Theorem 1259lem2 15839

Description: Lemma for 1259prm 15843. Calculate a power mod. In decimal, we calculate 2 ^ 3 4 = ( 2 ^ 1 7 ) ^ 2 == 1 3 6 ^ 2 == 1 4 N + 8 7 0. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 15-Sep-2021.)

Hypothesis

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

Assertion

Ref	Expression
1259lem2	|- ( ( 2 ^; 3 4 ) mod N ) = (;; 8 7 0 mod N )

Proof of Theorem 1259lem2

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		7nn0 11314	. . 3 |- 7 e. NN0
12	2, 11	deccl 11512	. 2 |- ; 1 7 e. NN0
13		4nn0 11311	. . . 4 |- 4 e. NN0
14	2, 13	deccl 11512	. . 3 |- ; 1 4 e. NN0
15	14	nn0zi 11402	. 2 |- ; 1 4 e. ZZ
16		3nn0 11310	. . . 4 |- 3 e. NN0
17	2, 16	deccl 11512	. . 3 |- ; 1 3 e. NN0
18		6nn0 11313	. . 3 |- 6 e. NN0
19	17, 18	deccl 11512	. 2 |- ;; 1 3 6 e. NN0
20		8nn0 11315	. . . 4 |- 8 e. NN0
21	20, 11	deccl 11512	. . 3 |- ; 8 7 e. NN0
22		0nn0 11307	. . 3 |- 0 e. NN0
23	21, 22	deccl 11512	. 2 |- ;; 8 7 0 e. NN0
24	1	1259lem1 15838	. 2 |- ( ( 2 ^; 1 7 ) mod N ) = (;; 1 3 6 mod N )
25		eqid 2622	. . 3 |- ; 1 7 = ; 1 7
26		2cn 11091	. . . . . 6 |- 2 e. CC
27	26	mulid1i 10042	. . . . 5 |- ( 2 x. 1 ) = 2
28	27	oveq1i 6660	. . . 4 |- ( ( 2 x. 1 ) + 1 ) = ( 2 + 1 )
29		2p1e3 11151	. . . 4 |- ( 2 + 1 ) = 3
30	28, 29	eqtri 2644	. . 3 |- ( ( 2 x. 1 ) + 1 ) = 3
31		7cn 11104	. . . 4 |- 7 e. CC
32		7t2e14 11648	. . . 4 |- ( 7 x. 2 ) = ; 1 4
33	31, 26, 32	mulcomli 10047	. . 3 |- ( 2 x. 7 ) = ; 1 4
34	3, 2, 11, 25, 13, 2, 30, 33	decmul2c 11589	. 2 |- ( 2 x. ; 1 7 ) = ; 3 4
35		9nn0 11316	. . . 4 |- 9 e. NN0
36		eqid 2622	. . . 4 |- ;; 8 7 0 = ;; 8 7 0
37		eqid 2622	. . . . 5 |- ;; 1 2 5 = ;; 1 2 5
38		eqid 2622	. . . . . 6 |- ; 8 7 = ; 8 7
39		eqid 2622	. . . . . 6 |- ; 1 2 = ; 1 2
40		8p1e9 11158	. . . . . 6 |- ( 8 + 1 ) = 9
41		7p2e9 11172	. . . . . 6 |- ( 7 + 2 ) = 9
42	20, 11, 2, 3, 38, 39, 40, 41	decadd 11570	. . . . 5 |- (; 8 7 + ; 1 2 ) = ; 9 9
43		9p7e16 11625	. . . . . 6 |- ( 9 + 7 ) = ; 1 6
44		eqid 2622	. . . . . . 7 |- ; 1 4 = ; 1 4
45		3cn 11095	. . . . . . . . 9 |- 3 e. CC
46		ax-1cn 9994	. . . . . . . . 9 |- 1 e. CC
47		3p1e4 11153	. . . . . . . . 9 |- ( 3 + 1 ) = 4
48	45, 46, 47	addcomli 10228	. . . . . . . 8 |- ( 1 + 3 ) = 4
49	13	dec0h 11522	. . . . . . . 8 |- 4 = ; 0 4
50	48, 49	eqtri 2644	. . . . . . 7 |- ( 1 + 3 ) = ; 0 4
51	46	mulid1i 10042	. . . . . . . . 9 |- ( 1 x. 1 ) = 1
52		00id 10211	. . . . . . . . 9 |- ( 0 + 0 ) = 0
53	51, 52	oveq12i 6662	. . . . . . . 8 |- ( ( 1 x. 1 ) + ( 0 + 0 ) ) = ( 1 + 0 )
54	46	addid1i 10223	. . . . . . . 8 |- ( 1 + 0 ) = 1
55	53, 54	eqtri 2644	. . . . . . 7 |- ( ( 1 x. 1 ) + ( 0 + 0 ) ) = 1
56		4cn 11098	. . . . . . . . . 10 |- 4 e. CC
57	56	mulid1i 10042	. . . . . . . . 9 |- ( 4 x. 1 ) = 4
58	57	oveq1i 6660	. . . . . . . 8 |- ( ( 4 x. 1 ) + 4 ) = ( 4 + 4 )
59		4p4e8 11164	. . . . . . . 8 |- ( 4 + 4 ) = 8
60	20	dec0h 11522	. . . . . . . 8 |- 8 = ; 0 8
61	58, 59, 60	3eqtri 2648	. . . . . . 7 |- ( ( 4 x. 1 ) + 4 ) = ; 0 8
62	2, 13, 22, 13, 44, 50, 2, 20, 22, 55, 61	decmac 11566	. . . . . 6 |- ( (; 1 4 x. 1 ) + ( 1 + 3 ) ) = ; 1 8
63	18	dec0h 11522	. . . . . . 7 |- 6 = ; 0 6
64	26	mulid2i 10043	. . . . . . . . 9 |- ( 1 x. 2 ) = 2
65	46	addid2i 10224	. . . . . . . . 9 |- ( 0 + 1 ) = 1
66	64, 65	oveq12i 6662	. . . . . . . 8 |- ( ( 1 x. 2 ) + ( 0 + 1 ) ) = ( 2 + 1 )
67	66, 29	eqtri 2644	. . . . . . 7 |- ( ( 1 x. 2 ) + ( 0 + 1 ) ) = 3
68		4t2e8 11181	. . . . . . . . 9 |- ( 4 x. 2 ) = 8
69	68	oveq1i 6660	. . . . . . . 8 |- ( ( 4 x. 2 ) + 6 ) = ( 8 + 6 )
70		8p6e14 11616	. . . . . . . 8 |- ( 8 + 6 ) = ; 1 4
71	69, 70	eqtri 2644	. . . . . . 7 |- ( ( 4 x. 2 ) + 6 ) = ; 1 4
72	2, 13, 22, 18, 44, 63, 3, 13, 2, 67, 71	decmac 11566	. . . . . 6 |- ( (; 1 4 x. 2 ) + 6 ) = ; 3 4
73	2, 3, 2, 18, 39, 43, 14, 13, 16, 62, 72	decma2c 11568	. . . . 5 |- ( (; 1 4 x. ; 1 2 ) + ( 9 + 7 ) ) = ;; 1 8 4
74	35	dec0h 11522	. . . . . 6 |- 9 = ; 0 9
75		5cn 11100	. . . . . . . . 9 |- 5 e. CC
76	75	mulid2i 10043	. . . . . . . 8 |- ( 1 x. 5 ) = 5
77	26	addid2i 10224	. . . . . . . 8 |- ( 0 + 2 ) = 2
78	76, 77	oveq12i 6662	. . . . . . 7 |- ( ( 1 x. 5 ) + ( 0 + 2 ) ) = ( 5 + 2 )
79		5p2e7 11165	. . . . . . 7 |- ( 5 + 2 ) = 7
80	78, 79	eqtri 2644	. . . . . 6 |- ( ( 1 x. 5 ) + ( 0 + 2 ) ) = 7
81		5t4e20 11637	. . . . . . . 8 |- ( 5 x. 4 ) = ; 2 0
82	75, 56, 81	mulcomli 10047	. . . . . . 7 |- ( 4 x. 5 ) = ; 2 0
83		9cn 11108	. . . . . . . 8 |- 9 e. CC
84	83	addid2i 10224	. . . . . . 7 |- ( 0 + 9 ) = 9
85	3, 22, 35, 82, 84	decaddi 11579	. . . . . 6 |- ( ( 4 x. 5 ) + 9 ) = ; 2 9
86	2, 13, 22, 35, 44, 74, 5, 35, 3, 80, 85	decmac 11566	. . . . 5 |- ( (; 1 4 x. 5 ) + 9 ) = ; 7 9
87	4, 5, 35, 35, 37, 42, 14, 35, 11, 73, 86	decma2c 11568	. . . 4 |- ( (; 1 4 x. ;; 1 2 5 ) + (; 8 7 + ; 1 2 ) ) = ;;; 1 8 4 9
88	83	mulid2i 10043	. . . . . . . . 9 |- ( 1 x. 9 ) = 9
89	88	oveq1i 6660	. . . . . . . 8 |- ( ( 1 x. 9 ) + 3 ) = ( 9 + 3 )
90		9p3e12 11621	. . . . . . . 8 |- ( 9 + 3 ) = ; 1 2
91	89, 90	eqtri 2644	. . . . . . 7 |- ( ( 1 x. 9 ) + 3 ) = ; 1 2
92		9t4e36 11665	. . . . . . . 8 |- ( 9 x. 4 ) = ; 3 6
93	83, 56, 92	mulcomli 10047	. . . . . . 7 |- ( 4 x. 9 ) = ; 3 6
94	35, 2, 13, 44, 18, 16, 91, 93	decmul1c 11587	. . . . . 6 |- (; 1 4 x. 9 ) = ;; 1 2 6
95	94	oveq1i 6660	. . . . 5 |- ( (; 1 4 x. 9 ) + 0 ) = (;; 1 2 6 + 0 )
96	4, 18	deccl 11512	. . . . . . 7 |- ;; 1 2 6 e. NN0
97	96	nn0cni 11304	. . . . . 6 |- ;; 1 2 6 e. CC
98	97	addid1i 10223	. . . . 5 |- (;; 1 2 6 + 0 ) = ;; 1 2 6
99	95, 98	eqtri 2644	. . . 4 |- ( (; 1 4 x. 9 ) + 0 ) = ;; 1 2 6
100	6, 35, 21, 22, 1, 36, 14, 18, 4, 87, 99	decma2c 11568	. . 3 |- ( (; 1 4 x. N ) + ;; 8 7 0 ) = ;;;; 1 8 4 9 6
101		eqid 2622	. . . 4 |- ;; 1 3 6 = ;; 1 3 6
102	20, 2	deccl 11512	. . . 4 |- ; 8 1 e. NN0
103		eqid 2622	. . . . 5 |- ; 1 3 = ; 1 3
104		eqid 2622	. . . . 5 |- ; 8 1 = ; 8 1
105	13, 22	deccl 11512	. . . . 5 |- ; 4 0 e. NN0
106		eqid 2622	. . . . . . 7 |- ; 4 0 = ; 4 0
107	56	addid2i 10224	. . . . . . 7 |- ( 0 + 4 ) = 4
108		8cn 11106	. . . . . . . 8 |- 8 e. CC
109	108	addid1i 10223	. . . . . . 7 |- ( 8 + 0 ) = 8
110	22, 20, 13, 22, 60, 106, 107, 109	decadd 11570	. . . . . 6 |- ( 8 + ; 4 0 ) = ; 4 8
111		4p1e5 11154	. . . . . . . 8 |- ( 4 + 1 ) = 5
112	5	dec0h 11522	. . . . . . . 8 |- 5 = ; 0 5
113	111, 112	eqtri 2644	. . . . . . 7 |- ( 4 + 1 ) = ; 0 5
114	45	mulid1i 10042	. . . . . . . . 9 |- ( 3 x. 1 ) = 3
115	114	oveq1i 6660	. . . . . . . 8 |- ( ( 3 x. 1 ) + 5 ) = ( 3 + 5 )
116		5p3e8 11166	. . . . . . . . 9 |- ( 5 + 3 ) = 8
117	75, 45, 116	addcomli 10228	. . . . . . . 8 |- ( 3 + 5 ) = 8
118	115, 117, 60	3eqtri 2648	. . . . . . 7 |- ( ( 3 x. 1 ) + 5 ) = ; 0 8
119	2, 16, 22, 5, 103, 113, 2, 20, 22, 55, 118	decmac 11566	. . . . . 6 |- ( (; 1 3 x. 1 ) + ( 4 + 1 ) ) = ; 1 8
120		6cn 11102	. . . . . . . . 9 |- 6 e. CC
121	120	mulid1i 10042	. . . . . . . 8 |- ( 6 x. 1 ) = 6
122	121	oveq1i 6660	. . . . . . 7 |- ( ( 6 x. 1 ) + 8 ) = ( 6 + 8 )
123	108, 120, 70	addcomli 10228	. . . . . . 7 |- ( 6 + 8 ) = ; 1 4
124	122, 123	eqtri 2644	. . . . . 6 |- ( ( 6 x. 1 ) + 8 ) = ; 1 4
125	17, 18, 13, 20, 101, 110, 2, 13, 2, 119, 124	decmac 11566	. . . . 5 |- ( (;; 1 3 6 x. 1 ) + ( 8 + ; 4 0 ) ) = ;; 1 8 4
126	2	dec0h 11522	. . . . . 6 |- 1 = ; 0 1
127	65, 126	eqtri 2644	. . . . . . 7 |- ( 0 + 1 ) = ; 0 1
128	45	mulid2i 10043	. . . . . . . . 9 |- ( 1 x. 3 ) = 3
129	128, 65	oveq12i 6662	. . . . . . . 8 |- ( ( 1 x. 3 ) + ( 0 + 1 ) ) = ( 3 + 1 )
130	129, 47	eqtri 2644	. . . . . . 7 |- ( ( 1 x. 3 ) + ( 0 + 1 ) ) = 4
131		3t3e9 11180	. . . . . . . . 9 |- ( 3 x. 3 ) = 9
132	131	oveq1i 6660	. . . . . . . 8 |- ( ( 3 x. 3 ) + 1 ) = ( 9 + 1 )
133		9p1e10 11496	. . . . . . . 8 |- ( 9 + 1 ) = ; 1 0
134	132, 133	eqtri 2644	. . . . . . 7 |- ( ( 3 x. 3 ) + 1 ) = ; 1 0
135	2, 16, 22, 2, 103, 127, 16, 22, 2, 130, 134	decmac 11566	. . . . . 6 |- ( (; 1 3 x. 3 ) + ( 0 + 1 ) ) = ; 4 0
136		6t3e18 11642	. . . . . . 7 |- ( 6 x. 3 ) = ; 1 8
137	2, 20, 2, 136, 40	decaddi 11579	. . . . . 6 |- ( ( 6 x. 3 ) + 1 ) = ; 1 9
138	17, 18, 22, 2, 101, 126, 16, 35, 2, 135, 137	decmac 11566	. . . . 5 |- ( (;; 1 3 6 x. 3 ) + 1 ) = ;; 4 0 9
139	2, 16, 20, 2, 103, 104, 19, 35, 105, 125, 138	decma2c 11568	. . . 4 |- ( (;; 1 3 6 x. ; 1 3 ) + ; 8 1 ) = ;;; 1 8 4 9
140	16	dec0h 11522	. . . . . 6 |- 3 = ; 0 3
141	120	mulid2i 10043	. . . . . . . 8 |- ( 1 x. 6 ) = 6
142	141, 77	oveq12i 6662	. . . . . . 7 |- ( ( 1 x. 6 ) + ( 0 + 2 ) ) = ( 6 + 2 )
143		6p2e8 11169	. . . . . . 7 |- ( 6 + 2 ) = 8
144	142, 143	eqtri 2644	. . . . . 6 |- ( ( 1 x. 6 ) + ( 0 + 2 ) ) = 8
145	120, 45, 136	mulcomli 10047	. . . . . . 7 |- ( 3 x. 6 ) = ; 1 8
146		1p1e2 11134	. . . . . . 7 |- ( 1 + 1 ) = 2
147		8p3e11 11612	. . . . . . 7 |- ( 8 + 3 ) = ; 1 1
148	2, 20, 16, 145, 146, 2, 147	decaddci 11580	. . . . . 6 |- ( ( 3 x. 6 ) + 3 ) = ; 2 1
149	2, 16, 22, 16, 103, 140, 18, 2, 3, 144, 148	decmac 11566	. . . . 5 |- ( (; 1 3 x. 6 ) + 3 ) = ; 8 1
150		6t6e36 11646	. . . . 5 |- ( 6 x. 6 ) = ; 3 6
151	18, 17, 18, 101, 18, 16, 149, 150	decmul1c 11587	. . . 4 |- (;; 1 3 6 x. 6 ) = ;; 8 1 6
152	19, 17, 18, 101, 18, 102, 139, 151	decmul2c 11589	. . 3 |- (;; 1 3 6 x. ;; 1 3 6 ) = ;;;; 1 8 4 9 6
153	100, 152	eqtr4i 2647	. 2 |- ( (; 1 4 x. N ) + ;; 8 7 0 ) = (;; 1 3 6 x. ;; 1 3 6 )
154	9, 10, 12, 15, 19, 23, 24, 34, 153	mod2xi 15773	1 |- ( ( 2 ^; 3 4 ) mod N ) = (;; 8 7 0 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:  1259lem3  15840  1259lem5  15842