Theorem 163prm 15832

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 11308	. . . 4 |- 1 e. NN0
2		6nn0 11313	. . . 4 |- 6 e. NN0
3	1, 2	deccl 11512	. . 3 |- ; 1 6 e. NN0
4		3nn 11186	. . 3 |- 3 e. NN
5	3, 4	decnncl 11518	. 2 |- ;; 1 6 3 e. NN
6		8nn0 11315	. . 3 |- 8 e. NN0
7		4nn0 11311	. . 3 |- 4 e. NN0
8		3nn0 11310	. . 3 |- 3 e. NN0
9		1lt8 11221	. . 3 |- 1 < 8
10		6lt10 11676	. . 3 |- 6 < ; 1 0
11		3lt10 11679	. . 3 |- 3 < ; 1 0
12	1, 6, 2, 7, 8, 1, 9, 10, 11	3decltc 11538	. 2 |- ;; 1 6 3 < ;; 8 4 1
13		6nn 11189	. . . 4 |- 6 e. NN
14	1, 13	decnncl 11518	. . 3 |- ; 1 6 e. NN
15		1lt10 11681	. . 3 |- 1 < ; 1 0
16	14, 8, 1, 15	declti 11546	. 2 |- 1 < ;; 1 6 3
17		2cn 11091	. . . 4 |- 2 e. CC
18	17	mulid2i 10043	. . 3 |- ( 1 x. 2 ) = 2
19		df-3 11080	. . 3 |- 3 = ( 2 + 1 )
20	3, 1, 18, 19	dec2dvds 15767	. 2 |- -. 2 || ;; 1 6 3
21		5nn0 11312	. . . 4 |- 5 e. NN0
22	21, 7	deccl 11512	. . 3 |- ; 5 4 e. NN0
23		1nn 11031	. . 3 |- 1 e. NN
24		0nn0 11307	. . . 4 |- 0 e. NN0
25		eqid 2622	. . . 4 |- ; 5 4 = ; 5 4
26	1	dec0h 11522	. . . 4 |- 1 = ; 0 1
27		ax-1cn 9994	. . . . . . 7 |- 1 e. CC
28	27	addid2i 10224	. . . . . 6 |- ( 0 + 1 ) = 1
29	28	oveq2i 6661	. . . . 5 |- ( ( 3 x. 5 ) + ( 0 + 1 ) ) = ( ( 3 x. 5 ) + 1 )
30		5p1e6 11155	. . . . . 6 |- ( 5 + 1 ) = 6
31		5cn 11100	. . . . . . 7 |- 5 e. CC
32		3cn 11095	. . . . . . 7 |- 3 e. CC
33		5t3e15 11635	. . . . . . 7 |- ( 5 x. 3 ) = ; 1 5
34	31, 32, 33	mulcomli 10047	. . . . . 6 |- ( 3 x. 5 ) = ; 1 5
35	1, 21, 30, 34	decsuc 11535	. . . . 5 |- ( ( 3 x. 5 ) + 1 ) = ; 1 6
36	29, 35	eqtri 2644	. . . 4 |- ( ( 3 x. 5 ) + ( 0 + 1 ) ) = ; 1 6
37		2nn0 11309	. . . . 5 |- 2 e. NN0
38		2p1e3 11151	. . . . 5 |- ( 2 + 1 ) = 3
39		4cn 11098	. . . . . 6 |- 4 e. CC
40		4t3e12 11632	. . . . . 6 |- ( 4 x. 3 ) = ; 1 2
41	39, 32, 40	mulcomli 10047	. . . . 5 |- ( 3 x. 4 ) = ; 1 2
42	1, 37, 38, 41	decsuc 11535	. . . 4 |- ( ( 3 x. 4 ) + 1 ) = ; 1 3
43	21, 7, 24, 1, 25, 26, 8, 8, 1, 36, 42	decma2c 11568	. . 3 |- ( ( 3 x. ; 5 4 ) + 1 ) = ;; 1 6 3
44		1lt3 11196	. . 3 |- 1 < 3
45	4, 22, 23, 43, 44	ndvdsi 15136	. 2 |- -. 3 || ;; 1 6 3
46		3lt5 11201	. . 3 |- 3 < 5
47	3, 4, 46	dec5dvds 15768	. 2 |- -. 5 || ;; 1 6 3
48		7nn 11190	. . 3 |- 7 e. NN
49	37, 8	deccl 11512	. . 3 |- ; 2 3 e. NN0
50		2nn 11185	. . 3 |- 2 e. NN
51		eqid 2622	. . . 4 |- ; 2 3 = ; 2 3
52	37	dec0h 11522	. . . 4 |- 2 = ; 0 2
53		7nn0 11314	. . . 4 |- 7 e. NN0
54	17	addid2i 10224	. . . . . 6 |- ( 0 + 2 ) = 2
55	54	oveq2i 6661	. . . . 5 |- ( ( 7 x. 2 ) + ( 0 + 2 ) ) = ( ( 7 x. 2 ) + 2 )
56		7t2e14 11648	. . . . . 6 |- ( 7 x. 2 ) = ; 1 4
57		4p2e6 11162	. . . . . 6 |- ( 4 + 2 ) = 6
58	1, 7, 37, 56, 57	decaddi 11579	. . . . 5 |- ( ( 7 x. 2 ) + 2 ) = ; 1 6
59	55, 58	eqtri 2644	. . . 4 |- ( ( 7 x. 2 ) + ( 0 + 2 ) ) = ; 1 6
60		7t3e21 11649	. . . . 5 |- ( 7 x. 3 ) = ; 2 1
61		1p2e3 11152	. . . . 5 |- ( 1 + 2 ) = 3
62	37, 1, 37, 60, 61	decaddi 11579	. . . 4 |- ( ( 7 x. 3 ) + 2 ) = ; 2 3
63	37, 8, 24, 37, 51, 52, 53, 8, 37, 59, 62	decma2c 11568	. . 3 |- ( ( 7 x. ; 2 3 ) + 2 ) = ;; 1 6 3
64		2lt7 11213	. . 3 |- 2 < 7
65	48, 49, 50, 63, 64	ndvdsi 15136	. 2 |- -. 7 || ;; 1 6 3
66	1, 23	decnncl 11518	. . 3 |- ; 1 1 e. NN
67	1, 7	deccl 11512	. . 3 |- ; 1 4 e. NN0
68		9nn 11192	. . 3 |- 9 e. NN
69		9nn0 11316	. . . 4 |- 9 e. NN0
70		eqid 2622	. . . 4 |- ; 1 4 = ; 1 4
71	69	dec0h 11522	. . . 4 |- 9 = ; 0 9
72	1, 1	deccl 11512	. . . 4 |- ; 1 1 e. NN0
73	31	addid2i 10224	. . . . . 6 |- ( 0 + 5 ) = 5
74	73	oveq2i 6661	. . . . 5 |- ( (; 1 1 x. 1 ) + ( 0 + 5 ) ) = ( (; 1 1 x. 1 ) + 5 )
75	66	nncni 11030	. . . . . . 7 |- ; 1 1 e. CC
76	75	mulid1i 10042	. . . . . 6 |- (; 1 1 x. 1 ) = ; 1 1
77	31, 27, 30	addcomli 10228	. . . . . 6 |- ( 1 + 5 ) = 6
78	1, 1, 21, 76, 77	decaddi 11579	. . . . 5 |- ( (; 1 1 x. 1 ) + 5 ) = ; 1 6
79	74, 78	eqtri 2644	. . . 4 |- ( (; 1 1 x. 1 ) + ( 0 + 5 ) ) = ; 1 6
80		eqid 2622	. . . . 5 |- ; 1 1 = ; 1 1
81	39	mulid2i 10043	. . . . . . 7 |- ( 1 x. 4 ) = 4
82	81, 28	oveq12i 6662	. . . . . 6 |- ( ( 1 x. 4 ) + ( 0 + 1 ) ) = ( 4 + 1 )
83		4p1e5 11154	. . . . . 6 |- ( 4 + 1 ) = 5
84	82, 83	eqtri 2644	. . . . 5 |- ( ( 1 x. 4 ) + ( 0 + 1 ) ) = 5
85	81	oveq1i 6660	. . . . . 6 |- ( ( 1 x. 4 ) + 9 ) = ( 4 + 9 )
86		9cn 11108	. . . . . . 7 |- 9 e. CC
87		9p4e13 11622	. . . . . . 7 |- ( 9 + 4 ) = ; 1 3
88	86, 39, 87	addcomli 10228	. . . . . 6 |- ( 4 + 9 ) = ; 1 3
89	85, 88	eqtri 2644	. . . . 5 |- ( ( 1 x. 4 ) + 9 ) = ; 1 3
90	1, 1, 24, 69, 80, 71, 7, 8, 1, 84, 89	decmac 11566	. . . 4 |- ( (; 1 1 x. 4 ) + 9 ) = ; 5 3
91	1, 7, 24, 69, 70, 71, 72, 8, 21, 79, 90	decma2c 11568	. . 3 |- ( (; 1 1 x. ; 1 4 ) + 9 ) = ;; 1 6 3
92		9lt10 11673	. . . 4 |- 9 < ; 1 0
93	23, 1, 69, 92	declti 11546	. . 3 |- 9 < ; 1 1
94	66, 67, 68, 91, 93	ndvdsi 15136	. 2 |- -. ; 1 1 || ;; 1 6 3
95	1, 4	decnncl 11518	. . 3 |- ; 1 3 e. NN
96	1, 37	deccl 11512	. . 3 |- ; 1 2 e. NN0
97		eqid 2622	. . . 4 |- ; 1 2 = ; 1 2
98	53	dec0h 11522	. . . 4 |- 7 = ; 0 7
99	1, 8	deccl 11512	. . . 4 |- ; 1 3 e. NN0
100		eqid 2622	. . . . 5 |- ; 1 3 = ; 1 3
101	32	addid2i 10224	. . . . . 6 |- ( 0 + 3 ) = 3
102	8	dec0h 11522	. . . . . 6 |- 3 = ; 0 3
103	101, 102	eqtri 2644	. . . . 5 |- ( 0 + 3 ) = ; 0 3
104	27	mulid1i 10042	. . . . . . 7 |- ( 1 x. 1 ) = 1
105		00id 10211	. . . . . . 7 |- ( 0 + 0 ) = 0
106	104, 105	oveq12i 6662	. . . . . 6 |- ( ( 1 x. 1 ) + ( 0 + 0 ) ) = ( 1 + 0 )
107	27	addid1i 10223	. . . . . 6 |- ( 1 + 0 ) = 1
108	106, 107	eqtri 2644	. . . . 5 |- ( ( 1 x. 1 ) + ( 0 + 0 ) ) = 1
109	32	mulid1i 10042	. . . . . . . 8 |- ( 3 x. 1 ) = 3
110	109	oveq1i 6660	. . . . . . 7 |- ( ( 3 x. 1 ) + 3 ) = ( 3 + 3 )
111		3p3e6 11161	. . . . . . 7 |- ( 3 + 3 ) = 6
112	110, 111	eqtri 2644	. . . . . 6 |- ( ( 3 x. 1 ) + 3 ) = 6
113	2	dec0h 11522	. . . . . 6 |- 6 = ; 0 6
114	112, 113	eqtri 2644	. . . . 5 |- ( ( 3 x. 1 ) + 3 ) = ; 0 6
115	1, 8, 24, 8, 100, 103, 1, 2, 24, 108, 114	decmac 11566	. . . 4 |- ( (; 1 3 x. 1 ) + ( 0 + 3 ) ) = ; 1 6
116	18, 28	oveq12i 6662	. . . . . 6 |- ( ( 1 x. 2 ) + ( 0 + 1 ) ) = ( 2 + 1 )
117	116, 38	eqtri 2644	. . . . 5 |- ( ( 1 x. 2 ) + ( 0 + 1 ) ) = 3
118		3t2e6 11179	. . . . . . 7 |- ( 3 x. 2 ) = 6
119	118	oveq1i 6660	. . . . . 6 |- ( ( 3 x. 2 ) + 7 ) = ( 6 + 7 )
120		7cn 11104	. . . . . . 7 |- 7 e. CC
121		6cn 11102	. . . . . . 7 |- 6 e. CC
122		7p6e13 11608	. . . . . . 7 |- ( 7 + 6 ) = ; 1 3
123	120, 121, 122	addcomli 10228	. . . . . 6 |- ( 6 + 7 ) = ; 1 3
124	119, 123	eqtri 2644	. . . . 5 |- ( ( 3 x. 2 ) + 7 ) = ; 1 3
125	1, 8, 24, 53, 100, 98, 37, 8, 1, 117, 124	decmac 11566	. . . 4 |- ( (; 1 3 x. 2 ) + 7 ) = ; 3 3
126	1, 37, 24, 53, 97, 98, 99, 8, 8, 115, 125	decma2c 11568	. . 3 |- ( (; 1 3 x. ; 1 2 ) + 7 ) = ;; 1 6 3
127		7lt10 11675	. . . 4 |- 7 < ; 1 0
128	23, 8, 53, 127	declti 11546	. . 3 |- 7 < ; 1 3
129	95, 96, 48, 126, 128	ndvdsi 15136	. 2 |- -. ; 1 3 || ;; 1 6 3
130	1, 48	decnncl 11518	. . 3 |- ; 1 7 e. NN
131		10nn 11514	. . 3 |- ; 1 0 e. NN
132		eqid 2622	. . . 4 |- ; 1 7 = ; 1 7
133		eqid 2622	. . . 4 |- ; 1 0 = ; 1 0
134	86	mulid2i 10043	. . . . . 6 |- ( 1 x. 9 ) = 9
135		6p1e7 11156	. . . . . . 7 |- ( 6 + 1 ) = 7
136	121, 27, 135	addcomli 10228	. . . . . 6 |- ( 1 + 6 ) = 7
137	134, 136	oveq12i 6662	. . . . 5 |- ( ( 1 x. 9 ) + ( 1 + 6 ) ) = ( 9 + 7 )
138		9p7e16 11625	. . . . 5 |- ( 9 + 7 ) = ; 1 6
139	137, 138	eqtri 2644	. . . 4 |- ( ( 1 x. 9 ) + ( 1 + 6 ) ) = ; 1 6
140		9t7e63 11668	. . . . . . 7 |- ( 9 x. 7 ) = ; 6 3
141	86, 120, 140	mulcomli 10047	. . . . . 6 |- ( 7 x. 9 ) = ; 6 3
142	141	oveq1i 6660	. . . . 5 |- ( ( 7 x. 9 ) + 0 ) = (; 6 3 + 0 )
143	2, 8	deccl 11512	. . . . . . 7 |- ; 6 3 e. NN0
144	143	nn0cni 11304	. . . . . 6 |- ; 6 3 e. CC
145	144	addid1i 10223	. . . . 5 |- (; 6 3 + 0 ) = ; 6 3
146	142, 145	eqtri 2644	. . . 4 |- ( ( 7 x. 9 ) + 0 ) = ; 6 3
147	1, 53, 1, 24, 132, 133, 69, 8, 2, 139, 146	decmac 11566	. . 3 |- ( (; 1 7 x. 9 ) + ; 1 0 ) = ;; 1 6 3
148		7pos 11120	. . . 4 |- 0 < 7
149	1, 24, 48, 148	declt 11530	. . 3 |- ; 1 0 < ; 1 7
150	130, 69, 131, 147, 149	ndvdsi 15136	. 2 |- -. ; 1 7 || ;; 1 6 3
151	1, 68	decnncl 11518	. . 3 |- ; 1 9 e. NN
152		eqid 2622	. . . 4 |- ; 1 9 = ; 1 9
153		8cn 11106	. . . . . . 7 |- 8 e. CC
154	153	mulid2i 10043	. . . . . 6 |- ( 1 x. 8 ) = 8
155		7p1e8 11157	. . . . . . 7 |- ( 7 + 1 ) = 8
156	120, 27, 155	addcomli 10228	. . . . . 6 |- ( 1 + 7 ) = 8
157	154, 156	oveq12i 6662	. . . . 5 |- ( ( 1 x. 8 ) + ( 1 + 7 ) ) = ( 8 + 8 )
158		8p8e16 11618	. . . . 5 |- ( 8 + 8 ) = ; 1 6
159	157, 158	eqtri 2644	. . . 4 |- ( ( 1 x. 8 ) + ( 1 + 7 ) ) = ; 1 6
160		9t8e72 11669	. . . . 5 |- ( 9 x. 8 ) = ; 7 2
161	53, 37, 38, 160	decsuc 11535	. . . 4 |- ( ( 9 x. 8 ) + 1 ) = ; 7 3
162	1, 69, 1, 1, 152, 80, 6, 8, 53, 159, 161	decmac 11566	. . 3 |- ( (; 1 9 x. 8 ) + ; 1 1 ) = ;; 1 6 3
163		1lt9 11229	. . . 4 |- 1 < 9
164	1, 1, 68, 163	declt 11530	. . 3 |- ; 1 1 < ; 1 9
165	151, 6, 66, 162, 164	ndvdsi 15136	. 2 |- -. ; 1 9 || ;; 1 6 3
166	37, 4	decnncl 11518	. . 3 |- ; 2 3 e. NN
167	120, 17, 56	mulcomli 10047	. . . . 5 |- ( 2 x. 7 ) = ; 1 4
168	1, 7, 37, 167, 57	decaddi 11579	. . . 4 |- ( ( 2 x. 7 ) + 2 ) = ; 1 6
169	120, 32, 60	mulcomli 10047	. . . . 5 |- ( 3 x. 7 ) = ; 2 1
170	37, 1, 37, 169, 61	decaddi 11579	. . . 4 |- ( ( 3 x. 7 ) + 2 ) = ; 2 3
171	37, 8, 37, 51, 53, 8, 37, 168, 170	decrmac 11577	. . 3 |- ( (; 2 3 x. 7 ) + 2 ) = ;; 1 6 3
172		2lt10 11680	. . . 4 |- 2 < ; 1 0
173	50, 8, 37, 172	declti 11546	. . 3 |- 2 < ; 2 3
174	166, 53, 50, 171, 173	ndvdsi 15136	. 2 |- -. ; 2 3 || ;; 1 6 3
175	5, 12, 16, 20, 45, 47, 65, 94, 129, 150, 165, 174	prmlem2 15827	1 |- ;; 1 6 3 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)