Theorem 317prm 13260

Description: 317 is a prime number. (Contributed by Mario Carneiro, 19-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)

Assertion

Ref	Expression
317prm	|- ;; 3 1 7 e. Prime

Proof of Theorem 317prm

Step	Hyp	Ref	Expression
1		3nn0 9585	. . . 4 |- 3 e. NN0
2		1nn0 9583	. . . 4 |- 1 e. NN0
3	1, 2	deccl 9795	. . 3 |- ; 3 1 e. NN0
4		7nn 9475	. . 3 |- 7 e. NN
5	3, 4	decnncl 9804	. 2 |- ;; 3 1 7 e. NN
6		8nn0 9590	. . 3 |- 8 e. NN0
7		4nn0 9586	. . 3 |- 4 e. NN0
8		7nn0 9589	. . 3 |- 7 e. NN0
9		3lt8 9503	. . 3 |- 3 < 8
10		1lt10 9924	. . 3 |- 1 < ; 1 0
11		7lt10 9918	. . 3 |- 7 < ; 1 0
12	1, 6, 2, 7, 8, 2, 9, 10, 11	3decltc 9818	. 2 |- ;; 3 1 7 < ;; 8 4 1
13		1nn 9317	. . . 4 |- 1 e. NN
14	1, 13	decnncl 9804	. . 3 |- ; 3 1 e. NN
15	14, 8, 2, 10	declti 9823	. 2 |- 1 < ;; 3 1 7
16		3t2e6 9463	. . 3 |- ( 3 x. 2 ) = 6
17		df-7 9370	. . 3 |- 7 = ( 6 + 1 )
18	3, 1, 16, 17	dec2dvds 13210	. 2 |- -. 2 || ;; 3 1 7
19		3nn 9471	. . 3 |- 3 e. NN
20		10nn0 9802	. . . 4 |- ; 1 0 e. NN0
21		5nn0 9587	. . . 4 |- 5 e. NN0
22	20, 21	deccl 9795	. . 3 |- ;; 1 0 5 e. NN0
23		2nn 9470	. . 3 |- 2 e. NN
24		0nn0 9582	. . . 4 |- 0 e. NN0
25		2nn0 9584	. . . 4 |- 2 e. NN0
26		eqid 2238	. . . 4 |- ;; 1 0 5 = ;; 1 0 5
27	25	dec0h 9807	. . . 4 |- 2 = ; 0 2
28		eqid 2238	. . . . 5 |- ; 1 0 = ; 1 0
29		ax-1cn 8272	. . . . . . 7 |- 1 e. CC
30	29	addlidi 8470	. . . . . 6 |- ( 0 + 1 ) = 1
31	2	dec0h 9807	. . . . . 6 |- 1 = ; 0 1
32	30, 31	eqtri 2259	. . . . 5 |- ( 0 + 1 ) = ; 0 1
33		3cn 9381	. . . . . . . 8 |- 3 e. CC
34	33	mulridi 8328	. . . . . . 7 |- ( 3 x. 1 ) = 3
35		00id 8468	. . . . . . 7 |- ( 0 + 0 ) = 0
36	34, 35	oveq12i 6097	. . . . . 6 |- ( ( 3 x. 1 ) + ( 0 + 0 ) ) = ( 3 + 0 )
37	33	addridi 8469	. . . . . 6 |- ( 3 + 0 ) = 3
38	36, 37	eqtri 2259	. . . . 5 |- ( ( 3 x. 1 ) + ( 0 + 0 ) ) = 3
39	33	mul01i 8719	. . . . . . . 8 |- ( 3 x. 0 ) = 0
40	39	oveq1i 6095	. . . . . . 7 |- ( ( 3 x. 0 ) + 1 ) = ( 0 + 1 )
41	40, 30	eqtri 2259	. . . . . 6 |- ( ( 3 x. 0 ) + 1 ) = 1
42	41, 31	eqtri 2259	. . . . 5 |- ( ( 3 x. 0 ) + 1 ) = ; 0 1
43	2, 24, 24, 2, 28, 32, 1, 2, 24, 38, 42	decma2c 9838	. . . 4 |- ( ( 3 x. ; 1 0 ) + ( 0 + 1 ) ) = ; 3 1
44		5cn 9386	. . . . . 6 |- 5 e. CC
45		5t3e15 9886	. . . . . 6 |- ( 5 x. 3 ) = ; 1 5
46	44, 33, 45	mulcomli 8333	. . . . 5 |- ( 3 x. 5 ) = ; 1 5
47		5p2e7 9453	. . . . 5 |- ( 5 + 2 ) = 7
48	2, 21, 25, 46, 47	decaddi 9845	. . . 4 |- ( ( 3 x. 5 ) + 2 ) = ; 1 7
49	20, 21, 24, 25, 26, 27, 1, 8, 2, 43, 48	decma2c 9838	. . 3 |- ( ( 3 x. ;; 1 0 5 ) + 2 ) = ;; 3 1 7
50		2lt3 9479	. . 3 |- 2 < 3
51	19, 22, 23, 49, 50	ndvdsi 12716	. 2 |- -. 3 || ;; 3 1 7
52		2lt5 9486	. . 3 |- 2 < 5
53	3, 23, 52, 47	dec5dvds2 13212	. 2 |- -. 5 || ;; 3 1 7
54	7, 21	deccl 9795	. . 3 |- ; 4 5 e. NN0
55		eqid 2238	. . . 4 |- ; 4 5 = ; 4 5
56	33	addlidi 8470	. . . . . 6 |- ( 0 + 3 ) = 3
57	56	oveq2i 6096	. . . . 5 |- ( ( 7 x. 4 ) + ( 0 + 3 ) ) = ( ( 7 x. 4 ) + 3 )
58		7t4e28 9896	. . . . . 6 |- ( 7 x. 4 ) = ; 2 8
59		2p1e3 9440	. . . . . 6 |- ( 2 + 1 ) = 3
60		8p3e11 9866	. . . . . 6 |- ( 8 + 3 ) = ; 1 1
61	25, 6, 1, 58, 59, 2, 60	decaddci 9846	. . . . 5 |- ( ( 7 x. 4 ) + 3 ) = ; 3 1
62	57, 61	eqtri 2259	. . . 4 |- ( ( 7 x. 4 ) + ( 0 + 3 ) ) = ; 3 1
63		7t5e35 9897	. . . . 5 |- ( 7 x. 5 ) = ; 3 5
64	1, 21, 25, 63, 47	decaddi 9845	. . . 4 |- ( ( 7 x. 5 ) + 2 ) = ; 3 7
65	7, 21, 24, 25, 55, 27, 8, 8, 1, 62, 64	decma2c 9838	. . 3 |- ( ( 7 x. ; 4 5 ) + 2 ) = ;; 3 1 7
66		2lt7 9497	. . 3 |- 2 < 7
67	4, 54, 23, 65, 66	ndvdsi 12716	. 2 |- -. 7 || ;; 3 1 7
68	2, 13	decnncl 9804	. . 3 |- ; 1 1 e. NN
69	25, 6	deccl 9795	. . 3 |- ; 2 8 e. NN0
70		9nn 9477	. . 3 |- 9 e. NN
71		9nn0 9591	. . . 4 |- 9 e. NN0
72		eqid 2238	. . . 4 |- ; 2 8 = ; 2 8
73	71	dec0h 9807	. . . 4 |- 9 = ; 0 9
74	2, 2	deccl 9795	. . . 4 |- ; 1 1 e. NN0
75		eqid 2238	. . . . 5 |- ; 1 1 = ; 1 1
76		9cn 9394	. . . . . . 7 |- 9 e. CC
77	76	addlidi 8470	. . . . . 6 |- ( 0 + 9 ) = 9
78	77, 73	eqtri 2259	. . . . 5 |- ( 0 + 9 ) = ; 0 9
79		2cn 9377	. . . . . . . 8 |- 2 e. CC
80	79	mullidi 8329	. . . . . . 7 |- ( 1 x. 2 ) = 2
81	80, 30	oveq12i 6097	. . . . . 6 |- ( ( 1 x. 2 ) + ( 0 + 1 ) ) = ( 2 + 1 )
82	81, 59	eqtri 2259	. . . . 5 |- ( ( 1 x. 2 ) + ( 0 + 1 ) ) = 3
83	80	oveq1i 6095	. . . . . 6 |- ( ( 1 x. 2 ) + 9 ) = ( 2 + 9 )
84		9p2e11 9872	. . . . . . 7 |- ( 9 + 2 ) = ; 1 1
85	76, 79, 84	addcomli 8472	. . . . . 6 |- ( 2 + 9 ) = ; 1 1
86	83, 85	eqtri 2259	. . . . 5 |- ( ( 1 x. 2 ) + 9 ) = ; 1 1
87	2, 2, 24, 71, 75, 78, 25, 2, 2, 82, 86	decmac 9837	. . . 4 |- ( (; 1 1 x. 2 ) + ( 0 + 9 ) ) = ; 3 1
88		8cn 9392	. . . . . . . 8 |- 8 e. CC
89	88	mullidi 8329	. . . . . . 7 |- ( 1 x. 8 ) = 8
90	89, 30	oveq12i 6097	. . . . . 6 |- ( ( 1 x. 8 ) + ( 0 + 1 ) ) = ( 8 + 1 )
91		8p1e9 9447	. . . . . 6 |- ( 8 + 1 ) = 9
92	90, 91	eqtri 2259	. . . . 5 |- ( ( 1 x. 8 ) + ( 0 + 1 ) ) = 9
93	89	oveq1i 6095	. . . . . 6 |- ( ( 1 x. 8 ) + 9 ) = ( 8 + 9 )
94		9p8e17 9878	. . . . . . 7 |- ( 9 + 8 ) = ; 1 7
95	76, 88, 94	addcomli 8472	. . . . . 6 |- ( 8 + 9 ) = ; 1 7
96	93, 95	eqtri 2259	. . . . 5 |- ( ( 1 x. 8 ) + 9 ) = ; 1 7
97	2, 2, 24, 71, 75, 73, 6, 8, 2, 92, 96	decmac 9837	. . . 4 |- ( (; 1 1 x. 8 ) + 9 ) = ; 9 7
98	25, 6, 24, 71, 72, 73, 74, 8, 71, 87, 97	decma2c 9838	. . 3 |- ( (; 1 1 x. ; 2 8 ) + 9 ) = ;; 3 1 7
99		9lt10 9916	. . . 4 |- 9 < ; 1 0
100	13, 2, 71, 99	declti 9823	. . 3 |- 9 < ; 1 1
101	68, 69, 70, 98, 100	ndvdsi 12716	. 2 |- -. ; 1 1 || ;; 3 1 7
102	2, 19	decnncl 9804	. . 3 |- ; 1 3 e. NN
103	25, 7	deccl 9795	. . 3 |- ; 2 4 e. NN0
104		5nn 9473	. . 3 |- 5 e. NN
105		eqid 2238	. . . 4 |- ; 2 4 = ; 2 4
106	21	dec0h 9807	. . . 4 |- 5 = ; 0 5
107	2, 1	deccl 9795	. . . 4 |- ; 1 3 e. NN0
108		eqid 2238	. . . . 5 |- ; 1 3 = ; 1 3
109	44	addlidi 8470	. . . . . 6 |- ( 0 + 5 ) = 5
110	109, 106	eqtri 2259	. . . . 5 |- ( 0 + 5 ) = ; 0 5
111	16	oveq1i 6095	. . . . . 6 |- ( ( 3 x. 2 ) + 5 ) = ( 6 + 5 )
112		6p5e11 9858	. . . . . 6 |- ( 6 + 5 ) = ; 1 1
113	111, 112	eqtri 2259	. . . . 5 |- ( ( 3 x. 2 ) + 5 ) = ; 1 1
114	2, 1, 24, 21, 108, 110, 25, 2, 2, 82, 113	decmac 9837	. . . 4 |- ( (; 1 3 x. 2 ) + ( 0 + 5 ) ) = ; 3 1
115		4cn 9384	. . . . . . . 8 |- 4 e. CC
116	115	mullidi 8329	. . . . . . 7 |- ( 1 x. 4 ) = 4
117	116, 30	oveq12i 6097	. . . . . 6 |- ( ( 1 x. 4 ) + ( 0 + 1 ) ) = ( 4 + 1 )
118		4p1e5 9443	. . . . . 6 |- ( 4 + 1 ) = 5
119	117, 118	eqtri 2259	. . . . 5 |- ( ( 1 x. 4 ) + ( 0 + 1 ) ) = 5
120		4t3e12 9883	. . . . . . 7 |- ( 4 x. 3 ) = ; 1 2
121	115, 33, 120	mulcomli 8333	. . . . . 6 |- ( 3 x. 4 ) = ; 1 2
122	44, 79, 47	addcomli 8472	. . . . . 6 |- ( 2 + 5 ) = 7
123	2, 25, 21, 121, 122	decaddi 9845	. . . . 5 |- ( ( 3 x. 4 ) + 5 ) = ; 1 7
124	2, 1, 24, 21, 108, 106, 7, 8, 2, 119, 123	decmac 9837	. . . 4 |- ( (; 1 3 x. 4 ) + 5 ) = ; 5 7
125	25, 7, 24, 21, 105, 106, 107, 8, 21, 114, 124	decma2c 9838	. . 3 |- ( (; 1 3 x. ; 2 4 ) + 5 ) = ;; 3 1 7
126		5lt10 9920	. . . 4 |- 5 < ; 1 0
127	13, 1, 21, 126	declti 9823	. . 3 |- 5 < ; 1 3
128	102, 103, 104, 125, 127	ndvdsi 12716	. 2 |- -. ; 1 3 || ;; 3 1 7
129	2, 4	decnncl 9804	. . 3 |- ; 1 7 e. NN
130	2, 6	deccl 9795	. . 3 |- ; 1 8 e. NN0
131		eqid 2238	. . . 4 |- ; 1 8 = ; 1 8
132	2, 8	deccl 9795	. . . 4 |- ; 1 7 e. NN0
133		eqid 2238	. . . . 5 |- ; 1 7 = ; 1 7
134		3p1e4 9442	. . . . . . 7 |- ( 3 + 1 ) = 4
135	33, 29, 134	addcomli 8472	. . . . . 6 |- ( 1 + 3 ) = 4
136	24, 2, 2, 1, 31, 108, 30, 135	decadd 9839	. . . . 5 |- ( 1 + ; 1 3 ) = ; 1 4
137	29	mulridi 8328	. . . . . . 7 |- ( 1 x. 1 ) = 1
138		1p1e2 9423	. . . . . . 7 |- ( 1 + 1 ) = 2
139	137, 138	oveq12i 6097	. . . . . 6 |- ( ( 1 x. 1 ) + ( 1 + 1 ) ) = ( 1 + 2 )
140		1p2e3 9441	. . . . . 6 |- ( 1 + 2 ) = 3
141	139, 140	eqtri 2259	. . . . 5 |- ( ( 1 x. 1 ) + ( 1 + 1 ) ) = 3
142		7cn 9390	. . . . . . . 8 |- 7 e. CC
143	142	mulridi 8328	. . . . . . 7 |- ( 7 x. 1 ) = 7
144	143	oveq1i 6095	. . . . . 6 |- ( ( 7 x. 1 ) + 4 ) = ( 7 + 4 )
145		7p4e11 9861	. . . . . 6 |- ( 7 + 4 ) = ; 1 1
146	144, 145	eqtri 2259	. . . . 5 |- ( ( 7 x. 1 ) + 4 ) = ; 1 1
147	2, 8, 2, 7, 133, 136, 2, 2, 2, 141, 146	decmac 9837	. . . 4 |- ( (; 1 7 x. 1 ) + ( 1 + ; 1 3 ) ) = ; 3 1
148	89, 109	oveq12i 6097	. . . . . 6 |- ( ( 1 x. 8 ) + ( 0 + 5 ) ) = ( 8 + 5 )
149		8p5e13 9868	. . . . . 6 |- ( 8 + 5 ) = ; 1 3
150	148, 149	eqtri 2259	. . . . 5 |- ( ( 1 x. 8 ) + ( 0 + 5 ) ) = ; 1 3
151		6nn0 9588	. . . . . 6 |- 6 e. NN0
152		6p1e7 9445	. . . . . 6 |- ( 6 + 1 ) = 7
153		8t7e56 9905	. . . . . . 7 |- ( 8 x. 7 ) = ; 5 6
154	88, 142, 153	mulcomli 8333	. . . . . 6 |- ( 7 x. 8 ) = ; 5 6
155	21, 151, 152, 154	decsuc 9816	. . . . 5 |- ( ( 7 x. 8 ) + 1 ) = ; 5 7
156	2, 8, 24, 2, 133, 31, 6, 8, 21, 150, 155	decmac 9837	. . . 4 |- ( (; 1 7 x. 8 ) + 1 ) = ;; 1 3 7
157	2, 6, 2, 2, 131, 75, 132, 8, 107, 147, 156	decma2c 9838	. . 3 |- ( (; 1 7 x. ; 1 8 ) + ; 1 1 ) = ;; 3 1 7
158		1lt7 9498	. . . 4 |- 1 < 7
159	2, 2, 4, 158	declt 9813	. . 3 |- ; 1 1 < ; 1 7
160	129, 130, 68, 157, 159	ndvdsi 12716	. 2 |- -. ; 1 7 || ;; 3 1 7
161	2, 70	decnncl 9804	. . 3 |- ; 1 9 e. NN
162	2, 151	deccl 9795	. . 3 |- ; 1 6 e. NN0
163		eqid 2238	. . . 4 |- ; 1 6 = ; 1 6
164	2, 71	deccl 9795	. . . 4 |- ; 1 9 e. NN0
165		eqid 2238	. . . . 5 |- ; 1 9 = ; 1 9
166	24, 2, 2, 2, 31, 75, 30, 138	decadd 9839	. . . . 5 |- ( 1 + ; 1 1 ) = ; 1 2
167	76	mulridi 8328	. . . . . . 7 |- ( 9 x. 1 ) = 9
168	167	oveq1i 6095	. . . . . 6 |- ( ( 9 x. 1 ) + 2 ) = ( 9 + 2 )
169	168, 84	eqtri 2259	. . . . 5 |- ( ( 9 x. 1 ) + 2 ) = ; 1 1
170	2, 71, 2, 25, 165, 166, 2, 2, 2, 141, 169	decmac 9837	. . . 4 |- ( (; 1 9 x. 1 ) + ( 1 + ; 1 1 ) ) = ; 3 1
171	1	dec0h 9807	. . . . 5 |- 3 = ; 0 3
172		6cn 9388	. . . . . . . 8 |- 6 e. CC
173	172	mullidi 8329	. . . . . . 7 |- ( 1 x. 6 ) = 6
174	173, 109	oveq12i 6097	. . . . . 6 |- ( ( 1 x. 6 ) + ( 0 + 5 ) ) = ( 6 + 5 )
175	174, 112	eqtri 2259	. . . . 5 |- ( ( 1 x. 6 ) + ( 0 + 5 ) ) = ; 1 1
176		9t6e54 9911	. . . . . 6 |- ( 9 x. 6 ) = ; 5 4
177		4p3e7 9451	. . . . . 6 |- ( 4 + 3 ) = 7
178	21, 7, 1, 176, 177	decaddi 9845	. . . . 5 |- ( ( 9 x. 6 ) + 3 ) = ; 5 7
179	2, 71, 24, 1, 165, 171, 151, 8, 21, 175, 178	decmac 9837	. . . 4 |- ( (; 1 9 x. 6 ) + 3 ) = ;; 1 1 7
180	2, 151, 2, 1, 163, 108, 164, 8, 74, 170, 179	decma2c 9838	. . 3 |- ( (; 1 9 x. ; 1 6 ) + ; 1 3 ) = ;; 3 1 7
181		3lt9 9511	. . . 4 |- 3 < 9
182	2, 1, 70, 181	declt 9813	. . 3 |- ; 1 3 < ; 1 9
183	161, 162, 102, 180, 182	ndvdsi 12716	. 2 |- -. ; 1 9 || ;; 3 1 7
184	25, 19	decnncl 9804	. . 3 |- ; 2 3 e. NN
185	102	nnnn0i 9575	. . 3 |- ; 1 3 e. NN0
186		8nn 9476	. . . 4 |- 8 e. NN
187	2, 186	decnncl 9804	. . 3 |- ; 1 8 e. NN
188	25, 1	deccl 9795	. . . 4 |- ; 2 3 e. NN0
189		eqid 2238	. . . . 5 |- ; 2 3 = ; 2 3
190		7p1e8 9446	. . . . . . 7 |- ( 7 + 1 ) = 8
191	142, 29, 190	addcomli 8472	. . . . . 6 |- ( 1 + 7 ) = 8
192	6	dec0h 9807	. . . . . 6 |- 8 = ; 0 8
193	191, 192	eqtri 2259	. . . . 5 |- ( 1 + 7 ) = ; 0 8
194	79	mulridi 8328	. . . . . . 7 |- ( 2 x. 1 ) = 2
195	194, 30	oveq12i 6097	. . . . . 6 |- ( ( 2 x. 1 ) + ( 0 + 1 ) ) = ( 2 + 1 )
196	195, 59	eqtri 2259	. . . . 5 |- ( ( 2 x. 1 ) + ( 0 + 1 ) ) = 3
197	34	oveq1i 6095	. . . . . 6 |- ( ( 3 x. 1 ) + 8 ) = ( 3 + 8 )
198	88, 33, 60	addcomli 8472	. . . . . 6 |- ( 3 + 8 ) = ; 1 1
199	197, 198	eqtri 2259	. . . . 5 |- ( ( 3 x. 1 ) + 8 ) = ; 1 1
200	25, 1, 24, 6, 189, 193, 2, 2, 2, 196, 199	decmac 9837	. . . 4 |- ( (; 2 3 x. 1 ) + ( 1 + 7 ) ) = ; 3 1
201	33, 79, 16	mulcomli 8333	. . . . . . 7 |- ( 2 x. 3 ) = 6
202	201, 30	oveq12i 6097	. . . . . 6 |- ( ( 2 x. 3 ) + ( 0 + 1 ) ) = ( 6 + 1 )
203	202, 152	eqtri 2259	. . . . 5 |- ( ( 2 x. 3 ) + ( 0 + 1 ) ) = 7
204		3t3e9 9465	. . . . . . 7 |- ( 3 x. 3 ) = 9
205	204	oveq1i 6095	. . . . . 6 |- ( ( 3 x. 3 ) + 8 ) = ( 9 + 8 )
206	205, 94	eqtri 2259	. . . . 5 |- ( ( 3 x. 3 ) + 8 ) = ; 1 7
207	25, 1, 24, 6, 189, 192, 1, 8, 2, 203, 206	decmac 9837	. . . 4 |- ( (; 2 3 x. 3 ) + 8 ) = ; 7 7
208	2, 1, 2, 6, 108, 131, 188, 8, 8, 200, 207	decma2c 9838	. . 3 |- ( (; 2 3 x. ; 1 3 ) + ; 1 8 ) = ;; 3 1 7
209		8lt10 9917	. . . 4 |- 8 < ; 1 0
210		1lt2 9478	. . . 4 |- 1 < 2
211	2, 25, 6, 1, 209, 210	decltc 9814	. . 3 |- ; 1 8 < ; 2 3
212	184, 185, 187, 208, 211	ndvdsi 12716	. 2 |- -. ; 2 3 || ;; 3 1 7
213	5, 12, 15, 18, 51, 53, 67, 101, 128, 160, 183, 212	prmlem2 13254	1 |- ;; 3 1 7 e. Prime

This proof depends on syntax axioms:   e. wcel 2209  (class class class)co 6085   0cc0 8179   1c1 8180   + caddc 8182   x. cmul 8184   2c2 9357   3c3 9358   4c4 9359   5c5 9360   6c6 9361   7c7 9362   8c8 9363   9c9 9364  ;cdc 9781   Primecprime 12901

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 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299

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 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-1o 6687  df-2o 6688  df-er 6807  df-en 7023  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8500  df-neg 8501  df-reap 8905  df-ap 8912  df-div 9005  df-inn 9307  df-2 9365  df-3 9366  df-4 9367  df-5 9368  df-6 9369  df-7 9370  df-8 9371  df-9 9372  df-n0 9568  df-z 9649  df-dec 9782  df-uz 9931  df-q 10029  df-rp 10065  df-fz 10422  df-fzo 10560  df-fl 10715  df-mod 10773  df-seqfrec 10898  df-exp 10989  df-cj 11621  df-re 11622  df-im 11623  df-rsqrt 11778  df-abs 11779  df-dvds 12571  df-prm 12902

This theorem is used by: (None)