Theorem 631prm 15834

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 11313	. . . 4 |- 6 e. NN0
2		3nn0 11310	. . . 4 |- 3 e. NN0
3	1, 2	deccl 11512	. . 3 |- ; 6 3 e. NN0
4		1nn 11031	. . 3 |- 1 e. NN
5	3, 4	decnncl 11518	. 2 |- ;; 6 3 1 e. NN
6		8nn0 11315	. . 3 |- 8 e. NN0
7		4nn0 11311	. . 3 |- 4 e. NN0
8		1nn0 11308	. . 3 |- 1 e. NN0
9		6lt8 11216	. . 3 |- 6 < 8
10		3lt10 11679	. . 3 |- 3 < ; 1 0
11		1lt10 11681	. . 3 |- 1 < ; 1 0
12	1, 6, 2, 7, 8, 8, 9, 10, 11	3decltc 11538	. 2 |- ;; 6 3 1 < ;; 8 4 1
13		3nn 11186	. . . 4 |- 3 e. NN
14	1, 13	decnncl 11518	. . 3 |- ; 6 3 e. NN
15	14, 8, 8, 11	declti 11546	. 2 |- 1 < ;; 6 3 1
16		0nn0 11307	. . 3 |- 0 e. NN0
17		2cn 11091	. . . 4 |- 2 e. CC
18	17	mul02i 10225	. . 3 |- ( 0 x. 2 ) = 0
19		1e0p1 11552	. . 3 |- 1 = ( 0 + 1 )
20	3, 16, 18, 19	dec2dvds 15767	. 2 |- -. 2 || ;; 6 3 1
21		2nn0 11309	. . . . 5 |- 2 e. NN0
22	21, 8	deccl 11512	. . . 4 |- ; 2 1 e. NN0
23	22, 16	deccl 11512	. . 3 |- ;; 2 1 0 e. NN0
24		eqid 2622	. . . 4 |- ;; 2 1 0 = ;; 2 1 0
25	8	dec0h 11522	. . . 4 |- 1 = ; 0 1
26		eqid 2622	. . . . 5 |- ; 2 1 = ; 2 1
27		00id 10211	. . . . . 6 |- ( 0 + 0 ) = 0
28	16	dec0h 11522	. . . . . 6 |- 0 = ; 0 0
29	27, 28	eqtri 2644	. . . . 5 |- ( 0 + 0 ) = ; 0 0
30		3t2e6 11179	. . . . . . 7 |- ( 3 x. 2 ) = 6
31	30, 27	oveq12i 6662	. . . . . 6 |- ( ( 3 x. 2 ) + ( 0 + 0 ) ) = ( 6 + 0 )
32		6cn 11102	. . . . . . 7 |- 6 e. CC
33	32	addid1i 10223	. . . . . 6 |- ( 6 + 0 ) = 6
34	31, 33	eqtri 2644	. . . . 5 |- ( ( 3 x. 2 ) + ( 0 + 0 ) ) = 6
35		3t1e3 11178	. . . . . . 7 |- ( 3 x. 1 ) = 3
36	35	oveq1i 6660	. . . . . 6 |- ( ( 3 x. 1 ) + 0 ) = ( 3 + 0 )
37		3cn 11095	. . . . . . 7 |- 3 e. CC
38	37	addid1i 10223	. . . . . 6 |- ( 3 + 0 ) = 3
39	2	dec0h 11522	. . . . . 6 |- 3 = ; 0 3
40	36, 38, 39	3eqtri 2648	. . . . 5 |- ( ( 3 x. 1 ) + 0 ) = ; 0 3
41	21, 8, 16, 16, 26, 29, 2, 2, 16, 34, 40	decma2c 11568	. . . 4 |- ( ( 3 x. ; 2 1 ) + ( 0 + 0 ) ) = ; 6 3
42	37	mul01i 10226	. . . . . 6 |- ( 3 x. 0 ) = 0
43	42	oveq1i 6660	. . . . 5 |- ( ( 3 x. 0 ) + 1 ) = ( 0 + 1 )
44		0p1e1 11132	. . . . 5 |- ( 0 + 1 ) = 1
45	43, 44, 25	3eqtri 2648	. . . 4 |- ( ( 3 x. 0 ) + 1 ) = ; 0 1
46	22, 16, 16, 8, 24, 25, 2, 8, 16, 41, 45	decma2c 11568	. . 3 |- ( ( 3 x. ;; 2 1 0 ) + 1 ) = ;; 6 3 1
47		1lt3 11196	. . 3 |- 1 < 3
48	13, 23, 4, 46, 47	ndvdsi 15136	. 2 |- -. 3 || ;; 6 3 1
49		1lt5 11203	. . 3 |- 1 < 5
50	3, 4, 49	dec5dvds 15768	. 2 |- -. 5 || ;; 6 3 1
51		7nn 11190	. . 3 |- 7 e. NN
52		9nn0 11316	. . . 4 |- 9 e. NN0
53	52, 16	deccl 11512	. . 3 |- ; 9 0 e. NN0
54		eqid 2622	. . . 4 |- ; 9 0 = ; 9 0
55		7nn0 11314	. . . 4 |- 7 e. NN0
56	27	oveq2i 6661	. . . . 5 |- ( ( 7 x. 9 ) + ( 0 + 0 ) ) = ( ( 7 x. 9 ) + 0 )
57		9cn 11108	. . . . . . 7 |- 9 e. CC
58		7cn 11104	. . . . . . 7 |- 7 e. CC
59		9t7e63 11668	. . . . . . 7 |- ( 9 x. 7 ) = ; 6 3
60	57, 58, 59	mulcomli 10047	. . . . . 6 |- ( 7 x. 9 ) = ; 6 3
61	60	oveq1i 6660	. . . . 5 |- ( ( 7 x. 9 ) + 0 ) = (; 6 3 + 0 )
62	3	nn0cni 11304	. . . . . 6 |- ; 6 3 e. CC
63	62	addid1i 10223	. . . . 5 |- (; 6 3 + 0 ) = ; 6 3
64	56, 61, 63	3eqtri 2648	. . . 4 |- ( ( 7 x. 9 ) + ( 0 + 0 ) ) = ; 6 3
65	58	mul01i 10226	. . . . . 6 |- ( 7 x. 0 ) = 0
66	65	oveq1i 6660	. . . . 5 |- ( ( 7 x. 0 ) + 1 ) = ( 0 + 1 )
67	66, 44, 25	3eqtri 2648	. . . 4 |- ( ( 7 x. 0 ) + 1 ) = ; 0 1
68	52, 16, 16, 8, 54, 25, 55, 8, 16, 64, 67	decma2c 11568	. . 3 |- ( ( 7 x. ; 9 0 ) + 1 ) = ;; 6 3 1
69		1lt7 11214	. . 3 |- 1 < 7
70	51, 53, 4, 68, 69	ndvdsi 15136	. 2 |- -. 7 || ;; 6 3 1
71	8, 4	decnncl 11518	. . 3 |- ; 1 1 e. NN
72		5nn0 11312	. . . 4 |- 5 e. NN0
73	72, 55	deccl 11512	. . 3 |- ; 5 7 e. NN0
74		4nn 11187	. . 3 |- 4 e. NN
75		eqid 2622	. . . 4 |- ; 5 7 = ; 5 7
76	7	dec0h 11522	. . . 4 |- 4 = ; 0 4
77	8, 8	deccl 11512	. . . 4 |- ; 1 1 e. NN0
78		eqid 2622	. . . . 5 |- ; 1 1 = ; 1 1
79		8cn 11106	. . . . . . 7 |- 8 e. CC
80	79	addid2i 10224	. . . . . 6 |- ( 0 + 8 ) = 8
81	6	dec0h 11522	. . . . . 6 |- 8 = ; 0 8
82	80, 81	eqtri 2644	. . . . 5 |- ( 0 + 8 ) = ; 0 8
83		5cn 11100	. . . . . . . 8 |- 5 e. CC
84	83	mulid2i 10043	. . . . . . 7 |- ( 1 x. 5 ) = 5
85	84, 44	oveq12i 6662	. . . . . 6 |- ( ( 1 x. 5 ) + ( 0 + 1 ) ) = ( 5 + 1 )
86		5p1e6 11155	. . . . . 6 |- ( 5 + 1 ) = 6
87	85, 86	eqtri 2644	. . . . 5 |- ( ( 1 x. 5 ) + ( 0 + 1 ) ) = 6
88	84	oveq1i 6660	. . . . . 6 |- ( ( 1 x. 5 ) + 8 ) = ( 5 + 8 )
89		8p5e13 11615	. . . . . . 7 |- ( 8 + 5 ) = ; 1 3
90	79, 83, 89	addcomli 10228	. . . . . 6 |- ( 5 + 8 ) = ; 1 3
91	88, 90	eqtri 2644	. . . . 5 |- ( ( 1 x. 5 ) + 8 ) = ; 1 3
92	8, 8, 16, 6, 78, 82, 72, 2, 8, 87, 91	decmac 11566	. . . 4 |- ( (; 1 1 x. 5 ) + ( 0 + 8 ) ) = ; 6 3
93	58	mulid2i 10043	. . . . . . 7 |- ( 1 x. 7 ) = 7
94	93, 44	oveq12i 6662	. . . . . 6 |- ( ( 1 x. 7 ) + ( 0 + 1 ) ) = ( 7 + 1 )
95		7p1e8 11157	. . . . . 6 |- ( 7 + 1 ) = 8
96	94, 95	eqtri 2644	. . . . 5 |- ( ( 1 x. 7 ) + ( 0 + 1 ) ) = 8
97	93	oveq1i 6660	. . . . . 6 |- ( ( 1 x. 7 ) + 4 ) = ( 7 + 4 )
98		7p4e11 11605	. . . . . 6 |- ( 7 + 4 ) = ; 1 1
99	97, 98	eqtri 2644	. . . . 5 |- ( ( 1 x. 7 ) + 4 ) = ; 1 1
100	8, 8, 16, 7, 78, 76, 55, 8, 8, 96, 99	decmac 11566	. . . 4 |- ( (; 1 1 x. 7 ) + 4 ) = ; 8 1
101	72, 55, 16, 7, 75, 76, 77, 8, 6, 92, 100	decma2c 11568	. . 3 |- ( (; 1 1 x. ; 5 7 ) + 4 ) = ;; 6 3 1
102		4lt10 11678	. . . 4 |- 4 < ; 1 0
103	4, 8, 7, 102	declti 11546	. . 3 |- 4 < ; 1 1
104	71, 73, 74, 101, 103	ndvdsi 15136	. 2 |- -. ; 1 1 || ;; 6 3 1
105	8, 13	decnncl 11518	. . 3 |- ; 1 3 e. NN
106	7, 6	deccl 11512	. . 3 |- ; 4 8 e. NN0
107		eqid 2622	. . . 4 |- ; 4 8 = ; 4 8
108	55	dec0h 11522	. . . 4 |- 7 = ; 0 7
109	8, 2	deccl 11512	. . . 4 |- ; 1 3 e. NN0
110		eqid 2622	. . . . 5 |- ; 1 3 = ; 1 3
111	77	nn0cni 11304	. . . . . 6 |- ; 1 1 e. CC
112	111	addid2i 10224	. . . . 5 |- ( 0 + ; 1 1 ) = ; 1 1
113		4cn 11098	. . . . . . . 8 |- 4 e. CC
114	113	mulid2i 10043	. . . . . . 7 |- ( 1 x. 4 ) = 4
115		1p1e2 11134	. . . . . . 7 |- ( 1 + 1 ) = 2
116	114, 115	oveq12i 6662	. . . . . 6 |- ( ( 1 x. 4 ) + ( 1 + 1 ) ) = ( 4 + 2 )
117		4p2e6 11162	. . . . . 6 |- ( 4 + 2 ) = 6
118	116, 117	eqtri 2644	. . . . 5 |- ( ( 1 x. 4 ) + ( 1 + 1 ) ) = 6
119		4t3e12 11632	. . . . . . 7 |- ( 4 x. 3 ) = ; 1 2
120	113, 37, 119	mulcomli 10047	. . . . . 6 |- ( 3 x. 4 ) = ; 1 2
121		2p1e3 11151	. . . . . 6 |- ( 2 + 1 ) = 3
122	8, 21, 8, 120, 121	decaddi 11579	. . . . 5 |- ( ( 3 x. 4 ) + 1 ) = ; 1 3
123	8, 2, 8, 8, 110, 112, 7, 2, 8, 118, 122	decmac 11566	. . . 4 |- ( (; 1 3 x. 4 ) + ( 0 + ; 1 1 ) ) = ; 6 3
124	79	mulid2i 10043	. . . . . . 7 |- ( 1 x. 8 ) = 8
125	37	addid2i 10224	. . . . . . 7 |- ( 0 + 3 ) = 3
126	124, 125	oveq12i 6662	. . . . . 6 |- ( ( 1 x. 8 ) + ( 0 + 3 ) ) = ( 8 + 3 )
127		8p3e11 11612	. . . . . 6 |- ( 8 + 3 ) = ; 1 1
128	126, 127	eqtri 2644	. . . . 5 |- ( ( 1 x. 8 ) + ( 0 + 3 ) ) = ; 1 1
129		8t3e24 11655	. . . . . . 7 |- ( 8 x. 3 ) = ; 2 4
130	79, 37, 129	mulcomli 10047	. . . . . 6 |- ( 3 x. 8 ) = ; 2 4
131	58, 113, 98	addcomli 10228	. . . . . 6 |- ( 4 + 7 ) = ; 1 1
132	21, 7, 55, 130, 121, 8, 131	decaddci 11580	. . . . 5 |- ( ( 3 x. 8 ) + 7 ) = ; 3 1
133	8, 2, 16, 55, 110, 108, 6, 8, 2, 128, 132	decmac 11566	. . . 4 |- ( (; 1 3 x. 8 ) + 7 ) = ;; 1 1 1
134	7, 6, 16, 55, 107, 108, 109, 8, 77, 123, 133	decma2c 11568	. . 3 |- ( (; 1 3 x. ; 4 8 ) + 7 ) = ;; 6 3 1
135		7lt10 11675	. . . 4 |- 7 < ; 1 0
136	4, 2, 55, 135	declti 11546	. . 3 |- 7 < ; 1 3
137	105, 106, 51, 134, 136	ndvdsi 15136	. 2 |- -. ; 1 3 || ;; 6 3 1
138	8, 51	decnncl 11518	. . 3 |- ; 1 7 e. NN
139	2, 55	deccl 11512	. . 3 |- ; 3 7 e. NN0
140		2nn 11185	. . 3 |- 2 e. NN
141		eqid 2622	. . . 4 |- ; 3 7 = ; 3 7
142	21	dec0h 11522	. . . 4 |- 2 = ; 0 2
143	8, 55	deccl 11512	. . . 4 |- ; 1 7 e. NN0
144	8, 21	deccl 11512	. . . 4 |- ; 1 2 e. NN0
145		eqid 2622	. . . . 5 |- ; 1 7 = ; 1 7
146	144	nn0cni 11304	. . . . . 6 |- ; 1 2 e. CC
147	146	addid2i 10224	. . . . 5 |- ( 0 + ; 1 2 ) = ; 1 2
148	37	mulid2i 10043	. . . . . . 7 |- ( 1 x. 3 ) = 3
149		1p2e3 11152	. . . . . . 7 |- ( 1 + 2 ) = 3
150	148, 149	oveq12i 6662	. . . . . 6 |- ( ( 1 x. 3 ) + ( 1 + 2 ) ) = ( 3 + 3 )
151		3p3e6 11161	. . . . . 6 |- ( 3 + 3 ) = 6
152	150, 151	eqtri 2644	. . . . 5 |- ( ( 1 x. 3 ) + ( 1 + 2 ) ) = 6
153		7t3e21 11649	. . . . . 6 |- ( 7 x. 3 ) = ; 2 1
154	21, 8, 21, 153, 149	decaddi 11579	. . . . 5 |- ( ( 7 x. 3 ) + 2 ) = ; 2 3
155	8, 55, 8, 21, 145, 147, 2, 2, 21, 152, 154	decmac 11566	. . . 4 |- ( (; 1 7 x. 3 ) + ( 0 + ; 1 2 ) ) = ; 6 3
156	83	addid2i 10224	. . . . . . 7 |- ( 0 + 5 ) = 5
157	93, 156	oveq12i 6662	. . . . . 6 |- ( ( 1 x. 7 ) + ( 0 + 5 ) ) = ( 7 + 5 )
158		7p5e12 11607	. . . . . 6 |- ( 7 + 5 ) = ; 1 2
159	157, 158	eqtri 2644	. . . . 5 |- ( ( 1 x. 7 ) + ( 0 + 5 ) ) = ; 1 2
160		7t7e49 11653	. . . . . 6 |- ( 7 x. 7 ) = ; 4 9
161		4p1e5 11154	. . . . . 6 |- ( 4 + 1 ) = 5
162		9p2e11 11619	. . . . . 6 |- ( 9 + 2 ) = ; 1 1
163	7, 52, 21, 160, 161, 8, 162	decaddci 11580	. . . . 5 |- ( ( 7 x. 7 ) + 2 ) = ; 5 1
164	8, 55, 16, 21, 145, 142, 55, 8, 72, 159, 163	decmac 11566	. . . 4 |- ( (; 1 7 x. 7 ) + 2 ) = ;; 1 2 1
165	2, 55, 16, 21, 141, 142, 143, 8, 144, 155, 164	decma2c 11568	. . 3 |- ( (; 1 7 x. ; 3 7 ) + 2 ) = ;; 6 3 1
166		2lt10 11680	. . . 4 |- 2 < ; 1 0
167	4, 55, 21, 166	declti 11546	. . 3 |- 2 < ; 1 7
168	138, 139, 140, 165, 167	ndvdsi 15136	. 2 |- -. ; 1 7 || ;; 6 3 1
169		9nn 11192	. . . 4 |- 9 e. NN
170	8, 169	decnncl 11518	. . 3 |- ; 1 9 e. NN
171	2, 2	deccl 11512	. . 3 |- ; 3 3 e. NN0
172		eqid 2622	. . . 4 |- ; 3 3 = ; 3 3
173	8, 52	deccl 11512	. . . 4 |- ; 1 9 e. NN0
174		eqid 2622	. . . . 5 |- ; 1 9 = ; 1 9
175	32	addid2i 10224	. . . . . 6 |- ( 0 + 6 ) = 6
176	1	dec0h 11522	. . . . . 6 |- 6 = ; 0 6
177	175, 176	eqtri 2644	. . . . 5 |- ( 0 + 6 ) = ; 0 6
178	148, 125	oveq12i 6662	. . . . . 6 |- ( ( 1 x. 3 ) + ( 0 + 3 ) ) = ( 3 + 3 )
179	178, 151	eqtri 2644	. . . . 5 |- ( ( 1 x. 3 ) + ( 0 + 3 ) ) = 6
180		9t3e27 11664	. . . . . 6 |- ( 9 x. 3 ) = ; 2 7
181		7p6e13 11608	. . . . . 6 |- ( 7 + 6 ) = ; 1 3
182	21, 55, 1, 180, 121, 2, 181	decaddci 11580	. . . . 5 |- ( ( 9 x. 3 ) + 6 ) = ; 3 3
183	8, 52, 16, 1, 174, 177, 2, 2, 2, 179, 182	decmac 11566	. . . 4 |- ( (; 1 9 x. 3 ) + ( 0 + 6 ) ) = ; 6 3
184	21, 55, 7, 180, 121, 8, 98	decaddci 11580	. . . . 5 |- ( ( 9 x. 3 ) + 4 ) = ; 3 1
185	8, 52, 16, 7, 174, 76, 2, 8, 2, 179, 184	decmac 11566	. . . 4 |- ( (; 1 9 x. 3 ) + 4 ) = ; 6 1
186	2, 2, 16, 7, 172, 76, 173, 8, 1, 183, 185	decma2c 11568	. . 3 |- ( (; 1 9 x. ; 3 3 ) + 4 ) = ;; 6 3 1
187	4, 52, 7, 102	declti 11546	. . 3 |- 4 < ; 1 9
188	170, 171, 74, 186, 187	ndvdsi 15136	. 2 |- -. ; 1 9 || ;; 6 3 1
189	21, 13	decnncl 11518	. . 3 |- ; 2 3 e. NN
190	21, 55	deccl 11512	. . 3 |- ; 2 7 e. NN0
191		10nn 11514	. . 3 |- ; 1 0 e. NN
192		eqid 2622	. . . 4 |- ; 2 7 = ; 2 7
193		eqid 2622	. . . 4 |- ; 1 0 = ; 1 0
194	21, 2	deccl 11512	. . . 4 |- ; 2 3 e. NN0
195	8, 1	deccl 11512	. . . 4 |- ; 1 6 e. NN0
196		eqid 2622	. . . . 5 |- ; 2 3 = ; 2 3
197		eqid 2622	. . . . . 6 |- ; 1 6 = ; 1 6
198		ax-1cn 9994	. . . . . . 7 |- 1 e. CC
199		6p1e7 11156	. . . . . . 7 |- ( 6 + 1 ) = 7
200	32, 198, 199	addcomli 10228	. . . . . 6 |- ( 1 + 6 ) = 7
201	16, 8, 8, 1, 25, 197, 44, 200	decadd 11570	. . . . 5 |- ( 1 + ; 1 6 ) = ; 1 7
202		2t2e4 11177	. . . . . . 7 |- ( 2 x. 2 ) = 4
203	202, 115	oveq12i 6662	. . . . . 6 |- ( ( 2 x. 2 ) + ( 1 + 1 ) ) = ( 4 + 2 )
204	203, 117	eqtri 2644	. . . . 5 |- ( ( 2 x. 2 ) + ( 1 + 1 ) ) = 6
205	30	oveq1i 6660	. . . . . 6 |- ( ( 3 x. 2 ) + 7 ) = ( 6 + 7 )
206	58, 32, 181	addcomli 10228	. . . . . 6 |- ( 6 + 7 ) = ; 1 3
207	205, 206	eqtri 2644	. . . . 5 |- ( ( 3 x. 2 ) + 7 ) = ; 1 3
208	21, 2, 8, 55, 196, 201, 21, 2, 8, 204, 207	decmac 11566	. . . 4 |- ( (; 2 3 x. 2 ) + ( 1 + ; 1 6 ) ) = ; 6 3
209		7t2e14 11648	. . . . . . . . 9 |- ( 7 x. 2 ) = ; 1 4
210	58, 17, 209	mulcomli 10047	. . . . . . . 8 |- ( 2 x. 7 ) = ; 1 4
211	8, 7, 21, 210, 117	decaddi 11579	. . . . . . 7 |- ( ( 2 x. 7 ) + 2 ) = ; 1 6
212	58, 37, 153	mulcomli 10047	. . . . . . 7 |- ( 3 x. 7 ) = ; 2 1
213	55, 21, 2, 196, 8, 21, 211, 212	decmul1c 11587	. . . . . 6 |- (; 2 3 x. 7 ) = ;; 1 6 1
214	213	oveq1i 6660	. . . . 5 |- ( (; 2 3 x. 7 ) + 0 ) = (;; 1 6 1 + 0 )
215	195, 8	deccl 11512	. . . . . . 7 |- ;; 1 6 1 e. NN0
216	215	nn0cni 11304	. . . . . 6 |- ;; 1 6 1 e. CC
217	216	addid1i 10223	. . . . 5 |- (;; 1 6 1 + 0 ) = ;; 1 6 1
218	214, 217	eqtri 2644	. . . 4 |- ( (; 2 3 x. 7 ) + 0 ) = ;; 1 6 1
219	21, 55, 8, 16, 192, 193, 194, 8, 195, 208, 218	decma2c 11568	. . 3 |- ( (; 2 3 x. ; 2 7 ) + ; 1 0 ) = ;; 6 3 1
220		10pos 11515	. . . 4 |- 0 < ; 1 0
221		1lt2 11194	. . . 4 |- 1 < 2
222	8, 21, 16, 2, 220, 221	decltc 11532	. . 3 |- ; 1 0 < ; 2 3
223	189, 190, 191, 219, 222	ndvdsi 15136	. 2 |- -. ; 2 3 || ;; 6 3 1
224	5, 12, 15, 20, 48, 50, 70, 104, 137, 168, 188, 223	prmlem2 15827	1 |- ;; 6 3 1 e. Prime

Syntax hints:   e. wcel 1990  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   2c2 11070   3c3 11071   4c4 11072   5c5 11073   6c6 11074   7c7 11075   8c8 11076   9c9 11077  ;cdc 11493   Primecprime 15385

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-rp 11833  df-fz 12327  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-prm 15386

This theorem is referenced by: (None)