Theorem 317prm 16451

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	⊢ ;;317 ∈ ℙ

Proof of Theorem 317prm

Step	Hyp	Ref	Expression
1		3nn0 11903	. . . 4 ⊢ 3 ∈ ℕ0
2		1nn0 11901	. . . 4 ⊢ 1 ∈ ℕ0
3	1, 2	deccl 12101	. . 3 ⊢ ;31 ∈ ℕ0
4		7nn 11717	. . 3 ⊢ 7 ∈ ℕ
5	3, 4	decnncl 12106	. 2 ⊢ ;;317 ∈ ℕ
6		8nn0 11908	. . 3 ⊢ 8 ∈ ℕ0
7		4nn0 11904	. . 3 ⊢ 4 ∈ ℕ0
8		7nn0 11907	. . 3 ⊢ 7 ∈ ℕ0
9		3lt8 11821	. . 3 ⊢ 3 < 8
10		1lt10 12225	. . 3 ⊢ 1 < ;10
11		7lt10 12219	. . 3 ⊢ 7 < ;10
12	1, 6, 2, 7, 8, 2, 9, 10, 11	3decltc 12119	. 2 ⊢ ;;317 < ;;841
13		1nn 11636	. . . 4 ⊢ 1 ∈ ℕ
14	1, 13	decnncl 12106	. . 3 ⊢ ;31 ∈ ℕ
15	14, 8, 2, 10	declti 12124	. 2 ⊢ 1 < ;;317
16		3t2e6 11791	. . 3 ⊢ (3 · 2) = 6
17		df-7 11693	. . 3 ⊢ 7 = (6 + 1)
18	3, 1, 16, 17	dec2dvds 16389	. 2 ⊢ ¬ 2 ∥ ;;317
19		3nn 11704	. . 3 ⊢ 3 ∈ ℕ
20		10nn0 12104	. . . 4 ⊢ ;10 ∈ ℕ0
21		5nn0 11905	. . . 4 ⊢ 5 ∈ ℕ0
22	20, 21	deccl 12101	. . 3 ⊢ ;;105 ∈ ℕ0
23		2nn 11698	. . 3 ⊢ 2 ∈ ℕ
24		0nn0 11900	. . . 4 ⊢ 0 ∈ ℕ0
25		2nn0 11902	. . . 4 ⊢ 2 ∈ ℕ0
26		eqid 2798	. . . 4 ⊢ ;;105 = ;;105
27	25	dec0h 12108	. . . 4 ⊢ 2 = ;02
28		eqid 2798	. . . . 5 ⊢ ;10 = ;10
29		ax-1cn 10584	. . . . . . 7 ⊢ 1 ∈ ℂ
30	29	addid2i 10817	. . . . . 6 ⊢ (0 + 1) = 1
31	2	dec0h 12108	. . . . . 6 ⊢ 1 = ;01
32	30, 31	eqtri 2821	. . . . 5 ⊢ (0 + 1) = ;01
33		3cn 11706	. . . . . . . 8 ⊢ 3 ∈ ℂ
34	33	mulid1i 10634	. . . . . . 7 ⊢ (3 · 1) = 3
35		00id 10804	. . . . . . 7 ⊢ (0 + 0) = 0
36	34, 35	oveq12i 7147	. . . . . 6 ⊢ ((3 · 1) + (0 + 0)) = (3 + 0)
37	33	addid1i 10816	. . . . . 6 ⊢ (3 + 0) = 3
38	36, 37	eqtri 2821	. . . . 5 ⊢ ((3 · 1) + (0 + 0)) = 3
39	33	mul01i 10819	. . . . . . . 8 ⊢ (3 · 0) = 0
40	39	oveq1i 7145	. . . . . . 7 ⊢ ((3 · 0) + 1) = (0 + 1)
41	40, 30	eqtri 2821	. . . . . 6 ⊢ ((3 · 0) + 1) = 1
42	41, 31	eqtri 2821	. . . . 5 ⊢ ((3 · 0) + 1) = ;01
43	2, 24, 24, 2, 28, 32, 1, 2, 24, 38, 42	decma2c 12139	. . . 4 ⊢ ((3 · ;10) + (0 + 1)) = ;31
44		5cn 11713	. . . . . 6 ⊢ 5 ∈ ℂ
45		5t3e15 12187	. . . . . 6 ⊢ (5 · 3) = ;15
46	44, 33, 45	mulcomli 10639	. . . . 5 ⊢ (3 · 5) = ;15
47		5p2e7 11781	. . . . 5 ⊢ (5 + 2) = 7
48	2, 21, 25, 46, 47	decaddi 12146	. . . 4 ⊢ ((3 · 5) + 2) = ;17
49	20, 21, 24, 25, 26, 27, 1, 8, 2, 43, 48	decma2c 12139	. . 3 ⊢ ((3 · ;;105) + 2) = ;;317
50		2lt3 11797	. . 3 ⊢ 2 < 3
51	19, 22, 23, 49, 50	ndvdsi 15753	. 2 ⊢ ¬ 3 ∥ ;;317
52		2lt5 11804	. . 3 ⊢ 2 < 5
53	3, 23, 52, 47	dec5dvds2 16391	. 2 ⊢ ¬ 5 ∥ ;;317
54	7, 21	deccl 12101	. . 3 ⊢ ;45 ∈ ℕ0
55		eqid 2798	. . . 4 ⊢ ;45 = ;45
56	33	addid2i 10817	. . . . . 6 ⊢ (0 + 3) = 3
57	56	oveq2i 7146	. . . . 5 ⊢ ((7 · 4) + (0 + 3)) = ((7 · 4) + 3)
58		7t4e28 12197	. . . . . 6 ⊢ (7 · 4) = ;28
59		2p1e3 11767	. . . . . 6 ⊢ (2 + 1) = 3
60		8p3e11 12167	. . . . . 6 ⊢ (8 + 3) = ;11
61	25, 6, 1, 58, 59, 2, 60	decaddci 12147	. . . . 5 ⊢ ((7 · 4) + 3) = ;31
62	57, 61	eqtri 2821	. . . 4 ⊢ ((7 · 4) + (0 + 3)) = ;31
63		7t5e35 12198	. . . . 5 ⊢ (7 · 5) = ;35
64	1, 21, 25, 63, 47	decaddi 12146	. . . 4 ⊢ ((7 · 5) + 2) = ;37
65	7, 21, 24, 25, 55, 27, 8, 8, 1, 62, 64	decma2c 12139	. . 3 ⊢ ((7 · ;45) + 2) = ;;317
66		2lt7 11815	. . 3 ⊢ 2 < 7
67	4, 54, 23, 65, 66	ndvdsi 15753	. 2 ⊢ ¬ 7 ∥ ;;317
68	2, 13	decnncl 12106	. . 3 ⊢ ;11 ∈ ℕ
69	25, 6	deccl 12101	. . 3 ⊢ ;28 ∈ ℕ0
70		9nn 11723	. . 3 ⊢ 9 ∈ ℕ
71		9nn0 11909	. . . 4 ⊢ 9 ∈ ℕ0
72		eqid 2798	. . . 4 ⊢ ;28 = ;28
73	71	dec0h 12108	. . . 4 ⊢ 9 = ;09
74	2, 2	deccl 12101	. . . 4 ⊢ ;11 ∈ ℕ0
75		eqid 2798	. . . . 5 ⊢ ;11 = ;11
76		9cn 11725	. . . . . . 7 ⊢ 9 ∈ ℂ
77	76	addid2i 10817	. . . . . 6 ⊢ (0 + 9) = 9
78	77, 73	eqtri 2821	. . . . 5 ⊢ (0 + 9) = ;09
79		2cn 11700	. . . . . . . 8 ⊢ 2 ∈ ℂ
80	79	mulid2i 10635	. . . . . . 7 ⊢ (1 · 2) = 2
81	80, 30	oveq12i 7147	. . . . . 6 ⊢ ((1 · 2) + (0 + 1)) = (2 + 1)
82	81, 59	eqtri 2821	. . . . 5 ⊢ ((1 · 2) + (0 + 1)) = 3
83	80	oveq1i 7145	. . . . . 6 ⊢ ((1 · 2) + 9) = (2 + 9)
84		9p2e11 12173	. . . . . . 7 ⊢ (9 + 2) = ;11
85	76, 79, 84	addcomli 10821	. . . . . 6 ⊢ (2 + 9) = ;11
86	83, 85	eqtri 2821	. . . . 5 ⊢ ((1 · 2) + 9) = ;11
87	2, 2, 24, 71, 75, 78, 25, 2, 2, 82, 86	decmac 12138	. . . 4 ⊢ ((;11 · 2) + (0 + 9)) = ;31
88		8cn 11722	. . . . . . . 8 ⊢ 8 ∈ ℂ
89	88	mulid2i 10635	. . . . . . 7 ⊢ (1 · 8) = 8
90	89, 30	oveq12i 7147	. . . . . 6 ⊢ ((1 · 8) + (0 + 1)) = (8 + 1)
91		8p1e9 11775	. . . . . 6 ⊢ (8 + 1) = 9
92	90, 91	eqtri 2821	. . . . 5 ⊢ ((1 · 8) + (0 + 1)) = 9
93	89	oveq1i 7145	. . . . . 6 ⊢ ((1 · 8) + 9) = (8 + 9)
94		9p8e17 12179	. . . . . . 7 ⊢ (9 + 8) = ;17
95	76, 88, 94	addcomli 10821	. . . . . 6 ⊢ (8 + 9) = ;17
96	93, 95	eqtri 2821	. . . . 5 ⊢ ((1 · 8) + 9) = ;17
97	2, 2, 24, 71, 75, 73, 6, 8, 2, 92, 96	decmac 12138	. . . 4 ⊢ ((;11 · 8) + 9) = ;97
98	25, 6, 24, 71, 72, 73, 74, 8, 71, 87, 97	decma2c 12139	. . 3 ⊢ ((;11 · ;28) + 9) = ;;317
99		9lt10 12217	. . . 4 ⊢ 9 < ;10
100	13, 2, 71, 99	declti 12124	. . 3 ⊢ 9 < ;11
101	68, 69, 70, 98, 100	ndvdsi 15753	. 2 ⊢ ¬ ;11 ∥ ;;317
102	2, 19	decnncl 12106	. . 3 ⊢ ;13 ∈ ℕ
103	25, 7	deccl 12101	. . 3 ⊢ ;24 ∈ ℕ0
104		5nn 11711	. . 3 ⊢ 5 ∈ ℕ
105		eqid 2798	. . . 4 ⊢ ;24 = ;24
106	21	dec0h 12108	. . . 4 ⊢ 5 = ;05
107	2, 1	deccl 12101	. . . 4 ⊢ ;13 ∈ ℕ0
108		eqid 2798	. . . . 5 ⊢ ;13 = ;13
109	44	addid2i 10817	. . . . . 6 ⊢ (0 + 5) = 5
110	109, 106	eqtri 2821	. . . . 5 ⊢ (0 + 5) = ;05
111	16	oveq1i 7145	. . . . . 6 ⊢ ((3 · 2) + 5) = (6 + 5)
112		6p5e11 12159	. . . . . 6 ⊢ (6 + 5) = ;11
113	111, 112	eqtri 2821	. . . . 5 ⊢ ((3 · 2) + 5) = ;11
114	2, 1, 24, 21, 108, 110, 25, 2, 2, 82, 113	decmac 12138	. . . 4 ⊢ ((;13 · 2) + (0 + 5)) = ;31
115		4cn 11710	. . . . . . . 8 ⊢ 4 ∈ ℂ
116	115	mulid2i 10635	. . . . . . 7 ⊢ (1 · 4) = 4
117	116, 30	oveq12i 7147	. . . . . 6 ⊢ ((1 · 4) + (0 + 1)) = (4 + 1)
118		4p1e5 11771	. . . . . 6 ⊢ (4 + 1) = 5
119	117, 118	eqtri 2821	. . . . 5 ⊢ ((1 · 4) + (0 + 1)) = 5
120		4t3e12 12184	. . . . . . 7 ⊢ (4 · 3) = ;12
121	115, 33, 120	mulcomli 10639	. . . . . 6 ⊢ (3 · 4) = ;12
122	44, 79, 47	addcomli 10821	. . . . . 6 ⊢ (2 + 5) = 7
123	2, 25, 21, 121, 122	decaddi 12146	. . . . 5 ⊢ ((3 · 4) + 5) = ;17
124	2, 1, 24, 21, 108, 106, 7, 8, 2, 119, 123	decmac 12138	. . . 4 ⊢ ((;13 · 4) + 5) = ;57
125	25, 7, 24, 21, 105, 106, 107, 8, 21, 114, 124	decma2c 12139	. . 3 ⊢ ((;13 · ;24) + 5) = ;;317
126		5lt10 12221	. . . 4 ⊢ 5 < ;10
127	13, 1, 21, 126	declti 12124	. . 3 ⊢ 5 < ;13
128	102, 103, 104, 125, 127	ndvdsi 15753	. 2 ⊢ ¬ ;13 ∥ ;;317
129	2, 4	decnncl 12106	. . 3 ⊢ ;17 ∈ ℕ
130	2, 6	deccl 12101	. . 3 ⊢ ;18 ∈ ℕ0
131		eqid 2798	. . . 4 ⊢ ;18 = ;18
132	2, 8	deccl 12101	. . . 4 ⊢ ;17 ∈ ℕ0
133		eqid 2798	. . . . 5 ⊢ ;17 = ;17
134		3p1e4 11770	. . . . . . 7 ⊢ (3 + 1) = 4
135	33, 29, 134	addcomli 10821	. . . . . 6 ⊢ (1 + 3) = 4
136	24, 2, 2, 1, 31, 108, 30, 135	decadd 12140	. . . . 5 ⊢ (1 + ;13) = ;14
137	29	mulid1i 10634	. . . . . . 7 ⊢ (1 · 1) = 1
138		1p1e2 11750	. . . . . . 7 ⊢ (1 + 1) = 2
139	137, 138	oveq12i 7147	. . . . . 6 ⊢ ((1 · 1) + (1 + 1)) = (1 + 2)
140		1p2e3 11768	. . . . . 6 ⊢ (1 + 2) = 3
141	139, 140	eqtri 2821	. . . . 5 ⊢ ((1 · 1) + (1 + 1)) = 3
142		7cn 11719	. . . . . . . 8 ⊢ 7 ∈ ℂ
143	142	mulid1i 10634	. . . . . . 7 ⊢ (7 · 1) = 7
144	143	oveq1i 7145	. . . . . 6 ⊢ ((7 · 1) + 4) = (7 + 4)
145		7p4e11 12162	. . . . . 6 ⊢ (7 + 4) = ;11
146	144, 145	eqtri 2821	. . . . 5 ⊢ ((7 · 1) + 4) = ;11
147	2, 8, 2, 7, 133, 136, 2, 2, 2, 141, 146	decmac 12138	. . . 4 ⊢ ((;17 · 1) + (1 + ;13)) = ;31
148	89, 109	oveq12i 7147	. . . . . 6 ⊢ ((1 · 8) + (0 + 5)) = (8 + 5)
149		8p5e13 12169	. . . . . 6 ⊢ (8 + 5) = ;13
150	148, 149	eqtri 2821	. . . . 5 ⊢ ((1 · 8) + (0 + 5)) = ;13
151		6nn0 11906	. . . . . 6 ⊢ 6 ∈ ℕ0
152		6p1e7 11773	. . . . . 6 ⊢ (6 + 1) = 7
153		8t7e56 12206	. . . . . . 7 ⊢ (8 · 7) = ;56
154	88, 142, 153	mulcomli 10639	. . . . . 6 ⊢ (7 · 8) = ;56
155	21, 151, 152, 154	decsuc 12117	. . . . 5 ⊢ ((7 · 8) + 1) = ;57
156	2, 8, 24, 2, 133, 31, 6, 8, 21, 150, 155	decmac 12138	. . . 4 ⊢ ((;17 · 8) + 1) = ;;137
157	2, 6, 2, 2, 131, 75, 132, 8, 107, 147, 156	decma2c 12139	. . 3 ⊢ ((;17 · ;18) + ;11) = ;;317
158		1lt7 11816	. . . 4 ⊢ 1 < 7
159	2, 2, 4, 158	declt 12114	. . 3 ⊢ ;11 < ;17
160	129, 130, 68, 157, 159	ndvdsi 15753	. 2 ⊢ ¬ ;17 ∥ ;;317
161	2, 70	decnncl 12106	. . 3 ⊢ ;19 ∈ ℕ
162	2, 151	deccl 12101	. . 3 ⊢ ;16 ∈ ℕ0
163		eqid 2798	. . . 4 ⊢ ;16 = ;16
164	2, 71	deccl 12101	. . . 4 ⊢ ;19 ∈ ℕ0
165		eqid 2798	. . . . 5 ⊢ ;19 = ;19
166	24, 2, 2, 2, 31, 75, 30, 138	decadd 12140	. . . . 5 ⊢ (1 + ;11) = ;12
167	76	mulid1i 10634	. . . . . . 7 ⊢ (9 · 1) = 9
168	167	oveq1i 7145	. . . . . 6 ⊢ ((9 · 1) + 2) = (9 + 2)
169	168, 84	eqtri 2821	. . . . 5 ⊢ ((9 · 1) + 2) = ;11
170	2, 71, 2, 25, 165, 166, 2, 2, 2, 141, 169	decmac 12138	. . . 4 ⊢ ((;19 · 1) + (1 + ;11)) = ;31
171	1	dec0h 12108	. . . . 5 ⊢ 3 = ;03
172		6cn 11716	. . . . . . . 8 ⊢ 6 ∈ ℂ
173	172	mulid2i 10635	. . . . . . 7 ⊢ (1 · 6) = 6
174	173, 109	oveq12i 7147	. . . . . 6 ⊢ ((1 · 6) + (0 + 5)) = (6 + 5)
175	174, 112	eqtri 2821	. . . . 5 ⊢ ((1 · 6) + (0 + 5)) = ;11
176		9t6e54 12212	. . . . . 6 ⊢ (9 · 6) = ;54
177		4p3e7 11779	. . . . . 6 ⊢ (4 + 3) = 7
178	21, 7, 1, 176, 177	decaddi 12146	. . . . 5 ⊢ ((9 · 6) + 3) = ;57
179	2, 71, 24, 1, 165, 171, 151, 8, 21, 175, 178	decmac 12138	. . . 4 ⊢ ((;19 · 6) + 3) = ;;117
180	2, 151, 2, 1, 163, 108, 164, 8, 74, 170, 179	decma2c 12139	. . 3 ⊢ ((;19 · ;16) + ;13) = ;;317
181		3lt9 11829	. . . 4 ⊢ 3 < 9
182	2, 1, 70, 181	declt 12114	. . 3 ⊢ ;13 < ;19
183	161, 162, 102, 180, 182	ndvdsi 15753	. 2 ⊢ ¬ ;19 ∥ ;;317
184	25, 19	decnncl 12106	. . 3 ⊢ ;23 ∈ ℕ
185	102	nnnn0i 11893	. . 3 ⊢ ;13 ∈ ℕ0
186		8nn 11720	. . . 4 ⊢ 8 ∈ ℕ
187	2, 186	decnncl 12106	. . 3 ⊢ ;18 ∈ ℕ
188	25, 1	deccl 12101	. . . 4 ⊢ ;23 ∈ ℕ0
189		eqid 2798	. . . . 5 ⊢ ;23 = ;23
190		7p1e8 11774	. . . . . . 7 ⊢ (7 + 1) = 8
191	142, 29, 190	addcomli 10821	. . . . . 6 ⊢ (1 + 7) = 8
192	6	dec0h 12108	. . . . . 6 ⊢ 8 = ;08
193	191, 192	eqtri 2821	. . . . 5 ⊢ (1 + 7) = ;08
194	79	mulid1i 10634	. . . . . . 7 ⊢ (2 · 1) = 2
195	194, 30	oveq12i 7147	. . . . . 6 ⊢ ((2 · 1) + (0 + 1)) = (2 + 1)
196	195, 59	eqtri 2821	. . . . 5 ⊢ ((2 · 1) + (0 + 1)) = 3
197	34	oveq1i 7145	. . . . . 6 ⊢ ((3 · 1) + 8) = (3 + 8)
198	88, 33, 60	addcomli 10821	. . . . . 6 ⊢ (3 + 8) = ;11
199	197, 198	eqtri 2821	. . . . 5 ⊢ ((3 · 1) + 8) = ;11
200	25, 1, 24, 6, 189, 193, 2, 2, 2, 196, 199	decmac 12138	. . . 4 ⊢ ((;23 · 1) + (1 + 7)) = ;31
201	33, 79, 16	mulcomli 10639	. . . . . . 7 ⊢ (2 · 3) = 6
202	201, 30	oveq12i 7147	. . . . . 6 ⊢ ((2 · 3) + (0 + 1)) = (6 + 1)
203	202, 152	eqtri 2821	. . . . 5 ⊢ ((2 · 3) + (0 + 1)) = 7
204		3t3e9 11792	. . . . . . 7 ⊢ (3 · 3) = 9
205	204	oveq1i 7145	. . . . . 6 ⊢ ((3 · 3) + 8) = (9 + 8)
206	205, 94	eqtri 2821	. . . . 5 ⊢ ((3 · 3) + 8) = ;17
207	25, 1, 24, 6, 189, 192, 1, 8, 2, 203, 206	decmac 12138	. . . 4 ⊢ ((;23 · 3) + 8) = ;77
208	2, 1, 2, 6, 108, 131, 188, 8, 8, 200, 207	decma2c 12139	. . 3 ⊢ ((;23 · ;13) + ;18) = ;;317
209		8lt10 12218	. . . 4 ⊢ 8 < ;10
210		1lt2 11796	. . . 4 ⊢ 1 < 2
211	2, 25, 6, 1, 209, 210	decltc 12115	. . 3 ⊢ ;18 < ;23
212	184, 185, 187, 208, 211	ndvdsi 15753	. 2 ⊢ ¬ ;23 ∥ ;;317
213	5, 12, 15, 18, 51, 53, 67, 101, 128, 160, 183, 212	prmlem2 16445	1 ⊢ ;;317 ∈ ℙ

Syntax hints:   ∈ wcel 2111  (class class class)co 7135  0cc0 10526  1c1 10527   + caddc 10529   · cmul 10531  2c2 11680  3c3 11681  4c4 11682  5c5 11683  6c6 11684  7c7 11685  8c8 11686  9c9 11687  ;cdc 12086  ℙcprime 16005

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-2o 8086  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-sup 8890  df-inf 8891  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11626  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-dec 12087  df-uz 12232  df-rp 12378  df-fz 12886  df-seq 13365  df-exp 13426  df-cj 14450  df-re 14451  df-im 14452  df-sqrt 14586  df-abs 14587  df-dvds 15600  df-prm 16006

This theorem is referenced by: (None)