Theorem 1259lem3 15840

Description: Lemma for 1259prm 15843. Calculate a power mod. In decimal, we calculate 2 ^ 3 8 = 2 ^ 3 4 x. 2 ^ 4 == 8 7 0 x. 1 6 = 1 1 N + 7 1 and 2 ^ 7 6 = ( 2 ^ 3 4 ) ^ 2 == 7 1 ^ 2 = 4 N + 5 == 5. (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
1259lem3	|- ( ( 2 ^; 7 6 ) mod N ) = ( 5 mod N )

Proof of Theorem 1259lem3

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		3nn0 11310	. . 3 |- 3 e. NN0
12		8nn0 11315	. . 3 |- 8 e. NN0
13	11, 12	deccl 11512	. 2 |- ; 3 8 e. NN0
14		4z 11411	. 2 |- 4 e. ZZ
15		7nn0 11314	. . 3 |- 7 e. NN0
16	15, 2	deccl 11512	. 2 |- ; 7 1 e. NN0
17		4nn0 11311	. . . 4 |- 4 e. NN0
18	11, 17	deccl 11512	. . 3 |- ; 3 4 e. NN0
19	2, 2	deccl 11512	. . . 4 |- ; 1 1 e. NN0
20	19	nn0zi 11402	. . 3 |- ; 1 1 e. ZZ
21	12, 15	deccl 11512	. . . 4 |- ; 8 7 e. NN0
22		0nn0 11307	. . . 4 |- 0 e. NN0
23	21, 22	deccl 11512	. . 3 |- ;; 8 7 0 e. NN0
24		6nn0 11313	. . . 4 |- 6 e. NN0
25	2, 24	deccl 11512	. . 3 |- ; 1 6 e. NN0
26	1	1259lem2 15839	. . 3 |- ( ( 2 ^; 3 4 ) mod N ) = (;; 8 7 0 mod N )
27		2exp4 15794	. . . 4 |- ( 2 ^ 4 ) = ; 1 6
28	27	oveq1i 6660	. . 3 |- ( ( 2 ^ 4 ) mod N ) = (; 1 6 mod N )
29		eqid 2622	. . . 4 |- ; 3 4 = ; 3 4
30		4p4e8 11164	. . . 4 |- ( 4 + 4 ) = 8
31	11, 17, 17, 29, 30	decaddi 11579	. . 3 |- (; 3 4 + 4 ) = ; 3 8
32		9nn0 11316	. . . . 5 |- 9 e. NN0
33		eqid 2622	. . . . 5 |- ; 7 1 = ; 7 1
34		10nn0 11516	. . . . 5 |- ; 1 0 e. NN0
35		eqid 2622	. . . . . 6 |- ; 1 1 = ; 1 1
36	34	nn0cni 11304	. . . . . . 7 |- ; 1 0 e. CC
37		7cn 11104	. . . . . . 7 |- 7 e. CC
38		dec10p 11553	. . . . . . 7 |- (; 1 0 + 7 ) = ; 1 7
39	36, 37, 38	addcomli 10228	. . . . . 6 |- ( 7 + ; 1 0 ) = ; 1 7
40	2, 11	deccl 11512	. . . . . 6 |- ; 1 3 e. NN0
41	6	nn0cni 11304	. . . . . . . 8 |- ;; 1 2 5 e. CC
42	41	mulid2i 10043	. . . . . . 7 |- ( 1 x. ;; 1 2 5 ) = ;; 1 2 5
43	2	dec0h 11522	. . . . . . . 8 |- 1 = ; 0 1
44		eqid 2622	. . . . . . . 8 |- ; 1 3 = ; 1 3
45		0p1e1 11132	. . . . . . . 8 |- ( 0 + 1 ) = 1
46		3cn 11095	. . . . . . . . 9 |- 3 e. CC
47		ax-1cn 9994	. . . . . . . . 9 |- 1 e. CC
48		3p1e4 11153	. . . . . . . . 9 |- ( 3 + 1 ) = 4
49	46, 47, 48	addcomli 10228	. . . . . . . 8 |- ( 1 + 3 ) = 4
50	22, 2, 2, 11, 43, 44, 45, 49	decadd 11570	. . . . . . 7 |- ( 1 + ; 1 3 ) = ; 1 4
51		2p1e3 11151	. . . . . . . 8 |- ( 2 + 1 ) = 3
52		eqid 2622	. . . . . . . 8 |- ; 1 2 = ; 1 2
53	2, 3, 51, 52	decsuc 11535	. . . . . . 7 |- (; 1 2 + 1 ) = ; 1 3
54		5p4e9 11167	. . . . . . 7 |- ( 5 + 4 ) = 9
55	4, 5, 2, 17, 42, 50, 53, 54	decadd 11570	. . . . . 6 |- ( ( 1 x. ;; 1 2 5 ) + ( 1 + ; 1 3 ) ) = ;; 1 3 9
56		5cn 11100	. . . . . . . 8 |- 5 e. CC
57		7p5e12 11607	. . . . . . . 8 |- ( 7 + 5 ) = ; 1 2
58	37, 56, 57	addcomli 10228	. . . . . . 7 |- ( 5 + 7 ) = ; 1 2
59	4, 5, 15, 42, 53, 3, 58	decaddci 11580	. . . . . 6 |- ( ( 1 x. ;; 1 2 5 ) + 7 ) = ;; 1 3 2
60	2, 2, 2, 15, 35, 39, 6, 3, 40, 55, 59	decmac 11566	. . . . 5 |- ( (; 1 1 x. ;; 1 2 5 ) + ( 7 + ; 1 0 ) ) = ;;; 1 3 9 2
61		9p1e10 11496	. . . . . 6 |- ( 9 + 1 ) = ; 1 0
62		9cn 11108	. . . . . . 7 |- 9 e. CC
63	19	nn0cni 11304	. . . . . . 7 |- ; 1 1 e. CC
64		9t11e99 11671	. . . . . . 7 |- ( 9 x. ; 1 1 ) = ; 9 9
65	62, 63, 64	mulcomli 10047	. . . . . 6 |- (; 1 1 x. 9 ) = ; 9 9
66	32, 61, 65	decsucc 11550	. . . . 5 |- ( (; 1 1 x. 9 ) + 1 ) = ;; 1 0 0
67	6, 32, 15, 2, 1, 33, 19, 22, 34, 60, 66	decma2c 11568	. . . 4 |- ( (; 1 1 x. N ) + ; 7 1 ) = ;;;; 1 3 9 2 0
68		eqid 2622	. . . . 5 |- ; 1 6 = ; 1 6
69	5, 3	deccl 11512	. . . . . 6 |- ; 5 2 e. NN0
70	69, 3	deccl 11512	. . . . 5 |- ;; 5 2 2 e. NN0
71		eqid 2622	. . . . . 6 |- ;; 8 7 0 = ;; 8 7 0
72		eqid 2622	. . . . . 6 |- ;; 5 2 2 = ;; 5 2 2
73		eqid 2622	. . . . . . 7 |- ; 8 7 = ; 8 7
74	69	nn0cni 11304	. . . . . . . 8 |- ; 5 2 e. CC
75	74	addid1i 10223	. . . . . . 7 |- (; 5 2 + 0 ) = ; 5 2
76		8cn 11106	. . . . . . . . . 10 |- 8 e. CC
77	76	mulid1i 10042	. . . . . . . . 9 |- ( 8 x. 1 ) = 8
78	56	addid1i 10223	. . . . . . . . 9 |- ( 5 + 0 ) = 5
79	77, 78	oveq12i 6662	. . . . . . . 8 |- ( ( 8 x. 1 ) + ( 5 + 0 ) ) = ( 8 + 5 )
80		8p5e13 11615	. . . . . . . 8 |- ( 8 + 5 ) = ; 1 3
81	79, 80	eqtri 2644	. . . . . . 7 |- ( ( 8 x. 1 ) + ( 5 + 0 ) ) = ; 1 3
82	37	mulid1i 10042	. . . . . . . . 9 |- ( 7 x. 1 ) = 7
83	82	oveq1i 6660	. . . . . . . 8 |- ( ( 7 x. 1 ) + 2 ) = ( 7 + 2 )
84		7p2e9 11172	. . . . . . . 8 |- ( 7 + 2 ) = 9
85	32	dec0h 11522	. . . . . . . 8 |- 9 = ; 0 9
86	83, 84, 85	3eqtri 2648	. . . . . . 7 |- ( ( 7 x. 1 ) + 2 ) = ; 0 9
87	12, 15, 5, 3, 73, 75, 2, 32, 22, 81, 86	decmac 11566	. . . . . 6 |- ( (; 8 7 x. 1 ) + (; 5 2 + 0 ) ) = ;; 1 3 9
88	47	mul02i 10225	. . . . . . . 8 |- ( 0 x. 1 ) = 0
89	88	oveq1i 6660	. . . . . . 7 |- ( ( 0 x. 1 ) + 2 ) = ( 0 + 2 )
90		2cn 11091	. . . . . . . 8 |- 2 e. CC
91	90	addid2i 10224	. . . . . . 7 |- ( 0 + 2 ) = 2
92	3	dec0h 11522	. . . . . . 7 |- 2 = ; 0 2
93	89, 91, 92	3eqtri 2648	. . . . . 6 |- ( ( 0 x. 1 ) + 2 ) = ; 0 2
94	21, 22, 69, 3, 71, 72, 2, 3, 22, 87, 93	decmac 11566	. . . . 5 |- ( (;; 8 7 0 x. 1 ) + ;; 5 2 2 ) = ;;; 1 3 9 2
95		8t6e48 11659	. . . . . . . 8 |- ( 8 x. 6 ) = ; 4 8
96		4p1e5 11154	. . . . . . . 8 |- ( 4 + 1 ) = 5
97		8p4e12 11614	. . . . . . . 8 |- ( 8 + 4 ) = ; 1 2
98	17, 12, 17, 95, 96, 3, 97	decaddci 11580	. . . . . . 7 |- ( ( 8 x. 6 ) + 4 ) = ; 5 2
99		7t6e42 11652	. . . . . . 7 |- ( 7 x. 6 ) = ; 4 2
100	24, 12, 15, 73, 3, 17, 98, 99	decmul1c 11587	. . . . . 6 |- (; 8 7 x. 6 ) = ;; 5 2 2
101		6cn 11102	. . . . . . 7 |- 6 e. CC
102	101	mul02i 10225	. . . . . 6 |- ( 0 x. 6 ) = 0
103	24, 21, 22, 71, 22, 100, 102	decmul1 11585	. . . . 5 |- (;; 8 7 0 x. 6 ) = ;;; 5 2 2 0
104	23, 2, 24, 68, 22, 70, 94, 103	decmul2c 11589	. . . 4 |- (;; 8 7 0 x. ; 1 6 ) = ;;;; 1 3 9 2 0
105	67, 104	eqtr4i 2647	. . 3 |- ( (; 1 1 x. N ) + ; 7 1 ) = (;; 8 7 0 x. ; 1 6 )
106	9, 10, 18, 20, 23, 16, 17, 25, 26, 28, 31, 105	modxai 15772	. 2 |- ( ( 2 ^; 3 8 ) mod N ) = (; 7 1 mod N )
107		eqid 2622	. . 3 |- ; 3 8 = ; 3 8
108		3t2e6 11179	. . . . . 6 |- ( 3 x. 2 ) = 6
109	46, 90, 108	mulcomli 10047	. . . . 5 |- ( 2 x. 3 ) = 6
110	109	oveq1i 6660	. . . 4 |- ( ( 2 x. 3 ) + 1 ) = ( 6 + 1 )
111		6p1e7 11156	. . . 4 |- ( 6 + 1 ) = 7
112	110, 111	eqtri 2644	. . 3 |- ( ( 2 x. 3 ) + 1 ) = 7
113		8t2e16 11654	. . . 4 |- ( 8 x. 2 ) = ; 1 6
114	76, 90, 113	mulcomli 10047	. . 3 |- ( 2 x. 8 ) = ; 1 6
115	3, 11, 12, 107, 24, 2, 112, 114	decmul2c 11589	. 2 |- ( 2 x. ; 3 8 ) = ; 7 6
116	5	dec0h 11522	. . . 4 |- 5 = ; 0 5
117		eqid 2622	. . . . 5 |- ;; 1 2 5 = ;; 1 2 5
118		4cn 11098	. . . . . . 7 |- 4 e. CC
119	118	addid2i 10224	. . . . . 6 |- ( 0 + 4 ) = 4
120	17	dec0h 11522	. . . . . 6 |- 4 = ; 0 4
121	119, 120	eqtri 2644	. . . . 5 |- ( 0 + 4 ) = ; 0 4
122	91, 92	eqtri 2644	. . . . . 6 |- ( 0 + 2 ) = ; 0 2
123	118	mulid1i 10042	. . . . . . . 8 |- ( 4 x. 1 ) = 4
124	123, 45	oveq12i 6662	. . . . . . 7 |- ( ( 4 x. 1 ) + ( 0 + 1 ) ) = ( 4 + 1 )
125	124, 96	eqtri 2644	. . . . . 6 |- ( ( 4 x. 1 ) + ( 0 + 1 ) ) = 5
126		4t2e8 11181	. . . . . . . 8 |- ( 4 x. 2 ) = 8
127	126	oveq1i 6660	. . . . . . 7 |- ( ( 4 x. 2 ) + 2 ) = ( 8 + 2 )
128		8p2e10 11610	. . . . . . 7 |- ( 8 + 2 ) = ; 1 0
129	127, 128	eqtri 2644	. . . . . 6 |- ( ( 4 x. 2 ) + 2 ) = ; 1 0
130	2, 3, 22, 3, 52, 122, 17, 22, 2, 125, 129	decma2c 11568	. . . . 5 |- ( ( 4 x. ; 1 2 ) + ( 0 + 2 ) ) = ; 5 0
131		5t4e20 11637	. . . . . . 7 |- ( 5 x. 4 ) = ; 2 0
132	56, 118, 131	mulcomli 10047	. . . . . 6 |- ( 4 x. 5 ) = ; 2 0
133	3, 22, 17, 132, 119	decaddi 11579	. . . . 5 |- ( ( 4 x. 5 ) + 4 ) = ; 2 4
134	4, 5, 22, 17, 117, 121, 17, 17, 3, 130, 133	decma2c 11568	. . . 4 |- ( ( 4 x. ;; 1 2 5 ) + ( 0 + 4 ) ) = ;; 5 0 4
135		9t4e36 11665	. . . . . 6 |- ( 9 x. 4 ) = ; 3 6
136	62, 118, 135	mulcomli 10047	. . . . 5 |- ( 4 x. 9 ) = ; 3 6
137		6p5e11 11600	. . . . 5 |- ( 6 + 5 ) = ; 1 1
138	11, 24, 5, 136, 48, 2, 137	decaddci 11580	. . . 4 |- ( ( 4 x. 9 ) + 5 ) = ; 4 1
139	6, 32, 22, 5, 1, 116, 17, 2, 17, 134, 138	decma2c 11568	. . 3 |- ( ( 4 x. N ) + 5 ) = ;;; 5 0 4 1
140		7t7e49 11653	. . . . . 6 |- ( 7 x. 7 ) = ; 4 9
141	17, 96, 140	decsucc 11550	. . . . 5 |- ( ( 7 x. 7 ) + 1 ) = ; 5 0
142	37	mulid2i 10043	. . . . . . 7 |- ( 1 x. 7 ) = 7
143	142	oveq1i 6660	. . . . . 6 |- ( ( 1 x. 7 ) + 7 ) = ( 7 + 7 )
144		7p7e14 11609	. . . . . 6 |- ( 7 + 7 ) = ; 1 4
145	143, 144	eqtri 2644	. . . . 5 |- ( ( 1 x. 7 ) + 7 ) = ; 1 4
146	15, 2, 15, 33, 15, 17, 2, 141, 145	decrmac 11577	. . . 4 |- ( (; 7 1 x. 7 ) + 7 ) = ;; 5 0 4
147	16	nn0cni 11304	. . . . 5 |- ; 7 1 e. CC
148	147	mulid1i 10042	. . . 4 |- (; 7 1 x. 1 ) = ; 7 1
149	16, 15, 2, 33, 2, 15, 146, 148	decmul2c 11589	. . 3 |- (; 7 1 x. ; 7 1 ) = ;;; 5 0 4 1
150	139, 149	eqtr4i 2647	. 2 |- ( ( 4 x. N ) + 5 ) = (; 7 1 x. ; 7 1 )
151	9, 10, 13, 14, 16, 5, 106, 115, 150	mod2xi 15773	1 |- ( ( 2 ^; 7 6 ) mod N ) = ( 5 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:  1259lem4  15841