Theorem 1259lem5 15842

Description: Lemma for 1259prm 15843. 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 11185	. . . 4 |- 2 e. NN
2		3nn0 11310	. . . . 5 |- 3 e. NN0
3		4nn0 11311	. . . . 5 |- 4 e. NN0
4	2, 3	deccl 11512	. . . 4 |- ; 3 4 e. NN0
5		nnexpcl 12873	. . . 4 |- ( ( 2 e. NN /\ ; 3 4 e. NN0 ) -> ( 2 ^; 3 4 ) e. NN )
6	1, 4, 5	mp2an 708	. . 3 |- ( 2 ^; 3 4 ) e. NN
7		nnm1nn0 11334	. . 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 11315	. . . 4 |- 8 e. NN0
10		6nn0 11313	. . . 4 |- 6 e. NN0
11	9, 10	deccl 11512	. . 3 |- ; 8 6 e. NN0
12		9nn0 11316	. . 3 |- 9 e. NN0
13	11, 12	deccl 11512	. 2 |- ;; 8 6 9 e. NN0
14		1259prm.1	. . 3 |- N = ;;; 1 2 5 9
15		1nn0 11308	. . . . . 6 |- 1 e. NN0
16		2nn0 11309	. . . . . 6 |- 2 e. NN0
17	15, 16	deccl 11512	. . . . 5 |- ; 1 2 e. NN0
18		5nn0 11312	. . . . 5 |- 5 e. NN0
19	17, 18	deccl 11512	. . . 4 |- ;; 1 2 5 e. NN0
20		9nn 11192	. . . 4 |- 9 e. NN
21	19, 20	decnncl 11518	. . 3 |- ;;; 1 2 5 9 e. NN
22	14, 21	eqeltri 2697	. 2 |- N e. NN
23	14	1259lem2 15839	. . 3 |- ( ( 2 ^; 3 4 ) mod N ) = (;; 8 7 0 mod N )
24		6p1e7 11156	. . . . 5 |- ( 6 + 1 ) = 7
25		eqid 2622	. . . . 5 |- ; 8 6 = ; 8 6
26	9, 10, 24, 25	decsuc 11535	. . . 4 |- (; 8 6 + 1 ) = ; 8 7
27		eqid 2622	. . . 4 |- ;; 8 6 9 = ;; 8 6 9
28	11, 26, 27	decsucc 11550	. . 3 |- (;; 8 6 9 + 1 ) = ;; 8 7 0
29	22, 6, 15, 13, 23, 28	modsubi 15776	. 2 |- ( ( ( 2 ^; 3 4 ) - 1 ) mod N ) = (;; 8 6 9 mod N )
30	2, 12	deccl 11512	. . . 4 |- ; 3 9 e. NN0
31		0nn0 11307	. . . 4 |- 0 e. NN0
32	30, 31	deccl 11512	. . 3 |- ;; 3 9 0 e. NN0
33	9, 12	deccl 11512	. . . 4 |- ; 8 9 e. NN0
34	16, 15	deccl 11512	. . . . . 6 |- ; 2 1 e. NN0
35	15, 2	deccl 11512	. . . . . . 7 |- ; 1 3 e. NN0
36	34	nn0zi 11402	. . . . . . . . 9 |- ; 2 1 e. ZZ
37	35	nn0zi 11402	. . . . . . . . 9 |- ; 1 3 e. ZZ
38		gcdcom 15235	. . . . . . . . 9 |- ( (; 2 1 e. ZZ /\ ; 1 3 e. ZZ ) -> (; 2 1 gcd ; 1 3 ) = (; 1 3 gcd ; 2 1 ) )
39	36, 37, 38	mp2an 708	. . . . . . . 8 |- (; 2 1 gcd ; 1 3 ) = (; 1 3 gcd ; 2 1 )
40		3nn 11186	. . . . . . . . . . 11 |- 3 e. NN
41	15, 40	decnncl 11518	. . . . . . . . . 10 |- ; 1 3 e. NN
42		8nn 11191	. . . . . . . . . 10 |- 8 e. NN
43		eqid 2622	. . . . . . . . . . 11 |- ; 1 3 = ; 1 3
44	9	dec0h 11522	. . . . . . . . . . 11 |- 8 = ; 0 8
45		ax-1cn 9994	. . . . . . . . . . . . . 14 |- 1 e. CC
46	45	mulid1i 10042	. . . . . . . . . . . . 13 |- ( 1 x. 1 ) = 1
47	45	addid2i 10224	. . . . . . . . . . . . 13 |- ( 0 + 1 ) = 1
48	46, 47	oveq12i 6662	. . . . . . . . . . . 12 |- ( ( 1 x. 1 ) + ( 0 + 1 ) ) = ( 1 + 1 )
49		1p1e2 11134	. . . . . . . . . . . 12 |- ( 1 + 1 ) = 2
50	48, 49	eqtri 2644	. . . . . . . . . . 11 |- ( ( 1 x. 1 ) + ( 0 + 1 ) ) = 2
51		3cn 11095	. . . . . . . . . . . . . 14 |- 3 e. CC
52	51	mulid1i 10042	. . . . . . . . . . . . 13 |- ( 3 x. 1 ) = 3
53	52	oveq1i 6660	. . . . . . . . . . . 12 |- ( ( 3 x. 1 ) + 8 ) = ( 3 + 8 )
54		8cn 11106	. . . . . . . . . . . . 13 |- 8 e. CC
55		8p3e11 11612	. . . . . . . . . . . . 13 |- ( 8 + 3 ) = ; 1 1
56	54, 51, 55	addcomli 10228	. . . . . . . . . . . 12 |- ( 3 + 8 ) = ; 1 1
57	53, 56	eqtri 2644	. . . . . . . . . . 11 |- ( ( 3 x. 1 ) + 8 ) = ; 1 1
58	15, 2, 31, 9, 43, 44, 15, 15, 15, 50, 57	decmac 11566	. . . . . . . . . 10 |- ( (; 1 3 x. 1 ) + 8 ) = ; 2 1
59		1nn 11031	. . . . . . . . . . 11 |- 1 e. NN
60		8lt10 11674	. . . . . . . . . . 11 |- 8 < ; 1 0
61	59, 2, 9, 60	declti 11546	. . . . . . . . . 10 |- 8 < ; 1 3
62	41, 15, 42, 58, 61	ndvdsi 15136	. . . . . . . . 9 |- -. ; 1 3 || ; 2 1
63		13prm 15823	. . . . . . . . . 10 |- ; 1 3 e. Prime
64		coprm 15423	. . . . . . . . . 10 |- ( (; 1 3 e. Prime /\ ; 2 1 e. ZZ ) -> ( -. ; 1 3 || ; 2 1 <-> (; 1 3 gcd ; 2 1 ) = 1 ) )
65	63, 36, 64	mp2an 708	. . . . . . . . 9 |- ( -. ; 1 3 || ; 2 1 <-> (; 1 3 gcd ; 2 1 ) = 1 )
66	62, 65	mpbi 220	. . . . . . . 8 |- (; 1 3 gcd ; 2 1 ) = 1
67	39, 66	eqtri 2644	. . . . . . 7 |- (; 2 1 gcd ; 1 3 ) = 1
68		eqid 2622	. . . . . . . 8 |- ; 2 1 = ; 2 1
69		2cn 11091	. . . . . . . . . . 11 |- 2 e. CC
70	69	mulid2i 10043	. . . . . . . . . 10 |- ( 1 x. 2 ) = 2
71	45	addid1i 10223	. . . . . . . . . 10 |- ( 1 + 0 ) = 1
72	70, 71	oveq12i 6662	. . . . . . . . 9 |- ( ( 1 x. 2 ) + ( 1 + 0 ) ) = ( 2 + 1 )
73		2p1e3 11151	. . . . . . . . 9 |- ( 2 + 1 ) = 3
74	72, 73	eqtri 2644	. . . . . . . 8 |- ( ( 1 x. 2 ) + ( 1 + 0 ) ) = 3
75	46	oveq1i 6660	. . . . . . . . 9 |- ( ( 1 x. 1 ) + 3 ) = ( 1 + 3 )
76		3p1e4 11153	. . . . . . . . . 10 |- ( 3 + 1 ) = 4
77	51, 45, 76	addcomli 10228	. . . . . . . . 9 |- ( 1 + 3 ) = 4
78	3	dec0h 11522	. . . . . . . . 9 |- 4 = ; 0 4
79	75, 77, 78	3eqtri 2648	. . . . . . . 8 |- ( ( 1 x. 1 ) + 3 ) = ; 0 4
80	16, 15, 15, 2, 68, 43, 15, 3, 31, 74, 79	decma2c 11568	. . . . . . 7 |- ( ( 1 x. ; 2 1 ) + ; 1 3 ) = ; 3 4
81	15, 35, 34, 67, 80	gcdi 15777	. . . . . 6 |- (; 3 4 gcd ; 2 1 ) = 1
82		eqid 2622	. . . . . . 7 |- ; 3 4 = ; 3 4
83		3t2e6 11179	. . . . . . . . . 10 |- ( 3 x. 2 ) = 6
84	51, 69, 83	mulcomli 10047	. . . . . . . . 9 |- ( 2 x. 3 ) = 6
85	69	addid1i 10223	. . . . . . . . 9 |- ( 2 + 0 ) = 2
86	84, 85	oveq12i 6662	. . . . . . . 8 |- ( ( 2 x. 3 ) + ( 2 + 0 ) ) = ( 6 + 2 )
87		6p2e8 11169	. . . . . . . 8 |- ( 6 + 2 ) = 8
88	86, 87	eqtri 2644	. . . . . . 7 |- ( ( 2 x. 3 ) + ( 2 + 0 ) ) = 8
89		4cn 11098	. . . . . . . . . 10 |- 4 e. CC
90		4t2e8 11181	. . . . . . . . . 10 |- ( 4 x. 2 ) = 8
91	89, 69, 90	mulcomli 10047	. . . . . . . . 9 |- ( 2 x. 4 ) = 8
92	91	oveq1i 6660	. . . . . . . 8 |- ( ( 2 x. 4 ) + 1 ) = ( 8 + 1 )
93		8p1e9 11158	. . . . . . . 8 |- ( 8 + 1 ) = 9
94	12	dec0h 11522	. . . . . . . 8 |- 9 = ; 0 9
95	92, 93, 94	3eqtri 2648	. . . . . . 7 |- ( ( 2 x. 4 ) + 1 ) = ; 0 9
96	2, 3, 16, 15, 82, 68, 16, 12, 31, 88, 95	decma2c 11568	. . . . . 6 |- ( ( 2 x. ; 3 4 ) + ; 2 1 ) = ; 8 9
97	16, 34, 4, 81, 96	gcdi 15777	. . . . 5 |- (; 8 9 gcd ; 3 4 ) = 1
98		eqid 2622	. . . . . 6 |- ; 8 9 = ; 8 9
99		4p3e7 11163	. . . . . . . . 9 |- ( 4 + 3 ) = 7
100	89, 51, 99	addcomli 10228	. . . . . . . 8 |- ( 3 + 4 ) = 7
101	100	oveq2i 6661	. . . . . . 7 |- ( ( 4 x. 8 ) + ( 3 + 4 ) ) = ( ( 4 x. 8 ) + 7 )
102		7nn0 11314	. . . . . . . 8 |- 7 e. NN0
103		8t4e32 11656	. . . . . . . . 9 |- ( 8 x. 4 ) = ; 3 2
104	54, 89, 103	mulcomli 10047	. . . . . . . 8 |- ( 4 x. 8 ) = ; 3 2
105		7cn 11104	. . . . . . . . 9 |- 7 e. CC
106		7p2e9 11172	. . . . . . . . 9 |- ( 7 + 2 ) = 9
107	105, 69, 106	addcomli 10228	. . . . . . . 8 |- ( 2 + 7 ) = 9
108	2, 16, 102, 104, 107	decaddi 11579	. . . . . . 7 |- ( ( 4 x. 8 ) + 7 ) = ; 3 9
109	101, 108	eqtri 2644	. . . . . 6 |- ( ( 4 x. 8 ) + ( 3 + 4 ) ) = ; 3 9
110		9cn 11108	. . . . . . . 8 |- 9 e. CC
111		9t4e36 11665	. . . . . . . 8 |- ( 9 x. 4 ) = ; 3 6
112	110, 89, 111	mulcomli 10047	. . . . . . 7 |- ( 4 x. 9 ) = ; 3 6
113		6p4e10 11598	. . . . . . 7 |- ( 6 + 4 ) = ; 1 0
114	2, 10, 3, 112, 76, 113	decaddci2 11581	. . . . . 6 |- ( ( 4 x. 9 ) + 4 ) = ; 4 0
115	9, 12, 2, 3, 98, 82, 3, 31, 3, 109, 114	decma2c 11568	. . . . 5 |- ( ( 4 x. ; 8 9 ) + ; 3 4 ) = ;; 3 9 0
116	3, 4, 33, 97, 115	gcdi 15777	. . . 4 |- (;; 3 9 0 gcd ; 8 9 ) = 1
117		eqid 2622	. . . . 5 |- ;; 3 9 0 = ;; 3 9 0
118		eqid 2622	. . . . . 6 |- ; 3 9 = ; 3 9
119	54	addid1i 10223	. . . . . . 7 |- ( 8 + 0 ) = 8
120	119, 44	eqtri 2644	. . . . . 6 |- ( 8 + 0 ) = ; 0 8
121	69	addid2i 10224	. . . . . . . 8 |- ( 0 + 2 ) = 2
122	84, 121	oveq12i 6662	. . . . . . 7 |- ( ( 2 x. 3 ) + ( 0 + 2 ) ) = ( 6 + 2 )
123	122, 87	eqtri 2644	. . . . . 6 |- ( ( 2 x. 3 ) + ( 0 + 2 ) ) = 8
124		9t2e18 11663	. . . . . . . 8 |- ( 9 x. 2 ) = ; 1 8
125	110, 69, 124	mulcomli 10047	. . . . . . 7 |- ( 2 x. 9 ) = ; 1 8
126		8p8e16 11618	. . . . . . 7 |- ( 8 + 8 ) = ; 1 6
127	15, 9, 9, 125, 49, 10, 126	decaddci 11580	. . . . . 6 |- ( ( 2 x. 9 ) + 8 ) = ; 2 6
128	2, 12, 31, 9, 118, 120, 16, 10, 16, 123, 127	decma2c 11568	. . . . 5 |- ( ( 2 x. ; 3 9 ) + ( 8 + 0 ) ) = ; 8 6
129		2t0e0 11183	. . . . . . 7 |- ( 2 x. 0 ) = 0
130	129	oveq1i 6660	. . . . . 6 |- ( ( 2 x. 0 ) + 9 ) = ( 0 + 9 )
131	110	addid2i 10224	. . . . . 6 |- ( 0 + 9 ) = 9
132	130, 131, 94	3eqtri 2648	. . . . 5 |- ( ( 2 x. 0 ) + 9 ) = ; 0 9
133	30, 31, 9, 12, 117, 98, 16, 12, 31, 128, 132	decma2c 11568	. . . 4 |- ( ( 2 x. ;; 3 9 0 ) + ; 8 9 ) = ;; 8 6 9
134	16, 33, 32, 116, 133	gcdi 15777	. . 3 |- (;; 8 6 9 gcd ;; 3 9 0 ) = 1
135	30	nn0cni 11304	. . . . . . 7 |- ; 3 9 e. CC
136	135	addid1i 10223	. . . . . 6 |- (; 3 9 + 0 ) = ; 3 9
137	54	mulid2i 10043	. . . . . . . 8 |- ( 1 x. 8 ) = 8
138	137, 76	oveq12i 6662	. . . . . . 7 |- ( ( 1 x. 8 ) + ( 3 + 1 ) ) = ( 8 + 4 )
139		8p4e12 11614	. . . . . . 7 |- ( 8 + 4 ) = ; 1 2
140	138, 139	eqtri 2644	. . . . . 6 |- ( ( 1 x. 8 ) + ( 3 + 1 ) ) = ; 1 2
141		6cn 11102	. . . . . . . . 9 |- 6 e. CC
142	141	mulid2i 10043	. . . . . . . 8 |- ( 1 x. 6 ) = 6
143	142	oveq1i 6660	. . . . . . 7 |- ( ( 1 x. 6 ) + 9 ) = ( 6 + 9 )
144		9p6e15 11624	. . . . . . . 8 |- ( 9 + 6 ) = ; 1 5
145	110, 141, 144	addcomli 10228	. . . . . . 7 |- ( 6 + 9 ) = ; 1 5
146	143, 145	eqtri 2644	. . . . . 6 |- ( ( 1 x. 6 ) + 9 ) = ; 1 5
147	9, 10, 2, 12, 25, 136, 15, 18, 15, 140, 146	decma2c 11568	. . . . 5 |- ( ( 1 x. ; 8 6 ) + (; 3 9 + 0 ) ) = ;; 1 2 5
148	110	mulid2i 10043	. . . . . . 7 |- ( 1 x. 9 ) = 9
149	148	oveq1i 6660	. . . . . 6 |- ( ( 1 x. 9 ) + 0 ) = ( 9 + 0 )
150	110	addid1i 10223	. . . . . 6 |- ( 9 + 0 ) = 9
151	149, 150, 94	3eqtri 2648	. . . . 5 |- ( ( 1 x. 9 ) + 0 ) = ; 0 9
152	11, 12, 30, 31, 27, 117, 15, 12, 31, 147, 151	decma2c 11568	. . . 4 |- ( ( 1 x. ;; 8 6 9 ) + ;; 3 9 0 ) = ;;; 1 2 5 9
153	152, 14	eqtr4i 2647	. . 3 |- ( ( 1 x. ;; 8 6 9 ) + ;; 3 9 0 ) = N
154	15, 32, 13, 134, 153	gcdi 15777	. 2 |- ( N gcd ;; 8 6 9 ) = 1
155	8, 13, 22, 29, 154	gcdmodi 15778	1 |- ( ( ( 2 ^; 3 4 ) - 1 ) gcd N ) = 1

Syntax hints:   -. wn 3   <-> wb 196   = wceq 1483   e. wcel 1990   class class class wbr 4653  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   NNcn 11020   2c2 11070   3c3 11071   4c4 11072   5c5 11073   6c6 11074   7c7 11075   8c8 11076   9c9 11077   NN0cn0 11292   ZZcz 11377  ;cdc 11493   ^cexp 12860   || cdvds 14983   gcd cgcd 15216   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-fl 12593  df-mod 12669  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-gcd 15217  df-prm 15386

This theorem is referenced by:  1259prm  15843