Theorem 1259lem1 13262

Description: Lemma for 1259prm 13267. 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 9583	. . . . . 6 |- 1 e. NN0
3		2nn0 9584	. . . . . 6 |- 2 e. NN0
4	2, 3	deccl 9795	. . . . 5 |- ; 1 2 e. NN0
5		5nn0 9587	. . . . 5 |- 5 e. NN0
6	4, 5	deccl 9795	. . . 4 |- ;; 1 2 5 e. NN0
7		9nn 9477	. . . 4 |- 9 e. NN
8	6, 7	decnncl 9804	. . 3 |- ;;; 1 2 5 9 e. NN
9	1, 8	eqeltri 2311	. 2 |- N e. NN
10		2nn 9470	. 2 |- 2 e. NN
11		6nn0 9588	. . 3 |- 6 e. NN0
12	2, 11	deccl 9795	. 2 |- ; 1 6 e. NN0
13		0z 9659	. 2 |- 0 e. ZZ
14		8nn0 9590	. . 3 |- 8 e. NN0
15	11, 14	deccl 9795	. 2 |- ; 6 8 e. NN0
16		3nn0 9585	. . . 4 |- 3 e. NN0
17	2, 16	deccl 9795	. . 3 |- ; 1 3 e. NN0
18	17, 11	deccl 9795	. 2 |- ;; 1 3 6 e. NN0
19	5, 3	deccl 9795	. . . 4 |- ; 5 2 e. NN0
20	19	nn0zi 9670	. . 3 |- ; 5 2 e. ZZ
21	3, 14	nn0expcli 11015	. . 3 |- ( 2 ^ 8 ) e. NN0
22		eqid 2238	. . 3 |- ( ( 2 ^ 8 ) mod N ) = ( ( 2 ^ 8 ) mod N )
23	14	nn0cni 9579	. . . 4 |- 8 e. CC
24		2cn 9377	. . . 4 |- 2 e. CC
25		8t2e16 9900	. . . 4 |- ( 8 x. 2 ) = ; 1 6
26	23, 24, 25	mulcomli 8333	. . 3 |- ( 2 x. 8 ) = ; 1 6
27		9nn0 9591	. . . . 5 |- 9 e. NN0
28		eqid 2238	. . . . 5 |- ; 6 8 = ; 6 8
29		4nn0 9586	. . . . . 6 |- 4 e. NN0
30		7nn0 9589	. . . . . 6 |- 7 e. NN0
31	29, 30	deccl 9795	. . . . 5 |- ; 4 7 e. NN0
32		eqid 2238	. . . . . 6 |- ;; 1 2 5 = ;; 1 2 5
33		0nn0 9582	. . . . . . 7 |- 0 e. NN0
34	11	dec0h 9807	. . . . . . 7 |- 6 = ; 0 6
35		eqid 2238	. . . . . . 7 |- ; 4 7 = ; 4 7
36		4cn 9384	. . . . . . . . . 10 |- 4 e. CC
37	36	addlidi 8470	. . . . . . . . 9 |- ( 0 + 4 ) = 4
38	37	oveq1i 6095	. . . . . . . 8 |- ( ( 0 + 4 ) + 1 ) = ( 4 + 1 )
39		4p1e5 9443	. . . . . . . 8 |- ( 4 + 1 ) = 5
40	38, 39	eqtri 2259	. . . . . . 7 |- ( ( 0 + 4 ) + 1 ) = 5
41		7cn 9390	. . . . . . . 8 |- 7 e. CC
42		6cn 9388	. . . . . . . 8 |- 6 e. CC
43		7p6e13 9863	. . . . . . . 8 |- ( 7 + 6 ) = ; 1 3
44	41, 42, 43	addcomli 8472	. . . . . . 7 |- ( 6 + 7 ) = ; 1 3
45	33, 11, 29, 30, 34, 35, 40, 16, 44	decaddc 9840	. . . . . 6 |- ( 6 + ; 4 7 ) = ; 5 3
46	3, 11	deccl 9795	. . . . . 6 |- ; 2 6 e. NN0
47		eqid 2238	. . . . . . 7 |- ; 1 2 = ; 1 2
48	5	dec0h 9807	. . . . . . . 8 |- 5 = ; 0 5
49		eqid 2238	. . . . . . . 8 |- ; 2 6 = ; 2 6
50	24	addlidi 8470	. . . . . . . . . 10 |- ( 0 + 2 ) = 2
51	50	oveq1i 6095	. . . . . . . . 9 |- ( ( 0 + 2 ) + 1 ) = ( 2 + 1 )
52		2p1e3 9440	. . . . . . . . 9 |- ( 2 + 1 ) = 3
53	51, 52	eqtri 2259	. . . . . . . 8 |- ( ( 0 + 2 ) + 1 ) = 3
54		5cn 9386	. . . . . . . . 9 |- 5 e. CC
55		6p5e11 9858	. . . . . . . . 9 |- ( 6 + 5 ) = ; 1 1
56	42, 54, 55	addcomli 8472	. . . . . . . 8 |- ( 5 + 6 ) = ; 1 1
57	33, 5, 3, 11, 48, 49, 53, 2, 56	decaddc 9840	. . . . . . 7 |- ( 5 + ; 2 6 ) = ; 3 1
58		10nn0 9802	. . . . . . 7 |- ; 1 0 e. NN0
59		eqid 2238	. . . . . . . 8 |- ; 5 2 = ; 5 2
60	58	nn0cni 9579	. . . . . . . . 9 |- ; 1 0 e. CC
61		3cn 9381	. . . . . . . . 9 |- 3 e. CC
62		dec10p 9828	. . . . . . . . 9 |- (; 1 0 + 3 ) = ; 1 3
63	60, 61, 62	addcomli 8472	. . . . . . . 8 |- ( 3 + ; 1 0 ) = ; 1 3
64	54	mulridi 8328	. . . . . . . . . 10 |- ( 5 x. 1 ) = 5
65		1p0e1 9422	. . . . . . . . . 10 |- ( 1 + 0 ) = 1
66	64, 65	oveq12i 6097	. . . . . . . . 9 |- ( ( 5 x. 1 ) + ( 1 + 0 ) ) = ( 5 + 1 )
67		5p1e6 9444	. . . . . . . . 9 |- ( 5 + 1 ) = 6
68	66, 67	eqtri 2259	. . . . . . . 8 |- ( ( 5 x. 1 ) + ( 1 + 0 ) ) = 6
69	24	mulridi 8328	. . . . . . . . . 10 |- ( 2 x. 1 ) = 2
70	69	oveq1i 6095	. . . . . . . . 9 |- ( ( 2 x. 1 ) + 3 ) = ( 2 + 3 )
71		3p2e5 9448	. . . . . . . . . 10 |- ( 3 + 2 ) = 5
72	61, 24, 71	addcomli 8472	. . . . . . . . 9 |- ( 2 + 3 ) = 5
73	70, 72, 48	3eqtri 2263	. . . . . . . 8 |- ( ( 2 x. 1 ) + 3 ) = ; 0 5
74	5, 3, 2, 16, 59, 63, 2, 5, 33, 68, 73	decmac 9837	. . . . . . 7 |- ( (; 5 2 x. 1 ) + ( 3 + ; 1 0 ) ) = ; 6 5
75	2	dec0h 9807	. . . . . . . 8 |- 1 = ; 0 1
76		5t2e10 9885	. . . . . . . . . 10 |- ( 5 x. 2 ) = ; 1 0
77		00id 8468	. . . . . . . . . 10 |- ( 0 + 0 ) = 0
78	76, 77	oveq12i 6097	. . . . . . . . 9 |- ( ( 5 x. 2 ) + ( 0 + 0 ) ) = (; 1 0 + 0 )
79		dec10p 9828	. . . . . . . . 9 |- (; 1 0 + 0 ) = ; 1 0
80	78, 79	eqtri 2259	. . . . . . . 8 |- ( ( 5 x. 2 ) + ( 0 + 0 ) ) = ; 1 0
81		2t2e4 9461	. . . . . . . . . 10 |- ( 2 x. 2 ) = 4
82	81	oveq1i 6095	. . . . . . . . 9 |- ( ( 2 x. 2 ) + 1 ) = ( 4 + 1 )
83	82, 39, 48	3eqtri 2263	. . . . . . . 8 |- ( ( 2 x. 2 ) + 1 ) = ; 0 5
84	5, 3, 33, 2, 59, 75, 3, 5, 33, 80, 83	decmac 9837	. . . . . . 7 |- ( (; 5 2 x. 2 ) + 1 ) = ;; 1 0 5
85	2, 3, 16, 2, 47, 57, 19, 5, 58, 74, 84	decma2c 9838	. . . . . 6 |- ( (; 5 2 x. ; 1 2 ) + ( 5 + ; 2 6 ) ) = ;; 6 5 5
86		5t5e25 9888	. . . . . . . 8 |- ( 5 x. 5 ) = ; 2 5
87	3, 5, 67, 86	decsuc 9816	. . . . . . 7 |- ( ( 5 x. 5 ) + 1 ) = ; 2 6
88	54, 24, 76	mulcomli 8333	. . . . . . . 8 |- ( 2 x. 5 ) = ; 1 0
89	61	addlidi 8470	. . . . . . . 8 |- ( 0 + 3 ) = 3
90	2, 33, 16, 88, 89	decaddi 9845	. . . . . . 7 |- ( ( 2 x. 5 ) + 3 ) = ; 1 3
91	5, 3, 16, 59, 5, 16, 2, 87, 90	decrmac 9843	. . . . . 6 |- ( (; 5 2 x. 5 ) + 3 ) = ;; 2 6 3
92	4, 5, 5, 16, 32, 45, 19, 16, 46, 85, 91	decma2c 9838	. . . . 5 |- ( (; 5 2 x. ;; 1 2 5 ) + ( 6 + ; 4 7 ) ) = ;;; 6 5 5 3
93		9cn 9394	. . . . . . . 8 |- 9 e. CC
94		9t5e45 9910	. . . . . . . 8 |- ( 9 x. 5 ) = ; 4 5
95	93, 54, 94	mulcomli 8333	. . . . . . 7 |- ( 5 x. 9 ) = ; 4 5
96		5p2e7 9453	. . . . . . 7 |- ( 5 + 2 ) = 7
97	29, 5, 3, 95, 96	decaddi 9845	. . . . . 6 |- ( ( 5 x. 9 ) + 2 ) = ; 4 7
98		9t2e18 9907	. . . . . . . 8 |- ( 9 x. 2 ) = ; 1 8
99	93, 24, 98	mulcomli 8333	. . . . . . 7 |- ( 2 x. 9 ) = ; 1 8
100		1p1e2 9423	. . . . . . 7 |- ( 1 + 1 ) = 2
101		8p8e16 9871	. . . . . . 7 |- ( 8 + 8 ) = ; 1 6
102	2, 14, 14, 99, 100, 11, 101	decaddci 9846	. . . . . 6 |- ( ( 2 x. 9 ) + 8 ) = ; 2 6
103	5, 3, 14, 59, 27, 11, 3, 97, 102	decrmac 9843	. . . . 5 |- ( (; 5 2 x. 9 ) + 8 ) = ;; 4 7 6
104	6, 27, 11, 14, 1, 28, 19, 11, 31, 92, 103	decma2c 9838	. . . 4 |- ( (; 5 2 x. N ) + ; 6 8 ) = ;;;; 6 5 5 3 6
105		2exp16 13237	. . . 4 |- ( 2 ^; 1 6 ) = ;;;; 6 5 5 3 6
106		eqid 2238	. . . . 5 |- ( 2 ^ 8 ) = ( 2 ^ 8 )
107		eqid 2238	. . . . 5 |- ( ( 2 ^ 8 ) x. ( 2 ^ 8 ) ) = ( ( 2 ^ 8 ) x. ( 2 ^ 8 ) )
108	3, 14, 26, 106, 107	numexp2x 13225	. . . 4 |- ( 2 ^; 1 6 ) = ( ( 2 ^ 8 ) x. ( 2 ^ 8 ) )
109	104, 105, 108	3eqtr2i 2265	. . 3 |- ( (; 5 2 x. N ) + ; 6 8 ) = ( ( 2 ^ 8 ) x. ( 2 ^ 8 ) )
110	9, 10, 14, 20, 21, 15, 22, 26, 109	mod2xi 13216	. 2 |- ( ( 2 ^; 1 6 ) mod N ) = (; 6 8 mod N )
111		6p1e7 9445	. . 3 |- ( 6 + 1 ) = 7
112		eqid 2238	. . 3 |- ; 1 6 = ; 1 6
113	2, 11, 111, 112	decsuc 9816	. 2 |- (; 1 6 + 1 ) = ; 1 7
114	18	nn0cni 9579	. . . 4 |- ;; 1 3 6 e. CC
115	114	addlidi 8470	. . 3 |- ( 0 + ;; 1 3 6 ) = ;; 1 3 6
116	9	nncni 9316	. . . . 5 |- N e. CC
117	116	mul02i 8718	. . . 4 |- ( 0 x. N ) = 0
118	117	oveq1i 6095	. . 3 |- ( ( 0 x. N ) + ;; 1 3 6 ) = ( 0 + ;; 1 3 6 )
119		6t2e12 9889	. . . . 5 |- ( 6 x. 2 ) = ; 1 2
120	2, 3, 52, 119	decsuc 9816	. . . 4 |- ( ( 6 x. 2 ) + 1 ) = ; 1 3
121	3, 11, 14, 28, 11, 2, 120, 25	decmul1c 9850	. . 3 |- (; 6 8 x. 2 ) = ;; 1 3 6
122	115, 118, 121	3eqtr4i 2269	. 2 |- ( ( 0 x. N ) + ;; 1 3 6 ) = (; 6 8 x. 2 )
123	9, 10, 12, 13, 15, 18, 110, 113, 122	modxp1i 13217	1 |- ( ( 2 ^; 1 7 ) mod N ) = (;; 1 3 6 mod N )

This proof depends on syntax axioms:   = wceq 1402  (class class class)co 6085   0cc0 8179   1c1 8180   + caddc 8182   x. cmul 8184   NNcn 9306   2c2 9357   3c3 9358   4c4 9359   5c5 9360   6c6 9361   7c7 9362   8c8 9363   9c9 9364  ;cdc 9781   mod cmo 10772   ^cexp 10988

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8500  df-neg 8501  df-reap 8905  df-ap 8912  df-div 9005  df-inn 9307  df-2 9365  df-3 9366  df-4 9367  df-5 9368  df-6 9369  df-7 9370  df-8 9371  df-9 9372  df-n0 9568  df-z 9649  df-dec 9782  df-uz 9931  df-q 10029  df-rp 10065  df-fl 10715  df-mod 10773  df-seqfrec 10898  df-exp 10989

This theorem is used by:  1259lem2  13263  1259lem4  13265