Theorem 317prm 15833

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 11310	. . . 4 |- 3 e. NN0
2		1nn0 11308	. . . 4 |- 1 e. NN0
3	1, 2	deccl 11512	. . 3 |- ; 3 1 e. NN0
4		7nn 11190	. . 3 |- 7 e. NN
5	3, 4	decnncl 11518	. 2 |- ;; 3 1 7 e. NN
6		8nn0 11315	. . 3 |- 8 e. NN0
7		4nn0 11311	. . 3 |- 4 e. NN0
8		7nn0 11314	. . 3 |- 7 e. NN0
9		3lt8 11219	. . 3 |- 3 < 8
10		1lt10 11681	. . 3 |- 1 < ; 1 0
11		7lt10 11675	. . 3 |- 7 < ; 1 0
12	1, 6, 2, 7, 8, 2, 9, 10, 11	3decltc 11538	. 2 |- ;; 3 1 7 < ;; 8 4 1
13		1nn 11031	. . . 4 |- 1 e. NN
14	1, 13	decnncl 11518	. . 3 |- ; 3 1 e. NN
15	14, 8, 2, 10	declti 11546	. 2 |- 1 < ;; 3 1 7
16		3t2e6 11179	. . 3 |- ( 3 x. 2 ) = 6
17		df-7 11084	. . 3 |- 7 = ( 6 + 1 )
18	3, 1, 16, 17	dec2dvds 15767	. 2 |- -. 2 || ;; 3 1 7
19		3nn 11186	. . 3 |- 3 e. NN
20		10nn0 11516	. . . 4 |- ; 1 0 e. NN0
21		5nn0 11312	. . . 4 |- 5 e. NN0
22	20, 21	deccl 11512	. . 3 |- ;; 1 0 5 e. NN0
23		2nn 11185	. . 3 |- 2 e. NN
24		0nn0 11307	. . . 4 |- 0 e. NN0
25		2nn0 11309	. . . 4 |- 2 e. NN0
26		eqid 2622	. . . 4 |- ;; 1 0 5 = ;; 1 0 5
27	25	dec0h 11522	. . . 4 |- 2 = ; 0 2
28		eqid 2622	. . . . 5 |- ; 1 0 = ; 1 0
29		ax-1cn 9994	. . . . . . 7 |- 1 e. CC
30	29	addid2i 10224	. . . . . 6 |- ( 0 + 1 ) = 1
31	2	dec0h 11522	. . . . . 6 |- 1 = ; 0 1
32	30, 31	eqtri 2644	. . . . 5 |- ( 0 + 1 ) = ; 0 1
33		3cn 11095	. . . . . . . 8 |- 3 e. CC
34	33	mulid1i 10042	. . . . . . 7 |- ( 3 x. 1 ) = 3
35		00id 10211	. . . . . . 7 |- ( 0 + 0 ) = 0
36	34, 35	oveq12i 6662	. . . . . 6 |- ( ( 3 x. 1 ) + ( 0 + 0 ) ) = ( 3 + 0 )
37	33	addid1i 10223	. . . . . 6 |- ( 3 + 0 ) = 3
38	36, 37	eqtri 2644	. . . . 5 |- ( ( 3 x. 1 ) + ( 0 + 0 ) ) = 3
39	33	mul01i 10226	. . . . . . . 8 |- ( 3 x. 0 ) = 0
40	39	oveq1i 6660	. . . . . . 7 |- ( ( 3 x. 0 ) + 1 ) = ( 0 + 1 )
41	40, 30	eqtri 2644	. . . . . 6 |- ( ( 3 x. 0 ) + 1 ) = 1
42	41, 31	eqtri 2644	. . . . 5 |- ( ( 3 x. 0 ) + 1 ) = ; 0 1
43	2, 24, 24, 2, 28, 32, 1, 2, 24, 38, 42	decma2c 11568	. . . 4 |- ( ( 3 x. ; 1 0 ) + ( 0 + 1 ) ) = ; 3 1
44		5cn 11100	. . . . . 6 |- 5 e. CC
45		5t3e15 11635	. . . . . 6 |- ( 5 x. 3 ) = ; 1 5
46	44, 33, 45	mulcomli 10047	. . . . 5 |- ( 3 x. 5 ) = ; 1 5
47		5p2e7 11165	. . . . 5 |- ( 5 + 2 ) = 7
48	2, 21, 25, 46, 47	decaddi 11579	. . . 4 |- ( ( 3 x. 5 ) + 2 ) = ; 1 7
49	20, 21, 24, 25, 26, 27, 1, 8, 2, 43, 48	decma2c 11568	. . 3 |- ( ( 3 x. ;; 1 0 5 ) + 2 ) = ;; 3 1 7
50		2lt3 11195	. . 3 |- 2 < 3
51	19, 22, 23, 49, 50	ndvdsi 15136	. 2 |- -. 3 || ;; 3 1 7
52		2lt5 11202	. . 3 |- 2 < 5
53	3, 23, 52, 47	dec5dvds2 15769	. 2 |- -. 5 || ;; 3 1 7
54	7, 21	deccl 11512	. . 3 |- ; 4 5 e. NN0
55		eqid 2622	. . . 4 |- ; 4 5 = ; 4 5
56	33	addid2i 10224	. . . . . 6 |- ( 0 + 3 ) = 3
57	56	oveq2i 6661	. . . . 5 |- ( ( 7 x. 4 ) + ( 0 + 3 ) ) = ( ( 7 x. 4 ) + 3 )
58		7t4e28 11650	. . . . . 6 |- ( 7 x. 4 ) = ; 2 8
59		2p1e3 11151	. . . . . 6 |- ( 2 + 1 ) = 3
60		8p3e11 11612	. . . . . 6 |- ( 8 + 3 ) = ; 1 1
61	25, 6, 1, 58, 59, 2, 60	decaddci 11580	. . . . 5 |- ( ( 7 x. 4 ) + 3 ) = ; 3 1
62	57, 61	eqtri 2644	. . . 4 |- ( ( 7 x. 4 ) + ( 0 + 3 ) ) = ; 3 1
63		7t5e35 11651	. . . . 5 |- ( 7 x. 5 ) = ; 3 5
64	1, 21, 25, 63, 47	decaddi 11579	. . . 4 |- ( ( 7 x. 5 ) + 2 ) = ; 3 7
65	7, 21, 24, 25, 55, 27, 8, 8, 1, 62, 64	decma2c 11568	. . 3 |- ( ( 7 x. ; 4 5 ) + 2 ) = ;; 3 1 7
66		2lt7 11213	. . 3 |- 2 < 7
67	4, 54, 23, 65, 66	ndvdsi 15136	. 2 |- -. 7 || ;; 3 1 7
68	2, 13	decnncl 11518	. . 3 |- ; 1 1 e. NN
69	25, 6	deccl 11512	. . 3 |- ; 2 8 e. NN0
70		9nn 11192	. . 3 |- 9 e. NN
71		9nn0 11316	. . . 4 |- 9 e. NN0
72		eqid 2622	. . . 4 |- ; 2 8 = ; 2 8
73	71	dec0h 11522	. . . 4 |- 9 = ; 0 9
74	2, 2	deccl 11512	. . . 4 |- ; 1 1 e. NN0
75		eqid 2622	. . . . 5 |- ; 1 1 = ; 1 1
76		9cn 11108	. . . . . . 7 |- 9 e. CC
77	76	addid2i 10224	. . . . . 6 |- ( 0 + 9 ) = 9
78	77, 73	eqtri 2644	. . . . 5 |- ( 0 + 9 ) = ; 0 9
79		2cn 11091	. . . . . . . 8 |- 2 e. CC
80	79	mulid2i 10043	. . . . . . 7 |- ( 1 x. 2 ) = 2
81	80, 30	oveq12i 6662	. . . . . 6 |- ( ( 1 x. 2 ) + ( 0 + 1 ) ) = ( 2 + 1 )
82	81, 59	eqtri 2644	. . . . 5 |- ( ( 1 x. 2 ) + ( 0 + 1 ) ) = 3
83	80	oveq1i 6660	. . . . . 6 |- ( ( 1 x. 2 ) + 9 ) = ( 2 + 9 )
84		9p2e11 11619	. . . . . . 7 |- ( 9 + 2 ) = ; 1 1
85	76, 79, 84	addcomli 10228	. . . . . 6 |- ( 2 + 9 ) = ; 1 1
86	83, 85	eqtri 2644	. . . . 5 |- ( ( 1 x. 2 ) + 9 ) = ; 1 1
87	2, 2, 24, 71, 75, 78, 25, 2, 2, 82, 86	decmac 11566	. . . 4 |- ( (; 1 1 x. 2 ) + ( 0 + 9 ) ) = ; 3 1
88		8cn 11106	. . . . . . . 8 |- 8 e. CC
89	88	mulid2i 10043	. . . . . . 7 |- ( 1 x. 8 ) = 8
90	89, 30	oveq12i 6662	. . . . . 6 |- ( ( 1 x. 8 ) + ( 0 + 1 ) ) = ( 8 + 1 )
91		8p1e9 11158	. . . . . 6 |- ( 8 + 1 ) = 9
92	90, 91	eqtri 2644	. . . . 5 |- ( ( 1 x. 8 ) + ( 0 + 1 ) ) = 9
93	89	oveq1i 6660	. . . . . 6 |- ( ( 1 x. 8 ) + 9 ) = ( 8 + 9 )
94		9p8e17 11626	. . . . . . 7 |- ( 9 + 8 ) = ; 1 7
95	76, 88, 94	addcomli 10228	. . . . . 6 |- ( 8 + 9 ) = ; 1 7
96	93, 95	eqtri 2644	. . . . 5 |- ( ( 1 x. 8 ) + 9 ) = ; 1 7
97	2, 2, 24, 71, 75, 73, 6, 8, 2, 92, 96	decmac 11566	. . . 4 |- ( (; 1 1 x. 8 ) + 9 ) = ; 9 7
98	25, 6, 24, 71, 72, 73, 74, 8, 71, 87, 97	decma2c 11568	. . 3 |- ( (; 1 1 x. ; 2 8 ) + 9 ) = ;; 3 1 7
99		9lt10 11673	. . . 4 |- 9 < ; 1 0
100	13, 2, 71, 99	declti 11546	. . 3 |- 9 < ; 1 1
101	68, 69, 70, 98, 100	ndvdsi 15136	. 2 |- -. ; 1 1 || ;; 3 1 7
102	2, 19	decnncl 11518	. . 3 |- ; 1 3 e. NN
103	25, 7	deccl 11512	. . 3 |- ; 2 4 e. NN0
104		5nn 11188	. . 3 |- 5 e. NN
105		eqid 2622	. . . 4 |- ; 2 4 = ; 2 4
106	21	dec0h 11522	. . . 4 |- 5 = ; 0 5
107	2, 1	deccl 11512	. . . 4 |- ; 1 3 e. NN0
108		eqid 2622	. . . . 5 |- ; 1 3 = ; 1 3
109	44	addid2i 10224	. . . . . 6 |- ( 0 + 5 ) = 5
110	109, 106	eqtri 2644	. . . . 5 |- ( 0 + 5 ) = ; 0 5
111	16	oveq1i 6660	. . . . . 6 |- ( ( 3 x. 2 ) + 5 ) = ( 6 + 5 )
112		6p5e11 11600	. . . . . 6 |- ( 6 + 5 ) = ; 1 1
113	111, 112	eqtri 2644	. . . . 5 |- ( ( 3 x. 2 ) + 5 ) = ; 1 1
114	2, 1, 24, 21, 108, 110, 25, 2, 2, 82, 113	decmac 11566	. . . 4 |- ( (; 1 3 x. 2 ) + ( 0 + 5 ) ) = ; 3 1
115		4cn 11098	. . . . . . . 8 |- 4 e. CC
116	115	mulid2i 10043	. . . . . . 7 |- ( 1 x. 4 ) = 4
117	116, 30	oveq12i 6662	. . . . . 6 |- ( ( 1 x. 4 ) + ( 0 + 1 ) ) = ( 4 + 1 )
118		4p1e5 11154	. . . . . 6 |- ( 4 + 1 ) = 5
119	117, 118	eqtri 2644	. . . . 5 |- ( ( 1 x. 4 ) + ( 0 + 1 ) ) = 5
120		4t3e12 11632	. . . . . . 7 |- ( 4 x. 3 ) = ; 1 2
121	115, 33, 120	mulcomli 10047	. . . . . 6 |- ( 3 x. 4 ) = ; 1 2
122	44, 79, 47	addcomli 10228	. . . . . 6 |- ( 2 + 5 ) = 7
123	2, 25, 21, 121, 122	decaddi 11579	. . . . 5 |- ( ( 3 x. 4 ) + 5 ) = ; 1 7
124	2, 1, 24, 21, 108, 106, 7, 8, 2, 119, 123	decmac 11566	. . . 4 |- ( (; 1 3 x. 4 ) + 5 ) = ; 5 7
125	25, 7, 24, 21, 105, 106, 107, 8, 21, 114, 124	decma2c 11568	. . 3 |- ( (; 1 3 x. ; 2 4 ) + 5 ) = ;; 3 1 7
126		5lt10 11677	. . . 4 |- 5 < ; 1 0
127	13, 1, 21, 126	declti 11546	. . 3 |- 5 < ; 1 3
128	102, 103, 104, 125, 127	ndvdsi 15136	. 2 |- -. ; 1 3 || ;; 3 1 7
129	2, 4	decnncl 11518	. . 3 |- ; 1 7 e. NN
130	2, 6	deccl 11512	. . 3 |- ; 1 8 e. NN0
131		eqid 2622	. . . 4 |- ; 1 8 = ; 1 8
132	2, 8	deccl 11512	. . . 4 |- ; 1 7 e. NN0
133		eqid 2622	. . . . 5 |- ; 1 7 = ; 1 7
134		3p1e4 11153	. . . . . . 7 |- ( 3 + 1 ) = 4
135	33, 29, 134	addcomli 10228	. . . . . 6 |- ( 1 + 3 ) = 4
136	24, 2, 2, 1, 31, 108, 30, 135	decadd 11570	. . . . 5 |- ( 1 + ; 1 3 ) = ; 1 4
137	29	mulid1i 10042	. . . . . . 7 |- ( 1 x. 1 ) = 1
138		1p1e2 11134	. . . . . . 7 |- ( 1 + 1 ) = 2
139	137, 138	oveq12i 6662	. . . . . 6 |- ( ( 1 x. 1 ) + ( 1 + 1 ) ) = ( 1 + 2 )
140		1p2e3 11152	. . . . . 6 |- ( 1 + 2 ) = 3
141	139, 140	eqtri 2644	. . . . 5 |- ( ( 1 x. 1 ) + ( 1 + 1 ) ) = 3
142		7cn 11104	. . . . . . . 8 |- 7 e. CC
143	142	mulid1i 10042	. . . . . . 7 |- ( 7 x. 1 ) = 7
144	143	oveq1i 6660	. . . . . 6 |- ( ( 7 x. 1 ) + 4 ) = ( 7 + 4 )
145		7p4e11 11605	. . . . . 6 |- ( 7 + 4 ) = ; 1 1
146	144, 145	eqtri 2644	. . . . 5 |- ( ( 7 x. 1 ) + 4 ) = ; 1 1
147	2, 8, 2, 7, 133, 136, 2, 2, 2, 141, 146	decmac 11566	. . . 4 |- ( (; 1 7 x. 1 ) + ( 1 + ; 1 3 ) ) = ; 3 1
148	89, 109	oveq12i 6662	. . . . . 6 |- ( ( 1 x. 8 ) + ( 0 + 5 ) ) = ( 8 + 5 )
149		8p5e13 11615	. . . . . 6 |- ( 8 + 5 ) = ; 1 3
150	148, 149	eqtri 2644	. . . . 5 |- ( ( 1 x. 8 ) + ( 0 + 5 ) ) = ; 1 3
151		6nn0 11313	. . . . . 6 |- 6 e. NN0
152		6p1e7 11156	. . . . . 6 |- ( 6 + 1 ) = 7
153		8t7e56 11661	. . . . . . 7 |- ( 8 x. 7 ) = ; 5 6
154	88, 142, 153	mulcomli 10047	. . . . . 6 |- ( 7 x. 8 ) = ; 5 6
155	21, 151, 152, 154	decsuc 11535	. . . . 5 |- ( ( 7 x. 8 ) + 1 ) = ; 5 7
156	2, 8, 24, 2, 133, 31, 6, 8, 21, 150, 155	decmac 11566	. . . 4 |- ( (; 1 7 x. 8 ) + 1 ) = ;; 1 3 7
157	2, 6, 2, 2, 131, 75, 132, 8, 107, 147, 156	decma2c 11568	. . 3 |- ( (; 1 7 x. ; 1 8 ) + ; 1 1 ) = ;; 3 1 7
158		1lt7 11214	. . . 4 |- 1 < 7
159	2, 2, 4, 158	declt 11530	. . 3 |- ; 1 1 < ; 1 7
160	129, 130, 68, 157, 159	ndvdsi 15136	. 2 |- -. ; 1 7 || ;; 3 1 7
161	2, 70	decnncl 11518	. . 3 |- ; 1 9 e. NN
162	2, 151	deccl 11512	. . 3 |- ; 1 6 e. NN0
163		eqid 2622	. . . 4 |- ; 1 6 = ; 1 6
164	2, 71	deccl 11512	. . . 4 |- ; 1 9 e. NN0
165		eqid 2622	. . . . 5 |- ; 1 9 = ; 1 9
166	24, 2, 2, 2, 31, 75, 30, 138	decadd 11570	. . . . 5 |- ( 1 + ; 1 1 ) = ; 1 2
167	76	mulid1i 10042	. . . . . . 7 |- ( 9 x. 1 ) = 9
168	167	oveq1i 6660	. . . . . 6 |- ( ( 9 x. 1 ) + 2 ) = ( 9 + 2 )
169	168, 84	eqtri 2644	. . . . 5 |- ( ( 9 x. 1 ) + 2 ) = ; 1 1
170	2, 71, 2, 25, 165, 166, 2, 2, 2, 141, 169	decmac 11566	. . . 4 |- ( (; 1 9 x. 1 ) + ( 1 + ; 1 1 ) ) = ; 3 1
171	1	dec0h 11522	. . . . 5 |- 3 = ; 0 3
172		6cn 11102	. . . . . . . 8 |- 6 e. CC
173	172	mulid2i 10043	. . . . . . 7 |- ( 1 x. 6 ) = 6
174	173, 109	oveq12i 6662	. . . . . 6 |- ( ( 1 x. 6 ) + ( 0 + 5 ) ) = ( 6 + 5 )
175	174, 112	eqtri 2644	. . . . 5 |- ( ( 1 x. 6 ) + ( 0 + 5 ) ) = ; 1 1
176		9t6e54 11667	. . . . . 6 |- ( 9 x. 6 ) = ; 5 4
177		4p3e7 11163	. . . . . 6 |- ( 4 + 3 ) = 7
178	21, 7, 1, 176, 177	decaddi 11579	. . . . 5 |- ( ( 9 x. 6 ) + 3 ) = ; 5 7
179	2, 71, 24, 1, 165, 171, 151, 8, 21, 175, 178	decmac 11566	. . . 4 |- ( (; 1 9 x. 6 ) + 3 ) = ;; 1 1 7
180	2, 151, 2, 1, 163, 108, 164, 8, 74, 170, 179	decma2c 11568	. . 3 |- ( (; 1 9 x. ; 1 6 ) + ; 1 3 ) = ;; 3 1 7
181		3lt9 11227	. . . 4 |- 3 < 9
182	2, 1, 70, 181	declt 11530	. . 3 |- ; 1 3 < ; 1 9
183	161, 162, 102, 180, 182	ndvdsi 15136	. 2 |- -. ; 1 9 || ;; 3 1 7
184	25, 19	decnncl 11518	. . 3 |- ; 2 3 e. NN
185	102	nnnn0i 11300	. . 3 |- ; 1 3 e. NN0
186		8nn 11191	. . . 4 |- 8 e. NN
187	2, 186	decnncl 11518	. . 3 |- ; 1 8 e. NN
188	25, 1	deccl 11512	. . . 4 |- ; 2 3 e. NN0
189		eqid 2622	. . . . 5 |- ; 2 3 = ; 2 3
190		7p1e8 11157	. . . . . . 7 |- ( 7 + 1 ) = 8
191	142, 29, 190	addcomli 10228	. . . . . 6 |- ( 1 + 7 ) = 8
192	6	dec0h 11522	. . . . . 6 |- 8 = ; 0 8
193	191, 192	eqtri 2644	. . . . 5 |- ( 1 + 7 ) = ; 0 8
194	79	mulid1i 10042	. . . . . . 7 |- ( 2 x. 1 ) = 2
195	194, 30	oveq12i 6662	. . . . . 6 |- ( ( 2 x. 1 ) + ( 0 + 1 ) ) = ( 2 + 1 )
196	195, 59	eqtri 2644	. . . . 5 |- ( ( 2 x. 1 ) + ( 0 + 1 ) ) = 3
197	34	oveq1i 6660	. . . . . 6 |- ( ( 3 x. 1 ) + 8 ) = ( 3 + 8 )
198	88, 33, 60	addcomli 10228	. . . . . 6 |- ( 3 + 8 ) = ; 1 1
199	197, 198	eqtri 2644	. . . . 5 |- ( ( 3 x. 1 ) + 8 ) = ; 1 1
200	25, 1, 24, 6, 189, 193, 2, 2, 2, 196, 199	decmac 11566	. . . 4 |- ( (; 2 3 x. 1 ) + ( 1 + 7 ) ) = ; 3 1
201	33, 79, 16	mulcomli 10047	. . . . . . 7 |- ( 2 x. 3 ) = 6
202	201, 30	oveq12i 6662	. . . . . 6 |- ( ( 2 x. 3 ) + ( 0 + 1 ) ) = ( 6 + 1 )
203	202, 152	eqtri 2644	. . . . 5 |- ( ( 2 x. 3 ) + ( 0 + 1 ) ) = 7
204		3t3e9 11180	. . . . . . 7 |- ( 3 x. 3 ) = 9
205	204	oveq1i 6660	. . . . . 6 |- ( ( 3 x. 3 ) + 8 ) = ( 9 + 8 )
206	205, 94	eqtri 2644	. . . . 5 |- ( ( 3 x. 3 ) + 8 ) = ; 1 7
207	25, 1, 24, 6, 189, 192, 1, 8, 2, 203, 206	decmac 11566	. . . 4 |- ( (; 2 3 x. 3 ) + 8 ) = ; 7 7
208	2, 1, 2, 6, 108, 131, 188, 8, 8, 200, 207	decma2c 11568	. . 3 |- ( (; 2 3 x. ; 1 3 ) + ; 1 8 ) = ;; 3 1 7
209		8lt10 11674	. . . 4 |- 8 < ; 1 0
210		1lt2 11194	. . . 4 |- 1 < 2
211	2, 25, 6, 1, 209, 210	decltc 11532	. . 3 |- ; 1 8 < ; 2 3
212	184, 185, 187, 208, 211	ndvdsi 15136	. 2 |- -. ; 2 3 || ;; 3 1 7
213	5, 12, 15, 18, 51, 53, 67, 101, 128, 160, 183, 212	prmlem2 15827	1 |- ;; 3 1 7 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)