Theorem 83prm 13257

Description: 83 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)

Assertion

Ref	Expression
83prm	⊢ ;83 ∈ ℙ

Proof of Theorem 83prm

Step	Hyp	Ref	Expression
1		8nn0 9590	. . 3 ⊢ 8 ∈ ℕ0
2		3nn 9471	. . 3 ⊢ 3 ∈ ℕ
3	1, 2	decnncl 9804	. 2 ⊢ ;83 ∈ ℕ
4		4nn0 9586	. . . 4 ⊢ 4 ∈ ℕ0
5	1, 4	deccl 9795	. . 3 ⊢ ;84 ∈ ℕ0
6		3nn0 9585	. . 3 ⊢ 3 ∈ ℕ0
7		1nn0 9583	. . 3 ⊢ 1 ∈ ℕ0
8		3lt10 9922	. . 3 ⊢ 3 < ;10
9		8nn 9476	. . . 4 ⊢ 8 ∈ ℕ
10		8lt10 9917	. . . 4 ⊢ 8 < ;10
11	9, 4, 1, 10	declti 9823	. . 3 ⊢ 8 < ;84
12	1, 5, 6, 7, 8, 11	decltc 9814	. 2 ⊢ ;83 < ;;841
13		1lt10 9924	. . 3 ⊢ 1 < ;10
14	9, 6, 7, 13	declti 9823	. 2 ⊢ 1 < ;83
15		2cn 9377	. . . 4 ⊢ 2 ∈ ℂ
16	15	mullidi 8329	. . 3 ⊢ (1 · 2) = 2
17		df-3 9366	. . 3 ⊢ 3 = (2 + 1)
18	1, 7, 16, 17	dec2dvds 13210	. 2 ⊢ ¬ 2 ∥ ;83
19		2nn0 9584	. . . 4 ⊢ 2 ∈ ℕ0
20		7nn0 9589	. . . 4 ⊢ 7 ∈ ℕ0
21	19, 20	deccl 9795	. . 3 ⊢ ;27 ∈ ℕ0
22		2nn 9470	. . 3 ⊢ 2 ∈ ℕ
23		0nn0 9582	. . . 4 ⊢ 0 ∈ ℕ0
24		eqid 2238	. . . 4 ⊢ ;27 = ;27
25	19	dec0h 9807	. . . 4 ⊢ 2 = ;02
26		3t2e6 9463	. . . . . 6 ⊢ (3 · 2) = 6
27	15	addlidi 8470	. . . . . 6 ⊢ (0 + 2) = 2
28	26, 27	oveq12i 6097	. . . . 5 ⊢ ((3 · 2) + (0 + 2)) = (6 + 2)
29		6p2e8 9456	. . . . 5 ⊢ (6 + 2) = 8
30	28, 29	eqtri 2259	. . . 4 ⊢ ((3 · 2) + (0 + 2)) = 8
31		7cn 9390	. . . . . 6 ⊢ 7 ∈ ℂ
32		3cn 9381	. . . . . 6 ⊢ 3 ∈ ℂ
33		7t3e21 9895	. . . . . 6 ⊢ (7 · 3) = ;21
34	31, 32, 33	mulcomli 8333	. . . . 5 ⊢ (3 · 7) = ;21
35		1p2e3 9441	. . . . 5 ⊢ (1 + 2) = 3
36	19, 7, 19, 34, 35	decaddi 9845	. . . 4 ⊢ ((3 · 7) + 2) = ;23
37	19, 20, 23, 19, 24, 25, 6, 6, 19, 30, 36	decma2c 9838	. . 3 ⊢ ((3 · ;27) + 2) = ;83
38		2lt3 9479	. . 3 ⊢ 2 < 3
39	2, 21, 22, 37, 38	ndvdsi 12716	. 2 ⊢ ¬ 3 ∥ ;83
40		3lt5 9485	. . 3 ⊢ 3 < 5
41	1, 2, 40	dec5dvds 13211	. 2 ⊢ ¬ 5 ∥ ;83
42		7nn 9475	. . 3 ⊢ 7 ∈ ℕ
43	7, 7	deccl 9795	. . 3 ⊢ ;11 ∈ ℕ0
44		6nn 9474	. . 3 ⊢ 6 ∈ ℕ
45		6nn0 9588	. . . 4 ⊢ 6 ∈ ℕ0
46		eqid 2238	. . . 4 ⊢ ;11 = ;11
47	45	dec0h 9807	. . . 4 ⊢ 6 = ;06
48	31	mulridi 8328	. . . . . 6 ⊢ (7 · 1) = 7
49		ax-1cn 8272	. . . . . . 7 ⊢ 1 ∈ ℂ
50	49	addlidi 8470	. . . . . 6 ⊢ (0 + 1) = 1
51	48, 50	oveq12i 6097	. . . . 5 ⊢ ((7 · 1) + (0 + 1)) = (7 + 1)
52		7p1e8 9446	. . . . 5 ⊢ (7 + 1) = 8
53	51, 52	eqtri 2259	. . . 4 ⊢ ((7 · 1) + (0 + 1)) = 8
54	48	oveq1i 6095	. . . . 5 ⊢ ((7 · 1) + 6) = (7 + 6)
55		7p6e13 9863	. . . . 5 ⊢ (7 + 6) = ;13
56	54, 55	eqtri 2259	. . . 4 ⊢ ((7 · 1) + 6) = ;13
57	7, 7, 23, 45, 46, 47, 20, 6, 7, 53, 56	decma2c 9838	. . 3 ⊢ ((7 · ;11) + 6) = ;83
58		6lt7 9493	. . 3 ⊢ 6 < 7
59	42, 43, 44, 57, 58	ndvdsi 12716	. 2 ⊢ ¬ 7 ∥ ;83
60		11nn 9805	. . 3 ⊢ ;11 ∈ ℕ
61		1nn 9317	. . . . . . . 8 ⊢ 1 ∈ ℕ
62	7, 61	decnncl 9804	. . . . . . 7 ⊢ ;11 ∈ ℕ
63	62	nncni 9316	. . . . . 6 ⊢ ;11 ∈ ℂ
64	63, 31	mulcomi 8332	. . . . 5 ⊢ (;11 · 7) = (7 · ;11)
65	64	oveq1i 6095	. . . 4 ⊢ ((;11 · 7) + 6) = ((7 · ;11) + 6)
66	65, 57	eqtri 2259	. . 3 ⊢ ((;11 · 7) + 6) = ;83
67		6lt10 9919	. . . 4 ⊢ 6 < ;10
68	61, 7, 45, 67	declti 9823	. . 3 ⊢ 6 < ;11
69	60, 20, 44, 66, 68	ndvdsi 12716	. 2 ⊢ ¬ ;11 ∥ ;83
70	7, 2	decnncl 9804	. . 3 ⊢ ;13 ∈ ℕ
71		5nn 9473	. . 3 ⊢ 5 ∈ ℕ
72		5nn0 9587	. . . 4 ⊢ 5 ∈ ℕ0
73		eqid 2238	. . . 4 ⊢ ;13 = ;13
74	72	dec0h 9807	. . . 4 ⊢ 5 = ;05
75		6cn 9388	. . . . . . 7 ⊢ 6 ∈ ℂ
76	75	mullidi 8329	. . . . . 6 ⊢ (1 · 6) = 6
77	76, 27	oveq12i 6097	. . . . 5 ⊢ ((1 · 6) + (0 + 2)) = (6 + 2)
78	77, 29	eqtri 2259	. . . 4 ⊢ ((1 · 6) + (0 + 2)) = 8
79		6t3e18 9890	. . . . . 6 ⊢ (6 · 3) = ;18
80	75, 32, 79	mulcomli 8333	. . . . 5 ⊢ (3 · 6) = ;18
81		1p1e2 9423	. . . . 5 ⊢ (1 + 1) = 2
82		8p5e13 9868	. . . . 5 ⊢ (8 + 5) = ;13
83	7, 1, 72, 80, 81, 6, 82	decaddci 9846	. . . 4 ⊢ ((3 · 6) + 5) = ;23
84	7, 6, 23, 72, 73, 74, 45, 6, 19, 78, 83	decmac 9837	. . 3 ⊢ ((;13 · 6) + 5) = ;83
85		5lt10 9920	. . . 4 ⊢ 5 < ;10
86	61, 6, 72, 85	declti 9823	. . 3 ⊢ 5 < ;13
87	70, 45, 71, 84, 86	ndvdsi 12716	. 2 ⊢ ¬ ;13 ∥ ;83
88	7, 42	decnncl 9804	. . 3 ⊢ ;17 ∈ ℕ
89	7, 71	decnncl 9804	. . 3 ⊢ ;15 ∈ ℕ
90		eqid 2238	. . . 4 ⊢ ;17 = ;17
91		eqid 2238	. . . 4 ⊢ ;15 = ;15
92		4cn 9384	. . . . . . 7 ⊢ 4 ∈ ℂ
93	92	mullidi 8329	. . . . . 6 ⊢ (1 · 4) = 4
94		3p1e4 9442	. . . . . . 7 ⊢ (3 + 1) = 4
95	32, 49, 94	addcomli 8472	. . . . . 6 ⊢ (1 + 3) = 4
96	93, 95	oveq12i 6097	. . . . 5 ⊢ ((1 · 4) + (1 + 3)) = (4 + 4)
97		4p4e8 9452	. . . . 5 ⊢ (4 + 4) = 8
98	96, 97	eqtri 2259	. . . 4 ⊢ ((1 · 4) + (1 + 3)) = 8
99		7t4e28 9896	. . . . 5 ⊢ (7 · 4) = ;28
100		2p1e3 9440	. . . . 5 ⊢ (2 + 1) = 3
101	19, 1, 72, 99, 100, 6, 82	decaddci 9846	. . . 4 ⊢ ((7 · 4) + 5) = ;33
102	7, 20, 7, 72, 90, 91, 4, 6, 6, 98, 101	decmac 9837	. . 3 ⊢ ((;17 · 4) + ;15) = ;83
103		5lt7 9494	. . . 4 ⊢ 5 < 7
104	7, 72, 42, 103	declt 9813	. . 3 ⊢ ;15 < ;17
105	88, 4, 89, 102, 104	ndvdsi 12716	. 2 ⊢ ¬ ;17 ∥ ;83
106		9nn 9477	. . . 4 ⊢ 9 ∈ ℕ
107	7, 106	decnncl 9804	. . 3 ⊢ ;19 ∈ ℕ
108		9nn0 9591	. . . 4 ⊢ 9 ∈ ℕ0
109		eqid 2238	. . . 4 ⊢ ;19 = ;19
110	20	dec0h 9807	. . . 4 ⊢ 7 = ;07
111	92	addlidi 8470	. . . . . 6 ⊢ (0 + 4) = 4
112	93, 111	oveq12i 6097	. . . . 5 ⊢ ((1 · 4) + (0 + 4)) = (4 + 4)
113	112, 97	eqtri 2259	. . . 4 ⊢ ((1 · 4) + (0 + 4)) = 8
114		9t4e36 9909	. . . . 5 ⊢ (9 · 4) = ;36
115	31, 75, 55	addcomli 8472	. . . . 5 ⊢ (6 + 7) = ;13
116	6, 45, 20, 114, 94, 6, 115	decaddci 9846	. . . 4 ⊢ ((9 · 4) + 7) = ;43
117	7, 108, 23, 20, 109, 110, 4, 6, 4, 113, 116	decmac 9837	. . 3 ⊢ ((;19 · 4) + 7) = ;83
118		7lt10 9918	. . . 4 ⊢ 7 < ;10
119	61, 108, 20, 118	declti 9823	. . 3 ⊢ 7 < ;19
120	107, 4, 42, 117, 119	ndvdsi 12716	. 2 ⊢ ¬ ;19 ∥ ;83
121	19, 2	decnncl 9804	. . 3 ⊢ ;23 ∈ ℕ
122		4nn 9472	. . . 4 ⊢ 4 ∈ ℕ
123	7, 122	decnncl 9804	. . 3 ⊢ ;14 ∈ ℕ
124		eqid 2238	. . . 4 ⊢ ;23 = ;23
125		eqid 2238	. . . 4 ⊢ ;14 = ;14
126		2t3e6 9464	. . . . . 6 ⊢ (2 · 3) = 6
127	126, 81	oveq12i 6097	. . . . 5 ⊢ ((2 · 3) + (1 + 1)) = (6 + 2)
128	127, 29	eqtri 2259	. . . 4 ⊢ ((2 · 3) + (1 + 1)) = 8
129		3t3e9 9465	. . . . . 6 ⊢ (3 · 3) = 9
130	129	oveq1i 6095	. . . . 5 ⊢ ((3 · 3) + 4) = (9 + 4)
131		9p4e13 9874	. . . . 5 ⊢ (9 + 4) = ;13
132	130, 131	eqtri 2259	. . . 4 ⊢ ((3 · 3) + 4) = ;13
133	19, 6, 7, 4, 124, 125, 6, 6, 7, 128, 132	decmac 9837	. . 3 ⊢ ((;23 · 3) + ;14) = ;83
134		4lt10 9921	. . . 4 ⊢ 4 < ;10
135		1lt2 9478	. . . 4 ⊢ 1 < 2
136	7, 19, 4, 6, 134, 135	decltc 9814	. . 3 ⊢ ;14 < ;23
137	121, 6, 123, 133, 136	ndvdsi 12716	. 2 ⊢ ¬ ;23 ∥ ;83
138	3, 12, 14, 18, 39, 41, 59, 69, 87, 105, 120, 137	prmlem2 13254	1 ⊢ ;83 ∈ ℙ

This proof depends on syntax axioms:   ∈ wcel 2209  (class class class)co 6085  0cc0 8179  1c1 8180   + caddc 8182   · cmul 8184  2c2 9357  3c3 9358  4c4 9359  5c5 9360  6c6 9361  7c7 9362  8c8 9363  9c9 9364  ;cdc 9781  ℙcprime 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:  bpos1  16208