Theorem 139prm 13258

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 9583	. . . 4 |- 1 e. NN0
2		3nn0 9585	. . . 4 |- 3 e. NN0
3	1, 2	deccl 9795	. . 3 |- ; 1 3 e. NN0
4		9nn 9477	. . 3 |- 9 e. NN
5	3, 4	decnncl 9804	. 2 |- ;; 1 3 9 e. NN
6		8nn0 9590	. . 3 |- 8 e. NN0
7		4nn0 9586	. . 3 |- 4 e. NN0
8		9nn0 9591	. . 3 |- 9 e. NN0
9		1lt8 9505	. . 3 |- 1 < 8
10		3lt10 9922	. . 3 |- 3 < ; 1 0
11		9lt10 9916	. . 3 |- 9 < ; 1 0
12	1, 6, 2, 7, 8, 1, 9, 10, 11	3decltc 9818	. 2 |- ;; 1 3 9 < ;; 8 4 1
13		3nn 9471	. . . 4 |- 3 e. NN
14	1, 13	decnncl 9804	. . 3 |- ; 1 3 e. NN
15		1lt10 9924	. . 3 |- 1 < ; 1 0
16	14, 8, 1, 15	declti 9823	. 2 |- 1 < ;; 1 3 9
17		4t2e8 9466	. . 3 |- ( 4 x. 2 ) = 8
18		df-9 9372	. . 3 |- 9 = ( 8 + 1 )
19	3, 7, 17, 18	dec2dvds 13210	. 2 |- -. 2 || ;; 1 3 9
20		6nn0 9588	. . . 4 |- 6 e. NN0
21	7, 20	deccl 9795	. . 3 |- ; 4 6 e. NN0
22		1nn 9317	. . 3 |- 1 e. NN
23		0nn0 9582	. . . 4 |- 0 e. NN0
24		eqid 2238	. . . 4 |- ; 4 6 = ; 4 6
25	1	dec0h 9807	. . . 4 |- 1 = ; 0 1
26		ax-1cn 8272	. . . . . . 7 |- 1 e. CC
27	26	addlidi 8470	. . . . . 6 |- ( 0 + 1 ) = 1
28	27	oveq2i 6096	. . . . 5 |- ( ( 3 x. 4 ) + ( 0 + 1 ) ) = ( ( 3 x. 4 ) + 1 )
29		2nn0 9584	. . . . . 6 |- 2 e. NN0
30		2p1e3 9440	. . . . . 6 |- ( 2 + 1 ) = 3
31	7	nn0cni 9579	. . . . . . 7 |- 4 e. CC
32		3cn 9381	. . . . . . 7 |- 3 e. CC
33		4t3e12 9883	. . . . . . 7 |- ( 4 x. 3 ) = ; 1 2
34	31, 32, 33	mulcomli 8333	. . . . . 6 |- ( 3 x. 4 ) = ; 1 2
35	1, 29, 30, 34	decsuc 9816	. . . . 5 |- ( ( 3 x. 4 ) + 1 ) = ; 1 3
36	28, 35	eqtri 2259	. . . 4 |- ( ( 3 x. 4 ) + ( 0 + 1 ) ) = ; 1 3
37		8p1e9 9447	. . . . 5 |- ( 8 + 1 ) = 9
38	20	nn0cni 9579	. . . . . 6 |- 6 e. CC
39		6t3e18 9890	. . . . . 6 |- ( 6 x. 3 ) = ; 1 8
40	38, 32, 39	mulcomli 8333	. . . . 5 |- ( 3 x. 6 ) = ; 1 8
41	1, 6, 37, 40	decsuc 9816	. . . 4 |- ( ( 3 x. 6 ) + 1 ) = ; 1 9
42	7, 20, 23, 1, 24, 25, 2, 8, 1, 36, 41	decma2c 9838	. . 3 |- ( ( 3 x. ; 4 6 ) + 1 ) = ;; 1 3 9
43		1lt3 9480	. . 3 |- 1 < 3
44	13, 21, 22, 42, 43	ndvdsi 12716	. 2 |- -. 3 || ;; 1 3 9
45		4nn 9472	. . 3 |- 4 e. NN
46		4lt5 9484	. . 3 |- 4 < 5
47		5p4e9 9455	. . 3 |- ( 5 + 4 ) = 9
48	3, 45, 46, 47	dec5dvds2 13212	. 2 |- -. 5 || ;; 1 3 9
49		7nn 9475	. . 3 |- 7 e. NN
50	1, 8	deccl 9795	. . 3 |- ; 1 9 e. NN0
51		6nn 9474	. . 3 |- 6 e. NN
52		eqid 2238	. . . 4 |- ; 1 9 = ; 1 9
53	20	dec0h 9807	. . . 4 |- 6 = ; 0 6
54		7nn0 9589	. . . 4 |- 7 e. NN0
55		7cn 9390	. . . . . . 7 |- 7 e. CC
56	55	mulridi 8328	. . . . . 6 |- ( 7 x. 1 ) = 7
57	38	addlidi 8470	. . . . . 6 |- ( 0 + 6 ) = 6
58	56, 57	oveq12i 6097	. . . . 5 |- ( ( 7 x. 1 ) + ( 0 + 6 ) ) = ( 7 + 6 )
59		7p6e13 9863	. . . . 5 |- ( 7 + 6 ) = ; 1 3
60	58, 59	eqtri 2259	. . . 4 |- ( ( 7 x. 1 ) + ( 0 + 6 ) ) = ; 1 3
61		9cn 9394	. . . . . 6 |- 9 e. CC
62		9t7e63 9912	. . . . . 6 |- ( 9 x. 7 ) = ; 6 3
63	61, 55, 62	mulcomli 8333	. . . . 5 |- ( 7 x. 9 ) = ; 6 3
64		6p3e9 9457	. . . . . 6 |- ( 6 + 3 ) = 9
65	38, 32, 64	addcomli 8472	. . . . 5 |- ( 3 + 6 ) = 9
66	20, 2, 20, 63, 65	decaddi 9845	. . . 4 |- ( ( 7 x. 9 ) + 6 ) = ; 6 9
67	1, 8, 23, 20, 52, 53, 54, 8, 20, 60, 66	decma2c 9838	. . 3 |- ( ( 7 x. ; 1 9 ) + 6 ) = ;; 1 3 9
68		6lt7 9493	. . 3 |- 6 < 7
69	49, 50, 51, 67, 68	ndvdsi 12716	. 2 |- -. 7 || ;; 1 3 9
70	1, 22	decnncl 9804	. . 3 |- ; 1 1 e. NN
71	1, 29	deccl 9795	. . 3 |- ; 1 2 e. NN0
72		eqid 2238	. . . 4 |- ; 1 2 = ; 1 2
73	54	dec0h 9807	. . . 4 |- 7 = ; 0 7
74	1, 1	deccl 9795	. . . 4 |- ; 1 1 e. NN0
75		2cn 9377	. . . . . . 7 |- 2 e. CC
76	75	addlidi 8470	. . . . . 6 |- ( 0 + 2 ) = 2
77	76	oveq2i 6096	. . . . 5 |- ( (; 1 1 x. 1 ) + ( 0 + 2 ) ) = ( (; 1 1 x. 1 ) + 2 )
78	70	nncni 9316	. . . . . . 7 |- ; 1 1 e. CC
79	78	mulridi 8328	. . . . . 6 |- (; 1 1 x. 1 ) = ; 1 1
80		1p2e3 9441	. . . . . 6 |- ( 1 + 2 ) = 3
81	1, 1, 29, 79, 80	decaddi 9845	. . . . 5 |- ( (; 1 1 x. 1 ) + 2 ) = ; 1 3
82	77, 81	eqtri 2259	. . . 4 |- ( (; 1 1 x. 1 ) + ( 0 + 2 ) ) = ; 1 3
83		eqid 2238	. . . . 5 |- ; 1 1 = ; 1 1
84	75	mullidi 8329	. . . . . . 7 |- ( 1 x. 2 ) = 2
85		00id 8468	. . . . . . 7 |- ( 0 + 0 ) = 0
86	84, 85	oveq12i 6097	. . . . . 6 |- ( ( 1 x. 2 ) + ( 0 + 0 ) ) = ( 2 + 0 )
87	75	addridi 8469	. . . . . 6 |- ( 2 + 0 ) = 2
88	86, 87	eqtri 2259	. . . . 5 |- ( ( 1 x. 2 ) + ( 0 + 0 ) ) = 2
89	84	oveq1i 6095	. . . . . 6 |- ( ( 1 x. 2 ) + 7 ) = ( 2 + 7 )
90		7p2e9 9458	. . . . . . 7 |- ( 7 + 2 ) = 9
91	55, 75, 90	addcomli 8472	. . . . . 6 |- ( 2 + 7 ) = 9
92	8	dec0h 9807	. . . . . 6 |- 9 = ; 0 9
93	89, 91, 92	3eqtri 2263	. . . . 5 |- ( ( 1 x. 2 ) + 7 ) = ; 0 9
94	1, 1, 23, 54, 83, 73, 29, 8, 23, 88, 93	decmac 9837	. . . 4 |- ( (; 1 1 x. 2 ) + 7 ) = ; 2 9
95	1, 29, 23, 54, 72, 73, 74, 8, 29, 82, 94	decma2c 9838	. . 3 |- ( (; 1 1 x. ; 1 2 ) + 7 ) = ;; 1 3 9
96		7lt10 9918	. . . 4 |- 7 < ; 1 0
97	22, 1, 54, 96	declti 9823	. . 3 |- 7 < ; 1 1
98	70, 71, 49, 95, 97	ndvdsi 12716	. 2 |- -. ; 1 1 || ;; 1 3 9
99		10nn0 9802	. . 3 |- ; 1 0 e. NN0
100		eqid 2238	. . . 4 |- ; 1 0 = ; 1 0
101		eqid 2238	. . . . 5 |- ; 1 3 = ; 1 3
102	23	dec0h 9807	. . . . . 6 |- 0 = ; 0 0
103	85, 102	eqtri 2259	. . . . 5 |- ( 0 + 0 ) = ; 0 0
104	26	mulridi 8328	. . . . . . 7 |- ( 1 x. 1 ) = 1
105	104, 85	oveq12i 6097	. . . . . 6 |- ( ( 1 x. 1 ) + ( 0 + 0 ) ) = ( 1 + 0 )
106	26	addridi 8469	. . . . . 6 |- ( 1 + 0 ) = 1
107	105, 106	eqtri 2259	. . . . 5 |- ( ( 1 x. 1 ) + ( 0 + 0 ) ) = 1
108	32	mulridi 8328	. . . . . . 7 |- ( 3 x. 1 ) = 3
109	108	oveq1i 6095	. . . . . 6 |- ( ( 3 x. 1 ) + 0 ) = ( 3 + 0 )
110	32	addridi 8469	. . . . . 6 |- ( 3 + 0 ) = 3
111	2	dec0h 9807	. . . . . 6 |- 3 = ; 0 3
112	109, 110, 111	3eqtri 2263	. . . . 5 |- ( ( 3 x. 1 ) + 0 ) = ; 0 3
113	1, 2, 23, 23, 101, 103, 1, 2, 23, 107, 112	decmac 9837	. . . 4 |- ( (; 1 3 x. 1 ) + ( 0 + 0 ) ) = ; 1 3
114	3	nn0cni 9579	. . . . . . 7 |- ; 1 3 e. CC
115	114	mul01i 8719	. . . . . 6 |- (; 1 3 x. 0 ) = 0
116	115	oveq1i 6095	. . . . 5 |- ( (; 1 3 x. 0 ) + 9 ) = ( 0 + 9 )
117	61	addlidi 8470	. . . . 5 |- ( 0 + 9 ) = 9
118	116, 117, 92	3eqtri 2263	. . . 4 |- ( (; 1 3 x. 0 ) + 9 ) = ; 0 9
119	1, 23, 23, 8, 100, 92, 3, 8, 23, 113, 118	decma2c 9838	. . 3 |- ( (; 1 3 x. ; 1 0 ) + 9 ) = ;; 1 3 9
120	22, 2, 8, 11	declti 9823	. . 3 |- 9 < ; 1 3
121	14, 99, 4, 119, 120	ndvdsi 12716	. 2 |- -. ; 1 3 || ;; 1 3 9
122	1, 49	decnncl 9804	. . 3 |- ; 1 7 e. NN
123		eqid 2238	. . . 4 |- ; 1 7 = ; 1 7
124		5nn0 9587	. . . 4 |- 5 e. NN0
125		8cn 9392	. . . . . . 7 |- 8 e. CC
126	125	mullidi 8329	. . . . . 6 |- ( 1 x. 8 ) = 8
127		5cn 9386	. . . . . . 7 |- 5 e. CC
128	127	addlidi 8470	. . . . . 6 |- ( 0 + 5 ) = 5
129	126, 128	oveq12i 6097	. . . . 5 |- ( ( 1 x. 8 ) + ( 0 + 5 ) ) = ( 8 + 5 )
130		8p5e13 9868	. . . . 5 |- ( 8 + 5 ) = ; 1 3
131	129, 130	eqtri 2259	. . . 4 |- ( ( 1 x. 8 ) + ( 0 + 5 ) ) = ; 1 3
132		8t7e56 9905	. . . . . 6 |- ( 8 x. 7 ) = ; 5 6
133	125, 55, 132	mulcomli 8333	. . . . 5 |- ( 7 x. 8 ) = ; 5 6
134	124, 20, 2, 133, 64	decaddi 9845	. . . 4 |- ( ( 7 x. 8 ) + 3 ) = ; 5 9
135	1, 54, 23, 2, 123, 111, 6, 8, 124, 131, 134	decmac 9837	. . 3 |- ( (; 1 7 x. 8 ) + 3 ) = ;; 1 3 9
136	22, 54, 2, 10	declti 9823	. . 3 |- 3 < ; 1 7
137	122, 6, 13, 135, 136	ndvdsi 12716	. 2 |- -. ; 1 7 || ;; 1 3 9
138	1, 4	decnncl 9804	. . 3 |- ; 1 9 e. NN
139	55	mullidi 8329	. . . . . 6 |- ( 1 x. 7 ) = 7
140	139, 57	oveq12i 6097	. . . . 5 |- ( ( 1 x. 7 ) + ( 0 + 6 ) ) = ( 7 + 6 )
141	140, 59	eqtri 2259	. . . 4 |- ( ( 1 x. 7 ) + ( 0 + 6 ) ) = ; 1 3
142	20, 2, 20, 62, 65	decaddi 9845	. . . 4 |- ( ( 9 x. 7 ) + 6 ) = ; 6 9
143	1, 8, 23, 20, 52, 53, 54, 8, 20, 141, 142	decmac 9837	. . 3 |- ( (; 1 9 x. 7 ) + 6 ) = ;; 1 3 9
144		6lt10 9919	. . . 4 |- 6 < ; 1 0
145	22, 8, 20, 144	declti 9823	. . 3 |- 6 < ; 1 9
146	138, 54, 51, 143, 145	ndvdsi 12716	. 2 |- -. ; 1 9 || ;; 1 3 9
147	29, 13	decnncl 9804	. . 3 |- ; 2 3 e. NN
148		eqid 2238	. . . 4 |- ; 2 3 = ; 2 3
149		6t2e12 9889	. . . . . 6 |- ( 6 x. 2 ) = ; 1 2
150	38, 75, 149	mulcomli 8333	. . . . 5 |- ( 2 x. 6 ) = ; 1 2
151	1, 29, 30, 150	decsuc 9816	. . . 4 |- ( ( 2 x. 6 ) + 1 ) = ; 1 3
152	29, 2, 1, 148, 20, 8, 1, 151, 41	decrmac 9843	. . 3 |- ( (; 2 3 x. 6 ) + 1 ) = ;; 1 3 9
153		2nn 9470	. . . 4 |- 2 e. NN
154	153, 2, 1, 15	declti 9823	. . 3 |- 1 < ; 2 3
155	147, 20, 22, 152, 154	ndvdsi 12716	. 2 |- -. ; 2 3 || ;; 1 3 9
156	5, 12, 16, 19, 44, 48, 69, 98, 121, 137, 146, 155	prmlem2 13254	1 |- ;; 1 3 9 e. Prime

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