Theorem 1259lem5 14281

Description: Lemma for 1259prm 14282. 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 10594	. . . 4 |- 2 e. NN
2		3nn0 10712	. . . . 5 |- 3 e. NN0
3		4nn0 10713	. . . . 5 |- 4 e. NN0
4	2, 3	deccl 10884	. . . 4 |- ; 3 4 e. NN0
5		nnexpcl 11999	. . . 4 |- ( ( 2 e. NN /\ ; 3 4 e. NN0 ) -> ( 2 ^; 3 4 ) e. NN )
6	1, 4, 5	mp2an 672	. . 3 |- ( 2 ^; 3 4 ) e. NN
7		nnm1nn0 10736	. . 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 10717	. . . 4 |- 8 e. NN0
10		6nn0 10715	. . . 4 |- 6 e. NN0
11	9, 10	deccl 10884	. . 3 |- ; 8 6 e. NN0
12		9nn0 10718	. . 3 |- 9 e. NN0
13	11, 12	deccl 10884	. 2 |- ;; 8 6 9 e. NN0
14		1259prm.1	. . 3 |- N = ;;; 1 2 5 9
15		1nn0 10710	. . . . . 6 |- 1 e. NN0
16		2nn0 10711	. . . . . 6 |- 2 e. NN0
17	15, 16	deccl 10884	. . . . 5 |- ; 1 2 e. NN0
18		5nn0 10714	. . . . 5 |- 5 e. NN0
19	17, 18	deccl 10884	. . . 4 |- ;; 1 2 5 e. NN0
20		9nn 10601	. . . 4 |- 9 e. NN
21	19, 20	decnncl 10883	. . 3 |- ;;; 1 2 5 9 e. NN
22	14, 21	eqeltri 2538	. 2 |- N e. NN
23	14	1259lem2 14278	. . 3 |- ( ( 2 ^; 3 4 ) mod N ) = (;; 8 7 0 mod N )
24		6p1e7 10565	. . . . 5 |- ( 6 + 1 ) = 7
25		eqid 2454	. . . . 5 |- ; 8 6 = ; 8 6
26	9, 10, 24, 25	decsuc 10893	. . . 4 |- (; 8 6 + 1 ) = ; 8 7
27		eqid 2454	. . . 4 |- ;; 8 6 9 = ;; 8 6 9
28	11, 26, 27	decsucc 10897	. . 3 |- (;; 8 6 9 + 1 ) = ;; 8 7 0
29	22, 6, 15, 13, 23, 28	modsubi 14223	. 2 |- ( ( ( 2 ^; 3 4 ) - 1 ) mod N ) = (;; 8 6 9 mod N )
30	2, 12	deccl 10884	. . . 4 |- ; 3 9 e. NN0
31		0nn0 10709	. . . 4 |- 0 e. NN0
32	30, 31	deccl 10884	. . 3 |- ;; 3 9 0 e. NN0
33	9, 12	deccl 10884	. . . 4 |- ; 8 9 e. NN0
34	16, 15	deccl 10884	. . . . . 6 |- ; 2 1 e. NN0
35	15, 2	deccl 10884	. . . . . . 7 |- ; 1 3 e. NN0
36	34	nn0zi 10786	. . . . . . . . 9 |- ; 2 1 e. ZZ
37	35	nn0zi 10786	. . . . . . . . 9 |- ; 1 3 e. ZZ
38		gcdcom 13826	. . . . . . . . 9 |- ( (; 2 1 e. ZZ /\ ; 1 3 e. ZZ ) -> (; 2 1 gcd ; 1 3 ) = (; 1 3 gcd ; 2 1 ) )
39	36, 37, 38	mp2an 672	. . . . . . . 8 |- (; 2 1 gcd ; 1 3 ) = (; 1 3 gcd ; 2 1 )
40		3nn 10595	. . . . . . . . . . 11 |- 3 e. NN
41	15, 40	decnncl 10883	. . . . . . . . . 10 |- ; 1 3 e. NN
42		8nn 10600	. . . . . . . . . 10 |- 8 e. NN
43		eqid 2454	. . . . . . . . . . 11 |- ; 1 3 = ; 1 3
44	9	dec0h 10886	. . . . . . . . . . 11 |- 8 = ; 0 8
45		ax-1cn 9455	. . . . . . . . . . . . . 14 |- 1 e. CC
46	45	mulid1i 9503	. . . . . . . . . . . . 13 |- ( 1 x. 1 ) = 1
47	45	addid2i 9672	. . . . . . . . . . . . 13 |- ( 0 + 1 ) = 1
48	46, 47	oveq12i 6215	. . . . . . . . . . . 12 |- ( ( 1 x. 1 ) + ( 0 + 1 ) ) = ( 1 + 1 )
49		1p1e2 10550	. . . . . . . . . . . 12 |- ( 1 + 1 ) = 2
50	48, 49	eqtri 2483	. . . . . . . . . . 11 |- ( ( 1 x. 1 ) + ( 0 + 1 ) ) = 2
51		3cn 10511	. . . . . . . . . . . . . 14 |- 3 e. CC
52	51	mulid1i 9503	. . . . . . . . . . . . 13 |- ( 3 x. 1 ) = 3
53	52	oveq1i 6213	. . . . . . . . . . . 12 |- ( ( 3 x. 1 ) + 8 ) = ( 3 + 8 )
54		8cn 10522	. . . . . . . . . . . . 13 |- 8 e. CC
55		8p3e11 10926	. . . . . . . . . . . . 13 |- ( 8 + 3 ) = ; 1 1
56	54, 51, 55	addcomli 9676	. . . . . . . . . . . 12 |- ( 3 + 8 ) = ; 1 1
57	53, 56	eqtri 2483	. . . . . . . . . . 11 |- ( ( 3 x. 1 ) + 8 ) = ; 1 1
58	15, 2, 31, 9, 43, 44, 15, 15, 15, 50, 57	decmac 10909	. . . . . . . . . 10 |- ( (; 1 3 x. 1 ) + 8 ) = ; 2 1
59		1nn 10448	. . . . . . . . . . 11 |- 1 e. NN
60		8lt10 10640	. . . . . . . . . . 11 |- 8 < 10
61	59, 2, 9, 60	declti 10895	. . . . . . . . . 10 |- 8 < ; 1 3
62	41, 15, 42, 58, 61	ndvdsi 13736	. . . . . . . . 9 |- -. ; 1 3 || ; 2 1
63		13prm 14265	. . . . . . . . . 10 |- ; 1 3 e. Prime
64		coprm 13908	. . . . . . . . . 10 |- ( (; 1 3 e. Prime /\ ; 2 1 e. ZZ ) -> ( -. ; 1 3 || ; 2 1 <-> (; 1 3 gcd ; 2 1 ) = 1 ) )
65	63, 36, 64	mp2an 672	. . . . . . . . 9 |- ( -. ; 1 3 || ; 2 1 <-> (; 1 3 gcd ; 2 1 ) = 1 )
66	62, 65	mpbi 208	. . . . . . . 8 |- (; 1 3 gcd ; 2 1 ) = 1
67	39, 66	eqtri 2483	. . . . . . 7 |- (; 2 1 gcd ; 1 3 ) = 1
68		eqid 2454	. . . . . . . 8 |- ; 2 1 = ; 2 1
69		2cn 10507	. . . . . . . . . . 11 |- 2 e. CC
70	69	mulid2i 9504	. . . . . . . . . 10 |- ( 1 x. 2 ) = 2
71	45	addid1i 9671	. . . . . . . . . 10 |- ( 1 + 0 ) = 1
72	70, 71	oveq12i 6215	. . . . . . . . 9 |- ( ( 1 x. 2 ) + ( 1 + 0 ) ) = ( 2 + 1 )
73		2p1e3 10560	. . . . . . . . 9 |- ( 2 + 1 ) = 3
74	72, 73	eqtri 2483	. . . . . . . 8 |- ( ( 1 x. 2 ) + ( 1 + 0 ) ) = 3
75	46	oveq1i 6213	. . . . . . . . 9 |- ( ( 1 x. 1 ) + 3 ) = ( 1 + 3 )
76		3p1e4 10562	. . . . . . . . . 10 |- ( 3 + 1 ) = 4
77	51, 45, 76	addcomli 9676	. . . . . . . . 9 |- ( 1 + 3 ) = 4
78	3	dec0h 10886	. . . . . . . . 9 |- 4 = ; 0 4
79	75, 77, 78	3eqtri 2487	. . . . . . . 8 |- ( ( 1 x. 1 ) + 3 ) = ; 0 4
80	16, 15, 15, 2, 68, 43, 15, 3, 31, 74, 79	decma2c 10910	. . . . . . 7 |- ( ( 1 x. ; 2 1 ) + ; 1 3 ) = ; 3 4
81	15, 35, 34, 67, 80	gcdi 14224	. . . . . 6 |- (; 3 4 gcd ; 2 1 ) = 1
82		eqid 2454	. . . . . . 7 |- ; 3 4 = ; 3 4
83		3t2e6 10588	. . . . . . . . . 10 |- ( 3 x. 2 ) = 6
84	51, 69, 83	mulcomli 9508	. . . . . . . . 9 |- ( 2 x. 3 ) = 6
85	69	addid1i 9671	. . . . . . . . 9 |- ( 2 + 0 ) = 2
86	84, 85	oveq12i 6215	. . . . . . . 8 |- ( ( 2 x. 3 ) + ( 2 + 0 ) ) = ( 6 + 2 )
87		6p2e8 10578	. . . . . . . 8 |- ( 6 + 2 ) = 8
88	86, 87	eqtri 2483	. . . . . . 7 |- ( ( 2 x. 3 ) + ( 2 + 0 ) ) = 8
89		4cn 10514	. . . . . . . . . 10 |- 4 e. CC
90		4t2e8 10590	. . . . . . . . . 10 |- ( 4 x. 2 ) = 8
91	89, 69, 90	mulcomli 9508	. . . . . . . . 9 |- ( 2 x. 4 ) = 8
92	91	oveq1i 6213	. . . . . . . 8 |- ( ( 2 x. 4 ) + 1 ) = ( 8 + 1 )
93		8p1e9 10567	. . . . . . . 8 |- ( 8 + 1 ) = 9
94	12	dec0h 10886	. . . . . . . 8 |- 9 = ; 0 9
95	92, 93, 94	3eqtri 2487	. . . . . . 7 |- ( ( 2 x. 4 ) + 1 ) = ; 0 9
96	2, 3, 16, 15, 82, 68, 16, 12, 31, 88, 95	decma2c 10910	. . . . . 6 |- ( ( 2 x. ; 3 4 ) + ; 2 1 ) = ; 8 9
97	16, 34, 4, 81, 96	gcdi 14224	. . . . 5 |- (; 8 9 gcd ; 3 4 ) = 1
98		eqid 2454	. . . . . 6 |- ; 8 9 = ; 8 9
99		4p3e7 10572	. . . . . . . . 9 |- ( 4 + 3 ) = 7
100	89, 51, 99	addcomli 9676	. . . . . . . 8 |- ( 3 + 4 ) = 7
101	100	oveq2i 6214	. . . . . . 7 |- ( ( 4 x. 8 ) + ( 3 + 4 ) ) = ( ( 4 x. 8 ) + 7 )
102		7nn0 10716	. . . . . . . 8 |- 7 e. NN0
103		8t4e32 10960	. . . . . . . . 9 |- ( 8 x. 4 ) = ; 3 2
104	54, 89, 103	mulcomli 9508	. . . . . . . 8 |- ( 4 x. 8 ) = ; 3 2
105		7cn 10520	. . . . . . . . 9 |- 7 e. CC
106		7p2e9 10581	. . . . . . . . 9 |- ( 7 + 2 ) = 9
107	105, 69, 106	addcomli 9676	. . . . . . . 8 |- ( 2 + 7 ) = 9
108	2, 16, 102, 104, 107	decaddi 10914	. . . . . . 7 |- ( ( 4 x. 8 ) + 7 ) = ; 3 9
109	101, 108	eqtri 2483	. . . . . 6 |- ( ( 4 x. 8 ) + ( 3 + 4 ) ) = ; 3 9
110		9cn 10524	. . . . . . . 8 |- 9 e. CC
111		9t4e36 10967	. . . . . . . 8 |- ( 9 x. 4 ) = ; 3 6
112	110, 89, 111	mulcomli 9508	. . . . . . 7 |- ( 4 x. 9 ) = ; 3 6
113		6p4e10 10580	. . . . . . 7 |- ( 6 + 4 ) = 10
114	2, 10, 3, 112, 76, 113	decaddci2 10916	. . . . . 6 |- ( ( 4 x. 9 ) + 4 ) = ; 4 0
115	9, 12, 2, 3, 98, 82, 3, 31, 3, 109, 114	decma2c 10910	. . . . 5 |- ( ( 4 x. ; 8 9 ) + ; 3 4 ) = ;; 3 9 0
116	3, 4, 33, 97, 115	gcdi 14224	. . . 4 |- (;; 3 9 0 gcd ; 8 9 ) = 1
117		eqid 2454	. . . . 5 |- ;; 3 9 0 = ;; 3 9 0
118		eqid 2454	. . . . . 6 |- ; 3 9 = ; 3 9
119	54	addid1i 9671	. . . . . . 7 |- ( 8 + 0 ) = 8
120	119, 44	eqtri 2483	. . . . . 6 |- ( 8 + 0 ) = ; 0 8
121	69	addid2i 9672	. . . . . . . 8 |- ( 0 + 2 ) = 2
122	84, 121	oveq12i 6215	. . . . . . 7 |- ( ( 2 x. 3 ) + ( 0 + 2 ) ) = ( 6 + 2 )
123	122, 87	eqtri 2483	. . . . . 6 |- ( ( 2 x. 3 ) + ( 0 + 2 ) ) = 8
124		9t2e18 10965	. . . . . . . 8 |- ( 9 x. 2 ) = ; 1 8
125	110, 69, 124	mulcomli 9508	. . . . . . 7 |- ( 2 x. 9 ) = ; 1 8
126		8p8e16 10931	. . . . . . 7 |- ( 8 + 8 ) = ; 1 6
127	15, 9, 9, 125, 49, 10, 126	decaddci 10915	. . . . . 6 |- ( ( 2 x. 9 ) + 8 ) = ; 2 6
128	2, 12, 31, 9, 118, 120, 16, 10, 16, 123, 127	decma2c 10910	. . . . 5 |- ( ( 2 x. ; 3 9 ) + ( 8 + 0 ) ) = ; 8 6
129		2t0e0 10592	. . . . . . 7 |- ( 2 x. 0 ) = 0
130	129	oveq1i 6213	. . . . . 6 |- ( ( 2 x. 0 ) + 9 ) = ( 0 + 9 )
131	110	addid2i 9672	. . . . . 6 |- ( 0 + 9 ) = 9
132	130, 131, 94	3eqtri 2487	. . . . 5 |- ( ( 2 x. 0 ) + 9 ) = ; 0 9
133	30, 31, 9, 12, 117, 98, 16, 12, 31, 128, 132	decma2c 10910	. . . 4 |- ( ( 2 x. ;; 3 9 0 ) + ; 8 9 ) = ;; 8 6 9
134	16, 33, 32, 116, 133	gcdi 14224	. . 3 |- (;; 8 6 9 gcd ;; 3 9 0 ) = 1
135	30	nn0cni 10706	. . . . . . 7 |- ; 3 9 e. CC
136	135	addid1i 9671	. . . . . 6 |- (; 3 9 + 0 ) = ; 3 9
137	54	mulid2i 9504	. . . . . . . 8 |- ( 1 x. 8 ) = 8
138	137, 76	oveq12i 6215	. . . . . . 7 |- ( ( 1 x. 8 ) + ( 3 + 1 ) ) = ( 8 + 4 )
139		8p4e12 10927	. . . . . . 7 |- ( 8 + 4 ) = ; 1 2
140	138, 139	eqtri 2483	. . . . . 6 |- ( ( 1 x. 8 ) + ( 3 + 1 ) ) = ; 1 2
141		6cn 10518	. . . . . . . . 9 |- 6 e. CC
142	141	mulid2i 9504	. . . . . . . 8 |- ( 1 x. 6 ) = 6
143	142	oveq1i 6213	. . . . . . 7 |- ( ( 1 x. 6 ) + 9 ) = ( 6 + 9 )
144		9p6e15 10936	. . . . . . . 8 |- ( 9 + 6 ) = ; 1 5
145	110, 141, 144	addcomli 9676	. . . . . . 7 |- ( 6 + 9 ) = ; 1 5
146	143, 145	eqtri 2483	. . . . . 6 |- ( ( 1 x. 6 ) + 9 ) = ; 1 5
147	9, 10, 2, 12, 25, 136, 15, 18, 15, 140, 146	decma2c 10910	. . . . 5 |- ( ( 1 x. ; 8 6 ) + (; 3 9 + 0 ) ) = ;; 1 2 5
148	110	mulid2i 9504	. . . . . . 7 |- ( 1 x. 9 ) = 9
149	148	oveq1i 6213	. . . . . 6 |- ( ( 1 x. 9 ) + 0 ) = ( 9 + 0 )
150	110	addid1i 9671	. . . . . 6 |- ( 9 + 0 ) = 9
151	149, 150, 94	3eqtri 2487	. . . . 5 |- ( ( 1 x. 9 ) + 0 ) = ; 0 9
152	11, 12, 30, 31, 27, 117, 15, 12, 31, 147, 151	decma2c 10910	. . . 4 |- ( ( 1 x. ;; 8 6 9 ) + ;; 3 9 0 ) = ;;; 1 2 5 9
153	152, 14	eqtr4i 2486	. . 3 |- ( ( 1 x. ;; 8 6 9 ) + ;; 3 9 0 ) = N
154	15, 32, 13, 134, 153	gcdi 14224	. 2 |- ( N gcd ;; 8 6 9 ) = 1
155	8, 13, 22, 29, 154	gcdmodi 14225	1 |- ( ( ( 2 ^; 3 4 ) - 1 ) gcd N ) = 1

Syntax hints:   -. wn 3   <-> wb 184   = wceq 1370   e. wcel 1758   class class class wbr 4403  (class class class)co 6203   0cc0 9397   1c1 9398   + caddc 9400   x. cmul 9402   - cmin 9710   NNcn 10437   2c2 10486   3c3 10487   4c4 10488   5c5 10489   6c6 10490   7c7 10491   8c8 10492   9c9 10493   NN0cn0 10694   ZZcz 10761  ;cdc 10870   ^cexp 11986   || cdivides 13657   gcd cgcd 13812   Primecprime 13885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474  ax-pre-sup 9475

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-2o 7034  df-oadd 7037  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-sup 7806  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-div 10109  df-nn 10438  df-2 10495  df-3 10496  df-4 10497  df-5 10498  df-6 10499  df-7 10500  df-8 10501  df-9 10502  df-10 10503  df-n0 10695  df-z 10762  df-dec 10871  df-uz 10977  df-rp 11107  df-fz 11559  df-fl 11763  df-mod 11830  df-seq 11928  df-exp 11987  df-cj 12710  df-re 12711  df-im 12712  df-sqr 12846  df-abs 12847  df-dvds 13658  df-gcd 13813  df-prm 13886

This theorem is referenced by:  1259prm  14282