Theorem 1259lem3 13264

Description: Lemma for 1259prm 13267. 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 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		3nn0 9585	. . 3 |- 3 e. NN0
12		8nn0 9590	. . 3 |- 8 e. NN0
13	11, 12	deccl 9795	. 2 |- ; 3 8 e. NN0
14		4z 9678	. 2 |- 4 e. ZZ
15		7nn0 9589	. . 3 |- 7 e. NN0
16	15, 2	deccl 9795	. 2 |- ; 7 1 e. NN0
17		4nn0 9586	. . . 4 |- 4 e. NN0
18	11, 17	deccl 9795	. . 3 |- ; 3 4 e. NN0
19	2, 2	deccl 9795	. . . 4 |- ; 1 1 e. NN0
20	19	nn0zi 9670	. . 3 |- ; 1 1 e. ZZ
21	12, 15	deccl 9795	. . . 4 |- ; 8 7 e. NN0
22		0nn0 9582	. . . 4 |- 0 e. NN0
23	21, 22	deccl 9795	. . 3 |- ;; 8 7 0 e. NN0
24		6nn0 9588	. . . 4 |- 6 e. NN0
25	2, 24	deccl 9795	. . 3 |- ; 1 6 e. NN0
26	1	1259lem2 13263	. . 3 |- ( ( 2 ^; 3 4 ) mod N ) = (;; 8 7 0 mod N )
27		2exp4 13231	. . . 4 |- ( 2 ^ 4 ) = ; 1 6
28	27	oveq1i 6095	. . 3 |- ( ( 2 ^ 4 ) mod N ) = (; 1 6 mod N )
29		eqid 2238	. . . 4 |- ; 3 4 = ; 3 4
30		4p4e8 9452	. . . 4 |- ( 4 + 4 ) = 8
31	11, 17, 17, 29, 30	decaddi 9845	. . 3 |- (; 3 4 + 4 ) = ; 3 8
32		9nn0 9591	. . . . 5 |- 9 e. NN0
33		eqid 2238	. . . . 5 |- ; 7 1 = ; 7 1
34		10nn0 9802	. . . . 5 |- ; 1 0 e. NN0
35		eqid 2238	. . . . . 6 |- ; 1 1 = ; 1 1
36	34	nn0cni 9579	. . . . . . 7 |- ; 1 0 e. CC
37		7cn 9390	. . . . . . 7 |- 7 e. CC
38		dec10p 9828	. . . . . . 7 |- (; 1 0 + 7 ) = ; 1 7
39	36, 37, 38	addcomli 8472	. . . . . 6 |- ( 7 + ; 1 0 ) = ; 1 7
40	2, 11	deccl 9795	. . . . . 6 |- ; 1 3 e. NN0
41	6	nn0cni 9579	. . . . . . . 8 |- ;; 1 2 5 e. CC
42	41	mullidi 8329	. . . . . . 7 |- ( 1 x. ;; 1 2 5 ) = ;; 1 2 5
43	2	dec0h 9807	. . . . . . . 8 |- 1 = ; 0 1
44		eqid 2238	. . . . . . . 8 |- ; 1 3 = ; 1 3
45		0p1e1 9420	. . . . . . . 8 |- ( 0 + 1 ) = 1
46		3cn 9381	. . . . . . . . 9 |- 3 e. CC
47		ax-1cn 8272	. . . . . . . . 9 |- 1 e. CC
48		3p1e4 9442	. . . . . . . . 9 |- ( 3 + 1 ) = 4
49	46, 47, 48	addcomli 8472	. . . . . . . 8 |- ( 1 + 3 ) = 4
50	22, 2, 2, 11, 43, 44, 45, 49	decadd 9839	. . . . . . 7 |- ( 1 + ; 1 3 ) = ; 1 4
51		2p1e3 9440	. . . . . . . 8 |- ( 2 + 1 ) = 3
52		eqid 2238	. . . . . . . 8 |- ; 1 2 = ; 1 2
53	2, 3, 51, 52	decsuc 9816	. . . . . . 7 |- (; 1 2 + 1 ) = ; 1 3
54		5p4e9 9455	. . . . . . 7 |- ( 5 + 4 ) = 9
55	4, 5, 2, 17, 42, 50, 53, 54	decadd 9839	. . . . . 6 |- ( ( 1 x. ;; 1 2 5 ) + ( 1 + ; 1 3 ) ) = ;; 1 3 9
56		5cn 9386	. . . . . . . 8 |- 5 e. CC
57		7p5e12 9862	. . . . . . . 8 |- ( 7 + 5 ) = ; 1 2
58	37, 56, 57	addcomli 8472	. . . . . . 7 |- ( 5 + 7 ) = ; 1 2
59	4, 5, 15, 42, 53, 3, 58	decaddci 9846	. . . . . 6 |- ( ( 1 x. ;; 1 2 5 ) + 7 ) = ;; 1 3 2
60	2, 2, 2, 15, 35, 39, 6, 3, 40, 55, 59	decmac 9837	. . . . 5 |- ( (; 1 1 x. ;; 1 2 5 ) + ( 7 + ; 1 0 ) ) = ;;; 1 3 9 2
61		9p1e10 9783	. . . . . 6 |- ( 9 + 1 ) = ; 1 0
62		9cn 9394	. . . . . . 7 |- 9 e. CC
63	19	nn0cni 9579	. . . . . . 7 |- ; 1 1 e. CC
64		9t11e99 9915	. . . . . . 7 |- ( 9 x. ; 1 1 ) = ; 9 9
65	62, 63, 64	mulcomli 8333	. . . . . 6 |- (; 1 1 x. 9 ) = ; 9 9
66	32, 61, 65	decsucc 9826	. . . . 5 |- ( (; 1 1 x. 9 ) + 1 ) = ;; 1 0 0
67	6, 32, 15, 2, 1, 33, 19, 22, 34, 60, 66	decma2c 9838	. . . 4 |- ( (; 1 1 x. N ) + ; 7 1 ) = ;;;; 1 3 9 2 0
68		eqid 2238	. . . . 5 |- ; 1 6 = ; 1 6
69	5, 3	deccl 9795	. . . . . 6 |- ; 5 2 e. NN0
70	69, 3	deccl 9795	. . . . 5 |- ;; 5 2 2 e. NN0
71		eqid 2238	. . . . . 6 |- ;; 8 7 0 = ;; 8 7 0
72		eqid 2238	. . . . . 6 |- ;; 5 2 2 = ;; 5 2 2
73		eqid 2238	. . . . . . 7 |- ; 8 7 = ; 8 7
74	69	nn0cni 9579	. . . . . . . 8 |- ; 5 2 e. CC
75	74	addridi 8469	. . . . . . 7 |- (; 5 2 + 0 ) = ; 5 2
76		8cn 9392	. . . . . . . . . 10 |- 8 e. CC
77	76	mulridi 8328	. . . . . . . . 9 |- ( 8 x. 1 ) = 8
78	56	addridi 8469	. . . . . . . . 9 |- ( 5 + 0 ) = 5
79	77, 78	oveq12i 6097	. . . . . . . 8 |- ( ( 8 x. 1 ) + ( 5 + 0 ) ) = ( 8 + 5 )
80		8p5e13 9868	. . . . . . . 8 |- ( 8 + 5 ) = ; 1 3
81	79, 80	eqtri 2259	. . . . . . 7 |- ( ( 8 x. 1 ) + ( 5 + 0 ) ) = ; 1 3
82	37	mulridi 8328	. . . . . . . . 9 |- ( 7 x. 1 ) = 7
83	82	oveq1i 6095	. . . . . . . 8 |- ( ( 7 x. 1 ) + 2 ) = ( 7 + 2 )
84		7p2e9 9458	. . . . . . . 8 |- ( 7 + 2 ) = 9
85	32	dec0h 9807	. . . . . . . 8 |- 9 = ; 0 9
86	83, 84, 85	3eqtri 2263	. . . . . . 7 |- ( ( 7 x. 1 ) + 2 ) = ; 0 9
87	12, 15, 5, 3, 73, 75, 2, 32, 22, 81, 86	decmac 9837	. . . . . 6 |- ( (; 8 7 x. 1 ) + (; 5 2 + 0 ) ) = ;; 1 3 9
88	47	mul02i 8718	. . . . . . . 8 |- ( 0 x. 1 ) = 0
89	88	oveq1i 6095	. . . . . . 7 |- ( ( 0 x. 1 ) + 2 ) = ( 0 + 2 )
90		2cn 9377	. . . . . . . 8 |- 2 e. CC
91	90	addlidi 8470	. . . . . . 7 |- ( 0 + 2 ) = 2
92	3	dec0h 9807	. . . . . . 7 |- 2 = ; 0 2
93	89, 91, 92	3eqtri 2263	. . . . . 6 |- ( ( 0 x. 1 ) + 2 ) = ; 0 2
94	21, 22, 69, 3, 71, 72, 2, 3, 22, 87, 93	decmac 9837	. . . . 5 |- ( (;; 8 7 0 x. 1 ) + ;; 5 2 2 ) = ;;; 1 3 9 2
95		8t6e48 9904	. . . . . . . 8 |- ( 8 x. 6 ) = ; 4 8
96		4p1e5 9443	. . . . . . . 8 |- ( 4 + 1 ) = 5
97		8p4e12 9867	. . . . . . . 8 |- ( 8 + 4 ) = ; 1 2
98	17, 12, 17, 95, 96, 3, 97	decaddci 9846	. . . . . . 7 |- ( ( 8 x. 6 ) + 4 ) = ; 5 2
99		7t6e42 9898	. . . . . . 7 |- ( 7 x. 6 ) = ; 4 2
100	24, 12, 15, 73, 3, 17, 98, 99	decmul1c 9850	. . . . . 6 |- (; 8 7 x. 6 ) = ;; 5 2 2
101		6cn 9388	. . . . . . 7 |- 6 e. CC
102	101	mul02i 8718	. . . . . 6 |- ( 0 x. 6 ) = 0
103	24, 21, 22, 71, 22, 100, 102	decmul1 9849	. . . . 5 |- (;; 8 7 0 x. 6 ) = ;;; 5 2 2 0
104	23, 2, 24, 68, 22, 70, 94, 103	decmul2c 9851	. . . 4 |- (;; 8 7 0 x. ; 1 6 ) = ;;;; 1 3 9 2 0
105	67, 104	eqtr4i 2262	. . 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 13215	. 2 |- ( ( 2 ^; 3 8 ) mod N ) = (; 7 1 mod N )
107		eqid 2238	. . 3 |- ; 3 8 = ; 3 8
108		2t3e6 9464	. . . . 5 |- ( 2 x. 3 ) = 6
109	108	oveq1i 6095	. . . 4 |- ( ( 2 x. 3 ) + 1 ) = ( 6 + 1 )
110		6p1e7 9445	. . . 4 |- ( 6 + 1 ) = 7
111	109, 110	eqtri 2259	. . 3 |- ( ( 2 x. 3 ) + 1 ) = 7
112		8t2e16 9900	. . . 4 |- ( 8 x. 2 ) = ; 1 6
113	76, 90, 112	mulcomli 8333	. . 3 |- ( 2 x. 8 ) = ; 1 6
114	3, 11, 12, 107, 24, 2, 111, 113	decmul2c 9851	. 2 |- ( 2 x. ; 3 8 ) = ; 7 6
115	5	dec0h 9807	. . . 4 |- 5 = ; 0 5
116		eqid 2238	. . . . 5 |- ;; 1 2 5 = ;; 1 2 5
117		4cn 9384	. . . . . . 7 |- 4 e. CC
118	117	addlidi 8470	. . . . . 6 |- ( 0 + 4 ) = 4
119	17	dec0h 9807	. . . . . 6 |- 4 = ; 0 4
120	118, 119	eqtri 2259	. . . . 5 |- ( 0 + 4 ) = ; 0 4
121	91, 92	eqtri 2259	. . . . . 6 |- ( 0 + 2 ) = ; 0 2
122	117	mulridi 8328	. . . . . . . 8 |- ( 4 x. 1 ) = 4
123	122, 45	oveq12i 6097	. . . . . . 7 |- ( ( 4 x. 1 ) + ( 0 + 1 ) ) = ( 4 + 1 )
124	123, 96	eqtri 2259	. . . . . 6 |- ( ( 4 x. 1 ) + ( 0 + 1 ) ) = 5
125		4t2e8 9466	. . . . . . . 8 |- ( 4 x. 2 ) = 8
126	125	oveq1i 6095	. . . . . . 7 |- ( ( 4 x. 2 ) + 2 ) = ( 8 + 2 )
127		8p2e10 9865	. . . . . . 7 |- ( 8 + 2 ) = ; 1 0
128	126, 127	eqtri 2259	. . . . . 6 |- ( ( 4 x. 2 ) + 2 ) = ; 1 0
129	2, 3, 22, 3, 52, 121, 17, 22, 2, 124, 128	decma2c 9838	. . . . 5 |- ( ( 4 x. ; 1 2 ) + ( 0 + 2 ) ) = ; 5 0
130		5t4e20 9887	. . . . . . 7 |- ( 5 x. 4 ) = ; 2 0
131	56, 117, 130	mulcomli 8333	. . . . . 6 |- ( 4 x. 5 ) = ; 2 0
132	3, 22, 17, 131, 118	decaddi 9845	. . . . 5 |- ( ( 4 x. 5 ) + 4 ) = ; 2 4
133	4, 5, 22, 17, 116, 120, 17, 17, 3, 129, 132	decma2c 9838	. . . 4 |- ( ( 4 x. ;; 1 2 5 ) + ( 0 + 4 ) ) = ;; 5 0 4
134		9t4e36 9909	. . . . . 6 |- ( 9 x. 4 ) = ; 3 6
135	62, 117, 134	mulcomli 8333	. . . . 5 |- ( 4 x. 9 ) = ; 3 6
136		6p5e11 9858	. . . . 5 |- ( 6 + 5 ) = ; 1 1
137	11, 24, 5, 135, 48, 2, 136	decaddci 9846	. . . 4 |- ( ( 4 x. 9 ) + 5 ) = ; 4 1
138	6, 32, 22, 5, 1, 115, 17, 2, 17, 133, 137	decma2c 9838	. . 3 |- ( ( 4 x. N ) + 5 ) = ;;; 5 0 4 1
139		7t7e49 9899	. . . . . 6 |- ( 7 x. 7 ) = ; 4 9
140	17, 96, 139	decsucc 9826	. . . . 5 |- ( ( 7 x. 7 ) + 1 ) = ; 5 0
141	37	mullidi 8329	. . . . . . 7 |- ( 1 x. 7 ) = 7
142	141	oveq1i 6095	. . . . . 6 |- ( ( 1 x. 7 ) + 7 ) = ( 7 + 7 )
143		7p7e14 9864	. . . . . 6 |- ( 7 + 7 ) = ; 1 4
144	142, 143	eqtri 2259	. . . . 5 |- ( ( 1 x. 7 ) + 7 ) = ; 1 4
145	15, 2, 15, 33, 15, 17, 2, 140, 144	decrmac 9843	. . . 4 |- ( (; 7 1 x. 7 ) + 7 ) = ;; 5 0 4
146	16	nn0cni 9579	. . . . 5 |- ; 7 1 e. CC
147	146	mulridi 8328	. . . 4 |- (; 7 1 x. 1 ) = ; 7 1
148	16, 15, 2, 33, 2, 15, 145, 147	decmul2c 9851	. . 3 |- (; 7 1 x. ; 7 1 ) = ;;; 5 0 4 1
149	138, 148	eqtr4i 2262	. 2 |- ( ( 4 x. N ) + 5 ) = (; 7 1 x. ; 7 1 )
150	9, 10, 13, 14, 16, 5, 106, 114, 149	mod2xi 13216	1 |- ( ( 2 ^; 7 6 ) mod N ) = ( 5 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:  1259lem4  13265