Theorem 1259lem5 15106

Description: Lemma for 1259prm 15107. 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 10767	. . . 4 |- 2 e. NN
2		3nn0 10887	. . . . 5 |- 3 e. NN0
3		4nn0 10888	. . . . 5 |- 4 e. NN0
4	2, 3	deccl 11065	. . . 4 |- ; 3 4 e. NN0
5		nnexpcl 12285	. . . 4 |- ( ( 2 e. NN /\ ; 3 4 e. NN0 ) -> ( 2 ^; 3 4 ) e. NN )
6	1, 4, 5	mp2an 678	. . 3 |- ( 2 ^; 3 4 ) e. NN
7		nnm1nn0 10911	. . 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 10892	. . . 4 |- 8 e. NN0
10		6nn0 10890	. . . 4 |- 6 e. NN0
11	9, 10	deccl 11065	. . 3 |- ; 8 6 e. NN0
12		9nn0 10893	. . 3 |- 9 e. NN0
13	11, 12	deccl 11065	. 2 |- ;; 8 6 9 e. NN0
14		1259prm.1	. . 3 |- N = ;;; 1 2 5 9
15		1nn0 10885	. . . . . 6 |- 1 e. NN0
16		2nn0 10886	. . . . . 6 |- 2 e. NN0
17	15, 16	deccl 11065	. . . . 5 |- ; 1 2 e. NN0
18		5nn0 10889	. . . . 5 |- 5 e. NN0
19	17, 18	deccl 11065	. . . 4 |- ;; 1 2 5 e. NN0
20		9nn 10774	. . . 4 |- 9 e. NN
21	19, 20	decnncl 11064	. . 3 |- ;;; 1 2 5 9 e. NN
22	14, 21	eqeltri 2525	. 2 |- N e. NN
23	14	1259lem2 15103	. . 3 |- ( ( 2 ^; 3 4 ) mod N ) = (;; 8 7 0 mod N )
24		6p1e7 10738	. . . . 5 |- ( 6 + 1 ) = 7
25		eqid 2451	. . . . 5 |- ; 8 6 = ; 8 6
26	9, 10, 24, 25	decsuc 11074	. . . 4 |- (; 8 6 + 1 ) = ; 8 7
27		eqid 2451	. . . 4 |- ;; 8 6 9 = ;; 8 6 9
28	11, 26, 27	decsucc 11078	. . 3 |- (;; 8 6 9 + 1 ) = ;; 8 7 0
29	22, 6, 15, 13, 23, 28	modsubi 15044	. 2 |- ( ( ( 2 ^; 3 4 ) - 1 ) mod N ) = (;; 8 6 9 mod N )
30	2, 12	deccl 11065	. . . 4 |- ; 3 9 e. NN0
31		0nn0 10884	. . . 4 |- 0 e. NN0
32	30, 31	deccl 11065	. . 3 |- ;; 3 9 0 e. NN0
33	9, 12	deccl 11065	. . . 4 |- ; 8 9 e. NN0
34	16, 15	deccl 11065	. . . . . 6 |- ; 2 1 e. NN0
35	15, 2	deccl 11065	. . . . . . 7 |- ; 1 3 e. NN0
36	34	nn0zi 10962	. . . . . . . . 9 |- ; 2 1 e. ZZ
37	35	nn0zi 10962	. . . . . . . . 9 |- ; 1 3 e. ZZ
38		gcdcom 14484	. . . . . . . . 9 |- ( (; 2 1 e. ZZ /\ ; 1 3 e. ZZ ) -> (; 2 1 gcd ; 1 3 ) = (; 1 3 gcd ; 2 1 ) )
39	36, 37, 38	mp2an 678	. . . . . . . 8 |- (; 2 1 gcd ; 1 3 ) = (; 1 3 gcd ; 2 1 )
40		3nn 10768	. . . . . . . . . . 11 |- 3 e. NN
41	15, 40	decnncl 11064	. . . . . . . . . 10 |- ; 1 3 e. NN
42		8nn 10773	. . . . . . . . . 10 |- 8 e. NN
43		eqid 2451	. . . . . . . . . . 11 |- ; 1 3 = ; 1 3
44	9	dec0h 11067	. . . . . . . . . . 11 |- 8 = ; 0 8
45		ax-1cn 9597	. . . . . . . . . . . . . 14 |- 1 e. CC
46	45	mulid1i 9645	. . . . . . . . . . . . 13 |- ( 1 x. 1 ) = 1
47	45	addid2i 9821	. . . . . . . . . . . . 13 |- ( 0 + 1 ) = 1
48	46, 47	oveq12i 6302	. . . . . . . . . . . 12 |- ( ( 1 x. 1 ) + ( 0 + 1 ) ) = ( 1 + 1 )
49		1p1e2 10723	. . . . . . . . . . . 12 |- ( 1 + 1 ) = 2
50	48, 49	eqtri 2473	. . . . . . . . . . 11 |- ( ( 1 x. 1 ) + ( 0 + 1 ) ) = 2
51		3cn 10684	. . . . . . . . . . . . . 14 |- 3 e. CC
52	51	mulid1i 9645	. . . . . . . . . . . . 13 |- ( 3 x. 1 ) = 3
53	52	oveq1i 6300	. . . . . . . . . . . 12 |- ( ( 3 x. 1 ) + 8 ) = ( 3 + 8 )
54		8cn 10695	. . . . . . . . . . . . 13 |- 8 e. CC
55		8p3e11 11107	. . . . . . . . . . . . 13 |- ( 8 + 3 ) = ; 1 1
56	54, 51, 55	addcomli 9825	. . . . . . . . . . . 12 |- ( 3 + 8 ) = ; 1 1
57	53, 56	eqtri 2473	. . . . . . . . . . 11 |- ( ( 3 x. 1 ) + 8 ) = ; 1 1
58	15, 2, 31, 9, 43, 44, 15, 15, 15, 50, 57	decmac 11090	. . . . . . . . . 10 |- ( (; 1 3 x. 1 ) + 8 ) = ; 2 1
59		1nn 10620	. . . . . . . . . . 11 |- 1 e. NN
60		8lt10 10813	. . . . . . . . . . 11 |- 8 < 10
61	59, 2, 9, 60	declti 11076	. . . . . . . . . 10 |- 8 < ; 1 3
62	41, 15, 42, 58, 61	ndvdsi 14391	. . . . . . . . 9 |- -. ; 1 3 || ; 2 1
63		13prm 15087	. . . . . . . . . 10 |- ; 1 3 e. Prime
64		coprm 14657	. . . . . . . . . 10 |- ( (; 1 3 e. Prime /\ ; 2 1 e. ZZ ) -> ( -. ; 1 3 || ; 2 1 <-> (; 1 3 gcd ; 2 1 ) = 1 ) )
65	63, 36, 64	mp2an 678	. . . . . . . . 9 |- ( -. ; 1 3 || ; 2 1 <-> (; 1 3 gcd ; 2 1 ) = 1 )
66	62, 65	mpbi 212	. . . . . . . 8 |- (; 1 3 gcd ; 2 1 ) = 1
67	39, 66	eqtri 2473	. . . . . . 7 |- (; 2 1 gcd ; 1 3 ) = 1
68		eqid 2451	. . . . . . . 8 |- ; 2 1 = ; 2 1
69		2cn 10680	. . . . . . . . . . 11 |- 2 e. CC
70	69	mulid2i 9646	. . . . . . . . . 10 |- ( 1 x. 2 ) = 2
71	45	addid1i 9820	. . . . . . . . . 10 |- ( 1 + 0 ) = 1
72	70, 71	oveq12i 6302	. . . . . . . . 9 |- ( ( 1 x. 2 ) + ( 1 + 0 ) ) = ( 2 + 1 )
73		2p1e3 10733	. . . . . . . . 9 |- ( 2 + 1 ) = 3
74	72, 73	eqtri 2473	. . . . . . . 8 |- ( ( 1 x. 2 ) + ( 1 + 0 ) ) = 3
75	46	oveq1i 6300	. . . . . . . . 9 |- ( ( 1 x. 1 ) + 3 ) = ( 1 + 3 )
76		3p1e4 10735	. . . . . . . . . 10 |- ( 3 + 1 ) = 4
77	51, 45, 76	addcomli 9825	. . . . . . . . 9 |- ( 1 + 3 ) = 4
78	3	dec0h 11067	. . . . . . . . 9 |- 4 = ; 0 4
79	75, 77, 78	3eqtri 2477	. . . . . . . 8 |- ( ( 1 x. 1 ) + 3 ) = ; 0 4
80	16, 15, 15, 2, 68, 43, 15, 3, 31, 74, 79	decma2c 11091	. . . . . . 7 |- ( ( 1 x. ; 2 1 ) + ; 1 3 ) = ; 3 4
81	15, 35, 34, 67, 80	gcdi 15045	. . . . . 6 |- (; 3 4 gcd ; 2 1 ) = 1
82		eqid 2451	. . . . . . 7 |- ; 3 4 = ; 3 4
83		3t2e6 10761	. . . . . . . . . 10 |- ( 3 x. 2 ) = 6
84	51, 69, 83	mulcomli 9650	. . . . . . . . 9 |- ( 2 x. 3 ) = 6
85	69	addid1i 9820	. . . . . . . . 9 |- ( 2 + 0 ) = 2
86	84, 85	oveq12i 6302	. . . . . . . 8 |- ( ( 2 x. 3 ) + ( 2 + 0 ) ) = ( 6 + 2 )
87		6p2e8 10751	. . . . . . . 8 |- ( 6 + 2 ) = 8
88	86, 87	eqtri 2473	. . . . . . 7 |- ( ( 2 x. 3 ) + ( 2 + 0 ) ) = 8
89		4cn 10687	. . . . . . . . . 10 |- 4 e. CC
90		4t2e8 10763	. . . . . . . . . 10 |- ( 4 x. 2 ) = 8
91	89, 69, 90	mulcomli 9650	. . . . . . . . 9 |- ( 2 x. 4 ) = 8
92	91	oveq1i 6300	. . . . . . . 8 |- ( ( 2 x. 4 ) + 1 ) = ( 8 + 1 )
93		8p1e9 10740	. . . . . . . 8 |- ( 8 + 1 ) = 9
94	12	dec0h 11067	. . . . . . . 8 |- 9 = ; 0 9
95	92, 93, 94	3eqtri 2477	. . . . . . 7 |- ( ( 2 x. 4 ) + 1 ) = ; 0 9
96	2, 3, 16, 15, 82, 68, 16, 12, 31, 88, 95	decma2c 11091	. . . . . 6 |- ( ( 2 x. ; 3 4 ) + ; 2 1 ) = ; 8 9
97	16, 34, 4, 81, 96	gcdi 15045	. . . . 5 |- (; 8 9 gcd ; 3 4 ) = 1
98		eqid 2451	. . . . . 6 |- ; 8 9 = ; 8 9
99		4p3e7 10745	. . . . . . . . 9 |- ( 4 + 3 ) = 7
100	89, 51, 99	addcomli 9825	. . . . . . . 8 |- ( 3 + 4 ) = 7
101	100	oveq2i 6301	. . . . . . 7 |- ( ( 4 x. 8 ) + ( 3 + 4 ) ) = ( ( 4 x. 8 ) + 7 )
102		7nn0 10891	. . . . . . . 8 |- 7 e. NN0
103		8t4e32 11141	. . . . . . . . 9 |- ( 8 x. 4 ) = ; 3 2
104	54, 89, 103	mulcomli 9650	. . . . . . . 8 |- ( 4 x. 8 ) = ; 3 2
105		7cn 10693	. . . . . . . . 9 |- 7 e. CC
106		7p2e9 10754	. . . . . . . . 9 |- ( 7 + 2 ) = 9
107	105, 69, 106	addcomli 9825	. . . . . . . 8 |- ( 2 + 7 ) = 9
108	2, 16, 102, 104, 107	decaddi 11095	. . . . . . 7 |- ( ( 4 x. 8 ) + 7 ) = ; 3 9
109	101, 108	eqtri 2473	. . . . . 6 |- ( ( 4 x. 8 ) + ( 3 + 4 ) ) = ; 3 9
110		9cn 10697	. . . . . . . 8 |- 9 e. CC
111		9t4e36 11148	. . . . . . . 8 |- ( 9 x. 4 ) = ; 3 6
112	110, 89, 111	mulcomli 9650	. . . . . . 7 |- ( 4 x. 9 ) = ; 3 6
113		6p4e10 10753	. . . . . . 7 |- ( 6 + 4 ) = 10
114	2, 10, 3, 112, 76, 113	decaddci2 11097	. . . . . 6 |- ( ( 4 x. 9 ) + 4 ) = ; 4 0
115	9, 12, 2, 3, 98, 82, 3, 31, 3, 109, 114	decma2c 11091	. . . . 5 |- ( ( 4 x. ; 8 9 ) + ; 3 4 ) = ;; 3 9 0
116	3, 4, 33, 97, 115	gcdi 15045	. . . 4 |- (;; 3 9 0 gcd ; 8 9 ) = 1
117		eqid 2451	. . . . 5 |- ;; 3 9 0 = ;; 3 9 0
118		eqid 2451	. . . . . 6 |- ; 3 9 = ; 3 9
119	54	addid1i 9820	. . . . . . 7 |- ( 8 + 0 ) = 8
120	119, 44	eqtri 2473	. . . . . 6 |- ( 8 + 0 ) = ; 0 8
121	69	addid2i 9821	. . . . . . . 8 |- ( 0 + 2 ) = 2
122	84, 121	oveq12i 6302	. . . . . . 7 |- ( ( 2 x. 3 ) + ( 0 + 2 ) ) = ( 6 + 2 )
123	122, 87	eqtri 2473	. . . . . 6 |- ( ( 2 x. 3 ) + ( 0 + 2 ) ) = 8
124		9t2e18 11146	. . . . . . . 8 |- ( 9 x. 2 ) = ; 1 8
125	110, 69, 124	mulcomli 9650	. . . . . . 7 |- ( 2 x. 9 ) = ; 1 8
126		8p8e16 11112	. . . . . . 7 |- ( 8 + 8 ) = ; 1 6
127	15, 9, 9, 125, 49, 10, 126	decaddci 11096	. . . . . 6 |- ( ( 2 x. 9 ) + 8 ) = ; 2 6
128	2, 12, 31, 9, 118, 120, 16, 10, 16, 123, 127	decma2c 11091	. . . . 5 |- ( ( 2 x. ; 3 9 ) + ( 8 + 0 ) ) = ; 8 6
129		2t0e0 10765	. . . . . . 7 |- ( 2 x. 0 ) = 0
130	129	oveq1i 6300	. . . . . 6 |- ( ( 2 x. 0 ) + 9 ) = ( 0 + 9 )
131	110	addid2i 9821	. . . . . 6 |- ( 0 + 9 ) = 9
132	130, 131, 94	3eqtri 2477	. . . . 5 |- ( ( 2 x. 0 ) + 9 ) = ; 0 9
133	30, 31, 9, 12, 117, 98, 16, 12, 31, 128, 132	decma2c 11091	. . . 4 |- ( ( 2 x. ;; 3 9 0 ) + ; 8 9 ) = ;; 8 6 9
134	16, 33, 32, 116, 133	gcdi 15045	. . 3 |- (;; 8 6 9 gcd ;; 3 9 0 ) = 1
135	30	nn0cni 10881	. . . . . . 7 |- ; 3 9 e. CC
136	135	addid1i 9820	. . . . . 6 |- (; 3 9 + 0 ) = ; 3 9
137	54	mulid2i 9646	. . . . . . . 8 |- ( 1 x. 8 ) = 8
138	137, 76	oveq12i 6302	. . . . . . 7 |- ( ( 1 x. 8 ) + ( 3 + 1 ) ) = ( 8 + 4 )
139		8p4e12 11108	. . . . . . 7 |- ( 8 + 4 ) = ; 1 2
140	138, 139	eqtri 2473	. . . . . 6 |- ( ( 1 x. 8 ) + ( 3 + 1 ) ) = ; 1 2
141		6cn 10691	. . . . . . . . 9 |- 6 e. CC
142	141	mulid2i 9646	. . . . . . . 8 |- ( 1 x. 6 ) = 6
143	142	oveq1i 6300	. . . . . . 7 |- ( ( 1 x. 6 ) + 9 ) = ( 6 + 9 )
144		9p6e15 11117	. . . . . . . 8 |- ( 9 + 6 ) = ; 1 5
145	110, 141, 144	addcomli 9825	. . . . . . 7 |- ( 6 + 9 ) = ; 1 5
146	143, 145	eqtri 2473	. . . . . 6 |- ( ( 1 x. 6 ) + 9 ) = ; 1 5
147	9, 10, 2, 12, 25, 136, 15, 18, 15, 140, 146	decma2c 11091	. . . . 5 |- ( ( 1 x. ; 8 6 ) + (; 3 9 + 0 ) ) = ;; 1 2 5
148	110	mulid2i 9646	. . . . . . 7 |- ( 1 x. 9 ) = 9
149	148	oveq1i 6300	. . . . . 6 |- ( ( 1 x. 9 ) + 0 ) = ( 9 + 0 )
150	110	addid1i 9820	. . . . . 6 |- ( 9 + 0 ) = 9
151	149, 150, 94	3eqtri 2477	. . . . 5 |- ( ( 1 x. 9 ) + 0 ) = ; 0 9
152	11, 12, 30, 31, 27, 117, 15, 12, 31, 147, 151	decma2c 11091	. . . 4 |- ( ( 1 x. ;; 8 6 9 ) + ;; 3 9 0 ) = ;;; 1 2 5 9
153	152, 14	eqtr4i 2476	. . 3 |- ( ( 1 x. ;; 8 6 9 ) + ;; 3 9 0 ) = N
154	15, 32, 13, 134, 153	gcdi 15045	. 2 |- ( N gcd ;; 8 6 9 ) = 1
155	8, 13, 22, 29, 154	gcdmodi 15046	1 |- ( ( ( 2 ^; 3 4 ) - 1 ) gcd N ) = 1

Syntax hints:   -. wn 3   <-> wb 188   = wceq 1444   e. wcel 1887   class class class wbr 4402  (class class class)co 6290   0cc0 9539   1c1 9540   + caddc 9542   x. cmul 9544   - cmin 9860   NNcn 10609   2c2 10659   3c3 10660   4c4 10661   5c5 10662   6c6 10663   7c7 10664   8c8 10665   9c9 10666   NN0cn0 10869   ZZcz 10937  ;cdc 11051   ^cexp 12272   || cdvds 14305   gcd cgcd 14468   Primecprime 14622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-sup 7956  df-inf 7957  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-rp 11303  df-fz 11785  df-fl 12028  df-mod 12097  df-seq 12214  df-exp 12273  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-dvds 14306  df-gcd 14469  df-prm 14623

This theorem is referenced by:  1259prm  15107