Theorem 139prm 15831

Description: 139 is a prime number. (Contributed by Mario Carneiro, 19-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)

Assertion

Ref	Expression
139prm	|- ;; 1 3 9 e. Prime

Proof of Theorem 139prm

Step	Hyp	Ref	Expression
1		1nn0 11308	. . . 4 |- 1 e. NN0
2		3nn0 11310	. . . 4 |- 3 e. NN0
3	1, 2	deccl 11512	. . 3 |- ; 1 3 e. NN0
4		9nn 11192	. . 3 |- 9 e. NN
5	3, 4	decnncl 11518	. 2 |- ;; 1 3 9 e. NN
6		8nn0 11315	. . 3 |- 8 e. NN0
7		4nn0 11311	. . 3 |- 4 e. NN0
8		9nn0 11316	. . 3 |- 9 e. NN0
9		1lt8 11221	. . 3 |- 1 < 8
10		3lt10 11679	. . 3 |- 3 < ; 1 0
11		9lt10 11673	. . 3 |- 9 < ; 1 0
12	1, 6, 2, 7, 8, 1, 9, 10, 11	3decltc 11538	. 2 |- ;; 1 3 9 < ;; 8 4 1
13		3nn 11186	. . . 4 |- 3 e. NN
14	1, 13	decnncl 11518	. . 3 |- ; 1 3 e. NN
15		1lt10 11681	. . 3 |- 1 < ; 1 0
16	14, 8, 1, 15	declti 11546	. 2 |- 1 < ;; 1 3 9
17		4t2e8 11181	. . 3 |- ( 4 x. 2 ) = 8
18		df-9 11086	. . 3 |- 9 = ( 8 + 1 )
19	3, 7, 17, 18	dec2dvds 15767	. 2 |- -. 2 || ;; 1 3 9
20		6nn0 11313	. . . 4 |- 6 e. NN0
21	7, 20	deccl 11512	. . 3 |- ; 4 6 e. NN0
22		1nn 11031	. . 3 |- 1 e. NN
23		0nn0 11307	. . . 4 |- 0 e. NN0
24		eqid 2622	. . . 4 |- ; 4 6 = ; 4 6
25	1	dec0h 11522	. . . 4 |- 1 = ; 0 1
26		ax-1cn 9994	. . . . . . 7 |- 1 e. CC
27	26	addid2i 10224	. . . . . 6 |- ( 0 + 1 ) = 1
28	27	oveq2i 6661	. . . . 5 |- ( ( 3 x. 4 ) + ( 0 + 1 ) ) = ( ( 3 x. 4 ) + 1 )
29		2nn0 11309	. . . . . 6 |- 2 e. NN0
30		2p1e3 11151	. . . . . 6 |- ( 2 + 1 ) = 3
31	7	nn0cni 11304	. . . . . . 7 |- 4 e. CC
32		3cn 11095	. . . . . . 7 |- 3 e. CC
33		4t3e12 11632	. . . . . . 7 |- ( 4 x. 3 ) = ; 1 2
34	31, 32, 33	mulcomli 10047	. . . . . 6 |- ( 3 x. 4 ) = ; 1 2
35	1, 29, 30, 34	decsuc 11535	. . . . 5 |- ( ( 3 x. 4 ) + 1 ) = ; 1 3
36	28, 35	eqtri 2644	. . . 4 |- ( ( 3 x. 4 ) + ( 0 + 1 ) ) = ; 1 3
37		8p1e9 11158	. . . . 5 |- ( 8 + 1 ) = 9
38	20	nn0cni 11304	. . . . . 6 |- 6 e. CC
39		6t3e18 11642	. . . . . 6 |- ( 6 x. 3 ) = ; 1 8
40	38, 32, 39	mulcomli 10047	. . . . 5 |- ( 3 x. 6 ) = ; 1 8
41	1, 6, 37, 40	decsuc 11535	. . . 4 |- ( ( 3 x. 6 ) + 1 ) = ; 1 9
42	7, 20, 23, 1, 24, 25, 2, 8, 1, 36, 41	decma2c 11568	. . 3 |- ( ( 3 x. ; 4 6 ) + 1 ) = ;; 1 3 9
43		1lt3 11196	. . 3 |- 1 < 3
44	13, 21, 22, 42, 43	ndvdsi 15136	. 2 |- -. 3 || ;; 1 3 9
45		4nn 11187	. . 3 |- 4 e. NN
46		4lt5 11200	. . 3 |- 4 < 5
47		5p4e9 11167	. . 3 |- ( 5 + 4 ) = 9
48	3, 45, 46, 47	dec5dvds2 15769	. 2 |- -. 5 || ;; 1 3 9
49		7nn 11190	. . 3 |- 7 e. NN
50	1, 8	deccl 11512	. . 3 |- ; 1 9 e. NN0
51		6nn 11189	. . 3 |- 6 e. NN
52		eqid 2622	. . . 4 |- ; 1 9 = ; 1 9
53	20	dec0h 11522	. . . 4 |- 6 = ; 0 6
54		7nn0 11314	. . . 4 |- 7 e. NN0
55		7cn 11104	. . . . . . 7 |- 7 e. CC
56	55	mulid1i 10042	. . . . . 6 |- ( 7 x. 1 ) = 7
57	38	addid2i 10224	. . . . . 6 |- ( 0 + 6 ) = 6
58	56, 57	oveq12i 6662	. . . . 5 |- ( ( 7 x. 1 ) + ( 0 + 6 ) ) = ( 7 + 6 )
59		7p6e13 11608	. . . . 5 |- ( 7 + 6 ) = ; 1 3
60	58, 59	eqtri 2644	. . . 4 |- ( ( 7 x. 1 ) + ( 0 + 6 ) ) = ; 1 3
61		9cn 11108	. . . . . 6 |- 9 e. CC
62		9t7e63 11668	. . . . . 6 |- ( 9 x. 7 ) = ; 6 3
63	61, 55, 62	mulcomli 10047	. . . . 5 |- ( 7 x. 9 ) = ; 6 3
64		6p3e9 11170	. . . . . 6 |- ( 6 + 3 ) = 9
65	38, 32, 64	addcomli 10228	. . . . 5 |- ( 3 + 6 ) = 9
66	20, 2, 20, 63, 65	decaddi 11579	. . . 4 |- ( ( 7 x. 9 ) + 6 ) = ; 6 9
67	1, 8, 23, 20, 52, 53, 54, 8, 20, 60, 66	decma2c 11568	. . 3 |- ( ( 7 x. ; 1 9 ) + 6 ) = ;; 1 3 9
68		6lt7 11209	. . 3 |- 6 < 7
69	49, 50, 51, 67, 68	ndvdsi 15136	. 2 |- -. 7 || ;; 1 3 9
70	1, 22	decnncl 11518	. . 3 |- ; 1 1 e. NN
71	1, 29	deccl 11512	. . 3 |- ; 1 2 e. NN0
72		eqid 2622	. . . 4 |- ; 1 2 = ; 1 2
73	54	dec0h 11522	. . . 4 |- 7 = ; 0 7
74	1, 1	deccl 11512	. . . 4 |- ; 1 1 e. NN0
75		2cn 11091	. . . . . . 7 |- 2 e. CC
76	75	addid2i 10224	. . . . . 6 |- ( 0 + 2 ) = 2
77	76	oveq2i 6661	. . . . 5 |- ( (; 1 1 x. 1 ) + ( 0 + 2 ) ) = ( (; 1 1 x. 1 ) + 2 )
78	70	nncni 11030	. . . . . . 7 |- ; 1 1 e. CC
79	78	mulid1i 10042	. . . . . 6 |- (; 1 1 x. 1 ) = ; 1 1
80		1p2e3 11152	. . . . . 6 |- ( 1 + 2 ) = 3
81	1, 1, 29, 79, 80	decaddi 11579	. . . . 5 |- ( (; 1 1 x. 1 ) + 2 ) = ; 1 3
82	77, 81	eqtri 2644	. . . 4 |- ( (; 1 1 x. 1 ) + ( 0 + 2 ) ) = ; 1 3
83		eqid 2622	. . . . 5 |- ; 1 1 = ; 1 1
84	75	mulid2i 10043	. . . . . . 7 |- ( 1 x. 2 ) = 2
85		00id 10211	. . . . . . 7 |- ( 0 + 0 ) = 0
86	84, 85	oveq12i 6662	. . . . . 6 |- ( ( 1 x. 2 ) + ( 0 + 0 ) ) = ( 2 + 0 )
87	75	addid1i 10223	. . . . . 6 |- ( 2 + 0 ) = 2
88	86, 87	eqtri 2644	. . . . 5 |- ( ( 1 x. 2 ) + ( 0 + 0 ) ) = 2
89	84	oveq1i 6660	. . . . . 6 |- ( ( 1 x. 2 ) + 7 ) = ( 2 + 7 )
90		7p2e9 11172	. . . . . . 7 |- ( 7 + 2 ) = 9
91	55, 75, 90	addcomli 10228	. . . . . 6 |- ( 2 + 7 ) = 9
92	8	dec0h 11522	. . . . . 6 |- 9 = ; 0 9
93	89, 91, 92	3eqtri 2648	. . . . 5 |- ( ( 1 x. 2 ) + 7 ) = ; 0 9
94	1, 1, 23, 54, 83, 73, 29, 8, 23, 88, 93	decmac 11566	. . . 4 |- ( (; 1 1 x. 2 ) + 7 ) = ; 2 9
95	1, 29, 23, 54, 72, 73, 74, 8, 29, 82, 94	decma2c 11568	. . 3 |- ( (; 1 1 x. ; 1 2 ) + 7 ) = ;; 1 3 9
96		7lt10 11675	. . . 4 |- 7 < ; 1 0
97	22, 1, 54, 96	declti 11546	. . 3 |- 7 < ; 1 1
98	70, 71, 49, 95, 97	ndvdsi 15136	. 2 |- -. ; 1 1 || ;; 1 3 9
99		10nn0 11516	. . 3 |- ; 1 0 e. NN0
100		eqid 2622	. . . 4 |- ; 1 0 = ; 1 0
101		eqid 2622	. . . . 5 |- ; 1 3 = ; 1 3
102	23	dec0h 11522	. . . . . 6 |- 0 = ; 0 0
103	85, 102	eqtri 2644	. . . . 5 |- ( 0 + 0 ) = ; 0 0
104	26	mulid1i 10042	. . . . . . 7 |- ( 1 x. 1 ) = 1
105	104, 85	oveq12i 6662	. . . . . 6 |- ( ( 1 x. 1 ) + ( 0 + 0 ) ) = ( 1 + 0 )
106	26	addid1i 10223	. . . . . 6 |- ( 1 + 0 ) = 1
107	105, 106	eqtri 2644	. . . . 5 |- ( ( 1 x. 1 ) + ( 0 + 0 ) ) = 1
108	32	mulid1i 10042	. . . . . . 7 |- ( 3 x. 1 ) = 3
109	108	oveq1i 6660	. . . . . 6 |- ( ( 3 x. 1 ) + 0 ) = ( 3 + 0 )
110	32	addid1i 10223	. . . . . 6 |- ( 3 + 0 ) = 3
111	2	dec0h 11522	. . . . . 6 |- 3 = ; 0 3
112	109, 110, 111	3eqtri 2648	. . . . 5 |- ( ( 3 x. 1 ) + 0 ) = ; 0 3
113	1, 2, 23, 23, 101, 103, 1, 2, 23, 107, 112	decmac 11566	. . . 4 |- ( (; 1 3 x. 1 ) + ( 0 + 0 ) ) = ; 1 3
114	3	nn0cni 11304	. . . . . . 7 |- ; 1 3 e. CC
115	114	mul01i 10226	. . . . . 6 |- (; 1 3 x. 0 ) = 0
116	115	oveq1i 6660	. . . . 5 |- ( (; 1 3 x. 0 ) + 9 ) = ( 0 + 9 )
117	61	addid2i 10224	. . . . 5 |- ( 0 + 9 ) = 9
118	116, 117, 92	3eqtri 2648	. . . 4 |- ( (; 1 3 x. 0 ) + 9 ) = ; 0 9
119	1, 23, 23, 8, 100, 92, 3, 8, 23, 113, 118	decma2c 11568	. . 3 |- ( (; 1 3 x. ; 1 0 ) + 9 ) = ;; 1 3 9
120	22, 2, 8, 11	declti 11546	. . 3 |- 9 < ; 1 3
121	14, 99, 4, 119, 120	ndvdsi 15136	. 2 |- -. ; 1 3 || ;; 1 3 9
122	1, 49	decnncl 11518	. . 3 |- ; 1 7 e. NN
123		eqid 2622	. . . 4 |- ; 1 7 = ; 1 7
124		5nn0 11312	. . . 4 |- 5 e. NN0
125		8cn 11106	. . . . . . 7 |- 8 e. CC
126	125	mulid2i 10043	. . . . . 6 |- ( 1 x. 8 ) = 8
127		5cn 11100	. . . . . . 7 |- 5 e. CC
128	127	addid2i 10224	. . . . . 6 |- ( 0 + 5 ) = 5
129	126, 128	oveq12i 6662	. . . . 5 |- ( ( 1 x. 8 ) + ( 0 + 5 ) ) = ( 8 + 5 )
130		8p5e13 11615	. . . . 5 |- ( 8 + 5 ) = ; 1 3
131	129, 130	eqtri 2644	. . . 4 |- ( ( 1 x. 8 ) + ( 0 + 5 ) ) = ; 1 3
132		8t7e56 11661	. . . . . 6 |- ( 8 x. 7 ) = ; 5 6
133	125, 55, 132	mulcomli 10047	. . . . 5 |- ( 7 x. 8 ) = ; 5 6
134	124, 20, 2, 133, 64	decaddi 11579	. . . 4 |- ( ( 7 x. 8 ) + 3 ) = ; 5 9
135	1, 54, 23, 2, 123, 111, 6, 8, 124, 131, 134	decmac 11566	. . 3 |- ( (; 1 7 x. 8 ) + 3 ) = ;; 1 3 9
136	22, 54, 2, 10	declti 11546	. . 3 |- 3 < ; 1 7
137	122, 6, 13, 135, 136	ndvdsi 15136	. 2 |- -. ; 1 7 || ;; 1 3 9
138	1, 4	decnncl 11518	. . 3 |- ; 1 9 e. NN
139	55	mulid2i 10043	. . . . . 6 |- ( 1 x. 7 ) = 7
140	139, 57	oveq12i 6662	. . . . 5 |- ( ( 1 x. 7 ) + ( 0 + 6 ) ) = ( 7 + 6 )
141	140, 59	eqtri 2644	. . . 4 |- ( ( 1 x. 7 ) + ( 0 + 6 ) ) = ; 1 3
142	20, 2, 20, 62, 65	decaddi 11579	. . . 4 |- ( ( 9 x. 7 ) + 6 ) = ; 6 9
143	1, 8, 23, 20, 52, 53, 54, 8, 20, 141, 142	decmac 11566	. . 3 |- ( (; 1 9 x. 7 ) + 6 ) = ;; 1 3 9
144		6lt10 11676	. . . 4 |- 6 < ; 1 0
145	22, 8, 20, 144	declti 11546	. . 3 |- 6 < ; 1 9
146	138, 54, 51, 143, 145	ndvdsi 15136	. 2 |- -. ; 1 9 || ;; 1 3 9
147	29, 13	decnncl 11518	. . 3 |- ; 2 3 e. NN
148		eqid 2622	. . . 4 |- ; 2 3 = ; 2 3
149		6t2e12 11641	. . . . . 6 |- ( 6 x. 2 ) = ; 1 2
150	38, 75, 149	mulcomli 10047	. . . . 5 |- ( 2 x. 6 ) = ; 1 2
151	1, 29, 30, 150	decsuc 11535	. . . 4 |- ( ( 2 x. 6 ) + 1 ) = ; 1 3
152	29, 2, 1, 148, 20, 8, 1, 151, 41	decrmac 11577	. . 3 |- ( (; 2 3 x. 6 ) + 1 ) = ;; 1 3 9
153		2nn 11185	. . . 4 |- 2 e. NN
154	153, 2, 1, 15	declti 11546	. . 3 |- 1 < ; 2 3
155	147, 20, 22, 152, 154	ndvdsi 15136	. 2 |- -. ; 2 3 || ;; 1 3 9
156	5, 12, 16, 19, 44, 48, 69, 98, 121, 137, 146, 155	prmlem2 15827	1 |- ;; 1 3 9 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:  2503prm  15847