Theorem 631prm 13270

Description: 631 is a prime number. (Contributed by Mario Carneiro, 1-Mar-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)

Assertion

Ref	Expression
631prm	|- ;; 6 3 1 e. Prime

Proof of Theorem 631prm

Step	Hyp	Ref	Expression
1		6nn0 9595	. . . 4 |- 6 e. NN0
2		3nn0 9592	. . . 4 |- 3 e. NN0
3	1, 2	deccl 9802	. . 3 |- ; 6 3 e. NN0
4		1nn 9324	. . 3 |- 1 e. NN
5	3, 4	decnncl 9811	. 2 |- ;; 6 3 1 e. NN
6		8nn0 9597	. . 3 |- 8 e. NN0
7		4nn0 9593	. . 3 |- 4 e. NN0
8		1nn0 9590	. . 3 |- 1 e. NN0
9		6lt8 9507	. . 3 |- 6 < 8
10		3lt10 9929	. . 3 |- 3 < ; 1 0
11		1lt10 9931	. . 3 |- 1 < ; 1 0
12	1, 6, 2, 7, 8, 8, 9, 10, 11	3decltc 9825	. 2 |- ;; 6 3 1 < ;; 8 4 1
13		3nn 9478	. . . 4 |- 3 e. NN
14	1, 13	decnncl 9811	. . 3 |- ; 6 3 e. NN
15	14, 8, 8, 11	declti 9830	. 2 |- 1 < ;; 6 3 1
16		0nn0 9589	. . 3 |- 0 e. NN0
17		2cn 9384	. . . 4 |- 2 e. CC
18	17	mul02i 8724	. . 3 |- ( 0 x. 2 ) = 0
19		1e0p1 9834	. . 3 |- 1 = ( 0 + 1 )
20	3, 16, 18, 19	dec2dvds 13219	. 2 |- -. 2 || ;; 6 3 1
21		2nn0 9591	. . . . 5 |- 2 e. NN0
22	21, 8	deccl 9802	. . . 4 |- ; 2 1 e. NN0
23	22, 16	deccl 9802	. . 3 |- ;; 2 1 0 e. NN0
24		eqid 2238	. . . 4 |- ;; 2 1 0 = ;; 2 1 0
25	8	dec0h 9814	. . . 4 |- 1 = ; 0 1
26		eqid 2238	. . . . 5 |- ; 2 1 = ; 2 1
27		00id 8474	. . . . . 6 |- ( 0 + 0 ) = 0
28	16	dec0h 9814	. . . . . 6 |- 0 = ; 0 0
29	27, 28	eqtri 2259	. . . . 5 |- ( 0 + 0 ) = ; 0 0
30		3t2e6 9470	. . . . . . 7 |- ( 3 x. 2 ) = 6
31	30, 27	oveq12i 6100	. . . . . 6 |- ( ( 3 x. 2 ) + ( 0 + 0 ) ) = ( 6 + 0 )
32		6cn 9395	. . . . . . 7 |- 6 e. CC
33	32	addridi 8475	. . . . . 6 |- ( 6 + 0 ) = 6
34	31, 33	eqtri 2259	. . . . 5 |- ( ( 3 x. 2 ) + ( 0 + 0 ) ) = 6
35		3t1e3 9469	. . . . . . 7 |- ( 3 x. 1 ) = 3
36	35	oveq1i 6098	. . . . . 6 |- ( ( 3 x. 1 ) + 0 ) = ( 3 + 0 )
37		3cn 9388	. . . . . . 7 |- 3 e. CC
38	37	addridi 8475	. . . . . 6 |- ( 3 + 0 ) = 3
39	2	dec0h 9814	. . . . . 6 |- 3 = ; 0 3
40	36, 38, 39	3eqtri 2263	. . . . 5 |- ( ( 3 x. 1 ) + 0 ) = ; 0 3
41	21, 8, 16, 16, 26, 29, 2, 2, 16, 34, 40	decma2c 9845	. . . 4 |- ( ( 3 x. ; 2 1 ) + ( 0 + 0 ) ) = ; 6 3
42	37	mul01i 8725	. . . . . 6 |- ( 3 x. 0 ) = 0
43	42	oveq1i 6098	. . . . 5 |- ( ( 3 x. 0 ) + 1 ) = ( 0 + 1 )
44		0p1e1 9427	. . . . 5 |- ( 0 + 1 ) = 1
45	43, 44, 25	3eqtri 2263	. . . 4 |- ( ( 3 x. 0 ) + 1 ) = ; 0 1
46	22, 16, 16, 8, 24, 25, 2, 8, 16, 41, 45	decma2c 9845	. . 3 |- ( ( 3 x. ;; 2 1 0 ) + 1 ) = ;; 6 3 1
47		1lt3 9487	. . 3 |- 1 < 3
48	13, 23, 4, 46, 47	ndvdsi 12725	. 2 |- -. 3 || ;; 6 3 1
49		1lt5 9494	. . 3 |- 1 < 5
50	3, 4, 49	dec5dvds 13220	. 2 |- -. 5 || ;; 6 3 1
51		7nn 9482	. . 3 |- 7 e. NN
52		9nn0 9598	. . . 4 |- 9 e. NN0
53	52, 16	deccl 9802	. . 3 |- ; 9 0 e. NN0
54		eqid 2238	. . . 4 |- ; 9 0 = ; 9 0
55		7nn0 9596	. . . 4 |- 7 e. NN0
56	27	oveq2i 6099	. . . . 5 |- ( ( 7 x. 9 ) + ( 0 + 0 ) ) = ( ( 7 x. 9 ) + 0 )
57		9cn 9401	. . . . . . 7 |- 9 e. CC
58		7cn 9397	. . . . . . 7 |- 7 e. CC
59		9t7e63 9919	. . . . . . 7 |- ( 9 x. 7 ) = ; 6 3
60	57, 58, 59	mulcomli 8339	. . . . . 6 |- ( 7 x. 9 ) = ; 6 3
61	60	oveq1i 6098	. . . . 5 |- ( ( 7 x. 9 ) + 0 ) = (; 6 3 + 0 )
62	3	nn0cni 9586	. . . . . 6 |- ; 6 3 e. CC
63	62	addridi 8475	. . . . 5 |- (; 6 3 + 0 ) = ; 6 3
64	56, 61, 63	3eqtri 2263	. . . 4 |- ( ( 7 x. 9 ) + ( 0 + 0 ) ) = ; 6 3
65	58	mul01i 8725	. . . . . 6 |- ( 7 x. 0 ) = 0
66	65	oveq1i 6098	. . . . 5 |- ( ( 7 x. 0 ) + 1 ) = ( 0 + 1 )
67	66, 44, 25	3eqtri 2263	. . . 4 |- ( ( 7 x. 0 ) + 1 ) = ; 0 1
68	52, 16, 16, 8, 54, 25, 55, 8, 16, 64, 67	decma2c 9845	. . 3 |- ( ( 7 x. ; 9 0 ) + 1 ) = ;; 6 3 1
69		1lt7 9505	. . 3 |- 1 < 7
70	51, 53, 4, 68, 69	ndvdsi 12725	. 2 |- -. 7 || ;; 6 3 1
71	8, 4	decnncl 9811	. . 3 |- ; 1 1 e. NN
72		5nn0 9594	. . . 4 |- 5 e. NN0
73	72, 55	deccl 9802	. . 3 |- ; 5 7 e. NN0
74		4nn 9479	. . 3 |- 4 e. NN
75		eqid 2238	. . . 4 |- ; 5 7 = ; 5 7
76	7	dec0h 9814	. . . 4 |- 4 = ; 0 4
77	8, 8	deccl 9802	. . . 4 |- ; 1 1 e. NN0
78		eqid 2238	. . . . 5 |- ; 1 1 = ; 1 1
79		8cn 9399	. . . . . . 7 |- 8 e. CC
80	79	addlidi 8476	. . . . . 6 |- ( 0 + 8 ) = 8
81	6	dec0h 9814	. . . . . 6 |- 8 = ; 0 8
82	80, 81	eqtri 2259	. . . . 5 |- ( 0 + 8 ) = ; 0 8
83		5cn 9393	. . . . . . . 8 |- 5 e. CC
84	83	mullidi 8335	. . . . . . 7 |- ( 1 x. 5 ) = 5
85	84, 44	oveq12i 6100	. . . . . 6 |- ( ( 1 x. 5 ) + ( 0 + 1 ) ) = ( 5 + 1 )
86		5p1e6 9451	. . . . . 6 |- ( 5 + 1 ) = 6
87	85, 86	eqtri 2259	. . . . 5 |- ( ( 1 x. 5 ) + ( 0 + 1 ) ) = 6
88	84	oveq1i 6098	. . . . . 6 |- ( ( 1 x. 5 ) + 8 ) = ( 5 + 8 )
89		8p5e13 9875	. . . . . . 7 |- ( 8 + 5 ) = ; 1 3
90	79, 83, 89	addcomli 8478	. . . . . 6 |- ( 5 + 8 ) = ; 1 3
91	88, 90	eqtri 2259	. . . . 5 |- ( ( 1 x. 5 ) + 8 ) = ; 1 3
92	8, 8, 16, 6, 78, 82, 72, 2, 8, 87, 91	decmac 9844	. . . 4 |- ( (; 1 1 x. 5 ) + ( 0 + 8 ) ) = ; 6 3
93	58	mullidi 8335	. . . . . . 7 |- ( 1 x. 7 ) = 7
94	93, 44	oveq12i 6100	. . . . . 6 |- ( ( 1 x. 7 ) + ( 0 + 1 ) ) = ( 7 + 1 )
95		7p1e8 9453	. . . . . 6 |- ( 7 + 1 ) = 8
96	94, 95	eqtri 2259	. . . . 5 |- ( ( 1 x. 7 ) + ( 0 + 1 ) ) = 8
97	93	oveq1i 6098	. . . . . 6 |- ( ( 1 x. 7 ) + 4 ) = ( 7 + 4 )
98		7p4e11 9868	. . . . . 6 |- ( 7 + 4 ) = ; 1 1
99	97, 98	eqtri 2259	. . . . 5 |- ( ( 1 x. 7 ) + 4 ) = ; 1 1
100	8, 8, 16, 7, 78, 76, 55, 8, 8, 96, 99	decmac 9844	. . . 4 |- ( (; 1 1 x. 7 ) + 4 ) = ; 8 1
101	72, 55, 16, 7, 75, 76, 77, 8, 6, 92, 100	decma2c 9845	. . 3 |- ( (; 1 1 x. ; 5 7 ) + 4 ) = ;; 6 3 1
102		4lt10 9928	. . . 4 |- 4 < ; 1 0
103	4, 8, 7, 102	declti 9830	. . 3 |- 4 < ; 1 1
104	71, 73, 74, 101, 103	ndvdsi 12725	. 2 |- -. ; 1 1 || ;; 6 3 1
105	8, 13	decnncl 9811	. . 3 |- ; 1 3 e. NN
106	7, 6	deccl 9802	. . 3 |- ; 4 8 e. NN0
107		eqid 2238	. . . 4 |- ; 4 8 = ; 4 8
108	55	dec0h 9814	. . . 4 |- 7 = ; 0 7
109	8, 2	deccl 9802	. . . 4 |- ; 1 3 e. NN0
110		eqid 2238	. . . . 5 |- ; 1 3 = ; 1 3
111	77	nn0cni 9586	. . . . . 6 |- ; 1 1 e. CC
112	111	addlidi 8476	. . . . 5 |- ( 0 + ; 1 1 ) = ; 1 1
113		4cn 9391	. . . . . . . 8 |- 4 e. CC
114	113	mullidi 8335	. . . . . . 7 |- ( 1 x. 4 ) = 4
115		1p1e2 9430	. . . . . . 7 |- ( 1 + 1 ) = 2
116	114, 115	oveq12i 6100	. . . . . 6 |- ( ( 1 x. 4 ) + ( 1 + 1 ) ) = ( 4 + 2 )
117		4p2e6 9457	. . . . . 6 |- ( 4 + 2 ) = 6
118	116, 117	eqtri 2259	. . . . 5 |- ( ( 1 x. 4 ) + ( 1 + 1 ) ) = 6
119		4t3e12 9890	. . . . . . 7 |- ( 4 x. 3 ) = ; 1 2
120	113, 37, 119	mulcomli 8339	. . . . . 6 |- ( 3 x. 4 ) = ; 1 2
121		2p1e3 9447	. . . . . 6 |- ( 2 + 1 ) = 3
122	8, 21, 8, 120, 121	decaddi 9852	. . . . 5 |- ( ( 3 x. 4 ) + 1 ) = ; 1 3
123	8, 2, 8, 8, 110, 112, 7, 2, 8, 118, 122	decmac 9844	. . . 4 |- ( (; 1 3 x. 4 ) + ( 0 + ; 1 1 ) ) = ; 6 3
124	79	mullidi 8335	. . . . . . 7 |- ( 1 x. 8 ) = 8
125	37	addlidi 8476	. . . . . . 7 |- ( 0 + 3 ) = 3
126	124, 125	oveq12i 6100	. . . . . 6 |- ( ( 1 x. 8 ) + ( 0 + 3 ) ) = ( 8 + 3 )
127		8p3e11 9873	. . . . . 6 |- ( 8 + 3 ) = ; 1 1
128	126, 127	eqtri 2259	. . . . 5 |- ( ( 1 x. 8 ) + ( 0 + 3 ) ) = ; 1 1
129		8t3e24 9908	. . . . . . 7 |- ( 8 x. 3 ) = ; 2 4
130	79, 37, 129	mulcomli 8339	. . . . . 6 |- ( 3 x. 8 ) = ; 2 4
131	58, 113, 98	addcomli 8478	. . . . . 6 |- ( 4 + 7 ) = ; 1 1
132	21, 7, 55, 130, 121, 8, 131	decaddci 9853	. . . . 5 |- ( ( 3 x. 8 ) + 7 ) = ; 3 1
133	8, 2, 16, 55, 110, 108, 6, 8, 2, 128, 132	decmac 9844	. . . 4 |- ( (; 1 3 x. 8 ) + 7 ) = ;; 1 1 1
134	7, 6, 16, 55, 107, 108, 109, 8, 77, 123, 133	decma2c 9845	. . 3 |- ( (; 1 3 x. ; 4 8 ) + 7 ) = ;; 6 3 1
135		7lt10 9925	. . . 4 |- 7 < ; 1 0
136	4, 2, 55, 135	declti 9830	. . 3 |- 7 < ; 1 3
137	105, 106, 51, 134, 136	ndvdsi 12725	. 2 |- -. ; 1 3 || ;; 6 3 1
138	8, 51	decnncl 9811	. . 3 |- ; 1 7 e. NN
139	2, 55	deccl 9802	. . 3 |- ; 3 7 e. NN0
140		2nn 9477	. . 3 |- 2 e. NN
141		eqid 2238	. . . 4 |- ; 3 7 = ; 3 7
142	21	dec0h 9814	. . . 4 |- 2 = ; 0 2
143	8, 55	deccl 9802	. . . 4 |- ; 1 7 e. NN0
144	8, 21	deccl 9802	. . . 4 |- ; 1 2 e. NN0
145		eqid 2238	. . . . 5 |- ; 1 7 = ; 1 7
146	144	nn0cni 9586	. . . . . 6 |- ; 1 2 e. CC
147	146	addlidi 8476	. . . . 5 |- ( 0 + ; 1 2 ) = ; 1 2
148	37	mullidi 8335	. . . . . . 7 |- ( 1 x. 3 ) = 3
149		1p2e3 9448	. . . . . . 7 |- ( 1 + 2 ) = 3
150	148, 149	oveq12i 6100	. . . . . 6 |- ( ( 1 x. 3 ) + ( 1 + 2 ) ) = ( 3 + 3 )
151		3p3e6 9456	. . . . . 6 |- ( 3 + 3 ) = 6
152	150, 151	eqtri 2259	. . . . 5 |- ( ( 1 x. 3 ) + ( 1 + 2 ) ) = 6
153		7t3e21 9902	. . . . . 6 |- ( 7 x. 3 ) = ; 2 1
154	21, 8, 21, 153, 149	decaddi 9852	. . . . 5 |- ( ( 7 x. 3 ) + 2 ) = ; 2 3
155	8, 55, 8, 21, 145, 147, 2, 2, 21, 152, 154	decmac 9844	. . . 4 |- ( (; 1 7 x. 3 ) + ( 0 + ; 1 2 ) ) = ; 6 3
156	83	addlidi 8476	. . . . . . 7 |- ( 0 + 5 ) = 5
157	93, 156	oveq12i 6100	. . . . . 6 |- ( ( 1 x. 7 ) + ( 0 + 5 ) ) = ( 7 + 5 )
158		7p5e12 9869	. . . . . 6 |- ( 7 + 5 ) = ; 1 2
159	157, 158	eqtri 2259	. . . . 5 |- ( ( 1 x. 7 ) + ( 0 + 5 ) ) = ; 1 2
160		7t7e49 9906	. . . . . 6 |- ( 7 x. 7 ) = ; 4 9
161		4p1e5 9450	. . . . . 6 |- ( 4 + 1 ) = 5
162		9p2e11 9879	. . . . . 6 |- ( 9 + 2 ) = ; 1 1
163	7, 52, 21, 160, 161, 8, 162	decaddci 9853	. . . . 5 |- ( ( 7 x. 7 ) + 2 ) = ; 5 1
164	8, 55, 16, 21, 145, 142, 55, 8, 72, 159, 163	decmac 9844	. . . 4 |- ( (; 1 7 x. 7 ) + 2 ) = ;; 1 2 1
165	2, 55, 16, 21, 141, 142, 143, 8, 144, 155, 164	decma2c 9845	. . 3 |- ( (; 1 7 x. ; 3 7 ) + 2 ) = ;; 6 3 1
166		2lt10 9930	. . . 4 |- 2 < ; 1 0
167	4, 55, 21, 166	declti 9830	. . 3 |- 2 < ; 1 7
168	138, 139, 140, 165, 167	ndvdsi 12725	. 2 |- -. ; 1 7 || ;; 6 3 1
169		9nn 9484	. . . 4 |- 9 e. NN
170	8, 169	decnncl 9811	. . 3 |- ; 1 9 e. NN
171	2, 2	deccl 9802	. . 3 |- ; 3 3 e. NN0
172		eqid 2238	. . . 4 |- ; 3 3 = ; 3 3
173	8, 52	deccl 9802	. . . 4 |- ; 1 9 e. NN0
174		eqid 2238	. . . . 5 |- ; 1 9 = ; 1 9
175	32	addlidi 8476	. . . . . 6 |- ( 0 + 6 ) = 6
176	1	dec0h 9814	. . . . . 6 |- 6 = ; 0 6
177	175, 176	eqtri 2259	. . . . 5 |- ( 0 + 6 ) = ; 0 6
178	148, 125	oveq12i 6100	. . . . . 6 |- ( ( 1 x. 3 ) + ( 0 + 3 ) ) = ( 3 + 3 )
179	178, 151	eqtri 2259	. . . . 5 |- ( ( 1 x. 3 ) + ( 0 + 3 ) ) = 6
180		9t3e27 9915	. . . . . 6 |- ( 9 x. 3 ) = ; 2 7
181		7p6e13 9870	. . . . . 6 |- ( 7 + 6 ) = ; 1 3
182	21, 55, 1, 180, 121, 2, 181	decaddci 9853	. . . . 5 |- ( ( 9 x. 3 ) + 6 ) = ; 3 3
183	8, 52, 16, 1, 174, 177, 2, 2, 2, 179, 182	decmac 9844	. . . 4 |- ( (; 1 9 x. 3 ) + ( 0 + 6 ) ) = ; 6 3
184	21, 55, 7, 180, 121, 8, 98	decaddci 9853	. . . . 5 |- ( ( 9 x. 3 ) + 4 ) = ; 3 1
185	8, 52, 16, 7, 174, 76, 2, 8, 2, 179, 184	decmac 9844	. . . 4 |- ( (; 1 9 x. 3 ) + 4 ) = ; 6 1
186	2, 2, 16, 7, 172, 76, 173, 8, 1, 183, 185	decma2c 9845	. . 3 |- ( (; 1 9 x. ; 3 3 ) + 4 ) = ;; 6 3 1
187	4, 52, 7, 102	declti 9830	. . 3 |- 4 < ; 1 9
188	170, 171, 74, 186, 187	ndvdsi 12725	. 2 |- -. ; 1 9 || ;; 6 3 1
189	21, 13	decnncl 9811	. . 3 |- ; 2 3 e. NN
190	21, 55	deccl 9802	. . 3 |- ; 2 7 e. NN0
191		10nn 9807	. . 3 |- ; 1 0 e. NN
192		eqid 2238	. . . 4 |- ; 2 7 = ; 2 7
193		eqid 2238	. . . 4 |- ; 1 0 = ; 1 0
194	21, 2	deccl 9802	. . . 4 |- ; 2 3 e. NN0
195	8, 1	deccl 9802	. . . 4 |- ; 1 6 e. NN0
196		eqid 2238	. . . . 5 |- ; 2 3 = ; 2 3
197		eqid 2238	. . . . . 6 |- ; 1 6 = ; 1 6
198		ax-1cn 8278	. . . . . . 7 |- 1 e. CC
199		6p1e7 9452	. . . . . . 7 |- ( 6 + 1 ) = 7
200	32, 198, 199	addcomli 8478	. . . . . 6 |- ( 1 + 6 ) = 7
201	16, 8, 8, 1, 25, 197, 44, 200	decadd 9846	. . . . 5 |- ( 1 + ; 1 6 ) = ; 1 7
202		2t2e4 9468	. . . . . . 7 |- ( 2 x. 2 ) = 4
203	202, 115	oveq12i 6100	. . . . . 6 |- ( ( 2 x. 2 ) + ( 1 + 1 ) ) = ( 4 + 2 )
204	203, 117	eqtri 2259	. . . . 5 |- ( ( 2 x. 2 ) + ( 1 + 1 ) ) = 6
205	30	oveq1i 6098	. . . . . 6 |- ( ( 3 x. 2 ) + 7 ) = ( 6 + 7 )
206	58, 32, 181	addcomli 8478	. . . . . 6 |- ( 6 + 7 ) = ; 1 3
207	205, 206	eqtri 2259	. . . . 5 |- ( ( 3 x. 2 ) + 7 ) = ; 1 3
208	21, 2, 8, 55, 196, 201, 21, 2, 8, 204, 207	decmac 9844	. . . 4 |- ( (; 2 3 x. 2 ) + ( 1 + ; 1 6 ) ) = ; 6 3
209		7t2e14 9901	. . . . . . . . 9 |- ( 7 x. 2 ) = ; 1 4
210	58, 17, 209	mulcomli 8339	. . . . . . . 8 |- ( 2 x. 7 ) = ; 1 4
211	8, 7, 21, 210, 117	decaddi 9852	. . . . . . 7 |- ( ( 2 x. 7 ) + 2 ) = ; 1 6
212	58, 37, 153	mulcomli 8339	. . . . . . 7 |- ( 3 x. 7 ) = ; 2 1
213	55, 21, 2, 196, 8, 21, 211, 212	decmul1c 9857	. . . . . 6 |- (; 2 3 x. 7 ) = ;; 1 6 1
214	213	oveq1i 6098	. . . . 5 |- ( (; 2 3 x. 7 ) + 0 ) = (;; 1 6 1 + 0 )
215	195, 8	deccl 9802	. . . . . . 7 |- ;; 1 6 1 e. NN0
216	215	nn0cni 9586	. . . . . 6 |- ;; 1 6 1 e. CC
217	216	addridi 8475	. . . . 5 |- (;; 1 6 1 + 0 ) = ;; 1 6 1
218	214, 217	eqtri 2259	. . . 4 |- ( (; 2 3 x. 7 ) + 0 ) = ;; 1 6 1
219	21, 55, 8, 16, 192, 193, 194, 8, 195, 208, 218	decma2c 9845	. . 3 |- ( (; 2 3 x. ; 2 7 ) + ; 1 0 ) = ;; 6 3 1
220		10pos 9808	. . . 4 |- 0 < ; 1 0
221		1lt2 9485	. . . 4 |- 1 < 2
222	8, 21, 16, 2, 220, 221	decltc 9821	. . 3 |- ; 1 0 < ; 2 3
223	189, 190, 191, 219, 222	ndvdsi 12725	. 2 |- -. ; 2 3 || ;; 6 3 1
224	5, 12, 15, 20, 48, 50, 70, 104, 137, 168, 188, 223	prmlem2 13263	1 |- ;; 6 3 1 e. Prime

This proof depends on syntax axioms:   e. wcel 2209  (class class class)co 6088   0cc0 8185   1c1 8186   + caddc 8188   x. cmul 8190   2c2 9364   3c3 9365   4c4 9366   5c5 9367   6c6 9368   7c7 9369   8c8 9370   9c9 9371  ;cdc 9788   Primecprime 12910

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303  ax-arch 8304  ax-caucvg 8305

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-xor 1425  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-1o 6691  df-2o 6692  df-er 6811  df-en 7027  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-5 9375  df-6 9376  df-7 9377  df-8 9378  df-9 9379  df-n0 9575  df-z 9656  df-dec 9789  df-uz 9938  df-q 10036  df-rp 10072  df-fz 10429  df-fzo 10567  df-fl 10722  df-mod 10781  df-seqfrec 10906  df-exp 10997  df-cj 11629  df-re 11630  df-im 11631  df-rsqrt 11786  df-abs 11787  df-dvds 12580  df-prm 12911

This theorem is used by: (None)