Theorem 1259lem1 15838

Description: Lemma for 1259prm 15843. Calculate a power mod. In decimal, we calculate 2 ^ 1 6 = 5 2 N + 6 8 == 6 8 and 2 ^ 1 7 == 6 8 x. 2 = 1 3 6 in this lemma. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.)

Hypothesis

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

Assertion

Ref	Expression
1259lem1	|- ( ( 2 ^; 1 7 ) mod N ) = (;; 1 3 6 mod N )

Proof of Theorem 1259lem1

Step	Hyp	Ref	Expression
1		1259prm.1	. . 3 |- N = ;;; 1 2 5 9
2		1nn0 11308	. . . . . 6 |- 1 e. NN0
3		2nn0 11309	. . . . . 6 |- 2 e. NN0
4	2, 3	deccl 11512	. . . . 5 |- ; 1 2 e. NN0
5		5nn0 11312	. . . . 5 |- 5 e. NN0
6	4, 5	deccl 11512	. . . 4 |- ;; 1 2 5 e. NN0
7		9nn 11192	. . . 4 |- 9 e. NN
8	6, 7	decnncl 11518	. . 3 |- ;;; 1 2 5 9 e. NN
9	1, 8	eqeltri 2697	. 2 |- N e. NN
10		2nn 11185	. 2 |- 2 e. NN
11		6nn0 11313	. . 3 |- 6 e. NN0
12	2, 11	deccl 11512	. 2 |- ; 1 6 e. NN0
13		0z 11388	. 2 |- 0 e. ZZ
14		8nn0 11315	. . 3 |- 8 e. NN0
15	11, 14	deccl 11512	. 2 |- ; 6 8 e. NN0
16		3nn0 11310	. . . 4 |- 3 e. NN0
17	2, 16	deccl 11512	. . 3 |- ; 1 3 e. NN0
18	17, 11	deccl 11512	. 2 |- ;; 1 3 6 e. NN0
19	5, 3	deccl 11512	. . . 4 |- ; 5 2 e. NN0
20	19	nn0zi 11402	. . 3 |- ; 5 2 e. ZZ
21	3, 14	nn0expcli 12886	. . 3 |- ( 2 ^ 8 ) e. NN0
22		eqid 2622	. . 3 |- ( ( 2 ^ 8 ) mod N ) = ( ( 2 ^ 8 ) mod N )
23	14	nn0cni 11304	. . . 4 |- 8 e. CC
24		2cn 11091	. . . 4 |- 2 e. CC
25		8t2e16 11654	. . . 4 |- ( 8 x. 2 ) = ; 1 6
26	23, 24, 25	mulcomli 10047	. . 3 |- ( 2 x. 8 ) = ; 1 6
27		9nn0 11316	. . . . 5 |- 9 e. NN0
28		eqid 2622	. . . . 5 |- ; 6 8 = ; 6 8
29		4nn0 11311	. . . . . 6 |- 4 e. NN0
30		7nn0 11314	. . . . . 6 |- 7 e. NN0
31	29, 30	deccl 11512	. . . . 5 |- ; 4 7 e. NN0
32		eqid 2622	. . . . . 6 |- ;; 1 2 5 = ;; 1 2 5
33		0nn0 11307	. . . . . . 7 |- 0 e. NN0
34	11	dec0h 11522	. . . . . . 7 |- 6 = ; 0 6
35		eqid 2622	. . . . . . 7 |- ; 4 7 = ; 4 7
36		4cn 11098	. . . . . . . . . 10 |- 4 e. CC
37	36	addid2i 10224	. . . . . . . . 9 |- ( 0 + 4 ) = 4
38	37	oveq1i 6660	. . . . . . . 8 |- ( ( 0 + 4 ) + 1 ) = ( 4 + 1 )
39		4p1e5 11154	. . . . . . . 8 |- ( 4 + 1 ) = 5
40	38, 39	eqtri 2644	. . . . . . 7 |- ( ( 0 + 4 ) + 1 ) = 5
41		7cn 11104	. . . . . . . 8 |- 7 e. CC
42		6cn 11102	. . . . . . . 8 |- 6 e. CC
43		7p6e13 11608	. . . . . . . 8 |- ( 7 + 6 ) = ; 1 3
44	41, 42, 43	addcomli 10228	. . . . . . 7 |- ( 6 + 7 ) = ; 1 3
45	33, 11, 29, 30, 34, 35, 40, 16, 44	decaddc 11572	. . . . . 6 |- ( 6 + ; 4 7 ) = ; 5 3
46	3, 11	deccl 11512	. . . . . 6 |- ; 2 6 e. NN0
47		eqid 2622	. . . . . . 7 |- ; 1 2 = ; 1 2
48	5	dec0h 11522	. . . . . . . 8 |- 5 = ; 0 5
49		eqid 2622	. . . . . . . 8 |- ; 2 6 = ; 2 6
50	24	addid2i 10224	. . . . . . . . . 10 |- ( 0 + 2 ) = 2
51	50	oveq1i 6660	. . . . . . . . 9 |- ( ( 0 + 2 ) + 1 ) = ( 2 + 1 )
52		2p1e3 11151	. . . . . . . . 9 |- ( 2 + 1 ) = 3
53	51, 52	eqtri 2644	. . . . . . . 8 |- ( ( 0 + 2 ) + 1 ) = 3
54		5cn 11100	. . . . . . . . 9 |- 5 e. CC
55		6p5e11 11600	. . . . . . . . 9 |- ( 6 + 5 ) = ; 1 1
56	42, 54, 55	addcomli 10228	. . . . . . . 8 |- ( 5 + 6 ) = ; 1 1
57	33, 5, 3, 11, 48, 49, 53, 2, 56	decaddc 11572	. . . . . . 7 |- ( 5 + ; 2 6 ) = ; 3 1
58		10nn0 11516	. . . . . . 7 |- ; 1 0 e. NN0
59		eqid 2622	. . . . . . . 8 |- ; 5 2 = ; 5 2
60	58	nn0cni 11304	. . . . . . . . 9 |- ; 1 0 e. CC
61		3cn 11095	. . . . . . . . 9 |- 3 e. CC
62		dec10p 11553	. . . . . . . . 9 |- (; 1 0 + 3 ) = ; 1 3
63	60, 61, 62	addcomli 10228	. . . . . . . 8 |- ( 3 + ; 1 0 ) = ; 1 3
64	54	mulid1i 10042	. . . . . . . . . 10 |- ( 5 x. 1 ) = 5
65		1p0e1 11133	. . . . . . . . . 10 |- ( 1 + 0 ) = 1
66	64, 65	oveq12i 6662	. . . . . . . . 9 |- ( ( 5 x. 1 ) + ( 1 + 0 ) ) = ( 5 + 1 )
67		5p1e6 11155	. . . . . . . . 9 |- ( 5 + 1 ) = 6
68	66, 67	eqtri 2644	. . . . . . . 8 |- ( ( 5 x. 1 ) + ( 1 + 0 ) ) = 6
69	24	mulid1i 10042	. . . . . . . . . 10 |- ( 2 x. 1 ) = 2
70	69	oveq1i 6660	. . . . . . . . 9 |- ( ( 2 x. 1 ) + 3 ) = ( 2 + 3 )
71		3p2e5 11160	. . . . . . . . . 10 |- ( 3 + 2 ) = 5
72	61, 24, 71	addcomli 10228	. . . . . . . . 9 |- ( 2 + 3 ) = 5
73	70, 72, 48	3eqtri 2648	. . . . . . . 8 |- ( ( 2 x. 1 ) + 3 ) = ; 0 5
74	5, 3, 2, 16, 59, 63, 2, 5, 33, 68, 73	decmac 11566	. . . . . . 7 |- ( (; 5 2 x. 1 ) + ( 3 + ; 1 0 ) ) = ; 6 5
75	2	dec0h 11522	. . . . . . . 8 |- 1 = ; 0 1
76		5t2e10 11634	. . . . . . . . . 10 |- ( 5 x. 2 ) = ; 1 0
77		00id 10211	. . . . . . . . . 10 |- ( 0 + 0 ) = 0
78	76, 77	oveq12i 6662	. . . . . . . . 9 |- ( ( 5 x. 2 ) + ( 0 + 0 ) ) = (; 1 0 + 0 )
79		dec10p 11553	. . . . . . . . 9 |- (; 1 0 + 0 ) = ; 1 0
80	78, 79	eqtri 2644	. . . . . . . 8 |- ( ( 5 x. 2 ) + ( 0 + 0 ) ) = ; 1 0
81		2t2e4 11177	. . . . . . . . . 10 |- ( 2 x. 2 ) = 4
82	81	oveq1i 6660	. . . . . . . . 9 |- ( ( 2 x. 2 ) + 1 ) = ( 4 + 1 )
83	82, 39, 48	3eqtri 2648	. . . . . . . 8 |- ( ( 2 x. 2 ) + 1 ) = ; 0 5
84	5, 3, 33, 2, 59, 75, 3, 5, 33, 80, 83	decmac 11566	. . . . . . 7 |- ( (; 5 2 x. 2 ) + 1 ) = ;; 1 0 5
85	2, 3, 16, 2, 47, 57, 19, 5, 58, 74, 84	decma2c 11568	. . . . . 6 |- ( (; 5 2 x. ; 1 2 ) + ( 5 + ; 2 6 ) ) = ;; 6 5 5
86		5t5e25 11639	. . . . . . . 8 |- ( 5 x. 5 ) = ; 2 5
87	3, 5, 67, 86	decsuc 11535	. . . . . . 7 |- ( ( 5 x. 5 ) + 1 ) = ; 2 6
88	54, 24, 76	mulcomli 10047	. . . . . . . 8 |- ( 2 x. 5 ) = ; 1 0
89	61	addid2i 10224	. . . . . . . 8 |- ( 0 + 3 ) = 3
90	2, 33, 16, 88, 89	decaddi 11579	. . . . . . 7 |- ( ( 2 x. 5 ) + 3 ) = ; 1 3
91	5, 3, 16, 59, 5, 16, 2, 87, 90	decrmac 11577	. . . . . 6 |- ( (; 5 2 x. 5 ) + 3 ) = ;; 2 6 3
92	4, 5, 5, 16, 32, 45, 19, 16, 46, 85, 91	decma2c 11568	. . . . 5 |- ( (; 5 2 x. ;; 1 2 5 ) + ( 6 + ; 4 7 ) ) = ;;; 6 5 5 3
93		9cn 11108	. . . . . . . 8 |- 9 e. CC
94		9t5e45 11666	. . . . . . . 8 |- ( 9 x. 5 ) = ; 4 5
95	93, 54, 94	mulcomli 10047	. . . . . . 7 |- ( 5 x. 9 ) = ; 4 5
96		5p2e7 11165	. . . . . . 7 |- ( 5 + 2 ) = 7
97	29, 5, 3, 95, 96	decaddi 11579	. . . . . 6 |- ( ( 5 x. 9 ) + 2 ) = ; 4 7
98		9t2e18 11663	. . . . . . . 8 |- ( 9 x. 2 ) = ; 1 8
99	93, 24, 98	mulcomli 10047	. . . . . . 7 |- ( 2 x. 9 ) = ; 1 8
100		1p1e2 11134	. . . . . . 7 |- ( 1 + 1 ) = 2
101		8p8e16 11618	. . . . . . 7 |- ( 8 + 8 ) = ; 1 6
102	2, 14, 14, 99, 100, 11, 101	decaddci 11580	. . . . . 6 |- ( ( 2 x. 9 ) + 8 ) = ; 2 6
103	5, 3, 14, 59, 27, 11, 3, 97, 102	decrmac 11577	. . . . 5 |- ( (; 5 2 x. 9 ) + 8 ) = ;; 4 7 6
104	6, 27, 11, 14, 1, 28, 19, 11, 31, 92, 103	decma2c 11568	. . . 4 |- ( (; 5 2 x. N ) + ; 6 8 ) = ;;;; 6 5 5 3 6
105		2exp16 15797	. . . 4 |- ( 2 ^; 1 6 ) = ;;;; 6 5 5 3 6
106		eqid 2622	. . . . 5 |- ( 2 ^ 8 ) = ( 2 ^ 8 )
107		eqid 2622	. . . . 5 |- ( ( 2 ^ 8 ) x. ( 2 ^ 8 ) ) = ( ( 2 ^ 8 ) x. ( 2 ^ 8 ) )
108	3, 14, 26, 106, 107	numexp2x 15783	. . . 4 |- ( 2 ^; 1 6 ) = ( ( 2 ^ 8 ) x. ( 2 ^ 8 ) )
109	104, 105, 108	3eqtr2i 2650	. . 3 |- ( (; 5 2 x. N ) + ; 6 8 ) = ( ( 2 ^ 8 ) x. ( 2 ^ 8 ) )
110	9, 10, 14, 20, 21, 15, 22, 26, 109	mod2xi 15773	. 2 |- ( ( 2 ^; 1 6 ) mod N ) = (; 6 8 mod N )
111		6p1e7 11156	. . 3 |- ( 6 + 1 ) = 7
112		eqid 2622	. . 3 |- ; 1 6 = ; 1 6
113	2, 11, 111, 112	decsuc 11535	. 2 |- (; 1 6 + 1 ) = ; 1 7
114	18	nn0cni 11304	. . . 4 |- ;; 1 3 6 e. CC
115	114	addid2i 10224	. . 3 |- ( 0 + ;; 1 3 6 ) = ;; 1 3 6
116	9	nncni 11030	. . . . 5 |- N e. CC
117	116	mul02i 10225	. . . 4 |- ( 0 x. N ) = 0
118	117	oveq1i 6660	. . 3 |- ( ( 0 x. N ) + ;; 1 3 6 ) = ( 0 + ;; 1 3 6 )
119		6t2e12 11641	. . . . 5 |- ( 6 x. 2 ) = ; 1 2
120	2, 3, 52, 119	decsuc 11535	. . . 4 |- ( ( 6 x. 2 ) + 1 ) = ; 1 3
121	3, 11, 14, 28, 11, 2, 120, 25	decmul1c 11587	. . 3 |- (; 6 8 x. 2 ) = ;; 1 3 6
122	115, 118, 121	3eqtr4i 2654	. 2 |- ( ( 0 x. N ) + ;; 1 3 6 ) = (; 6 8 x. 2 )
123	9, 10, 12, 13, 15, 18, 110, 113, 122	modxp1i 15774	1 |- ( ( 2 ^; 1 7 ) mod N ) = (;; 1 3 6 mod N )

Syntax hints:   = wceq 1483  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   NNcn 11020   2c2 11070   3c3 11071   4c4 11072   5c5 11073   6c6 11074   7c7 11075   8c8 11076   9c9 11077  ;cdc 11493   mod cmo 12668   ^cexp 12860

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-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  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-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861

This theorem is referenced by:  1259lem2  15839  1259lem4  15841