Theorem 1259lem2 13266

Description: Lemma for 1259prm 13270. 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		12nn0 9798	. . . . 5 |- ; 1 2 e. NN0
3		5nn0 9588	. . . . 5 |- 5 e. NN0
4	2, 3	deccl 9796	. . . 4 |- ;; 1 2 5 e. NN0
5		9nn 9478	. . . 4 |- 9 e. NN
6	4, 5	decnncl 9805	. . 3 |- ;;; 1 2 5 9 e. NN
7	1, 6	eqeltri 2311	. 2 |- N e. NN
8		2nn 9471	. 2 |- 2 e. NN
9		1nn0 9584	. . 3 |- 1 e. NN0
10		7nn0 9590	. . 3 |- 7 e. NN0
11	9, 10	deccl 9796	. 2 |- ; 1 7 e. NN0
12		4nn0 9587	. . . 4 |- 4 e. NN0
13	9, 12	deccl 9796	. . 3 |- ; 1 4 e. NN0
14	13	nn0zi 9671	. 2 |- ; 1 4 e. ZZ
15		3nn0 9586	. . . 4 |- 3 e. NN0
16	9, 15	deccl 9796	. . 3 |- ; 1 3 e. NN0
17		6nn0 9589	. . 3 |- 6 e. NN0
18	16, 17	deccl 9796	. 2 |- ;; 1 3 6 e. NN0
19		8nn0 9591	. . . 4 |- 8 e. NN0
20	19, 10	deccl 9796	. . 3 |- ; 8 7 e. NN0
21		0nn0 9583	. . 3 |- 0 e. NN0
22	20, 21	deccl 9796	. 2 |- ;; 8 7 0 e. NN0
23	1	1259lem1 13265	. 2 |- ( ( 2 ^; 1 7 ) mod N ) = (;; 1 3 6 mod N )
24		2nn0 9585	. . 3 |- 2 e. NN0
25		eqid 2238	. . 3 |- ; 1 7 = ; 1 7
26		2cn 9378	. . . . . 6 |- 2 e. CC
27	26	mulridi 8329	. . . . 5 |- ( 2 x. 1 ) = 2
28	27	oveq1i 6095	. . . 4 |- ( ( 2 x. 1 ) + 1 ) = ( 2 + 1 )
29		2p1e3 9441	. . . 4 |- ( 2 + 1 ) = 3
30	28, 29	eqtri 2259	. . 3 |- ( ( 2 x. 1 ) + 1 ) = 3
31		7cn 9391	. . . 4 |- 7 e. CC
32		7t2e14 9895	. . . 4 |- ( 7 x. 2 ) = ; 1 4
33	31, 26, 32	mulcomli 8334	. . 3 |- ( 2 x. 7 ) = ; 1 4
34	24, 9, 10, 25, 12, 9, 30, 33	decmul2c 9852	. 2 |- ( 2 x. ; 1 7 ) = ; 3 4
35		9nn0 9592	. . . 4 |- 9 e. NN0
36		eqid 2238	. . . 4 |- ;; 8 7 0 = ;; 8 7 0
37		eqid 2238	. . . . 5 |- ;; 1 2 5 = ;; 1 2 5
38		eqid 2238	. . . . . 6 |- ; 8 7 = ; 8 7
39		eqid 2238	. . . . . 6 |- ; 1 2 = ; 1 2
40		8p1e9 9448	. . . . . 6 |- ( 8 + 1 ) = 9
41		7p2e9 9459	. . . . . 6 |- ( 7 + 2 ) = 9
42	19, 10, 9, 24, 38, 39, 40, 41	decadd 9840	. . . . 5 |- (; 8 7 + ; 1 2 ) = ; 9 9
43		9p7e16 9878	. . . . . 6 |- ( 9 + 7 ) = ; 1 6
44		eqid 2238	. . . . . . 7 |- ; 1 4 = ; 1 4
45		3cn 9382	. . . . . . . . 9 |- 3 e. CC
46		ax-1cn 8273	. . . . . . . . 9 |- 1 e. CC
47		3p1e4 9443	. . . . . . . . 9 |- ( 3 + 1 ) = 4
48	45, 46, 47	addcomli 8473	. . . . . . . 8 |- ( 1 + 3 ) = 4
49	12	dec0h 9808	. . . . . . . 8 |- 4 = ; 0 4
50	48, 49	eqtri 2259	. . . . . . 7 |- ( 1 + 3 ) = ; 0 4
51	46	mulridi 8329	. . . . . . . . 9 |- ( 1 x. 1 ) = 1
52		00id 8469	. . . . . . . . 9 |- ( 0 + 0 ) = 0
53	51, 52	oveq12i 6097	. . . . . . . 8 |- ( ( 1 x. 1 ) + ( 0 + 0 ) ) = ( 1 + 0 )
54	46	addridi 8470	. . . . . . . 8 |- ( 1 + 0 ) = 1
55	53, 54	eqtri 2259	. . . . . . 7 |- ( ( 1 x. 1 ) + ( 0 + 0 ) ) = 1
56		4cn 9385	. . . . . . . . . 10 |- 4 e. CC
57	56	mulridi 8329	. . . . . . . . 9 |- ( 4 x. 1 ) = 4
58	57	oveq1i 6095	. . . . . . . 8 |- ( ( 4 x. 1 ) + 4 ) = ( 4 + 4 )
59		4p4e8 9453	. . . . . . . 8 |- ( 4 + 4 ) = 8
60	19	dec0h 9808	. . . . . . . 8 |- 8 = ; 0 8
61	58, 59, 60	3eqtri 2263	. . . . . . 7 |- ( ( 4 x. 1 ) + 4 ) = ; 0 8
62	9, 12, 21, 12, 44, 50, 9, 19, 21, 55, 61	decmac 9838	. . . . . 6 |- ( (; 1 4 x. 1 ) + ( 1 + 3 ) ) = ; 1 8
63	17	dec0h 9808	. . . . . . 7 |- 6 = ; 0 6
64	26	mullidi 8330	. . . . . . . . 9 |- ( 1 x. 2 ) = 2
65	46	addlidi 8471	. . . . . . . . 9 |- ( 0 + 1 ) = 1
66	64, 65	oveq12i 6097	. . . . . . . 8 |- ( ( 1 x. 2 ) + ( 0 + 1 ) ) = ( 2 + 1 )
67	66, 29	eqtri 2259	. . . . . . 7 |- ( ( 1 x. 2 ) + ( 0 + 1 ) ) = 3
68		4t2e8 9467	. . . . . . . . 9 |- ( 4 x. 2 ) = 8
69	68	oveq1i 6095	. . . . . . . 8 |- ( ( 4 x. 2 ) + 6 ) = ( 8 + 6 )
70		8p6e14 9870	. . . . . . . 8 |- ( 8 + 6 ) = ; 1 4
71	69, 70	eqtri 2259	. . . . . . 7 |- ( ( 4 x. 2 ) + 6 ) = ; 1 4
72	9, 12, 21, 17, 44, 63, 24, 12, 9, 67, 71	decmac 9838	. . . . . 6 |- ( (; 1 4 x. 2 ) + 6 ) = ; 3 4
73	9, 24, 9, 17, 39, 43, 13, 12, 15, 62, 72	decma2c 9839	. . . . 5 |- ( (; 1 4 x. ; 1 2 ) + ( 9 + 7 ) ) = ;; 1 8 4
74	35	dec0h 9808	. . . . . 6 |- 9 = ; 0 9
75		5cn 9387	. . . . . . . . 9 |- 5 e. CC
76	75	mullidi 8330	. . . . . . . 8 |- ( 1 x. 5 ) = 5
77	26	addlidi 8471	. . . . . . . 8 |- ( 0 + 2 ) = 2
78	76, 77	oveq12i 6097	. . . . . . 7 |- ( ( 1 x. 5 ) + ( 0 + 2 ) ) = ( 5 + 2 )
79		5p2e7 9454	. . . . . . 7 |- ( 5 + 2 ) = 7
80	78, 79	eqtri 2259	. . . . . 6 |- ( ( 1 x. 5 ) + ( 0 + 2 ) ) = 7
81		5t4e20 9888	. . . . . . . 8 |- ( 5 x. 4 ) = ; 2 0
82	75, 56, 81	mulcomli 8334	. . . . . . 7 |- ( 4 x. 5 ) = ; 2 0
83		9cn 9395	. . . . . . . 8 |- 9 e. CC
84	83	addlidi 8471	. . . . . . 7 |- ( 0 + 9 ) = 9
85	24, 21, 35, 82, 84	decaddi 9846	. . . . . 6 |- ( ( 4 x. 5 ) + 9 ) = ; 2 9
86	9, 12, 21, 35, 44, 74, 3, 35, 24, 80, 85	decmac 9838	. . . . 5 |- ( (; 1 4 x. 5 ) + 9 ) = ; 7 9
87	2, 3, 35, 35, 37, 42, 13, 35, 10, 73, 86	decma2c 9839	. . . 4 |- ( (; 1 4 x. ;; 1 2 5 ) + (; 8 7 + ; 1 2 ) ) = ;;; 1 8 4 9
88	83	mullidi 8330	. . . . . . . . 9 |- ( 1 x. 9 ) = 9
89	88	oveq1i 6095	. . . . . . . 8 |- ( ( 1 x. 9 ) + 3 ) = ( 9 + 3 )
90		9p3e12 9874	. . . . . . . 8 |- ( 9 + 3 ) = ; 1 2
91	89, 90	eqtri 2259	. . . . . . 7 |- ( ( 1 x. 9 ) + 3 ) = ; 1 2
92		9t4e36 9910	. . . . . . . 8 |- ( 9 x. 4 ) = ; 3 6
93	83, 56, 92	mulcomli 8334	. . . . . . 7 |- ( 4 x. 9 ) = ; 3 6
94	35, 9, 12, 44, 17, 15, 91, 93	decmul1c 9851	. . . . . 6 |- (; 1 4 x. 9 ) = ;; 1 2 6
95	94	oveq1i 6095	. . . . 5 |- ( (; 1 4 x. 9 ) + 0 ) = (;; 1 2 6 + 0 )
96	2, 17	deccl 9796	. . . . . . 7 |- ;; 1 2 6 e. NN0
97	96	nn0cni 9580	. . . . . 6 |- ;; 1 2 6 e. CC
98	97	addridi 8470	. . . . 5 |- (;; 1 2 6 + 0 ) = ;; 1 2 6
99	95, 98	eqtri 2259	. . . 4 |- ( (; 1 4 x. 9 ) + 0 ) = ;; 1 2 6
100	4, 35, 20, 21, 1, 36, 13, 17, 2, 87, 99	decma2c 9839	. . 3 |- ( (; 1 4 x. N ) + ;; 8 7 0 ) = ;;;; 1 8 4 9 6
101		eqid 2238	. . . 4 |- ;; 1 3 6 = ;; 1 3 6
102	19, 9	deccl 9796	. . . 4 |- ; 8 1 e. NN0
103		eqid 2238	. . . . 5 |- ; 1 3 = ; 1 3
104		eqid 2238	. . . . 5 |- ; 8 1 = ; 8 1
105	12, 21	deccl 9796	. . . . 5 |- ; 4 0 e. NN0
106		eqid 2238	. . . . . . 7 |- ; 4 0 = ; 4 0
107	56	addlidi 8471	. . . . . . 7 |- ( 0 + 4 ) = 4
108		8cn 9393	. . . . . . . 8 |- 8 e. CC
109	108	addridi 8470	. . . . . . 7 |- ( 8 + 0 ) = 8
110	21, 19, 12, 21, 60, 106, 107, 109	decadd 9840	. . . . . 6 |- ( 8 + ; 4 0 ) = ; 4 8
111		4p1e5 9444	. . . . . . . 8 |- ( 4 + 1 ) = 5
112	3	dec0h 9808	. . . . . . . 8 |- 5 = ; 0 5
113	111, 112	eqtri 2259	. . . . . . 7 |- ( 4 + 1 ) = ; 0 5
114	45	mulridi 8329	. . . . . . . . 9 |- ( 3 x. 1 ) = 3
115	114	oveq1i 6095	. . . . . . . 8 |- ( ( 3 x. 1 ) + 5 ) = ( 3 + 5 )
116		5p3e8 9455	. . . . . . . . 9 |- ( 5 + 3 ) = 8
117	75, 45, 116	addcomli 8473	. . . . . . . 8 |- ( 3 + 5 ) = 8
118	115, 117, 60	3eqtri 2263	. . . . . . 7 |- ( ( 3 x. 1 ) + 5 ) = ; 0 8
119	9, 15, 21, 3, 103, 113, 9, 19, 21, 55, 118	decmac 9838	. . . . . 6 |- ( (; 1 3 x. 1 ) + ( 4 + 1 ) ) = ; 1 8
120		6cn 9389	. . . . . . . . 9 |- 6 e. CC
121	120	mulridi 8329	. . . . . . . 8 |- ( 6 x. 1 ) = 6
122	121	oveq1i 6095	. . . . . . 7 |- ( ( 6 x. 1 ) + 8 ) = ( 6 + 8 )
123	108, 120, 70	addcomli 8473	. . . . . . 7 |- ( 6 + 8 ) = ; 1 4
124	122, 123	eqtri 2259	. . . . . 6 |- ( ( 6 x. 1 ) + 8 ) = ; 1 4
125	16, 17, 12, 19, 101, 110, 9, 12, 9, 119, 124	decmac 9838	. . . . 5 |- ( (;; 1 3 6 x. 1 ) + ( 8 + ; 4 0 ) ) = ;; 1 8 4
126	9	dec0h 9808	. . . . . 6 |- 1 = ; 0 1
127	65, 126	eqtri 2259	. . . . . . 7 |- ( 0 + 1 ) = ; 0 1
128	45	mullidi 8330	. . . . . . . . 9 |- ( 1 x. 3 ) = 3
129	128, 65	oveq12i 6097	. . . . . . . 8 |- ( ( 1 x. 3 ) + ( 0 + 1 ) ) = ( 3 + 1 )
130	129, 47	eqtri 2259	. . . . . . 7 |- ( ( 1 x. 3 ) + ( 0 + 1 ) ) = 4
131		3t3e9 9466	. . . . . . . . 9 |- ( 3 x. 3 ) = 9
132	131	oveq1i 6095	. . . . . . . 8 |- ( ( 3 x. 3 ) + 1 ) = ( 9 + 1 )
133		9p1e10 9784	. . . . . . . 8 |- ( 9 + 1 ) = ; 1 0
134	132, 133	eqtri 2259	. . . . . . 7 |- ( ( 3 x. 3 ) + 1 ) = ; 1 0
135	9, 15, 21, 9, 103, 127, 15, 21, 9, 130, 134	decmac 9838	. . . . . 6 |- ( (; 1 3 x. 3 ) + ( 0 + 1 ) ) = ; 4 0
136		6t3e18 9891	. . . . . . 7 |- ( 6 x. 3 ) = ; 1 8
137	9, 19, 9, 136, 40	decaddi 9846	. . . . . 6 |- ( ( 6 x. 3 ) + 1 ) = ; 1 9
138	16, 17, 21, 9, 101, 126, 15, 35, 9, 135, 137	decmac 9838	. . . . 5 |- ( (;; 1 3 6 x. 3 ) + 1 ) = ;; 4 0 9
139	9, 15, 19, 9, 103, 104, 18, 35, 105, 125, 138	decma2c 9839	. . . 4 |- ( (;; 1 3 6 x. ; 1 3 ) + ; 8 1 ) = ;;; 1 8 4 9
140	15	dec0h 9808	. . . . . 6 |- 3 = ; 0 3
141	120	mullidi 8330	. . . . . . . 8 |- ( 1 x. 6 ) = 6
142	141, 77	oveq12i 6097	. . . . . . 7 |- ( ( 1 x. 6 ) + ( 0 + 2 ) ) = ( 6 + 2 )
143		6p2e8 9457	. . . . . . 7 |- ( 6 + 2 ) = 8
144	142, 143	eqtri 2259	. . . . . 6 |- ( ( 1 x. 6 ) + ( 0 + 2 ) ) = 8
145	120, 45, 136	mulcomli 8334	. . . . . . 7 |- ( 3 x. 6 ) = ; 1 8
146		1p1e2 9424	. . . . . . 7 |- ( 1 + 1 ) = 2
147		8p3e11 9867	. . . . . . 7 |- ( 8 + 3 ) = ; 1 1
148	9, 19, 15, 145, 146, 9, 147	decaddci 9847	. . . . . 6 |- ( ( 3 x. 6 ) + 3 ) = ; 2 1
149	9, 15, 21, 15, 103, 140, 17, 9, 24, 144, 148	decmac 9838	. . . . 5 |- ( (; 1 3 x. 6 ) + 3 ) = ; 8 1
150		6t6e36 9894	. . . . 5 |- ( 6 x. 6 ) = ; 3 6
151	17, 16, 17, 101, 17, 15, 149, 150	decmul1c 9851	. . . 4 |- (;; 1 3 6 x. 6 ) = ;; 8 1 6
152	18, 16, 17, 101, 17, 102, 139, 151	decmul2c 9852	. . 3 |- (;; 1 3 6 x. ;; 1 3 6 ) = ;;;; 1 8 4 9 6
153	100, 152	eqtr4i 2262	. 2 |- ( (; 1 4 x. N ) + ;; 8 7 0 ) = (;; 1 3 6 x. ;; 1 3 6 )
154	7, 8, 11, 14, 18, 22, 23, 34, 153	mod2xi 13219	1 |- ( ( 2 ^; 3 4 ) mod N ) = (;; 8 7 0 mod N )

This proof depends on syntax axioms:   = wceq 1402  (class class class)co 6085   0cc0 8180   1c1 8181   + caddc 8183   x. cmul 8185   NNcn 9307   2c2 9358   3c3 9359   4c4 9360   5c5 9361   6c6 9362   7c7 9363   8c8 9364   9c9 9365  ;cdc 9782   mod cmo 10774   ^cexp 10990

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 8271  ax-resscn 8272  ax-1cn 8273  ax-1re 8274  ax-icn 8275  ax-addcl 8276  ax-addrcl 8277  ax-mulcl 8278  ax-mulrcl 8279  ax-addcom 8280  ax-mulcom 8281  ax-addass 8282  ax-mulass 8283  ax-distr 8284  ax-i2m1 8285  ax-0lt1 8286  ax-1rid 8287  ax-0id 8288  ax-rnegex 8289  ax-precex 8290  ax-cnre 8291  ax-pre-ltirr 8292  ax-pre-ltwlin 8293  ax-pre-lttrn 8294  ax-pre-apti 8295  ax-pre-ltadd 8296  ax-pre-mulgt0 8297  ax-pre-mulext 8298  ax-arch 8299

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 8363  df-mnf 8364  df-xr 8365  df-ltxr 8366  df-le 8367  df-sub 8501  df-neg 8502  df-reap 8906  df-ap 8913  df-div 9006  df-inn 9308  df-2 9366  df-3 9367  df-4 9368  df-5 9369  df-6 9370  df-7 9371  df-8 9372  df-9 9373  df-n0 9569  df-z 9650  df-dec 9783  df-uz 9932  df-q 10030  df-rp 10066  df-fl 10716  df-mod 10775  df-seqfrec 10900  df-exp 10991

This theorem is used by:  1259lem3  13267  1259lem5  13269