Theorem 139prmALT 41511

Description: 139 is a prime number. In contrast to 139prm 15831, the proof of this theorem uses 3dvds2dec 15056 for checking the divisibility by 3. Although the proof using 3dvds2dec 15056 is longer (regarding size: 1849 characters compared with 1809 for 139prm 15831), the number of essential steps is smaller (301 compared with 327 for 139prm 15831). (Contributed by Mario Carneiro, 19-Feb-2014.) (Revised by AV, 18-Aug-2021.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

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

Proof of Theorem 139prmALT

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		3ndvds4 41510	. . . 4 |- -. 3 || 4
21	1, 2	3dvdsdec 15054	. . . . 5 |- ( 3 || ; 1 3 <-> 3 || ( 1 + 3 ) )
22		3cn 11095	. . . . . . 7 |- 3 e. CC
23		ax-1cn 9994	. . . . . . 7 |- 1 e. CC
24		3p1e4 11153	. . . . . . 7 |- ( 3 + 1 ) = 4
25	22, 23, 24	addcomli 10228	. . . . . 6 |- ( 1 + 3 ) = 4
26	25	breq2i 4661	. . . . 5 |- ( 3 || ( 1 + 3 ) <-> 3 || 4 )
27	21, 26	bitri 264	. . . 4 |- ( 3 || ; 1 3 <-> 3 || 4 )
28	20, 27	mtbir 313	. . 3 |- -. 3 || ; 1 3
29	1, 2, 8	3dvds2dec 15056	. . . 4 |- ( 3 || ;; 1 3 9 <-> 3 || ( ( 1 + 3 ) + 9 ) )
30	25	oveq1i 6660	. . . . . 6 |- ( ( 1 + 3 ) + 9 ) = ( 4 + 9 )
31		9cn 11108	. . . . . . 7 |- 9 e. CC
32		4cn 11098	. . . . . . 7 |- 4 e. CC
33		9p4e13 11622	. . . . . . 7 |- ( 9 + 4 ) = ; 1 3
34	31, 32, 33	addcomli 10228	. . . . . 6 |- ( 4 + 9 ) = ; 1 3
35	30, 34	eqtri 2644	. . . . 5 |- ( ( 1 + 3 ) + 9 ) = ; 1 3
36	35	breq2i 4661	. . . 4 |- ( 3 || ( ( 1 + 3 ) + 9 ) <-> 3 || ; 1 3 )
37	29, 36	bitri 264	. . 3 |- ( 3 || ;; 1 3 9 <-> 3 || ; 1 3 )
38	28, 37	mtbir 313	. 2 |- -. 3 || ;; 1 3 9
39		4nn 11187	. . 3 |- 4 e. NN
40		4lt5 11200	. . 3 |- 4 < 5
41		5p4e9 11167	. . 3 |- ( 5 + 4 ) = 9
42	3, 39, 40, 41	dec5dvds2 15769	. 2 |- -. 5 || ;; 1 3 9
43		7nn 11190	. . 3 |- 7 e. NN
44	1, 8	deccl 11512	. . 3 |- ; 1 9 e. NN0
45		6nn 11189	. . 3 |- 6 e. NN
46		0nn0 11307	. . . 4 |- 0 e. NN0
47		6nn0 11313	. . . 4 |- 6 e. NN0
48		eqid 2622	. . . 4 |- ; 1 9 = ; 1 9
49	47	dec0h 11522	. . . 4 |- 6 = ; 0 6
50		7nn0 11314	. . . 4 |- 7 e. NN0
51		7cn 11104	. . . . . . 7 |- 7 e. CC
52	51	mulid1i 10042	. . . . . 6 |- ( 7 x. 1 ) = 7
53		6cn 11102	. . . . . . 7 |- 6 e. CC
54	53	addid2i 10224	. . . . . 6 |- ( 0 + 6 ) = 6
55	52, 54	oveq12i 6662	. . . . 5 |- ( ( 7 x. 1 ) + ( 0 + 6 ) ) = ( 7 + 6 )
56		7p6e13 11608	. . . . 5 |- ( 7 + 6 ) = ; 1 3
57	55, 56	eqtri 2644	. . . 4 |- ( ( 7 x. 1 ) + ( 0 + 6 ) ) = ; 1 3
58		9t7e63 11668	. . . . . 6 |- ( 9 x. 7 ) = ; 6 3
59	31, 51, 58	mulcomli 10047	. . . . 5 |- ( 7 x. 9 ) = ; 6 3
60		6p3e9 11170	. . . . . 6 |- ( 6 + 3 ) = 9
61	53, 22, 60	addcomli 10228	. . . . 5 |- ( 3 + 6 ) = 9
62	47, 2, 47, 59, 61	decaddi 11579	. . . 4 |- ( ( 7 x. 9 ) + 6 ) = ; 6 9
63	1, 8, 46, 47, 48, 49, 50, 8, 47, 57, 62	decma2c 11568	. . 3 |- ( ( 7 x. ; 1 9 ) + 6 ) = ;; 1 3 9
64		6lt7 11209	. . 3 |- 6 < 7
65	43, 44, 45, 63, 64	ndvdsi 15136	. 2 |- -. 7 || ;; 1 3 9
66		1nn 11031	. . . 4 |- 1 e. NN
67	1, 66	decnncl 11518	. . 3 |- ; 1 1 e. NN
68		2nn0 11309	. . . 4 |- 2 e. NN0
69	1, 68	deccl 11512	. . 3 |- ; 1 2 e. NN0
70		eqid 2622	. . . 4 |- ; 1 2 = ; 1 2
71	50	dec0h 11522	. . . 4 |- 7 = ; 0 7
72	1, 1	deccl 11512	. . . 4 |- ; 1 1 e. NN0
73		2cn 11091	. . . . . . 7 |- 2 e. CC
74	73	addid2i 10224	. . . . . 6 |- ( 0 + 2 ) = 2
75	74	oveq2i 6661	. . . . 5 |- ( (; 1 1 x. 1 ) + ( 0 + 2 ) ) = ( (; 1 1 x. 1 ) + 2 )
76	67	nncni 11030	. . . . . . 7 |- ; 1 1 e. CC
77	76	mulid1i 10042	. . . . . 6 |- (; 1 1 x. 1 ) = ; 1 1
78		1p2e3 11152	. . . . . 6 |- ( 1 + 2 ) = 3
79	1, 1, 68, 77, 78	decaddi 11579	. . . . 5 |- ( (; 1 1 x. 1 ) + 2 ) = ; 1 3
80	75, 79	eqtri 2644	. . . 4 |- ( (; 1 1 x. 1 ) + ( 0 + 2 ) ) = ; 1 3
81		eqid 2622	. . . . 5 |- ; 1 1 = ; 1 1
82	73	mulid2i 10043	. . . . . . 7 |- ( 1 x. 2 ) = 2
83		00id 10211	. . . . . . 7 |- ( 0 + 0 ) = 0
84	82, 83	oveq12i 6662	. . . . . 6 |- ( ( 1 x. 2 ) + ( 0 + 0 ) ) = ( 2 + 0 )
85	73	addid1i 10223	. . . . . 6 |- ( 2 + 0 ) = 2
86	84, 85	eqtri 2644	. . . . 5 |- ( ( 1 x. 2 ) + ( 0 + 0 ) ) = 2
87	82	oveq1i 6660	. . . . . 6 |- ( ( 1 x. 2 ) + 7 ) = ( 2 + 7 )
88		7p2e9 11172	. . . . . . 7 |- ( 7 + 2 ) = 9
89	51, 73, 88	addcomli 10228	. . . . . 6 |- ( 2 + 7 ) = 9
90	8	dec0h 11522	. . . . . 6 |- 9 = ; 0 9
91	87, 89, 90	3eqtri 2648	. . . . 5 |- ( ( 1 x. 2 ) + 7 ) = ; 0 9
92	1, 1, 46, 50, 81, 71, 68, 8, 46, 86, 91	decmac 11566	. . . 4 |- ( (; 1 1 x. 2 ) + 7 ) = ; 2 9
93	1, 68, 46, 50, 70, 71, 72, 8, 68, 80, 92	decma2c 11568	. . 3 |- ( (; 1 1 x. ; 1 2 ) + 7 ) = ;; 1 3 9
94		7lt10 11675	. . . 4 |- 7 < ; 1 0
95	66, 1, 50, 94	declti 11546	. . 3 |- 7 < ; 1 1
96	67, 69, 43, 93, 95	ndvdsi 15136	. 2 |- -. ; 1 1 || ;; 1 3 9
97	1, 46	deccl 11512	. . 3 |- ; 1 0 e. NN0
98		eqid 2622	. . . 4 |- ; 1 0 = ; 1 0
99	3	nn0cni 11304	. . . . . . 7 |- ; 1 3 e. CC
100	99	mulid1i 10042	. . . . . 6 |- (; 1 3 x. 1 ) = ; 1 3
101	100, 83	oveq12i 6662	. . . . 5 |- ( (; 1 3 x. 1 ) + ( 0 + 0 ) ) = (; 1 3 + 0 )
102	99	addid1i 10223	. . . . 5 |- (; 1 3 + 0 ) = ; 1 3
103	101, 102	eqtri 2644	. . . 4 |- ( (; 1 3 x. 1 ) + ( 0 + 0 ) ) = ; 1 3
104	99	mul01i 10226	. . . . . 6 |- (; 1 3 x. 0 ) = 0
105	104	oveq1i 6660	. . . . 5 |- ( (; 1 3 x. 0 ) + 9 ) = ( 0 + 9 )
106	31	addid2i 10224	. . . . 5 |- ( 0 + 9 ) = 9
107	105, 106, 90	3eqtri 2648	. . . 4 |- ( (; 1 3 x. 0 ) + 9 ) = ; 0 9
108	1, 46, 46, 8, 98, 90, 3, 8, 46, 103, 107	decma2c 11568	. . 3 |- ( (; 1 3 x. ; 1 0 ) + 9 ) = ;; 1 3 9
109	66, 2, 8, 11	declti 11546	. . 3 |- 9 < ; 1 3
110	14, 97, 4, 108, 109	ndvdsi 15136	. 2 |- -. ; 1 3 || ;; 1 3 9
111	1, 43	decnncl 11518	. . 3 |- ; 1 7 e. NN
112		eqid 2622	. . . 4 |- ; 1 7 = ; 1 7
113	2	dec0h 11522	. . . 4 |- 3 = ; 0 3
114		5nn0 11312	. . . 4 |- 5 e. NN0
115		8cn 11106	. . . . . . 7 |- 8 e. CC
116	115	mulid2i 10043	. . . . . 6 |- ( 1 x. 8 ) = 8
117		5cn 11100	. . . . . . 7 |- 5 e. CC
118	117	addid2i 10224	. . . . . 6 |- ( 0 + 5 ) = 5
119	116, 118	oveq12i 6662	. . . . 5 |- ( ( 1 x. 8 ) + ( 0 + 5 ) ) = ( 8 + 5 )
120		8p5e13 11615	. . . . 5 |- ( 8 + 5 ) = ; 1 3
121	119, 120	eqtri 2644	. . . 4 |- ( ( 1 x. 8 ) + ( 0 + 5 ) ) = ; 1 3
122		8t7e56 11661	. . . . . 6 |- ( 8 x. 7 ) = ; 5 6
123	115, 51, 122	mulcomli 10047	. . . . 5 |- ( 7 x. 8 ) = ; 5 6
124	114, 47, 2, 123, 60	decaddi 11579	. . . 4 |- ( ( 7 x. 8 ) + 3 ) = ; 5 9
125	1, 50, 46, 2, 112, 113, 6, 8, 114, 121, 124	decmac 11566	. . 3 |- ( (; 1 7 x. 8 ) + 3 ) = ;; 1 3 9
126	66, 50, 2, 10	declti 11546	. . 3 |- 3 < ; 1 7
127	111, 6, 13, 125, 126	ndvdsi 15136	. 2 |- -. ; 1 7 || ;; 1 3 9
128	1, 4	decnncl 11518	. . 3 |- ; 1 9 e. NN
129	51	mulid2i 10043	. . . . . 6 |- ( 1 x. 7 ) = 7
130	129, 54	oveq12i 6662	. . . . 5 |- ( ( 1 x. 7 ) + ( 0 + 6 ) ) = ( 7 + 6 )
131	130, 56	eqtri 2644	. . . 4 |- ( ( 1 x. 7 ) + ( 0 + 6 ) ) = ; 1 3
132	47, 2, 47, 58, 61	decaddi 11579	. . . 4 |- ( ( 9 x. 7 ) + 6 ) = ; 6 9
133	1, 8, 46, 47, 48, 49, 50, 8, 47, 131, 132	decmac 11566	. . 3 |- ( (; 1 9 x. 7 ) + 6 ) = ;; 1 3 9
134		6lt10 11676	. . . 4 |- 6 < ; 1 0
135	66, 8, 47, 134	declti 11546	. . 3 |- 6 < ; 1 9
136	128, 50, 45, 133, 135	ndvdsi 15136	. 2 |- -. ; 1 9 || ;; 1 3 9
137	68, 13	decnncl 11518	. . 3 |- ; 2 3 e. NN
138		eqid 2622	. . . 4 |- ; 2 3 = ; 2 3
139		2p1e3 11151	. . . . 5 |- ( 2 + 1 ) = 3
140		6t2e12 11641	. . . . . 6 |- ( 6 x. 2 ) = ; 1 2
141	53, 73, 140	mulcomli 10047	. . . . 5 |- ( 2 x. 6 ) = ; 1 2
142	1, 68, 139, 141	decsuc 11535	. . . 4 |- ( ( 2 x. 6 ) + 1 ) = ; 1 3
143		8p1e9 11158	. . . . 5 |- ( 8 + 1 ) = 9
144		6t3e18 11642	. . . . . 6 |- ( 6 x. 3 ) = ; 1 8
145	53, 22, 144	mulcomli 10047	. . . . 5 |- ( 3 x. 6 ) = ; 1 8
146	1, 6, 143, 145	decsuc 11535	. . . 4 |- ( ( 3 x. 6 ) + 1 ) = ; 1 9
147	68, 2, 1, 138, 47, 8, 1, 142, 146	decrmac 11577	. . 3 |- ( (; 2 3 x. 6 ) + 1 ) = ;; 1 3 9
148		2nn 11185	. . . 4 |- 2 e. NN
149	148, 2, 1, 15	declti 11546	. . 3 |- 1 < ; 2 3
150	137, 47, 66, 147, 149	ndvdsi 15136	. 2 |- -. ; 2 3 || ;; 1 3 9
151	5, 12, 16, 19, 38, 42, 65, 96, 110, 127, 136, 150	prmlem2 15827	1 |- ;; 1 3 9 e. Prime

Syntax hints:   e. wcel 1990   class class class wbr 4653  (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   || cdvds 14983   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)