Theorem 631prm 13261

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	⊢ ;;631 ∈ ℙ

Proof of Theorem 631prm

Step	Hyp	Ref	Expression
1		6nn0 9588	. . . 4 ⊢ 6 ∈ ℕ0
2		3nn0 9585	. . . 4 ⊢ 3 ∈ ℕ0
3	1, 2	deccl 9795	. . 3 ⊢ ;63 ∈ ℕ0
4		1nn 9317	. . 3 ⊢ 1 ∈ ℕ
5	3, 4	decnncl 9804	. 2 ⊢ ;;631 ∈ ℕ
6		8nn0 9590	. . 3 ⊢ 8 ∈ ℕ0
7		4nn0 9586	. . 3 ⊢ 4 ∈ ℕ0
8		1nn0 9583	. . 3 ⊢ 1 ∈ ℕ0
9		6lt8 9500	. . 3 ⊢ 6 < 8
10		3lt10 9922	. . 3 ⊢ 3 < ;10
11		1lt10 9924	. . 3 ⊢ 1 < ;10
12	1, 6, 2, 7, 8, 8, 9, 10, 11	3decltc 9818	. 2 ⊢ ;;631 < ;;841
13		3nn 9471	. . . 4 ⊢ 3 ∈ ℕ
14	1, 13	decnncl 9804	. . 3 ⊢ ;63 ∈ ℕ
15	14, 8, 8, 11	declti 9823	. 2 ⊢ 1 < ;;631
16		0nn0 9582	. . 3 ⊢ 0 ∈ ℕ0
17		2cn 9377	. . . 4 ⊢ 2 ∈ ℂ
18	17	mul02i 8718	. . 3 ⊢ (0 · 2) = 0
19		1e0p1 9827	. . 3 ⊢ 1 = (0 + 1)
20	3, 16, 18, 19	dec2dvds 13210	. 2 ⊢ ¬ 2 ∥ ;;631
21		2nn0 9584	. . . . 5 ⊢ 2 ∈ ℕ0
22	21, 8	deccl 9795	. . . 4 ⊢ ;21 ∈ ℕ0
23	22, 16	deccl 9795	. . 3 ⊢ ;;210 ∈ ℕ0
24		eqid 2238	. . . 4 ⊢ ;;210 = ;;210
25	8	dec0h 9807	. . . 4 ⊢ 1 = ;01
26		eqid 2238	. . . . 5 ⊢ ;21 = ;21
27		00id 8468	. . . . . 6 ⊢ (0 + 0) = 0
28	16	dec0h 9807	. . . . . 6 ⊢ 0 = ;00
29	27, 28	eqtri 2259	. . . . 5 ⊢ (0 + 0) = ;00
30		3t2e6 9463	. . . . . . 7 ⊢ (3 · 2) = 6
31	30, 27	oveq12i 6097	. . . . . 6 ⊢ ((3 · 2) + (0 + 0)) = (6 + 0)
32		6cn 9388	. . . . . . 7 ⊢ 6 ∈ ℂ
33	32	addridi 8469	. . . . . 6 ⊢ (6 + 0) = 6
34	31, 33	eqtri 2259	. . . . 5 ⊢ ((3 · 2) + (0 + 0)) = 6
35		3t1e3 9462	. . . . . . 7 ⊢ (3 · 1) = 3
36	35	oveq1i 6095	. . . . . 6 ⊢ ((3 · 1) + 0) = (3 + 0)
37		3cn 9381	. . . . . . 7 ⊢ 3 ∈ ℂ
38	37	addridi 8469	. . . . . 6 ⊢ (3 + 0) = 3
39	2	dec0h 9807	. . . . . 6 ⊢ 3 = ;03
40	36, 38, 39	3eqtri 2263	. . . . 5 ⊢ ((3 · 1) + 0) = ;03
41	21, 8, 16, 16, 26, 29, 2, 2, 16, 34, 40	decma2c 9838	. . . 4 ⊢ ((3 · ;21) + (0 + 0)) = ;63
42	37	mul01i 8719	. . . . . 6 ⊢ (3 · 0) = 0
43	42	oveq1i 6095	. . . . 5 ⊢ ((3 · 0) + 1) = (0 + 1)
44		0p1e1 9420	. . . . 5 ⊢ (0 + 1) = 1
45	43, 44, 25	3eqtri 2263	. . . 4 ⊢ ((3 · 0) + 1) = ;01
46	22, 16, 16, 8, 24, 25, 2, 8, 16, 41, 45	decma2c 9838	. . 3 ⊢ ((3 · ;;210) + 1) = ;;631
47		1lt3 9480	. . 3 ⊢ 1 < 3
48	13, 23, 4, 46, 47	ndvdsi 12716	. 2 ⊢ ¬ 3 ∥ ;;631
49		1lt5 9487	. . 3 ⊢ 1 < 5
50	3, 4, 49	dec5dvds 13211	. 2 ⊢ ¬ 5 ∥ ;;631
51		7nn 9475	. . 3 ⊢ 7 ∈ ℕ
52		9nn0 9591	. . . 4 ⊢ 9 ∈ ℕ0
53	52, 16	deccl 9795	. . 3 ⊢ ;90 ∈ ℕ0
54		eqid 2238	. . . 4 ⊢ ;90 = ;90
55		7nn0 9589	. . . 4 ⊢ 7 ∈ ℕ0
56	27	oveq2i 6096	. . . . 5 ⊢ ((7 · 9) + (0 + 0)) = ((7 · 9) + 0)
57		9cn 9394	. . . . . . 7 ⊢ 9 ∈ ℂ
58		7cn 9390	. . . . . . 7 ⊢ 7 ∈ ℂ
59		9t7e63 9912	. . . . . . 7 ⊢ (9 · 7) = ;63
60	57, 58, 59	mulcomli 8333	. . . . . 6 ⊢ (7 · 9) = ;63
61	60	oveq1i 6095	. . . . 5 ⊢ ((7 · 9) + 0) = (;63 + 0)
62	3	nn0cni 9579	. . . . . 6 ⊢ ;63 ∈ ℂ
63	62	addridi 8469	. . . . 5 ⊢ (;63 + 0) = ;63
64	56, 61, 63	3eqtri 2263	. . . 4 ⊢ ((7 · 9) + (0 + 0)) = ;63
65	58	mul01i 8719	. . . . . 6 ⊢ (7 · 0) = 0
66	65	oveq1i 6095	. . . . 5 ⊢ ((7 · 0) + 1) = (0 + 1)
67	66, 44, 25	3eqtri 2263	. . . 4 ⊢ ((7 · 0) + 1) = ;01
68	52, 16, 16, 8, 54, 25, 55, 8, 16, 64, 67	decma2c 9838	. . 3 ⊢ ((7 · ;90) + 1) = ;;631
69		1lt7 9498	. . 3 ⊢ 1 < 7
70	51, 53, 4, 68, 69	ndvdsi 12716	. 2 ⊢ ¬ 7 ∥ ;;631
71	8, 4	decnncl 9804	. . 3 ⊢ ;11 ∈ ℕ
72		5nn0 9587	. . . 4 ⊢ 5 ∈ ℕ0
73	72, 55	deccl 9795	. . 3 ⊢ ;57 ∈ ℕ0
74		4nn 9472	. . 3 ⊢ 4 ∈ ℕ
75		eqid 2238	. . . 4 ⊢ ;57 = ;57
76	7	dec0h 9807	. . . 4 ⊢ 4 = ;04
77	8, 8	deccl 9795	. . . 4 ⊢ ;11 ∈ ℕ0
78		eqid 2238	. . . . 5 ⊢ ;11 = ;11
79		8cn 9392	. . . . . . 7 ⊢ 8 ∈ ℂ
80	79	addlidi 8470	. . . . . 6 ⊢ (0 + 8) = 8
81	6	dec0h 9807	. . . . . 6 ⊢ 8 = ;08
82	80, 81	eqtri 2259	. . . . 5 ⊢ (0 + 8) = ;08
83		5cn 9386	. . . . . . . 8 ⊢ 5 ∈ ℂ
84	83	mullidi 8329	. . . . . . 7 ⊢ (1 · 5) = 5
85	84, 44	oveq12i 6097	. . . . . 6 ⊢ ((1 · 5) + (0 + 1)) = (5 + 1)
86		5p1e6 9444	. . . . . 6 ⊢ (5 + 1) = 6
87	85, 86	eqtri 2259	. . . . 5 ⊢ ((1 · 5) + (0 + 1)) = 6
88	84	oveq1i 6095	. . . . . 6 ⊢ ((1 · 5) + 8) = (5 + 8)
89		8p5e13 9868	. . . . . . 7 ⊢ (8 + 5) = ;13
90	79, 83, 89	addcomli 8472	. . . . . 6 ⊢ (5 + 8) = ;13
91	88, 90	eqtri 2259	. . . . 5 ⊢ ((1 · 5) + 8) = ;13
92	8, 8, 16, 6, 78, 82, 72, 2, 8, 87, 91	decmac 9837	. . . 4 ⊢ ((;11 · 5) + (0 + 8)) = ;63
93	58	mullidi 8329	. . . . . . 7 ⊢ (1 · 7) = 7
94	93, 44	oveq12i 6097	. . . . . 6 ⊢ ((1 · 7) + (0 + 1)) = (7 + 1)
95		7p1e8 9446	. . . . . 6 ⊢ (7 + 1) = 8
96	94, 95	eqtri 2259	. . . . 5 ⊢ ((1 · 7) + (0 + 1)) = 8
97	93	oveq1i 6095	. . . . . 6 ⊢ ((1 · 7) + 4) = (7 + 4)
98		7p4e11 9861	. . . . . 6 ⊢ (7 + 4) = ;11
99	97, 98	eqtri 2259	. . . . 5 ⊢ ((1 · 7) + 4) = ;11
100	8, 8, 16, 7, 78, 76, 55, 8, 8, 96, 99	decmac 9837	. . . 4 ⊢ ((;11 · 7) + 4) = ;81
101	72, 55, 16, 7, 75, 76, 77, 8, 6, 92, 100	decma2c 9838	. . 3 ⊢ ((;11 · ;57) + 4) = ;;631
102		4lt10 9921	. . . 4 ⊢ 4 < ;10
103	4, 8, 7, 102	declti 9823	. . 3 ⊢ 4 < ;11
104	71, 73, 74, 101, 103	ndvdsi 12716	. 2 ⊢ ¬ ;11 ∥ ;;631
105	8, 13	decnncl 9804	. . 3 ⊢ ;13 ∈ ℕ
106	7, 6	deccl 9795	. . 3 ⊢ ;48 ∈ ℕ0
107		eqid 2238	. . . 4 ⊢ ;48 = ;48
108	55	dec0h 9807	. . . 4 ⊢ 7 = ;07
109	8, 2	deccl 9795	. . . 4 ⊢ ;13 ∈ ℕ0
110		eqid 2238	. . . . 5 ⊢ ;13 = ;13
111	77	nn0cni 9579	. . . . . 6 ⊢ ;11 ∈ ℂ
112	111	addlidi 8470	. . . . 5 ⊢ (0 + ;11) = ;11
113		4cn 9384	. . . . . . . 8 ⊢ 4 ∈ ℂ
114	113	mullidi 8329	. . . . . . 7 ⊢ (1 · 4) = 4
115		1p1e2 9423	. . . . . . 7 ⊢ (1 + 1) = 2
116	114, 115	oveq12i 6097	. . . . . 6 ⊢ ((1 · 4) + (1 + 1)) = (4 + 2)
117		4p2e6 9450	. . . . . 6 ⊢ (4 + 2) = 6
118	116, 117	eqtri 2259	. . . . 5 ⊢ ((1 · 4) + (1 + 1)) = 6
119		4t3e12 9883	. . . . . . 7 ⊢ (4 · 3) = ;12
120	113, 37, 119	mulcomli 8333	. . . . . 6 ⊢ (3 · 4) = ;12
121		2p1e3 9440	. . . . . 6 ⊢ (2 + 1) = 3
122	8, 21, 8, 120, 121	decaddi 9845	. . . . 5 ⊢ ((3 · 4) + 1) = ;13
123	8, 2, 8, 8, 110, 112, 7, 2, 8, 118, 122	decmac 9837	. . . 4 ⊢ ((;13 · 4) + (0 + ;11)) = ;63
124	79	mullidi 8329	. . . . . . 7 ⊢ (1 · 8) = 8
125	37	addlidi 8470	. . . . . . 7 ⊢ (0 + 3) = 3
126	124, 125	oveq12i 6097	. . . . . 6 ⊢ ((1 · 8) + (0 + 3)) = (8 + 3)
127		8p3e11 9866	. . . . . 6 ⊢ (8 + 3) = ;11
128	126, 127	eqtri 2259	. . . . 5 ⊢ ((1 · 8) + (0 + 3)) = ;11
129		8t3e24 9901	. . . . . . 7 ⊢ (8 · 3) = ;24
130	79, 37, 129	mulcomli 8333	. . . . . 6 ⊢ (3 · 8) = ;24
131	58, 113, 98	addcomli 8472	. . . . . 6 ⊢ (4 + 7) = ;11
132	21, 7, 55, 130, 121, 8, 131	decaddci 9846	. . . . 5 ⊢ ((3 · 8) + 7) = ;31
133	8, 2, 16, 55, 110, 108, 6, 8, 2, 128, 132	decmac 9837	. . . 4 ⊢ ((;13 · 8) + 7) = ;;111
134	7, 6, 16, 55, 107, 108, 109, 8, 77, 123, 133	decma2c 9838	. . 3 ⊢ ((;13 · ;48) + 7) = ;;631
135		7lt10 9918	. . . 4 ⊢ 7 < ;10
136	4, 2, 55, 135	declti 9823	. . 3 ⊢ 7 < ;13
137	105, 106, 51, 134, 136	ndvdsi 12716	. 2 ⊢ ¬ ;13 ∥ ;;631
138	8, 51	decnncl 9804	. . 3 ⊢ ;17 ∈ ℕ
139	2, 55	deccl 9795	. . 3 ⊢ ;37 ∈ ℕ0
140		2nn 9470	. . 3 ⊢ 2 ∈ ℕ
141		eqid 2238	. . . 4 ⊢ ;37 = ;37
142	21	dec0h 9807	. . . 4 ⊢ 2 = ;02
143	8, 55	deccl 9795	. . . 4 ⊢ ;17 ∈ ℕ0
144	8, 21	deccl 9795	. . . 4 ⊢ ;12 ∈ ℕ0
145		eqid 2238	. . . . 5 ⊢ ;17 = ;17
146	144	nn0cni 9579	. . . . . 6 ⊢ ;12 ∈ ℂ
147	146	addlidi 8470	. . . . 5 ⊢ (0 + ;12) = ;12
148	37	mullidi 8329	. . . . . . 7 ⊢ (1 · 3) = 3
149		1p2e3 9441	. . . . . . 7 ⊢ (1 + 2) = 3
150	148, 149	oveq12i 6097	. . . . . 6 ⊢ ((1 · 3) + (1 + 2)) = (3 + 3)
151		3p3e6 9449	. . . . . 6 ⊢ (3 + 3) = 6
152	150, 151	eqtri 2259	. . . . 5 ⊢ ((1 · 3) + (1 + 2)) = 6
153		7t3e21 9895	. . . . . 6 ⊢ (7 · 3) = ;21
154	21, 8, 21, 153, 149	decaddi 9845	. . . . 5 ⊢ ((7 · 3) + 2) = ;23
155	8, 55, 8, 21, 145, 147, 2, 2, 21, 152, 154	decmac 9837	. . . 4 ⊢ ((;17 · 3) + (0 + ;12)) = ;63
156	83	addlidi 8470	. . . . . . 7 ⊢ (0 + 5) = 5
157	93, 156	oveq12i 6097	. . . . . 6 ⊢ ((1 · 7) + (0 + 5)) = (7 + 5)
158		7p5e12 9862	. . . . . 6 ⊢ (7 + 5) = ;12
159	157, 158	eqtri 2259	. . . . 5 ⊢ ((1 · 7) + (0 + 5)) = ;12
160		7t7e49 9899	. . . . . 6 ⊢ (7 · 7) = ;49
161		4p1e5 9443	. . . . . 6 ⊢ (4 + 1) = 5
162		9p2e11 9872	. . . . . 6 ⊢ (9 + 2) = ;11
163	7, 52, 21, 160, 161, 8, 162	decaddci 9846	. . . . 5 ⊢ ((7 · 7) + 2) = ;51
164	8, 55, 16, 21, 145, 142, 55, 8, 72, 159, 163	decmac 9837	. . . 4 ⊢ ((;17 · 7) + 2) = ;;121
165	2, 55, 16, 21, 141, 142, 143, 8, 144, 155, 164	decma2c 9838	. . 3 ⊢ ((;17 · ;37) + 2) = ;;631
166		2lt10 9923	. . . 4 ⊢ 2 < ;10
167	4, 55, 21, 166	declti 9823	. . 3 ⊢ 2 < ;17
168	138, 139, 140, 165, 167	ndvdsi 12716	. 2 ⊢ ¬ ;17 ∥ ;;631
169		9nn 9477	. . . 4 ⊢ 9 ∈ ℕ
170	8, 169	decnncl 9804	. . 3 ⊢ ;19 ∈ ℕ
171	2, 2	deccl 9795	. . 3 ⊢ ;33 ∈ ℕ0
172		eqid 2238	. . . 4 ⊢ ;33 = ;33
173	8, 52	deccl 9795	. . . 4 ⊢ ;19 ∈ ℕ0
174		eqid 2238	. . . . 5 ⊢ ;19 = ;19
175	32	addlidi 8470	. . . . . 6 ⊢ (0 + 6) = 6
176	1	dec0h 9807	. . . . . 6 ⊢ 6 = ;06
177	175, 176	eqtri 2259	. . . . 5 ⊢ (0 + 6) = ;06
178	148, 125	oveq12i 6097	. . . . . 6 ⊢ ((1 · 3) + (0 + 3)) = (3 + 3)
179	178, 151	eqtri 2259	. . . . 5 ⊢ ((1 · 3) + (0 + 3)) = 6
180		9t3e27 9908	. . . . . 6 ⊢ (9 · 3) = ;27
181		7p6e13 9863	. . . . . 6 ⊢ (7 + 6) = ;13
182	21, 55, 1, 180, 121, 2, 181	decaddci 9846	. . . . 5 ⊢ ((9 · 3) + 6) = ;33
183	8, 52, 16, 1, 174, 177, 2, 2, 2, 179, 182	decmac 9837	. . . 4 ⊢ ((;19 · 3) + (0 + 6)) = ;63
184	21, 55, 7, 180, 121, 8, 98	decaddci 9846	. . . . 5 ⊢ ((9 · 3) + 4) = ;31
185	8, 52, 16, 7, 174, 76, 2, 8, 2, 179, 184	decmac 9837	. . . 4 ⊢ ((;19 · 3) + 4) = ;61
186	2, 2, 16, 7, 172, 76, 173, 8, 1, 183, 185	decma2c 9838	. . 3 ⊢ ((;19 · ;33) + 4) = ;;631
187	4, 52, 7, 102	declti 9823	. . 3 ⊢ 4 < ;19
188	170, 171, 74, 186, 187	ndvdsi 12716	. 2 ⊢ ¬ ;19 ∥ ;;631
189	21, 13	decnncl 9804	. . 3 ⊢ ;23 ∈ ℕ
190	21, 55	deccl 9795	. . 3 ⊢ ;27 ∈ ℕ0
191		10nn 9800	. . 3 ⊢ ;10 ∈ ℕ
192		eqid 2238	. . . 4 ⊢ ;27 = ;27
193		eqid 2238	. . . 4 ⊢ ;10 = ;10
194	21, 2	deccl 9795	. . . 4 ⊢ ;23 ∈ ℕ0
195	8, 1	deccl 9795	. . . 4 ⊢ ;16 ∈ ℕ0
196		eqid 2238	. . . . 5 ⊢ ;23 = ;23
197		eqid 2238	. . . . . 6 ⊢ ;16 = ;16
198		ax-1cn 8272	. . . . . . 7 ⊢ 1 ∈ ℂ
199		6p1e7 9445	. . . . . . 7 ⊢ (6 + 1) = 7
200	32, 198, 199	addcomli 8472	. . . . . 6 ⊢ (1 + 6) = 7
201	16, 8, 8, 1, 25, 197, 44, 200	decadd 9839	. . . . 5 ⊢ (1 + ;16) = ;17
202		2t2e4 9461	. . . . . . 7 ⊢ (2 · 2) = 4
203	202, 115	oveq12i 6097	. . . . . 6 ⊢ ((2 · 2) + (1 + 1)) = (4 + 2)
204	203, 117	eqtri 2259	. . . . 5 ⊢ ((2 · 2) + (1 + 1)) = 6
205	30	oveq1i 6095	. . . . . 6 ⊢ ((3 · 2) + 7) = (6 + 7)
206	58, 32, 181	addcomli 8472	. . . . . 6 ⊢ (6 + 7) = ;13
207	205, 206	eqtri 2259	. . . . 5 ⊢ ((3 · 2) + 7) = ;13
208	21, 2, 8, 55, 196, 201, 21, 2, 8, 204, 207	decmac 9837	. . . 4 ⊢ ((;23 · 2) + (1 + ;16)) = ;63
209		7t2e14 9894	. . . . . . . . 9 ⊢ (7 · 2) = ;14
210	58, 17, 209	mulcomli 8333	. . . . . . . 8 ⊢ (2 · 7) = ;14
211	8, 7, 21, 210, 117	decaddi 9845	. . . . . . 7 ⊢ ((2 · 7) + 2) = ;16
212	58, 37, 153	mulcomli 8333	. . . . . . 7 ⊢ (3 · 7) = ;21
213	55, 21, 2, 196, 8, 21, 211, 212	decmul1c 9850	. . . . . 6 ⊢ (;23 · 7) = ;;161
214	213	oveq1i 6095	. . . . 5 ⊢ ((;23 · 7) + 0) = (;;161 + 0)
215	195, 8	deccl 9795	. . . . . . 7 ⊢ ;;161 ∈ ℕ0
216	215	nn0cni 9579	. . . . . 6 ⊢ ;;161 ∈ ℂ
217	216	addridi 8469	. . . . 5 ⊢ (;;161 + 0) = ;;161
218	214, 217	eqtri 2259	. . . 4 ⊢ ((;23 · 7) + 0) = ;;161
219	21, 55, 8, 16, 192, 193, 194, 8, 195, 208, 218	decma2c 9838	. . 3 ⊢ ((;23 · ;27) + ;10) = ;;631
220		10pos 9801	. . . 4 ⊢ 0 < ;10
221		1lt2 9478	. . . 4 ⊢ 1 < 2
222	8, 21, 16, 2, 220, 221	decltc 9814	. . 3 ⊢ ;10 < ;23
223	189, 190, 191, 219, 222	ndvdsi 12716	. 2 ⊢ ¬ ;23 ∥ ;;631
224	5, 12, 15, 20, 48, 50, 70, 104, 137, 168, 188, 223	prmlem2 13254	1 ⊢ ;;631 ∈ ℙ

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