48, 67, 68	mulcomli 10649	. . . . . . . . . . . . . 14 ⊢ (2 · 4) = 8
70	69	oveq1i 7165	. . . . . . . . . . . . 13 ⊢ ((2 · 4) + 3) = (8 + 3)
71		8p3e11 12178	. . . . . . . . . . . . 13 ⊢ (8 + 3) = ;11
72	70, 71	eqtri 2844	. . . . . . . . . . . 12 ⊢ ((2 · 4) + 3) = ;11
73	13, 17, 3, 11, 61, 62, 10, 13, 13, 66, 72	decma2c 12150	. . . . . . . . . . 11 ⊢ ((2 · ;14) + 3) = ;31
74	10, 11, 42, 60, 73	gcdi 16408	. . . . . . . . . 10 ⊢ (;31 gcd ;14) = 1
75		eqid 2821	. . . . . . . . . . 11 ⊢ ;31 = ;31
76	49	mulid2i 10645	. . . . . . . . . . . . 13 ⊢ (1 · 3) = 3
77		ax-1cn 10594	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℂ
78	77	addid1i 10826	. . . . . . . . . . . . 13 ⊢ (1 + 0) = 1
79	76, 78	oveq12i 7167	. . . . . . . . . . . 12 ⊢ ((1 · 3) + (1 + 0)) = (3 + 1)
80		3p1e4 11781	. . . . . . . . . . . 12 ⊢ (3 + 1) = 4
81	79, 80	eqtri 2844	. . . . . . . . . . 11 ⊢ ((1 · 3) + (1 + 0)) = 4
82		1t1e1 11798	. . . . . . . . . . . . 13 ⊢ (1 · 1) = 1
83	82	oveq1i 7165	. . . . . . . . . . . 12 ⊢ ((1 · 1) + 4) = (1 + 4)
84		4p1e5 11782	. . . . . . . . . . . . 13 ⊢ (4 + 1) = 5
85	48, 77, 84	addcomli 10831	. . . . . . . . . . . 12 ⊢ (1 + 4) = 5
86	35	dec0h 12119	. . . . . . . . . . . 12 ⊢ 5 = ;05
87	83, 85, 86	3eqtri 2848	. . . . . . . . . . 11 ⊢ ((1 · 1) + 4) = ;05
88	11, 13, 13, 17, 75, 61, 13, 35, 3, 81, 87	decma2c 12150	. . . . . . . . . 10 ⊢ ((1 · ;31) + ;14) = ;45
89	13, 42, 41, 74, 88	gcdi 16408	. . . . . . . . 9 ⊢ (;45 gcd ;31) = 1
90		eqid 2821	. . . . . . . . . 10 ⊢ ;45 = ;45
91	69, 80	oveq12i 7167	. . . . . . . . . . 11 ⊢ ((2 · 4) + (3 + 1)) = (8 + 4)
92		8p4e12 12179	. . . . . . . . . . 11 ⊢ (8 + 4) = ;12
93	91, 92	eqtri 2844	. . . . . . . . . 10 ⊢ ((2 · 4) + (3 + 1)) = ;12
94		5cn 11724	. . . . . . . . . . . 12 ⊢ 5 ∈ ℂ
95		5t2e10 12197	. . . . . . . . . . . 12 ⊢ (5 · 2) = ;10
96	94, 67, 95	mulcomli 10649	. . . . . . . . . . 11 ⊢ (2 · 5) = ;10
97	13, 3, 24, 96	decsuc 12128	. . . . . . . . . 10 ⊢ ((2 · 5) + 1) = ;11
98	17, 35, 11, 13, 90, 75, 10, 13, 13, 93, 97	decma2c 12150	. . . . . . . . 9 ⊢ ((2 · ;45) + ;31) = ;;121
99	10, 41, 36, 89, 98	gcdi 16408	. . . . . . . 8 ⊢ (;;121 gcd ;45) = 1
100		eqid 2821	. . . . . . . . 9 ⊢ ;;121 = ;;121
101		eqid 2821	. . . . . . . . . 10 ⊢ ;12 = ;12
102	48	addid1i 10826	. . . . . . . . . . 11 ⊢ (4 + 0) = 4
103	17	dec0h 12119	. . . . . . . . . . 11 ⊢ 4 = ;04
104	102, 103	eqtri 2844	. . . . . . . . . 10 ⊢ (4 + 0) = ;04
105		00id 10814	. . . . . . . . . . . 12 ⊢ (0 + 0) = 0
106	82, 105	oveq12i 7167	. . . . . . . . . . 11 ⊢ ((1 · 1) + (0 + 0)) = (1 + 0)
107	106, 78	eqtri 2844	. . . . . . . . . 10 ⊢ ((1 · 1) + (0 + 0)) = 1
108	67	mulid2i 10645	. . . . . . . . . . . 12 ⊢ (1 · 2) = 2
109	108	oveq1i 7165	. . . . . . . . . . 11 ⊢ ((1 · 2) + 4) = (2 + 4)
110		4p2e6 11789	. . . . . . . . . . . 12 ⊢ (4 + 2) = 6
111	48, 67, 110	addcomli 10831	. . . . . . . . . . 11 ⊢ (2 + 4) = 6
112	28	dec0h 12119	. . . . . . . . . . 11 ⊢ 6 = ;06
113	109, 111, 112	3eqtri 2848	. . . . . . . . . 10 ⊢ ((1 · 2) + 4) = ;06
114	13, 10, 3, 17, 101, 104, 13, 28, 3, 107, 113	decma2c 12150	. . . . . . . . 9 ⊢ ((1 · ;12) + (4 + 0)) = ;16
115	82	oveq1i 7165	. . . . . . . . . 10 ⊢ ((1 · 1) + 5) = (1 + 5)
116		5p1e6 11783	. . . . . . . . . . 11 ⊢ (5 + 1) = 6
117	94, 77, 116	addcomli 10831	. . . . . . . . . 10 ⊢ (1 + 5) = 6
118	115, 117, 112	3eqtri 2848	. . . . . . . . 9 ⊢ ((1 · 1) + 5) = ;06
119	39, 13, 17, 35, 100, 90, 13, 28, 3, 114, 118	decma2c 12150	. . . . . . . 8 ⊢ ((1 · ;;121) + ;45) = ;;166
120	13, 36, 40, 99, 119	gcdi 16408	. . . . . . 7 ⊢ (;;166 gcd ;;121) = 1
121		eqid 2821	. . . . . . . 8 ⊢ ;;166 = ;;166
122		eqid 2821	. . . . . . . . 9 ⊢ ;16 = ;16
123	13, 10, 65, 101	decsuc 12128	. . . . . . . . 9 ⊢ (;12 + 1) = ;13
124		1p1e2 11761	. . . . . . . . . . 11 ⊢ (1 + 1) = 2
125	63, 124	oveq12i 7167	. . . . . . . . . 10 ⊢ ((2 · 1) + (1 + 1)) = (2 + 2)
126	125, 52	eqtri 2844	. . . . . . . . 9 ⊢ ((2 · 1) + (1 + 1)) = 4
127		6cn 11727	. . . . . . . . . . 11 ⊢ 6 ∈ ℂ
128		6t2e12 12201	. . . . . . . . . . 11 ⊢ (6 · 2) = ;12
129	127, 67, 128	mulcomli 10649	. . . . . . . . . 10 ⊢ (2 · 6) = ;12
130		3p2e5 11787	. . . . . . . . . . 11 ⊢ (3 + 2) = 5
131	49, 67, 130	addcomli 10831	. . . . . . . . . 10 ⊢ (2 + 3) = 5
132	13, 10, 11, 129, 131	decaddi 12157	. . . . . . . . 9 ⊢ ((2 · 6) + 3) = ;15
133	13, 28, 13, 11, 122, 123, 10, 35, 13, 126, 132	decma2c 12150	. . . . . . . 8 ⊢ ((2 · ;16) + (;12 + 1)) = ;45
134	13, 10, 65, 129	decsuc 12128	. . . . . . . 8 ⊢ ((2 · 6) + 1) = ;13
135	29, 28, 39, 13, 121, 100, 10, 11, 13, 133, 134	decma2c 12150	. . . . . . 7 ⊢ ((2 · ;;166) + ;;121) = ;;453
136	10, 40, 38, 120, 135	gcdi 16408	. . . . . 6 ⊢ (;;453 gcd ;;166) = 1
137		eqid 2821	. . . . . . 7 ⊢ ;;453 = ;;453
138	29	nn0cni 11908	. . . . . . . . 9 ⊢ ;16 ∈ ℂ
139	138	addid1i 10826	. . . . . . . 8 ⊢ (;16 + 0) = ;16
140	48	mulid2i 10645	. . . . . . . . . 10 ⊢ (1 · 4) = 4
141	140, 124	oveq12i 7167	. . . . . . . . 9 ⊢ ((1 · 4) + (1 + 1)) = (4 + 2)
142	141, 110	eqtri 2844	. . . . . . . 8 ⊢ ((1 · 4) + (1 + 1)) = 6
143	94	mulid2i 10645	. . . . . . . . . 10 ⊢ (1 · 5) = 5
144	143	oveq1i 7165	. . . . . . . . 9 ⊢ ((1 · 5) + 6) = (5 + 6)
145		6p5e11 12170	. . . . . . . . . 10 ⊢ (6 + 5) = ;11
146	127, 94, 145	addcomli 10831	. . . . . . . . 9 ⊢ (5 + 6) = ;11
147	144, 146	eqtri 2844	. . . . . . . 8 ⊢ ((1 · 5) + 6) = ;11
148	17, 35, 13, 28, 90, 139, 13, 13, 13, 142, 147	decma2c 12150	. . . . . . 7 ⊢ ((1 · ;45) + (;16 + 0)) = ;61
149	76	oveq1i 7165	. . . . . . . 8 ⊢ ((1 · 3) + 6) = (3 + 6)
150		6p3e9 11796	. . . . . . . . 9 ⊢ (6 + 3) = 9
151	127, 49, 150	addcomli 10831	. . . . . . . 8 ⊢ (3 + 6) = 9
152	30	dec0h 12119	. . . . . . . 8 ⊢ 9 = ;09
153	149, 151, 152	3eqtri 2848	. . . . . . 7 ⊢ ((1 · 3) + 6) = ;09
154	36, 11, 29, 28, 137, 121, 13, 30, 3, 148, 153	decma2c 12150	. . . . . 6 ⊢ ((1 · ;;453) + ;;166) = ;;619
155	13, 38, 37, 136, 154	gcdi 16408	. . . . 5 ⊢ (;;619 gcd ;;453) = 1
156		eqid 2821	. . . . . 6 ⊢ ;;619 = ;;619
157		7nn0 11918	. . . . . . 7 ⊢ 7 ∈ ℕ₀
158		eqid 2821	. . . . . . 7 ⊢ ;61 = ;61
159		5p2e7 11792	. . . . . . . 8 ⊢ (5 + 2) = 7
160	17, 35, 10, 90, 159	decaddi 12157	. . . . . . 7 ⊢ (;45 + 2) = ;47
161	102	oveq2i 7166	. . . . . . . 8 ⊢ ((2 · 6) + (4 + 0)) = ((2 · 6) + 4)
162	13, 10, 17, 129, 111	decaddi 12157	. . . . . . . 8 ⊢ ((2 · 6) + 4) = ;16
163	161, 162	eqtri 2844	. . . . . . 7 ⊢ ((2 · 6) + (4 + 0)) = ;16
164	63	oveq1i 7165	. . . . . . . 8 ⊢ ((2 · 1) + 7) = (2 + 7)
165		7cn 11730	. . . . . . . . 9 ⊢ 7 ∈ ℂ
166		7p2e9 11797	. . . . . . . . 9 ⊢ (7 + 2) = 9
167	165, 67, 166	addcomli 10831	. . . . . . . 8 ⊢ (2 + 7) = 9
168	164, 167, 152	3eqtri 2848	. . . . . . 7 ⊢ ((2 · 1) + 7) = ;09
169	28, 13, 17, 157, 158, 160, 10, 30, 3, 163, 168	decma2c 12150	. . . . . 6 ⊢ ((2 · ;61) + (;45 + 2)) = ;;169
170		9cn 11736	. . . . . . . 8 ⊢ 9 ∈ ℂ
171		9t2e18 12219	. . . . . . . 8 ⊢ (9 · 2) = ;18
172	170, 67, 171	mulcomli 10649	. . . . . . 7 ⊢ (2 · 9) = ;18
173	13, 2, 11, 172, 124, 13, 71	decaddci 12158	. . . . . 6 ⊢ ((2 · 9) + 3) = ;21
174	33, 30, 36, 11, 156, 137, 10, 13, 10, 169, 173	decma2c 12150	. . . . 5 ⊢ ((2 · ;;619) + ;;453) = ;;;1691
175	10, 37, 34, 155, 174	gcdi 16408	. . . 4 ⊢ (;;;1691 gcd ;;619) = 1
176		eqid 2821	. . . . 5 ⊢ ;;;1691 = ;;;1691
177		eqid 2821	. . . . . 6 ⊢ ;;169 = ;;169
178	28, 13, 124, 158	decsuc 12128	. . . . . 6 ⊢ (;61 + 1) = ;62
179		6p1e7 11784	. . . . . . . 8 ⊢ (6 + 1) = 7
180	157	dec0h 12119	. . . . . . . 8 ⊢ 7 = ;07
181	179, 180	eqtri 2844	. . . . . . 7 ⊢ (6 + 1) = ;07
182	82, 24	oveq12i 7167	. . . . . . . 8 ⊢ ((1 · 1) + (0 + 1)) = (1 + 1)
183	182, 124	eqtri 2844	. . . . . . 7 ⊢ ((1 · 1) + (0 + 1)) = 2
184	127	mulid2i 10645	. . . . . . . . 9 ⊢ (1 · 6) = 6
185	184	oveq1i 7165	. . . . . . . 8 ⊢ ((1 · 6) + 7) = (6 + 7)
186		7p6e13 12175	. . . . . . . . 9 ⊢ (7 + 6) = ;13
187	165, 127, 186	addcomli 10831	. . . . . . . 8 ⊢ (6 + 7) = ;13
188	185, 187	eqtri 2844	. . . . . . 7 ⊢ ((1 · 6) + 7) = ;13
189	13, 28, 3, 157, 122, 181, 13, 11, 13, 183, 188	decma2c 12150	. . . . . 6 ⊢ ((1 · ;16) + (6 + 1)) = ;23
190	170	mulid2i 10645	. . . . . . . 8 ⊢ (1 · 9) = 9
191	190	oveq1i 7165	. . . . . . 7 ⊢ ((1 · 9) + 2) = (9 + 2)
192		9p2e11 12184	. . . . . . 7 ⊢ (9 + 2) = ;11
193	191, 192	eqtri 2844	. . . . . 6 ⊢ ((1 · 9) + 2) = ;11
194	29, 30, 28, 10, 177, 178, 13, 13, 13, 189, 193	decma2c 12150	. . . . 5 ⊢ ((1 · ;;169) + (;61 + 1)) = ;;231
195	82	oveq1i 7165	. . . . . 6 ⊢ ((1 · 1) + 9) = (1 + 9)
196		9p1e10 12099	. . . . . . 7 ⊢ (9 + 1) = ;10
197	170, 77, 196	addcomli 10831	. . . . . 6 ⊢ (1 + 9) = ;10
198	195, 197	eqtri 2844	. . . . 5 ⊢ ((1 · 1) + 9) = ;10
199	31, 13, 33, 30, 176, 156, 13, 3, 13, 194, 198	decma2c 12150	. . . 4 ⊢ ((1 · ;;;1691) + ;;619) = ;;;2310
200	13, 34, 32, 175, 199	gcdi 16408	. . 3 ⊢ (;;;2310 gcd ;;;1691) = 1
201		eqid 2821	. . . . . 6 ⊢ ;;231 = ;;231
202	31	nn0cni 11908	. . . . . . 7 ⊢ ;;169 ∈ ℂ
203	202	addid1i 10826	. . . . . 6 ⊢ (;;169 + 0) = ;;169
204		eqid 2821	. . . . . . 7 ⊢ ;23 = ;23
205	13, 28, 179, 122	decsuc 12128	. . . . . . 7 ⊢ (;16 + 1) = ;17
206	108, 124	oveq12i 7167	. . . . . . . 8 ⊢ ((1 · 2) + (1 + 1)) = (2 + 2)
207	206, 52	eqtri 2844	. . . . . . 7 ⊢ ((1 · 2) + (1 + 1)) = 4
208	76	oveq1i 7165	. . . . . . . 8 ⊢ ((1 · 3) + 7) = (3 + 7)
209		7p3e10 12172	. . . . . . . . 9 ⊢ (7 + 3) = ;10
210	165, 49, 209	addcomli 10831	. . . . . . . 8 ⊢ (3 + 7) = ;10
211	208, 210	eqtri 2844	. . . . . . 7 ⊢ ((1 · 3) + 7) = ;10
212	10, 11, 13, 157, 204, 205, 13, 3, 13, 207, 211	decma2c 12150	. . . . . 6 ⊢ ((1 · ;23) + (;16 + 1)) = ;40
213	12, 13, 29, 30, 201, 203, 13, 3, 13, 212, 198	decma2c 12150	. . . . 5 ⊢ ((1 · ;;231) + (;;169 + 0)) = ;;400
214	77	mul01i 10829	. . . . . . 7 ⊢ (1 · 0) = 0
215	214	oveq1i 7165	. . . . . 6 ⊢ ((1 · 0) + 1) = (0 + 1)
216	13	dec0h 12119	. . . . . 6 ⊢ 1 = ;01
217	215, 24, 216	3eqtri 2848	. . . . 5 ⊢ ((1 · 0) + 1) = ;01
218	14, 3, 31, 13, 25, 176, 13, 13, 3, 213, 217	decma2c 12150	. . . 4 ⊢ ((1 · ;;;2310) + ;;;1691) = ;;;4001
219	218, 16	eqtr4i 2847	. . 3 ⊢ ((1 · ;;;2310) + ;;;1691) = 𝑁
220	13, 32, 15, 200, 219	gcdi 16408	. 2 ⊢ (𝑁 gcd ;;;2310) = 1
221	9, 15, 22, 27, 220	gcdmodi 16409	1 ⊢ (((2↑;;800) − 1) gcd 𝑁) = 1

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 208 = wceq 1533 ∈ wcel 2110 class class class wbr 5065 (class class class)co 7155 0cc0 10536 1c1 10537 + caddc 10539 · cmul 10541 − cmin 10869 ℕcn 11637 2c2 11691 3c3 11692 4c4 11693 5c5 11694 6c6 11695 7c7 11696 8c8 11697 9c9 11698 ℕ₀cn0 11896 ℤcz 11980 cdc 12097 ↑cexp 13428 ∥ cdvds 15606 gcd cgcd 15842 ℙcprime 16014

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-2o 8102 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-sup 8905 df-inf 8906 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-z 11981 df-dec 12098 df-uz 12243 df-rp 12389 df-fz 12892 df-fl 13161 df-mod 13237 df-seq 13369 df-exp 13429 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 df-dvds 15607 df-gcd 15843 df-prm 16015

This theorem is referenced by: 4001prm 16477

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