Theorem 317prm 13269

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 9592	. . . 4 |- 3 e. NN0
2		1nn0 9590	. . . 4 |- 1 e. NN0
3	1, 2	deccl 9802	. . 3 |- ; 3 1 e. NN0
4		7nn 9482	. . 3 |- 7 e. NN
5	3, 4	decnncl 9811	. 2 |- ;; 3 1 7 e. NN
6		8nn0 9597	. . 3 |- 8 e. NN0
7		4nn0 9593	. . 3 |- 4 e. NN0
8		7nn0 9596	. . 3 |- 7 e. NN0
9		3lt8 9510	. . 3 |- 3 < 8
10		1lt10 9931	. . 3 |- 1 < ; 1 0
11		7lt10 9925	. . 3 |- 7 < ; 1 0
12	1, 6, 2, 7, 8, 2, 9, 10, 11	3decltc 9825	. 2 |- ;; 3 1 7 < ;; 8 4 1
13		1nn 9324	. . . 4 |- 1 e. NN
14	1, 13	decnncl 9811	. . 3 |- ; 3 1 e. NN
15	14, 8, 2, 10	declti 9830	. 2 |- 1 < ;; 3 1 7
16		3t2e6 9470	. . 3 |- ( 3 x. 2 ) = 6
17		df-7 9377	. . 3 |- 7 = ( 6 + 1 )
18	3, 1, 16, 17	dec2dvds 13219	. 2 |- -. 2 || ;; 3 1 7
19		3nn 9478	. . 3 |- 3 e. NN
20		10nn0 9809	. . . 4 |- ; 1 0 e. NN0
21		5nn0 9594	. . . 4 |- 5 e. NN0
22	20, 21	deccl 9802	. . 3 |- ;; 1 0 5 e. NN0
23		2nn 9477	. . 3 |- 2 e. NN
24		0nn0 9589	. . . 4 |- 0 e. NN0
25		2nn0 9591	. . . 4 |- 2 e. NN0
26		eqid 2238	. . . 4 |- ;; 1 0 5 = ;; 1 0 5
27	25	dec0h 9814	. . . 4 |- 2 = ; 0 2
28		eqid 2238	. . . . 5 |- ; 1 0 = ; 1 0
29		ax-1cn 8278	. . . . . . 7 |- 1 e. CC
30	29	addlidi 8476	. . . . . 6 |- ( 0 + 1 ) = 1
31	2	dec0h 9814	. . . . . 6 |- 1 = ; 0 1
32	30, 31	eqtri 2259	. . . . 5 |- ( 0 + 1 ) = ; 0 1
33		3cn 9388	. . . . . . . 8 |- 3 e. CC
34	33	mulridi 8334	. . . . . . 7 |- ( 3 x. 1 ) = 3
35		00id 8474	. . . . . . 7 |- ( 0 + 0 ) = 0
36	34, 35	oveq12i 6100	. . . . . 6 |- ( ( 3 x. 1 ) + ( 0 + 0 ) ) = ( 3 + 0 )
37	33	addridi 8475	. . . . . 6 |- ( 3 + 0 ) = 3
38	36, 37	eqtri 2259	. . . . 5 |- ( ( 3 x. 1 ) + ( 0 + 0 ) ) = 3
39	33	mul01i 8725	. . . . . . . 8 |- ( 3 x. 0 ) = 0
40	39	oveq1i 6098	. . . . . . 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 9845	. . . 4 |- ( ( 3 x. ; 1 0 ) + ( 0 + 1 ) ) = ; 3 1
44		5cn 9393	. . . . . 6 |- 5 e. CC
45		5t3e15 9893	. . . . . 6 |- ( 5 x. 3 ) = ; 1 5
46	44, 33, 45	mulcomli 8339	. . . . 5 |- ( 3 x. 5 ) = ; 1 5
47		5p2e7 9460	. . . . 5 |- ( 5 + 2 ) = 7
48	2, 21, 25, 46, 47	decaddi 9852	. . . 4 |- ( ( 3 x. 5 ) + 2 ) = ; 1 7
49	20, 21, 24, 25, 26, 27, 1, 8, 2, 43, 48	decma2c 9845	. . 3 |- ( ( 3 x. ;; 1 0 5 ) + 2 ) = ;; 3 1 7
50		2lt3 9486	. . 3 |- 2 < 3
51	19, 22, 23, 49, 50	ndvdsi 12725	. 2 |- -. 3 || ;; 3 1 7
52		2lt5 9493	. . 3 |- 2 < 5
53	3, 23, 52, 47	dec5dvds2 13221	. 2 |- -. 5 || ;; 3 1 7
54	7, 21	deccl 9802	. . 3 |- ; 4 5 e. NN0
55		eqid 2238	. . . 4 |- ; 4 5 = ; 4 5
56	33	addlidi 8476	. . . . . 6 |- ( 0 + 3 ) = 3
57	56	oveq2i 6099	. . . . 5 |- ( ( 7 x. 4 ) + ( 0 + 3 ) ) = ( ( 7 x. 4 ) + 3 )
58		7t4e28 9903	. . . . . 6 |- ( 7 x. 4 ) = ; 2 8
59		2p1e3 9447	. . . . . 6 |- ( 2 + 1 ) = 3
60		8p3e11 9873	. . . . . 6 |- ( 8 + 3 ) = ; 1 1
61	25, 6, 1, 58, 59, 2, 60	decaddci 9853	. . . . 5 |- ( ( 7 x. 4 ) + 3 ) = ; 3 1
62	57, 61	eqtri 2259	. . . 4 |- ( ( 7 x. 4 ) + ( 0 + 3 ) ) = ; 3 1
63		7t5e35 9904	. . . . 5 |- ( 7 x. 5 ) = ; 3 5
64	1, 21, 25, 63, 47	decaddi 9852	. . . 4 |- ( ( 7 x. 5 ) + 2 ) = ; 3 7
65	7, 21, 24, 25, 55, 27, 8, 8, 1, 62, 64	decma2c 9845	. . 3 |- ( ( 7 x. ; 4 5 ) + 2 ) = ;; 3 1 7
66		2lt7 9504	. . 3 |- 2 < 7
67	4, 54, 23, 65, 66	ndvdsi 12725	. 2 |- -. 7 || ;; 3 1 7
68	2, 13	decnncl 9811	. . 3 |- ; 1 1 e. NN
69	25, 6	deccl 9802	. . 3 |- ; 2 8 e. NN0
70		9nn 9484	. . 3 |- 9 e. NN
71		9nn0 9598	. . . 4 |- 9 e. NN0
72		eqid 2238	. . . 4 |- ; 2 8 = ; 2 8
73	71	dec0h 9814	. . . 4 |- 9 = ; 0 9
74	2, 2	deccl 9802	. . . 4 |- ; 1 1 e. NN0
75		eqid 2238	. . . . 5 |- ; 1 1 = ; 1 1
76		9cn 9401	. . . . . . 7 |- 9 e. CC
77	76	addlidi 8476	. . . . . 6 |- ( 0 + 9 ) = 9
78	77, 73	eqtri 2259	. . . . 5 |- ( 0 + 9 ) = ; 0 9
79		2cn 9384	. . . . . . . 8 |- 2 e. CC
80	79	mullidi 8335	. . . . . . 7 |- ( 1 x. 2 ) = 2
81	80, 30	oveq12i 6100	. . . . . 6 |- ( ( 1 x. 2 ) + ( 0 + 1 ) ) = ( 2 + 1 )
82	81, 59	eqtri 2259	. . . . 5 |- ( ( 1 x. 2 ) + ( 0 + 1 ) ) = 3
83	80	oveq1i 6098	. . . . . 6 |- ( ( 1 x. 2 ) + 9 ) = ( 2 + 9 )
84		9p2e11 9879	. . . . . . 7 |- ( 9 + 2 ) = ; 1 1
85	76, 79, 84	addcomli 8478	. . . . . 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 9844	. . . 4 |- ( (; 1 1 x. 2 ) + ( 0 + 9 ) ) = ; 3 1
88		8cn 9399	. . . . . . . 8 |- 8 e. CC
89	88	mullidi 8335	. . . . . . 7 |- ( 1 x. 8 ) = 8
90	89, 30	oveq12i 6100	. . . . . 6 |- ( ( 1 x. 8 ) + ( 0 + 1 ) ) = ( 8 + 1 )
91		8p1e9 9454	. . . . . 6 |- ( 8 + 1 ) = 9
92	90, 91	eqtri 2259	. . . . 5 |- ( ( 1 x. 8 ) + ( 0 + 1 ) ) = 9
93	89	oveq1i 6098	. . . . . 6 |- ( ( 1 x. 8 ) + 9 ) = ( 8 + 9 )
94		9p8e17 9885	. . . . . . 7 |- ( 9 + 8 ) = ; 1 7
95	76, 88, 94	addcomli 8478	. . . . . 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 9844	. . . 4 |- ( (; 1 1 x. 8 ) + 9 ) = ; 9 7
98	25, 6, 24, 71, 72, 73, 74, 8, 71, 87, 97	decma2c 9845	. . 3 |- ( (; 1 1 x. ; 2 8 ) + 9 ) = ;; 3 1 7
99		9lt10 9923	. . . 4 |- 9 < ; 1 0
100	13, 2, 71, 99	declti 9830	. . 3 |- 9 < ; 1 1
101	68, 69, 70, 98, 100	ndvdsi 12725	. 2 |- -. ; 1 1 || ;; 3 1 7
102	2, 19	decnncl 9811	. . 3 |- ; 1 3 e. NN
103	25, 7	deccl 9802	. . 3 |- ; 2 4 e. NN0
104		5nn 9480	. . 3 |- 5 e. NN
105		eqid 2238	. . . 4 |- ; 2 4 = ; 2 4
106	21	dec0h 9814	. . . 4 |- 5 = ; 0 5
107	2, 1	deccl 9802	. . . 4 |- ; 1 3 e. NN0
108		eqid 2238	. . . . 5 |- ; 1 3 = ; 1 3
109	44	addlidi 8476	. . . . . 6 |- ( 0 + 5 ) = 5
110	109, 106	eqtri 2259	. . . . 5 |- ( 0 + 5 ) = ; 0 5
111	16	oveq1i 6098	. . . . . 6 |- ( ( 3 x. 2 ) + 5 ) = ( 6 + 5 )
112		6p5e11 9865	. . . . . 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 9844	. . . 4 |- ( (; 1 3 x. 2 ) + ( 0 + 5 ) ) = ; 3 1
115		4cn 9391	. . . . . . . 8 |- 4 e. CC
116	115	mullidi 8335	. . . . . . 7 |- ( 1 x. 4 ) = 4
117	116, 30	oveq12i 6100	. . . . . 6 |- ( ( 1 x. 4 ) + ( 0 + 1 ) ) = ( 4 + 1 )
118		4p1e5 9450	. . . . . 6 |- ( 4 + 1 ) = 5
119	117, 118	eqtri 2259	. . . . 5 |- ( ( 1 x. 4 ) + ( 0 + 1 ) ) = 5
120		4t3e12 9890	. . . . . . 7 |- ( 4 x. 3 ) = ; 1 2
121	115, 33, 120	mulcomli 8339	. . . . . 6 |- ( 3 x. 4 ) = ; 1 2
122	44, 79, 47	addcomli 8478	. . . . . 6 |- ( 2 + 5 ) = 7
123	2, 25, 21, 121, 122	decaddi 9852	. . . . 5 |- ( ( 3 x. 4 ) + 5 ) = ; 1 7
124	2, 1, 24, 21, 108, 106, 7, 8, 2, 119, 123	decmac 9844	. . . 4 |- ( (; 1 3 x. 4 ) + 5 ) = ; 5 7
125	25, 7, 24, 21, 105, 106, 107, 8, 21, 114, 124	decma2c 9845	. . 3 |- ( (; 1 3 x. ; 2 4 ) + 5 ) = ;; 3 1 7
126		5lt10 9927	. . . 4 |- 5 < ; 1 0
127	13, 1, 21, 126	declti 9830	. . 3 |- 5 < ; 1 3
128	102, 103, 104, 125, 127	ndvdsi 12725	. 2 |- -. ; 1 3 || ;; 3 1 7
129	2, 4	decnncl 9811	. . 3 |- ; 1 7 e. NN
130	2, 6	deccl 9802	. . 3 |- ; 1 8 e. NN0
131		eqid 2238	. . . 4 |- ; 1 8 = ; 1 8
132	2, 8	deccl 9802	. . . 4 |- ; 1 7 e. NN0
133		eqid 2238	. . . . 5 |- ; 1 7 = ; 1 7
134		3p1e4 9449	. . . . . . 7 |- ( 3 + 1 ) = 4
135	33, 29, 134	addcomli 8478	. . . . . 6 |- ( 1 + 3 ) = 4
136	24, 2, 2, 1, 31, 108, 30, 135	decadd 9846	. . . . 5 |- ( 1 + ; 1 3 ) = ; 1 4
137	29	mulridi 8334	. . . . . . 7 |- ( 1 x. 1 ) = 1
138		1p1e2 9430	. . . . . . 7 |- ( 1 + 1 ) = 2
139	137, 138	oveq12i 6100	. . . . . 6 |- ( ( 1 x. 1 ) + ( 1 + 1 ) ) = ( 1 + 2 )
140		1p2e3 9448	. . . . . 6 |- ( 1 + 2 ) = 3
141	139, 140	eqtri 2259	. . . . 5 |- ( ( 1 x. 1 ) + ( 1 + 1 ) ) = 3
142		7cn 9397	. . . . . . . 8 |- 7 e. CC
143	142	mulridi 8334	. . . . . . 7 |- ( 7 x. 1 ) = 7
144	143	oveq1i 6098	. . . . . 6 |- ( ( 7 x. 1 ) + 4 ) = ( 7 + 4 )
145		7p4e11 9868	. . . . . 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 9844	. . . 4 |- ( (; 1 7 x. 1 ) + ( 1 + ; 1 3 ) ) = ; 3 1
148	89, 109	oveq12i 6100	. . . . . 6 |- ( ( 1 x. 8 ) + ( 0 + 5 ) ) = ( 8 + 5 )
149		8p5e13 9875	. . . . . 6 |- ( 8 + 5 ) = ; 1 3
150	148, 149	eqtri 2259	. . . . 5 |- ( ( 1 x. 8 ) + ( 0 + 5 ) ) = ; 1 3
151		6nn0 9595	. . . . . 6 |- 6 e. NN0
152		6p1e7 9452	. . . . . 6 |- ( 6 + 1 ) = 7
153		8t7e56 9912	. . . . . . 7 |- ( 8 x. 7 ) = ; 5 6
154	88, 142, 153	mulcomli 8339	. . . . . 6 |- ( 7 x. 8 ) = ; 5 6
155	21, 151, 152, 154	decsuc 9823	. . . . 5 |- ( ( 7 x. 8 ) + 1 ) = ; 5 7
156	2, 8, 24, 2, 133, 31, 6, 8, 21, 150, 155	decmac 9844	. . . 4 |- ( (; 1 7 x. 8 ) + 1 ) = ;; 1 3 7
157	2, 6, 2, 2, 131, 75, 132, 8, 107, 147, 156	decma2c 9845	. . 3 |- ( (; 1 7 x. ; 1 8 ) + ; 1 1 ) = ;; 3 1 7
158		1lt7 9505	. . . 4 |- 1 < 7
159	2, 2, 4, 158	declt 9820	. . 3 |- ; 1 1 < ; 1 7
160	129, 130, 68, 157, 159	ndvdsi 12725	. 2 |- -. ; 1 7 || ;; 3 1 7
161	2, 70	decnncl 9811	. . 3 |- ; 1 9 e. NN
162	2, 151	deccl 9802	. . 3 |- ; 1 6 e. NN0
163		eqid 2238	. . . 4 |- ; 1 6 = ; 1 6
164	2, 71	deccl 9802	. . . 4 |- ; 1 9 e. NN0
165		eqid 2238	. . . . 5 |- ; 1 9 = ; 1 9
166	24, 2, 2, 2, 31, 75, 30, 138	decadd 9846	. . . . 5 |- ( 1 + ; 1 1 ) = ; 1 2
167	76	mulridi 8334	. . . . . . 7 |- ( 9 x. 1 ) = 9
168	167	oveq1i 6098	. . . . . 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 9844	. . . 4 |- ( (; 1 9 x. 1 ) + ( 1 + ; 1 1 ) ) = ; 3 1
171	1	dec0h 9814	. . . . 5 |- 3 = ; 0 3
172		6cn 9395	. . . . . . . 8 |- 6 e. CC
173	172	mullidi 8335	. . . . . . 7 |- ( 1 x. 6 ) = 6
174	173, 109	oveq12i 6100	. . . . . 6 |- ( ( 1 x. 6 ) + ( 0 + 5 ) ) = ( 6 + 5 )
175	174, 112	eqtri 2259	. . . . 5 |- ( ( 1 x. 6 ) + ( 0 + 5 ) ) = ; 1 1
176		9t6e54 9918	. . . . . 6 |- ( 9 x. 6 ) = ; 5 4
177		4p3e7 9458	. . . . . 6 |- ( 4 + 3 ) = 7
178	21, 7, 1, 176, 177	decaddi 9852	. . . . 5 |- ( ( 9 x. 6 ) + 3 ) = ; 5 7
179	2, 71, 24, 1, 165, 171, 151, 8, 21, 175, 178	decmac 9844	. . . 4 |- ( (; 1 9 x. 6 ) + 3 ) = ;; 1 1 7
180	2, 151, 2, 1, 163, 108, 164, 8, 74, 170, 179	decma2c 9845	. . 3 |- ( (; 1 9 x. ; 1 6 ) + ; 1 3 ) = ;; 3 1 7
181		3lt9 9518	. . . 4 |- 3 < 9
182	2, 1, 70, 181	declt 9820	. . 3 |- ; 1 3 < ; 1 9
183	161, 162, 102, 180, 182	ndvdsi 12725	. 2 |- -. ; 1 9 || ;; 3 1 7
184	25, 19	decnncl 9811	. . 3 |- ; 2 3 e. NN
185	102	nnnn0i 9582	. . 3 |- ; 1 3 e. NN0
186		8nn 9483	. . . 4 |- 8 e. NN
187	2, 186	decnncl 9811	. . 3 |- ; 1 8 e. NN
188	25, 1	deccl 9802	. . . 4 |- ; 2 3 e. NN0
189		eqid 2238	. . . . 5 |- ; 2 3 = ; 2 3
190		7p1e8 9453	. . . . . . 7 |- ( 7 + 1 ) = 8
191	142, 29, 190	addcomli 8478	. . . . . 6 |- ( 1 + 7 ) = 8
192	6	dec0h 9814	. . . . . 6 |- 8 = ; 0 8
193	191, 192	eqtri 2259	. . . . 5 |- ( 1 + 7 ) = ; 0 8
194	79	mulridi 8334	. . . . . . 7 |- ( 2 x. 1 ) = 2
195	194, 30	oveq12i 6100	. . . . . 6 |- ( ( 2 x. 1 ) + ( 0 + 1 ) ) = ( 2 + 1 )
196	195, 59	eqtri 2259	. . . . 5 |- ( ( 2 x. 1 ) + ( 0 + 1 ) ) = 3
197	34	oveq1i 6098	. . . . . 6 |- ( ( 3 x. 1 ) + 8 ) = ( 3 + 8 )
198	88, 33, 60	addcomli 8478	. . . . . 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 9844	. . . 4 |- ( (; 2 3 x. 1 ) + ( 1 + 7 ) ) = ; 3 1
201	33, 79, 16	mulcomli 8339	. . . . . . 7 |- ( 2 x. 3 ) = 6
202	201, 30	oveq12i 6100	. . . . . 6 |- ( ( 2 x. 3 ) + ( 0 + 1 ) ) = ( 6 + 1 )
203	202, 152	eqtri 2259	. . . . 5 |- ( ( 2 x. 3 ) + ( 0 + 1 ) ) = 7
204		3t3e9 9472	. . . . . . 7 |- ( 3 x. 3 ) = 9
205	204	oveq1i 6098	. . . . . 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 9844	. . . 4 |- ( (; 2 3 x. 3 ) + 8 ) = ; 7 7
208	2, 1, 2, 6, 108, 131, 188, 8, 8, 200, 207	decma2c 9845	. . 3 |- ( (; 2 3 x. ; 1 3 ) + ; 1 8 ) = ;; 3 1 7
209		8lt10 9924	. . . 4 |- 8 < ; 1 0
210		1lt2 9485	. . . 4 |- 1 < 2
211	2, 25, 6, 1, 209, 210	decltc 9821	. . 3 |- ; 1 8 < ; 2 3
212	184, 185, 187, 208, 211	ndvdsi 12725	. 2 |- -. ; 2 3 || ;; 3 1 7
213	5, 12, 15, 18, 51, 53, 67, 101, 128, 160, 183, 212	prmlem2 13263	1 |- ;; 3 1 7 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)