Theorem 1259lem5 13266

Description: Lemma for 1259prm 13267. Calculate the GCD of 2 ^ 3 4 - 1 == 8 6 9 with N = 1 2 5 9. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)

Hypothesis

Ref	Expression
1259prm.1	|- N = ;;; 1 2 5 9

Assertion

Ref	Expression
1259lem5	|- ( ( ( 2 ^; 3 4 ) - 1 ) gcd N ) = 1

Proof of Theorem 1259lem5

Step	Hyp	Ref	Expression
1		2nn 9470	. . . 4 |- 2 e. NN
2		3nn0 9585	. . . . 5 |- 3 e. NN0
3		4nn0 9586	. . . . 5 |- 4 e. NN0
4	2, 3	deccl 9795	. . . 4 |- ; 3 4 e. NN0
5		nnexpcl 11002	. . . 4 |- ( ( 2 e. NN /\ ; 3 4 e. NN0 ) -> ( 2 ^; 3 4 ) e. NN )
6	1, 4, 5	mp2an 430	. . 3 |- ( 2 ^; 3 4 ) e. NN
7		nnm1nn0 9608	. . 3 |- ( ( 2 ^; 3 4 ) e. NN -> ( ( 2 ^; 3 4 ) - 1 ) e. NN0 )
8	6, 7	ax-mp 5	. 2 |- ( ( 2 ^; 3 4 ) - 1 ) e. NN0
9		8nn0 9590	. . . 4 |- 8 e. NN0
10		6nn0 9588	. . . 4 |- 6 e. NN0
11	9, 10	deccl 9795	. . 3 |- ; 8 6 e. NN0
12		9nn0 9591	. . 3 |- 9 e. NN0
13	11, 12	deccl 9795	. 2 |- ;; 8 6 9 e. NN0
14		1259prm.1	. . 3 |- N = ;;; 1 2 5 9
15		1nn0 9583	. . . . . 6 |- 1 e. NN0
16		2nn0 9584	. . . . . 6 |- 2 e. NN0
17	15, 16	deccl 9795	. . . . 5 |- ; 1 2 e. NN0
18		5nn0 9587	. . . . 5 |- 5 e. NN0
19	17, 18	deccl 9795	. . . 4 |- ;; 1 2 5 e. NN0
20		9nn 9477	. . . 4 |- 9 e. NN
21	19, 20	decnncl 9804	. . 3 |- ;;; 1 2 5 9 e. NN
22	14, 21	eqeltri 2311	. 2 |- N e. NN
23	14	1259lem2 13263	. . 3 |- ( ( 2 ^; 3 4 ) mod N ) = (;; 8 7 0 mod N )
24		6p1e7 9445	. . . . 5 |- ( 6 + 1 ) = 7
25		eqid 2238	. . . . 5 |- ; 8 6 = ; 8 6
26	9, 10, 24, 25	decsuc 9816	. . . 4 |- (; 8 6 + 1 ) = ; 8 7
27		eqid 2238	. . . 4 |- ;; 8 6 9 = ;; 8 6 9
28	11, 26, 27	decsucc 9826	. . 3 |- (;; 8 6 9 + 1 ) = ;; 8 7 0
29	22, 6, 15, 13, 23, 28	modsubi 13219	. 2 |- ( ( ( 2 ^; 3 4 ) - 1 ) mod N ) = (;; 8 6 9 mod N )
30	2, 12	deccl 9795	. . . 4 |- ; 3 9 e. NN0
31		0nn0 9582	. . . 4 |- 0 e. NN0
32	30, 31	deccl 9795	. . 3 |- ;; 3 9 0 e. NN0
33	9, 12	deccl 9795	. . . 4 |- ; 8 9 e. NN0
34	16, 15	deccl 9795	. . . . . 6 |- ; 2 1 e. NN0
35	15, 2	deccl 9795	. . . . . . 7 |- ; 1 3 e. NN0
36	34	nn0zi 9670	. . . . . . . . 9 |- ; 2 1 e. ZZ
37	35	nn0zi 9670	. . . . . . . . 9 |- ; 1 3 e. ZZ
38		gcdcom 12766	. . . . . . . . 9 |- ( (; 2 1 e. ZZ /\ ; 1 3 e. ZZ ) -> (; 2 1 gcd ; 1 3 ) = (; 1 3 gcd ; 2 1 ) )
39	36, 37, 38	mp2an 430	. . . . . . . 8 |- (; 2 1 gcd ; 1 3 ) = (; 1 3 gcd ; 2 1 )
40		3nn 9471	. . . . . . . . . . 11 |- 3 e. NN
41	15, 40	decnncl 9804	. . . . . . . . . 10 |- ; 1 3 e. NN
42		8nn 9476	. . . . . . . . . 10 |- 8 e. NN
43		eqid 2238	. . . . . . . . . . 11 |- ; 1 3 = ; 1 3
44	9	dec0h 9807	. . . . . . . . . . 11 |- 8 = ; 0 8
45		ax-1cn 8272	. . . . . . . . . . . . . 14 |- 1 e. CC
46	45	mulridi 8328	. . . . . . . . . . . . 13 |- ( 1 x. 1 ) = 1
47	45	addlidi 8470	. . . . . . . . . . . . 13 |- ( 0 + 1 ) = 1
48	46, 47	oveq12i 6097	. . . . . . . . . . . 12 |- ( ( 1 x. 1 ) + ( 0 + 1 ) ) = ( 1 + 1 )
49		1p1e2 9423	. . . . . . . . . . . 12 |- ( 1 + 1 ) = 2
50	48, 49	eqtri 2259	. . . . . . . . . . 11 |- ( ( 1 x. 1 ) + ( 0 + 1 ) ) = 2
51		3cn 9381	. . . . . . . . . . . . . 14 |- 3 e. CC
52	51	mulridi 8328	. . . . . . . . . . . . 13 |- ( 3 x. 1 ) = 3
53	52	oveq1i 6095	. . . . . . . . . . . 12 |- ( ( 3 x. 1 ) + 8 ) = ( 3 + 8 )
54		8cn 9392	. . . . . . . . . . . . 13 |- 8 e. CC
55		8p3e11 9866	. . . . . . . . . . . . 13 |- ( 8 + 3 ) = ; 1 1
56	54, 51, 55	addcomli 8472	. . . . . . . . . . . 12 |- ( 3 + 8 ) = ; 1 1
57	53, 56	eqtri 2259	. . . . . . . . . . 11 |- ( ( 3 x. 1 ) + 8 ) = ; 1 1
58	15, 2, 31, 9, 43, 44, 15, 15, 15, 50, 57	decmac 9837	. . . . . . . . . 10 |- ( (; 1 3 x. 1 ) + 8 ) = ; 2 1
59		1nn 9317	. . . . . . . . . . 11 |- 1 e. NN
60		8lt10 9917	. . . . . . . . . . 11 |- 8 < ; 1 0
61	59, 2, 9, 60	declti 9823	. . . . . . . . . 10 |- 8 < ; 1 3
62	41, 15, 42, 58, 61	ndvdsi 12716	. . . . . . . . 9 |- -. ; 1 3 || ; 2 1
63		13prm 13250	. . . . . . . . . 10 |- ; 1 3 e. Prime
64		coprm 12939	. . . . . . . . . 10 |- ( (; 1 3 e. Prime /\ ; 2 1 e. ZZ ) -> ( -. ; 1 3 || ; 2 1 <-> (; 1 3 gcd ; 2 1 ) = 1 ) )
65	63, 36, 64	mp2an 430	. . . . . . . . 9 |- ( -. ; 1 3 || ; 2 1 <-> (; 1 3 gcd ; 2 1 ) = 1 )
66	62, 65	mpbi 145	. . . . . . . 8 |- (; 1 3 gcd ; 2 1 ) = 1
67	39, 66	eqtri 2259	. . . . . . 7 |- (; 2 1 gcd ; 1 3 ) = 1
68		eqid 2238	. . . . . . . 8 |- ; 2 1 = ; 2 1
69		2cn 9377	. . . . . . . . . . 11 |- 2 e. CC
70	69	mullidi 8329	. . . . . . . . . 10 |- ( 1 x. 2 ) = 2
71	45	addridi 8469	. . . . . . . . . 10 |- ( 1 + 0 ) = 1
72	70, 71	oveq12i 6097	. . . . . . . . 9 |- ( ( 1 x. 2 ) + ( 1 + 0 ) ) = ( 2 + 1 )
73		2p1e3 9440	. . . . . . . . 9 |- ( 2 + 1 ) = 3
74	72, 73	eqtri 2259	. . . . . . . 8 |- ( ( 1 x. 2 ) + ( 1 + 0 ) ) = 3
75	46	oveq1i 6095	. . . . . . . . 9 |- ( ( 1 x. 1 ) + 3 ) = ( 1 + 3 )
76		3p1e4 9442	. . . . . . . . . 10 |- ( 3 + 1 ) = 4
77	51, 45, 76	addcomli 8472	. . . . . . . . 9 |- ( 1 + 3 ) = 4
78	3	dec0h 9807	. . . . . . . . 9 |- 4 = ; 0 4
79	75, 77, 78	3eqtri 2263	. . . . . . . 8 |- ( ( 1 x. 1 ) + 3 ) = ; 0 4
80	16, 15, 15, 2, 68, 43, 15, 3, 31, 74, 79	decma2c 9838	. . . . . . 7 |- ( ( 1 x. ; 2 1 ) + ; 1 3 ) = ; 3 4
81	15, 35, 34, 67, 80	gcdi 13220	. . . . . 6 |- (; 3 4 gcd ; 2 1 ) = 1
82		eqid 2238	. . . . . . 7 |- ; 3 4 = ; 3 4
83		2t3e6 9464	. . . . . . . . 9 |- ( 2 x. 3 ) = 6
84	69	addridi 8469	. . . . . . . . 9 |- ( 2 + 0 ) = 2
85	83, 84	oveq12i 6097	. . . . . . . 8 |- ( ( 2 x. 3 ) + ( 2 + 0 ) ) = ( 6 + 2 )
86		6p2e8 9456	. . . . . . . 8 |- ( 6 + 2 ) = 8
87	85, 86	eqtri 2259	. . . . . . 7 |- ( ( 2 x. 3 ) + ( 2 + 0 ) ) = 8
88		2t4e8 9467	. . . . . . . . 9 |- ( 2 x. 4 ) = 8
89	88	oveq1i 6095	. . . . . . . 8 |- ( ( 2 x. 4 ) + 1 ) = ( 8 + 1 )
90		8p1e9 9447	. . . . . . . 8 |- ( 8 + 1 ) = 9
91	12	dec0h 9807	. . . . . . . 8 |- 9 = ; 0 9
92	89, 90, 91	3eqtri 2263	. . . . . . 7 |- ( ( 2 x. 4 ) + 1 ) = ; 0 9
93	2, 3, 16, 15, 82, 68, 16, 12, 31, 87, 92	decma2c 9838	. . . . . 6 |- ( ( 2 x. ; 3 4 ) + ; 2 1 ) = ; 8 9
94	16, 34, 4, 81, 93	gcdi 13220	. . . . 5 |- (; 8 9 gcd ; 3 4 ) = 1
95		eqid 2238	. . . . . 6 |- ; 8 9 = ; 8 9
96		4cn 9384	. . . . . . . . 9 |- 4 e. CC
97		4p3e7 9451	. . . . . . . . 9 |- ( 4 + 3 ) = 7
98	96, 51, 97	addcomli 8472	. . . . . . . 8 |- ( 3 + 4 ) = 7
99	98	oveq2i 6096	. . . . . . 7 |- ( ( 4 x. 8 ) + ( 3 + 4 ) ) = ( ( 4 x. 8 ) + 7 )
100		7nn0 9589	. . . . . . . 8 |- 7 e. NN0
101		8t4e32 9902	. . . . . . . . 9 |- ( 8 x. 4 ) = ; 3 2
102	54, 96, 101	mulcomli 8333	. . . . . . . 8 |- ( 4 x. 8 ) = ; 3 2
103		7cn 9390	. . . . . . . . 9 |- 7 e. CC
104		7p2e9 9458	. . . . . . . . 9 |- ( 7 + 2 ) = 9
105	103, 69, 104	addcomli 8472	. . . . . . . 8 |- ( 2 + 7 ) = 9
106	2, 16, 100, 102, 105	decaddi 9845	. . . . . . 7 |- ( ( 4 x. 8 ) + 7 ) = ; 3 9
107	99, 106	eqtri 2259	. . . . . 6 |- ( ( 4 x. 8 ) + ( 3 + 4 ) ) = ; 3 9
108		9cn 9394	. . . . . . . 8 |- 9 e. CC
109		9t4e36 9909	. . . . . . . 8 |- ( 9 x. 4 ) = ; 3 6
110	108, 96, 109	mulcomli 8333	. . . . . . 7 |- ( 4 x. 9 ) = ; 3 6
111		6p4e10 9857	. . . . . . 7 |- ( 6 + 4 ) = ; 1 0
112	2, 10, 3, 110, 76, 111	decaddci2 9847	. . . . . 6 |- ( ( 4 x. 9 ) + 4 ) = ; 4 0
113	9, 12, 2, 3, 95, 82, 3, 31, 3, 107, 112	decma2c 9838	. . . . 5 |- ( ( 4 x. ; 8 9 ) + ; 3 4 ) = ;; 3 9 0
114	3, 4, 33, 94, 113	gcdi 13220	. . . 4 |- (;; 3 9 0 gcd ; 8 9 ) = 1
115		eqid 2238	. . . . 5 |- ;; 3 9 0 = ;; 3 9 0
116		eqid 2238	. . . . . 6 |- ; 3 9 = ; 3 9
117	54	addridi 8469	. . . . . . 7 |- ( 8 + 0 ) = 8
118	117, 44	eqtri 2259	. . . . . 6 |- ( 8 + 0 ) = ; 0 8
119	69	addlidi 8470	. . . . . . . 8 |- ( 0 + 2 ) = 2
120	83, 119	oveq12i 6097	. . . . . . 7 |- ( ( 2 x. 3 ) + ( 0 + 2 ) ) = ( 6 + 2 )
121	120, 86	eqtri 2259	. . . . . 6 |- ( ( 2 x. 3 ) + ( 0 + 2 ) ) = 8
122		9t2e18 9907	. . . . . . . 8 |- ( 9 x. 2 ) = ; 1 8
123	108, 69, 122	mulcomli 8333	. . . . . . 7 |- ( 2 x. 9 ) = ; 1 8
124		8p8e16 9871	. . . . . . 7 |- ( 8 + 8 ) = ; 1 6
125	15, 9, 9, 123, 49, 10, 124	decaddci 9846	. . . . . 6 |- ( ( 2 x. 9 ) + 8 ) = ; 2 6
126	2, 12, 31, 9, 116, 118, 16, 10, 16, 121, 125	decma2c 9838	. . . . 5 |- ( ( 2 x. ; 3 9 ) + ( 8 + 0 ) ) = ; 8 6
127		2t0e0 9468	. . . . . . 7 |- ( 2 x. 0 ) = 0
128	127	oveq1i 6095	. . . . . 6 |- ( ( 2 x. 0 ) + 9 ) = ( 0 + 9 )
129	108	addlidi 8470	. . . . . 6 |- ( 0 + 9 ) = 9
130	128, 129, 91	3eqtri 2263	. . . . 5 |- ( ( 2 x. 0 ) + 9 ) = ; 0 9
131	30, 31, 9, 12, 115, 95, 16, 12, 31, 126, 130	decma2c 9838	. . . 4 |- ( ( 2 x. ;; 3 9 0 ) + ; 8 9 ) = ;; 8 6 9
132	16, 33, 32, 114, 131	gcdi 13220	. . 3 |- (;; 8 6 9 gcd ;; 3 9 0 ) = 1
133	30	nn0cni 9579	. . . . . . 7 |- ; 3 9 e. CC
134	133	addridi 8469	. . . . . 6 |- (; 3 9 + 0 ) = ; 3 9
135	54	mullidi 8329	. . . . . . . 8 |- ( 1 x. 8 ) = 8
136	135, 76	oveq12i 6097	. . . . . . 7 |- ( ( 1 x. 8 ) + ( 3 + 1 ) ) = ( 8 + 4 )
137		8p4e12 9867	. . . . . . 7 |- ( 8 + 4 ) = ; 1 2
138	136, 137	eqtri 2259	. . . . . 6 |- ( ( 1 x. 8 ) + ( 3 + 1 ) ) = ; 1 2
139		6cn 9388	. . . . . . . . 9 |- 6 e. CC
140	139	mullidi 8329	. . . . . . . 8 |- ( 1 x. 6 ) = 6
141	140	oveq1i 6095	. . . . . . 7 |- ( ( 1 x. 6 ) + 9 ) = ( 6 + 9 )
142		9p6e15 9876	. . . . . . . 8 |- ( 9 + 6 ) = ; 1 5
143	108, 139, 142	addcomli 8472	. . . . . . 7 |- ( 6 + 9 ) = ; 1 5
144	141, 143	eqtri 2259	. . . . . 6 |- ( ( 1 x. 6 ) + 9 ) = ; 1 5
145	9, 10, 2, 12, 25, 134, 15, 18, 15, 138, 144	decma2c 9838	. . . . 5 |- ( ( 1 x. ; 8 6 ) + (; 3 9 + 0 ) ) = ;; 1 2 5
146	108	mullidi 8329	. . . . . . 7 |- ( 1 x. 9 ) = 9
147	146	oveq1i 6095	. . . . . 6 |- ( ( 1 x. 9 ) + 0 ) = ( 9 + 0 )
148	108	addridi 8469	. . . . . 6 |- ( 9 + 0 ) = 9
149	147, 148, 91	3eqtri 2263	. . . . 5 |- ( ( 1 x. 9 ) + 0 ) = ; 0 9
150	11, 12, 30, 31, 27, 115, 15, 12, 31, 145, 149	decma2c 9838	. . . 4 |- ( ( 1 x. ;; 8 6 9 ) + ;; 3 9 0 ) = ;;; 1 2 5 9
151	150, 14	eqtr4i 2262	. . 3 |- ( ( 1 x. ;; 8 6 9 ) + ;; 3 9 0 ) = N
152	15, 32, 13, 132, 151	gcdi 13220	. 2 |- ( N gcd ;; 8 6 9 ) = 1
153	8, 13, 22, 29, 152	gcdmodi 13221	1 |- ( ( ( 2 ^; 3 4 ) - 1 ) gcd N ) = 1

This proof depends on syntax axioms:   -. wn 3   <-> wb 105   = wceq 1402   e. wcel 2209   class class class wbr 4130  (class class class)co 6085   0cc0 8179   1c1 8180   + caddc 8182   x. cmul 8184   - cmin 8498   NNcn 9306   2c2 9357   3c3 9358   4c4 9359   5c5 9360   6c6 9361   7c7 9362   8c8 9363   9c9 9364   NN0cn0 9567   ZZcz 9648  ;cdc 9781   ^cexp 10988   || cdvds 12570   gcd cgcd 12746   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-sup 7324  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-gcd 12747  df-prm 12902

This theorem is used by:  1259prm  13267