Theorem 163prm 13268

Description: 163 is a prime number. (Contributed by Mario Carneiro, 19-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)

Assertion

Ref	Expression
163prm	|- ;; 1 6 3 e. Prime

Proof of Theorem 163prm

Step	Hyp	Ref	Expression
1		1nn0 9590	. . . 4 |- 1 e. NN0
2		6nn0 9595	. . . 4 |- 6 e. NN0
3	1, 2	deccl 9802	. . 3 |- ; 1 6 e. NN0
4		3nn 9478	. . 3 |- 3 e. NN
5	3, 4	decnncl 9811	. 2 |- ;; 1 6 3 e. NN
6		8nn0 9597	. . 3 |- 8 e. NN0
7		4nn0 9593	. . 3 |- 4 e. NN0
8		3nn0 9592	. . 3 |- 3 e. NN0
9		1lt8 9512	. . 3 |- 1 < 8
10		6lt10 9926	. . 3 |- 6 < ; 1 0
11		3lt10 9929	. . 3 |- 3 < ; 1 0
12	1, 6, 2, 7, 8, 1, 9, 10, 11	3decltc 9825	. 2 |- ;; 1 6 3 < ;; 8 4 1
13		6nn 9481	. . . 4 |- 6 e. NN
14	1, 13	decnncl 9811	. . 3 |- ; 1 6 e. NN
15		1lt10 9931	. . 3 |- 1 < ; 1 0
16	14, 8, 1, 15	declti 9830	. 2 |- 1 < ;; 1 6 3
17		2cn 9384	. . . 4 |- 2 e. CC
18	17	mullidi 8335	. . 3 |- ( 1 x. 2 ) = 2
19		df-3 9373	. . 3 |- 3 = ( 2 + 1 )
20	3, 1, 18, 19	dec2dvds 13219	. 2 |- -. 2 || ;; 1 6 3
21		5nn0 9594	. . . 4 |- 5 e. NN0
22	21, 7	deccl 9802	. . 3 |- ; 5 4 e. NN0
23		1nn 9324	. . 3 |- 1 e. NN
24		0nn0 9589	. . . 4 |- 0 e. NN0
25		eqid 2238	. . . 4 |- ; 5 4 = ; 5 4
26	1	dec0h 9814	. . . 4 |- 1 = ; 0 1
27		ax-1cn 8278	. . . . . . 7 |- 1 e. CC
28	27	addlidi 8476	. . . . . 6 |- ( 0 + 1 ) = 1
29	28	oveq2i 6099	. . . . 5 |- ( ( 3 x. 5 ) + ( 0 + 1 ) ) = ( ( 3 x. 5 ) + 1 )
30		5p1e6 9451	. . . . . 6 |- ( 5 + 1 ) = 6
31		5cn 9393	. . . . . . 7 |- 5 e. CC
32		3cn 9388	. . . . . . 7 |- 3 e. CC
33		5t3e15 9893	. . . . . . 7 |- ( 5 x. 3 ) = ; 1 5
34	31, 32, 33	mulcomli 8339	. . . . . 6 |- ( 3 x. 5 ) = ; 1 5
35	1, 21, 30, 34	decsuc 9823	. . . . 5 |- ( ( 3 x. 5 ) + 1 ) = ; 1 6
36	29, 35	eqtri 2259	. . . 4 |- ( ( 3 x. 5 ) + ( 0 + 1 ) ) = ; 1 6
37		2nn0 9591	. . . . 5 |- 2 e. NN0
38		2p1e3 9447	. . . . 5 |- ( 2 + 1 ) = 3
39		4cn 9391	. . . . . 6 |- 4 e. CC
40		4t3e12 9890	. . . . . 6 |- ( 4 x. 3 ) = ; 1 2
41	39, 32, 40	mulcomli 8339	. . . . 5 |- ( 3 x. 4 ) = ; 1 2
42	1, 37, 38, 41	decsuc 9823	. . . 4 |- ( ( 3 x. 4 ) + 1 ) = ; 1 3
43	21, 7, 24, 1, 25, 26, 8, 8, 1, 36, 42	decma2c 9845	. . 3 |- ( ( 3 x. ; 5 4 ) + 1 ) = ;; 1 6 3
44		1lt3 9487	. . 3 |- 1 < 3
45	4, 22, 23, 43, 44	ndvdsi 12725	. 2 |- -. 3 || ;; 1 6 3
46		3lt5 9492	. . 3 |- 3 < 5
47	3, 4, 46	dec5dvds 13220	. 2 |- -. 5 || ;; 1 6 3
48		7nn 9482	. . 3 |- 7 e. NN
49	37, 8	deccl 9802	. . 3 |- ; 2 3 e. NN0
50		2nn 9477	. . 3 |- 2 e. NN
51		eqid 2238	. . . 4 |- ; 2 3 = ; 2 3
52	37	dec0h 9814	. . . 4 |- 2 = ; 0 2
53		7nn0 9596	. . . 4 |- 7 e. NN0
54	17	addlidi 8476	. . . . . 6 |- ( 0 + 2 ) = 2
55	54	oveq2i 6099	. . . . 5 |- ( ( 7 x. 2 ) + ( 0 + 2 ) ) = ( ( 7 x. 2 ) + 2 )
56		7t2e14 9901	. . . . . 6 |- ( 7 x. 2 ) = ; 1 4
57		4p2e6 9457	. . . . . 6 |- ( 4 + 2 ) = 6
58	1, 7, 37, 56, 57	decaddi 9852	. . . . 5 |- ( ( 7 x. 2 ) + 2 ) = ; 1 6
59	55, 58	eqtri 2259	. . . 4 |- ( ( 7 x. 2 ) + ( 0 + 2 ) ) = ; 1 6
60		7t3e21 9902	. . . . 5 |- ( 7 x. 3 ) = ; 2 1
61		1p2e3 9448	. . . . 5 |- ( 1 + 2 ) = 3
62	37, 1, 37, 60, 61	decaddi 9852	. . . 4 |- ( ( 7 x. 3 ) + 2 ) = ; 2 3
63	37, 8, 24, 37, 51, 52, 53, 8, 37, 59, 62	decma2c 9845	. . 3 |- ( ( 7 x. ; 2 3 ) + 2 ) = ;; 1 6 3
64		2lt7 9504	. . 3 |- 2 < 7
65	48, 49, 50, 63, 64	ndvdsi 12725	. 2 |- -. 7 || ;; 1 6 3
66	1, 23	decnncl 9811	. . 3 |- ; 1 1 e. NN
67	1, 7	deccl 9802	. . 3 |- ; 1 4 e. NN0
68		9nn 9484	. . 3 |- 9 e. NN
69		9nn0 9598	. . . 4 |- 9 e. NN0
70		eqid 2238	. . . 4 |- ; 1 4 = ; 1 4
71	69	dec0h 9814	. . . 4 |- 9 = ; 0 9
72	1, 1	deccl 9802	. . . 4 |- ; 1 1 e. NN0
73	31	addlidi 8476	. . . . . 6 |- ( 0 + 5 ) = 5
74	73	oveq2i 6099	. . . . 5 |- ( (; 1 1 x. 1 ) + ( 0 + 5 ) ) = ( (; 1 1 x. 1 ) + 5 )
75	66	nncni 9323	. . . . . . 7 |- ; 1 1 e. CC
76	75	mulridi 8334	. . . . . 6 |- (; 1 1 x. 1 ) = ; 1 1
77	31, 27, 30	addcomli 8478	. . . . . 6 |- ( 1 + 5 ) = 6
78	1, 1, 21, 76, 77	decaddi 9852	. . . . 5 |- ( (; 1 1 x. 1 ) + 5 ) = ; 1 6
79	74, 78	eqtri 2259	. . . 4 |- ( (; 1 1 x. 1 ) + ( 0 + 5 ) ) = ; 1 6
80		eqid 2238	. . . . 5 |- ; 1 1 = ; 1 1
81	39	mullidi 8335	. . . . . . 7 |- ( 1 x. 4 ) = 4
82	81, 28	oveq12i 6100	. . . . . 6 |- ( ( 1 x. 4 ) + ( 0 + 1 ) ) = ( 4 + 1 )
83		4p1e5 9450	. . . . . 6 |- ( 4 + 1 ) = 5
84	82, 83	eqtri 2259	. . . . 5 |- ( ( 1 x. 4 ) + ( 0 + 1 ) ) = 5
85	81	oveq1i 6098	. . . . . 6 |- ( ( 1 x. 4 ) + 9 ) = ( 4 + 9 )
86		9cn 9401	. . . . . . 7 |- 9 e. CC
87		9p4e13 9881	. . . . . . 7 |- ( 9 + 4 ) = ; 1 3
88	86, 39, 87	addcomli 8478	. . . . . 6 |- ( 4 + 9 ) = ; 1 3
89	85, 88	eqtri 2259	. . . . 5 |- ( ( 1 x. 4 ) + 9 ) = ; 1 3
90	1, 1, 24, 69, 80, 71, 7, 8, 1, 84, 89	decmac 9844	. . . 4 |- ( (; 1 1 x. 4 ) + 9 ) = ; 5 3
91	1, 7, 24, 69, 70, 71, 72, 8, 21, 79, 90	decma2c 9845	. . 3 |- ( (; 1 1 x. ; 1 4 ) + 9 ) = ;; 1 6 3
92		9lt10 9923	. . . 4 |- 9 < ; 1 0
93	23, 1, 69, 92	declti 9830	. . 3 |- 9 < ; 1 1
94	66, 67, 68, 91, 93	ndvdsi 12725	. 2 |- -. ; 1 1 || ;; 1 6 3
95	1, 4	decnncl 9811	. . 3 |- ; 1 3 e. NN
96	1, 37	deccl 9802	. . 3 |- ; 1 2 e. NN0
97		eqid 2238	. . . 4 |- ; 1 2 = ; 1 2
98	53	dec0h 9814	. . . 4 |- 7 = ; 0 7
99	1, 8	deccl 9802	. . . 4 |- ; 1 3 e. NN0
100		eqid 2238	. . . . 5 |- ; 1 3 = ; 1 3
101	32	addlidi 8476	. . . . . 6 |- ( 0 + 3 ) = 3
102	8	dec0h 9814	. . . . . 6 |- 3 = ; 0 3
103	101, 102	eqtri 2259	. . . . 5 |- ( 0 + 3 ) = ; 0 3
104	27	mulridi 8334	. . . . . . 7 |- ( 1 x. 1 ) = 1
105		00id 8474	. . . . . . 7 |- ( 0 + 0 ) = 0
106	104, 105	oveq12i 6100	. . . . . 6 |- ( ( 1 x. 1 ) + ( 0 + 0 ) ) = ( 1 + 0 )
107	27	addridi 8475	. . . . . 6 |- ( 1 + 0 ) = 1
108	106, 107	eqtri 2259	. . . . 5 |- ( ( 1 x. 1 ) + ( 0 + 0 ) ) = 1
109	32	mulridi 8334	. . . . . . . 8 |- ( 3 x. 1 ) = 3
110	109	oveq1i 6098	. . . . . . 7 |- ( ( 3 x. 1 ) + 3 ) = ( 3 + 3 )
111		3p3e6 9456	. . . . . . 7 |- ( 3 + 3 ) = 6
112	110, 111	eqtri 2259	. . . . . 6 |- ( ( 3 x. 1 ) + 3 ) = 6
113	2	dec0h 9814	. . . . . 6 |- 6 = ; 0 6
114	112, 113	eqtri 2259	. . . . 5 |- ( ( 3 x. 1 ) + 3 ) = ; 0 6
115	1, 8, 24, 8, 100, 103, 1, 2, 24, 108, 114	decmac 9844	. . . 4 |- ( (; 1 3 x. 1 ) + ( 0 + 3 ) ) = ; 1 6
116	18, 28	oveq12i 6100	. . . . . 6 |- ( ( 1 x. 2 ) + ( 0 + 1 ) ) = ( 2 + 1 )
117	116, 38	eqtri 2259	. . . . 5 |- ( ( 1 x. 2 ) + ( 0 + 1 ) ) = 3
118		3t2e6 9470	. . . . . . 7 |- ( 3 x. 2 ) = 6
119	118	oveq1i 6098	. . . . . 6 |- ( ( 3 x. 2 ) + 7 ) = ( 6 + 7 )
120		7cn 9397	. . . . . . 7 |- 7 e. CC
121		6cn 9395	. . . . . . 7 |- 6 e. CC
122		7p6e13 9870	. . . . . . 7 |- ( 7 + 6 ) = ; 1 3
123	120, 121, 122	addcomli 8478	. . . . . 6 |- ( 6 + 7 ) = ; 1 3
124	119, 123	eqtri 2259	. . . . 5 |- ( ( 3 x. 2 ) + 7 ) = ; 1 3
125	1, 8, 24, 53, 100, 98, 37, 8, 1, 117, 124	decmac 9844	. . . 4 |- ( (; 1 3 x. 2 ) + 7 ) = ; 3 3
126	1, 37, 24, 53, 97, 98, 99, 8, 8, 115, 125	decma2c 9845	. . 3 |- ( (; 1 3 x. ; 1 2 ) + 7 ) = ;; 1 6 3
127		7lt10 9925	. . . 4 |- 7 < ; 1 0
128	23, 8, 53, 127	declti 9830	. . 3 |- 7 < ; 1 3
129	95, 96, 48, 126, 128	ndvdsi 12725	. 2 |- -. ; 1 3 || ;; 1 6 3
130	1, 48	decnncl 9811	. . 3 |- ; 1 7 e. NN
131		10nn 9807	. . 3 |- ; 1 0 e. NN
132		eqid 2238	. . . 4 |- ; 1 7 = ; 1 7
133		eqid 2238	. . . 4 |- ; 1 0 = ; 1 0
134	86	mullidi 8335	. . . . . 6 |- ( 1 x. 9 ) = 9
135		6p1e7 9452	. . . . . . 7 |- ( 6 + 1 ) = 7
136	121, 27, 135	addcomli 8478	. . . . . 6 |- ( 1 + 6 ) = 7
137	134, 136	oveq12i 6100	. . . . 5 |- ( ( 1 x. 9 ) + ( 1 + 6 ) ) = ( 9 + 7 )
138		9p7e16 9884	. . . . 5 |- ( 9 + 7 ) = ; 1 6
139	137, 138	eqtri 2259	. . . 4 |- ( ( 1 x. 9 ) + ( 1 + 6 ) ) = ; 1 6
140		9t7e63 9919	. . . . . . 7 |- ( 9 x. 7 ) = ; 6 3
141	86, 120, 140	mulcomli 8339	. . . . . 6 |- ( 7 x. 9 ) = ; 6 3
142	141	oveq1i 6098	. . . . 5 |- ( ( 7 x. 9 ) + 0 ) = (; 6 3 + 0 )
143	2, 8	deccl 9802	. . . . . . 7 |- ; 6 3 e. NN0
144	143	nn0cni 9586	. . . . . 6 |- ; 6 3 e. CC
145	144	addridi 8475	. . . . 5 |- (; 6 3 + 0 ) = ; 6 3
146	142, 145	eqtri 2259	. . . 4 |- ( ( 7 x. 9 ) + 0 ) = ; 6 3
147	1, 53, 1, 24, 132, 133, 69, 8, 2, 139, 146	decmac 9844	. . 3 |- ( (; 1 7 x. 9 ) + ; 1 0 ) = ;; 1 6 3
148		7pos 9415	. . . 4 |- 0 < 7
149	1, 24, 48, 148	declt 9820	. . 3 |- ; 1 0 < ; 1 7
150	130, 69, 131, 147, 149	ndvdsi 12725	. 2 |- -. ; 1 7 || ;; 1 6 3
151	1, 68	decnncl 9811	. . 3 |- ; 1 9 e. NN
152		eqid 2238	. . . 4 |- ; 1 9 = ; 1 9
153		8cn 9399	. . . . . . 7 |- 8 e. CC
154	153	mullidi 8335	. . . . . 6 |- ( 1 x. 8 ) = 8
155		7p1e8 9453	. . . . . . 7 |- ( 7 + 1 ) = 8
156	120, 27, 155	addcomli 8478	. . . . . 6 |- ( 1 + 7 ) = 8
157	154, 156	oveq12i 6100	. . . . 5 |- ( ( 1 x. 8 ) + ( 1 + 7 ) ) = ( 8 + 8 )
158		8p8e16 9878	. . . . 5 |- ( 8 + 8 ) = ; 1 6
159	157, 158	eqtri 2259	. . . 4 |- ( ( 1 x. 8 ) + ( 1 + 7 ) ) = ; 1 6
160		9t8e72 9920	. . . . 5 |- ( 9 x. 8 ) = ; 7 2
161	53, 37, 38, 160	decsuc 9823	. . . 4 |- ( ( 9 x. 8 ) + 1 ) = ; 7 3
162	1, 69, 1, 1, 152, 80, 6, 8, 53, 159, 161	decmac 9844	. . 3 |- ( (; 1 9 x. 8 ) + ; 1 1 ) = ;; 1 6 3
163		1lt9 9520	. . . 4 |- 1 < 9
164	1, 1, 68, 163	declt 9820	. . 3 |- ; 1 1 < ; 1 9
165	151, 6, 66, 162, 164	ndvdsi 12725	. 2 |- -. ; 1 9 || ;; 1 6 3
166	37, 4	decnncl 9811	. . 3 |- ; 2 3 e. NN
167	120, 17, 56	mulcomli 8339	. . . . 5 |- ( 2 x. 7 ) = ; 1 4
168	1, 7, 37, 167, 57	decaddi 9852	. . . 4 |- ( ( 2 x. 7 ) + 2 ) = ; 1 6
169	120, 32, 60	mulcomli 8339	. . . . 5 |- ( 3 x. 7 ) = ; 2 1
170	37, 1, 37, 169, 61	decaddi 9852	. . . 4 |- ( ( 3 x. 7 ) + 2 ) = ; 2 3
171	37, 8, 37, 51, 53, 8, 37, 168, 170	decrmac 9850	. . 3 |- ( (; 2 3 x. 7 ) + 2 ) = ;; 1 6 3
172		2lt10 9930	. . . 4 |- 2 < ; 1 0
173	50, 8, 37, 172	declti 9830	. . 3 |- 2 < ; 2 3
174	166, 53, 50, 171, 173	ndvdsi 12725	. 2 |- -. ; 2 3 || ;; 1 6 3
175	5, 12, 16, 20, 45, 47, 65, 94, 129, 150, 165, 174	prmlem2 13263	1 |- ;; 1 6 3 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)