Theorem 139prm 13267

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	⊢ ;;139 ∈ ℙ

Proof of Theorem 139prm

Step	Hyp	Ref	Expression
1		1nn0 9590	. . . 4 ⊢ 1 ∈ ℕ0
2		3nn0 9592	. . . 4 ⊢ 3 ∈ ℕ0
3	1, 2	deccl 9802	. . 3 ⊢ ;13 ∈ ℕ0
4		9nn 9484	. . 3 ⊢ 9 ∈ ℕ
5	3, 4	decnncl 9811	. 2 ⊢ ;;139 ∈ ℕ
6		8nn0 9597	. . 3 ⊢ 8 ∈ ℕ0
7		4nn0 9593	. . 3 ⊢ 4 ∈ ℕ0
8		9nn0 9598	. . 3 ⊢ 9 ∈ ℕ0
9		1lt8 9512	. . 3 ⊢ 1 < 8
10		3lt10 9929	. . 3 ⊢ 3 < ;10
11		9lt10 9923	. . 3 ⊢ 9 < ;10
12	1, 6, 2, 7, 8, 1, 9, 10, 11	3decltc 9825	. 2 ⊢ ;;139 < ;;841
13		3nn 9478	. . . 4 ⊢ 3 ∈ ℕ
14	1, 13	decnncl 9811	. . 3 ⊢ ;13 ∈ ℕ
15		1lt10 9931	. . 3 ⊢ 1 < ;10
16	14, 8, 1, 15	declti 9830	. 2 ⊢ 1 < ;;139
17		4t2e8 9473	. . 3 ⊢ (4 · 2) = 8
18		df-9 9379	. . 3 ⊢ 9 = (8 + 1)
19	3, 7, 17, 18	dec2dvds 13219	. 2 ⊢ ¬ 2 ∥ ;;139
20		6nn0 9595	. . . 4 ⊢ 6 ∈ ℕ0
21	7, 20	deccl 9802	. . 3 ⊢ ;46 ∈ ℕ0
22		1nn 9324	. . 3 ⊢ 1 ∈ ℕ
23		0nn0 9589	. . . 4 ⊢ 0 ∈ ℕ0
24		eqid 2238	. . . 4 ⊢ ;46 = ;46
25	1	dec0h 9814	. . . 4 ⊢ 1 = ;01
26		ax-1cn 8278	. . . . . . 7 ⊢ 1 ∈ ℂ
27	26	addlidi 8476	. . . . . 6 ⊢ (0 + 1) = 1
28	27	oveq2i 6099	. . . . 5 ⊢ ((3 · 4) + (0 + 1)) = ((3 · 4) + 1)
29		2nn0 9591	. . . . . 6 ⊢ 2 ∈ ℕ0
30		2p1e3 9447	. . . . . 6 ⊢ (2 + 1) = 3
31	7	nn0cni 9586	. . . . . . 7 ⊢ 4 ∈ ℂ
32		3cn 9388	. . . . . . 7 ⊢ 3 ∈ ℂ
33		4t3e12 9890	. . . . . . 7 ⊢ (4 · 3) = ;12
34	31, 32, 33	mulcomli 8339	. . . . . 6 ⊢ (3 · 4) = ;12
35	1, 29, 30, 34	decsuc 9823	. . . . 5 ⊢ ((3 · 4) + 1) = ;13
36	28, 35	eqtri 2259	. . . 4 ⊢ ((3 · 4) + (0 + 1)) = ;13
37		8p1e9 9454	. . . . 5 ⊢ (8 + 1) = 9
38	20	nn0cni 9586	. . . . . 6 ⊢ 6 ∈ ℂ
39		6t3e18 9897	. . . . . 6 ⊢ (6 · 3) = ;18
40	38, 32, 39	mulcomli 8339	. . . . 5 ⊢ (3 · 6) = ;18
41	1, 6, 37, 40	decsuc 9823	. . . 4 ⊢ ((3 · 6) + 1) = ;19
42	7, 20, 23, 1, 24, 25, 2, 8, 1, 36, 41	decma2c 9845	. . 3 ⊢ ((3 · ;46) + 1) = ;;139
43		1lt3 9487	. . 3 ⊢ 1 < 3
44	13, 21, 22, 42, 43	ndvdsi 12725	. 2 ⊢ ¬ 3 ∥ ;;139
45		4nn 9479	. . 3 ⊢ 4 ∈ ℕ
46		4lt5 9491	. . 3 ⊢ 4 < 5
47		5p4e9 9462	. . 3 ⊢ (5 + 4) = 9
48	3, 45, 46, 47	dec5dvds2 13221	. 2 ⊢ ¬ 5 ∥ ;;139
49		7nn 9482	. . 3 ⊢ 7 ∈ ℕ
50	1, 8	deccl 9802	. . 3 ⊢ ;19 ∈ ℕ0
51		6nn 9481	. . 3 ⊢ 6 ∈ ℕ
52		eqid 2238	. . . 4 ⊢ ;19 = ;19
53	20	dec0h 9814	. . . 4 ⊢ 6 = ;06
54		7nn0 9596	. . . 4 ⊢ 7 ∈ ℕ0
55		7cn 9397	. . . . . . 7 ⊢ 7 ∈ ℂ
56	55	mulridi 8334	. . . . . 6 ⊢ (7 · 1) = 7
57	38	addlidi 8476	. . . . . 6 ⊢ (0 + 6) = 6
58	56, 57	oveq12i 6100	. . . . 5 ⊢ ((7 · 1) + (0 + 6)) = (7 + 6)
59		7p6e13 9870	. . . . 5 ⊢ (7 + 6) = ;13
60	58, 59	eqtri 2259	. . . 4 ⊢ ((7 · 1) + (0 + 6)) = ;13
61		9cn 9401	. . . . . 6 ⊢ 9 ∈ ℂ
62		9t7e63 9919	. . . . . 6 ⊢ (9 · 7) = ;63
63	61, 55, 62	mulcomli 8339	. . . . 5 ⊢ (7 · 9) = ;63
64		6p3e9 9464	. . . . . 6 ⊢ (6 + 3) = 9
65	38, 32, 64	addcomli 8478	. . . . 5 ⊢ (3 + 6) = 9
66	20, 2, 20, 63, 65	decaddi 9852	. . . 4 ⊢ ((7 · 9) + 6) = ;69
67	1, 8, 23, 20, 52, 53, 54, 8, 20, 60, 66	decma2c 9845	. . 3 ⊢ ((7 · ;19) + 6) = ;;139
68		6lt7 9500	. . 3 ⊢ 6 < 7
69	49, 50, 51, 67, 68	ndvdsi 12725	. 2 ⊢ ¬ 7 ∥ ;;139
70	1, 22	decnncl 9811	. . 3 ⊢ ;11 ∈ ℕ
71	1, 29	deccl 9802	. . 3 ⊢ ;12 ∈ ℕ0
72		eqid 2238	. . . 4 ⊢ ;12 = ;12
73	54	dec0h 9814	. . . 4 ⊢ 7 = ;07
74	1, 1	deccl 9802	. . . 4 ⊢ ;11 ∈ ℕ0
75		2cn 9384	. . . . . . 7 ⊢ 2 ∈ ℂ
76	75	addlidi 8476	. . . . . 6 ⊢ (0 + 2) = 2
77	76	oveq2i 6099	. . . . 5 ⊢ ((;11 · 1) + (0 + 2)) = ((;11 · 1) + 2)
78	70	nncni 9323	. . . . . . 7 ⊢ ;11 ∈ ℂ
79	78	mulridi 8334	. . . . . 6 ⊢ (;11 · 1) = ;11
80		1p2e3 9448	. . . . . 6 ⊢ (1 + 2) = 3
81	1, 1, 29, 79, 80	decaddi 9852	. . . . 5 ⊢ ((;11 · 1) + 2) = ;13
82	77, 81	eqtri 2259	. . . 4 ⊢ ((;11 · 1) + (0 + 2)) = ;13
83		eqid 2238	. . . . 5 ⊢ ;11 = ;11
84	75	mullidi 8335	. . . . . . 7 ⊢ (1 · 2) = 2
85		00id 8474	. . . . . . 7 ⊢ (0 + 0) = 0
86	84, 85	oveq12i 6100	. . . . . 6 ⊢ ((1 · 2) + (0 + 0)) = (2 + 0)
87	75	addridi 8475	. . . . . 6 ⊢ (2 + 0) = 2
88	86, 87	eqtri 2259	. . . . 5 ⊢ ((1 · 2) + (0 + 0)) = 2
89	84	oveq1i 6098	. . . . . 6 ⊢ ((1 · 2) + 7) = (2 + 7)
90		7p2e9 9465	. . . . . . 7 ⊢ (7 + 2) = 9
91	55, 75, 90	addcomli 8478	. . . . . 6 ⊢ (2 + 7) = 9
92	8	dec0h 9814	. . . . . 6 ⊢ 9 = ;09
93	89, 91, 92	3eqtri 2263	. . . . 5 ⊢ ((1 · 2) + 7) = ;09
94	1, 1, 23, 54, 83, 73, 29, 8, 23, 88, 93	decmac 9844	. . . 4 ⊢ ((;11 · 2) + 7) = ;29
95	1, 29, 23, 54, 72, 73, 74, 8, 29, 82, 94	decma2c 9845	. . 3 ⊢ ((;11 · ;12) + 7) = ;;139
96		7lt10 9925	. . . 4 ⊢ 7 < ;10
97	22, 1, 54, 96	declti 9830	. . 3 ⊢ 7 < ;11
98	70, 71, 49, 95, 97	ndvdsi 12725	. 2 ⊢ ¬ ;11 ∥ ;;139
99		10nn0 9809	. . 3 ⊢ ;10 ∈ ℕ0
100		eqid 2238	. . . 4 ⊢ ;10 = ;10
101		eqid 2238	. . . . 5 ⊢ ;13 = ;13
102	23	dec0h 9814	. . . . . 6 ⊢ 0 = ;00
103	85, 102	eqtri 2259	. . . . 5 ⊢ (0 + 0) = ;00
104	26	mulridi 8334	. . . . . . 7 ⊢ (1 · 1) = 1
105	104, 85	oveq12i 6100	. . . . . 6 ⊢ ((1 · 1) + (0 + 0)) = (1 + 0)
106	26	addridi 8475	. . . . . 6 ⊢ (1 + 0) = 1
107	105, 106	eqtri 2259	. . . . 5 ⊢ ((1 · 1) + (0 + 0)) = 1
108	32	mulridi 8334	. . . . . . 7 ⊢ (3 · 1) = 3
109	108	oveq1i 6098	. . . . . 6 ⊢ ((3 · 1) + 0) = (3 + 0)
110	32	addridi 8475	. . . . . 6 ⊢ (3 + 0) = 3
111	2	dec0h 9814	. . . . . 6 ⊢ 3 = ;03
112	109, 110, 111	3eqtri 2263	. . . . 5 ⊢ ((3 · 1) + 0) = ;03
113	1, 2, 23, 23, 101, 103, 1, 2, 23, 107, 112	decmac 9844	. . . 4 ⊢ ((;13 · 1) + (0 + 0)) = ;13
114	3	nn0cni 9586	. . . . . . 7 ⊢ ;13 ∈ ℂ
115	114	mul01i 8725	. . . . . 6 ⊢ (;13 · 0) = 0
116	115	oveq1i 6098	. . . . 5 ⊢ ((;13 · 0) + 9) = (0 + 9)
117	61	addlidi 8476	. . . . 5 ⊢ (0 + 9) = 9
118	116, 117, 92	3eqtri 2263	. . . 4 ⊢ ((;13 · 0) + 9) = ;09
119	1, 23, 23, 8, 100, 92, 3, 8, 23, 113, 118	decma2c 9845	. . 3 ⊢ ((;13 · ;10) + 9) = ;;139
120	22, 2, 8, 11	declti 9830	. . 3 ⊢ 9 < ;13
121	14, 99, 4, 119, 120	ndvdsi 12725	. 2 ⊢ ¬ ;13 ∥ ;;139
122	1, 49	decnncl 9811	. . 3 ⊢ ;17 ∈ ℕ
123		eqid 2238	. . . 4 ⊢ ;17 = ;17
124		5nn0 9594	. . . 4 ⊢ 5 ∈ ℕ0
125		8cn 9399	. . . . . . 7 ⊢ 8 ∈ ℂ
126	125	mullidi 8335	. . . . . 6 ⊢ (1 · 8) = 8
127		5cn 9393	. . . . . . 7 ⊢ 5 ∈ ℂ
128	127	addlidi 8476	. . . . . 6 ⊢ (0 + 5) = 5
129	126, 128	oveq12i 6100	. . . . 5 ⊢ ((1 · 8) + (0 + 5)) = (8 + 5)
130		8p5e13 9875	. . . . 5 ⊢ (8 + 5) = ;13
131	129, 130	eqtri 2259	. . . 4 ⊢ ((1 · 8) + (0 + 5)) = ;13
132		8t7e56 9912	. . . . . 6 ⊢ (8 · 7) = ;56
133	125, 55, 132	mulcomli 8339	. . . . 5 ⊢ (7 · 8) = ;56
134	124, 20, 2, 133, 64	decaddi 9852	. . . 4 ⊢ ((7 · 8) + 3) = ;59
135	1, 54, 23, 2, 123, 111, 6, 8, 124, 131, 134	decmac 9844	. . 3 ⊢ ((;17 · 8) + 3) = ;;139
136	22, 54, 2, 10	declti 9830	. . 3 ⊢ 3 < ;17
137	122, 6, 13, 135, 136	ndvdsi 12725	. 2 ⊢ ¬ ;17 ∥ ;;139
138	1, 4	decnncl 9811	. . 3 ⊢ ;19 ∈ ℕ
139	55	mullidi 8335	. . . . . 6 ⊢ (1 · 7) = 7
140	139, 57	oveq12i 6100	. . . . 5 ⊢ ((1 · 7) + (0 + 6)) = (7 + 6)
141	140, 59	eqtri 2259	. . . 4 ⊢ ((1 · 7) + (0 + 6)) = ;13
142	20, 2, 20, 62, 65	decaddi 9852	. . . 4 ⊢ ((9 · 7) + 6) = ;69
143	1, 8, 23, 20, 52, 53, 54, 8, 20, 141, 142	decmac 9844	. . 3 ⊢ ((;19 · 7) + 6) = ;;139
144		6lt10 9926	. . . 4 ⊢ 6 < ;10
145	22, 8, 20, 144	declti 9830	. . 3 ⊢ 6 < ;19
146	138, 54, 51, 143, 145	ndvdsi 12725	. 2 ⊢ ¬ ;19 ∥ ;;139
147	29, 13	decnncl 9811	. . 3 ⊢ ;23 ∈ ℕ
148		eqid 2238	. . . 4 ⊢ ;23 = ;23
149		6t2e12 9896	. . . . . 6 ⊢ (6 · 2) = ;12
150	38, 75, 149	mulcomli 8339	. . . . 5 ⊢ (2 · 6) = ;12
151	1, 29, 30, 150	decsuc 9823	. . . 4 ⊢ ((2 · 6) + 1) = ;13
152	29, 2, 1, 148, 20, 8, 1, 151, 41	decrmac 9850	. . 3 ⊢ ((;23 · 6) + 1) = ;;139
153		2nn 9477	. . . 4 ⊢ 2 ∈ ℕ
154	153, 2, 1, 15	declti 9830	. . 3 ⊢ 1 < ;23
155	147, 20, 22, 152, 154	ndvdsi 12725	. 2 ⊢ ¬ ;23 ∥ ;;139
156	5, 12, 16, 19, 44, 48, 69, 98, 121, 137, 146, 155	prmlem2 13263	1 ⊢ ;;139 ∈ ℙ

This proof depends on syntax axioms:   ∈ wcel 2209  (class class class)co 6088  0cc0 8185  1c1 8186   + caddc 8188   · cmul 8190  2c2 9364  3c3 9365  4c4 9366  5c5 9367  6c6 9368  7c7 9369  8c8 9370  9c9 9371  ;cdc 9788  ℙcprime 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)