Theorem 2503lem1 15844

Description: Lemma for 2503prm 15847. Calculate a power mod. In decimal, we calculate 2 ^ 1 8 = 5 1 2 ^ 2 = 1 0 4 N + 1 8 3 2 == 1 8 3 2. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.)

Hypothesis

Ref	Expression
2503prm.1	|- N = ;;; 2 5 0 3

Assertion

Ref	Expression
2503lem1	|- ( ( 2 ^; 1 8 ) mod N ) = (;;; 1 8 3 2 mod N )

Proof of Theorem 2503lem1

Step	Hyp	Ref	Expression
1		2503prm.1	. . 3 |- N = ;;; 2 5 0 3
2		2nn0 11309	. . . . . 6 |- 2 e. NN0
3		5nn0 11312	. . . . . 6 |- 5 e. NN0
4	2, 3	deccl 11512	. . . . 5 |- ; 2 5 e. NN0
5		0nn0 11307	. . . . 5 |- 0 e. NN0
6	4, 5	deccl 11512	. . . 4 |- ;; 2 5 0 e. NN0
7		3nn 11186	. . . 4 |- 3 e. NN
8	6, 7	decnncl 11518	. . 3 |- ;;; 2 5 0 3 e. NN
9	1, 8	eqeltri 2697	. 2 |- N e. NN
10		2nn 11185	. 2 |- 2 e. NN
11		9nn0 11316	. 2 |- 9 e. NN0
12		10nn0 11516	. . . 4 |- ; 1 0 e. NN0
13		4nn0 11311	. . . 4 |- 4 e. NN0
14	12, 13	deccl 11512	. . 3 |- ;; 1 0 4 e. NN0
15	14	nn0zi 11402	. 2 |- ;; 1 0 4 e. ZZ
16		1nn0 11308	. . . 4 |- 1 e. NN0
17	3, 16	deccl 11512	. . 3 |- ; 5 1 e. NN0
18	17, 2	deccl 11512	. 2 |- ;; 5 1 2 e. NN0
19		8nn0 11315	. . . . 5 |- 8 e. NN0
20	16, 19	deccl 11512	. . . 4 |- ; 1 8 e. NN0
21		3nn0 11310	. . . 4 |- 3 e. NN0
22	20, 21	deccl 11512	. . 3 |- ;; 1 8 3 e. NN0
23	22, 2	deccl 11512	. 2 |- ;;; 1 8 3 2 e. NN0
24		8p1e9 11158	. . . 4 |- ( 8 + 1 ) = 9
25		6nn0 11313	. . . . 5 |- 6 e. NN0
26		2exp8 15796	. . . . 5 |- ( 2 ^ 8 ) = ;; 2 5 6
27		eqid 2622	. . . . . 6 |- ; 2 5 = ; 2 5
28	16	dec0h 11522	. . . . . 6 |- 1 = ; 0 1
29		2t2e4 11177	. . . . . . . 8 |- ( 2 x. 2 ) = 4
30		ax-1cn 9994	. . . . . . . . 9 |- 1 e. CC
31	30	addid2i 10224	. . . . . . . 8 |- ( 0 + 1 ) = 1
32	29, 31	oveq12i 6662	. . . . . . 7 |- ( ( 2 x. 2 ) + ( 0 + 1 ) ) = ( 4 + 1 )
33		4p1e5 11154	. . . . . . 7 |- ( 4 + 1 ) = 5
34	32, 33	eqtri 2644	. . . . . 6 |- ( ( 2 x. 2 ) + ( 0 + 1 ) ) = 5
35		5t2e10 11634	. . . . . . 7 |- ( 5 x. 2 ) = ; 1 0
36	16, 5, 31, 35	decsuc 11535	. . . . . 6 |- ( ( 5 x. 2 ) + 1 ) = ; 1 1
37	2, 3, 5, 16, 27, 28, 2, 16, 16, 34, 36	decmac 11566	. . . . 5 |- ( (; 2 5 x. 2 ) + 1 ) = ; 5 1
38		6t2e12 11641	. . . . 5 |- ( 6 x. 2 ) = ; 1 2
39	2, 4, 25, 26, 2, 16, 37, 38	decmul1c 11587	. . . 4 |- ( ( 2 ^ 8 ) x. 2 ) = ;; 5 1 2
40	2, 19, 24, 39	numexpp1 15782	. . 3 |- ( 2 ^ 9 ) = ;; 5 1 2
41	40	oveq1i 6660	. 2 |- ( ( 2 ^ 9 ) mod N ) = (;; 5 1 2 mod N )
42		9cn 11108	. . 3 |- 9 e. CC
43		2cn 11091	. . 3 |- 2 e. CC
44		9t2e18 11663	. . 3 |- ( 9 x. 2 ) = ; 1 8
45	42, 43, 44	mulcomli 10047	. 2 |- ( 2 x. 9 ) = ; 1 8
46		eqid 2622	. . . 4 |- ;;; 1 8 3 2 = ;;; 1 8 3 2
47	21, 16	deccl 11512	. . . 4 |- ; 3 1 e. NN0
48	2, 16	deccl 11512	. . . . 5 |- ; 2 1 e. NN0
49		eqid 2622	. . . . 5 |- ;; 2 5 0 = ;; 2 5 0
50		eqid 2622	. . . . . 6 |- ;; 1 8 3 = ;; 1 8 3
51		eqid 2622	. . . . . 6 |- ; 3 1 = ; 3 1
52		eqid 2622	. . . . . . 7 |- ; 1 8 = ; 1 8
53		1p1e2 11134	. . . . . . 7 |- ( 1 + 1 ) = 2
54		8p3e11 11612	. . . . . . 7 |- ( 8 + 3 ) = ; 1 1
55	16, 19, 21, 52, 53, 16, 54	decaddci 11580	. . . . . 6 |- (; 1 8 + 3 ) = ; 2 1
56		3p1e4 11153	. . . . . 6 |- ( 3 + 1 ) = 4
57	20, 21, 21, 16, 50, 51, 55, 56	decadd 11570	. . . . 5 |- (;; 1 8 3 + ; 3 1 ) = ;; 2 1 4
58	48	nn0cni 11304	. . . . . . 7 |- ; 2 1 e. CC
59	58	addid1i 10223	. . . . . 6 |- (; 2 1 + 0 ) = ; 2 1
60	3, 2	deccl 11512	. . . . . 6 |- ; 5 2 e. NN0
61		eqid 2622	. . . . . . 7 |- ;; 1 0 4 = ;; 1 0 4
62	60	nn0cni 11304	. . . . . . . 8 |- ; 5 2 e. CC
63		eqid 2622	. . . . . . . . 9 |- ; 5 2 = ; 5 2
64		2p2e4 11144	. . . . . . . . 9 |- ( 2 + 2 ) = 4
65	3, 2, 2, 63, 64	decaddi 11579	. . . . . . . 8 |- (; 5 2 + 2 ) = ; 5 4
66	62, 43, 65	addcomli 10228	. . . . . . 7 |- ( 2 + ; 5 2 ) = ; 5 4
67	2	dec0u 11520	. . . . . . . . 9 |- (; 1 0 x. 2 ) = ; 2 0
68		5p1e6 11155	. . . . . . . . 9 |- ( 5 + 1 ) = 6
69	67, 68	oveq12i 6662	. . . . . . . 8 |- ( (; 1 0 x. 2 ) + ( 5 + 1 ) ) = (; 2 0 + 6 )
70		eqid 2622	. . . . . . . . 9 |- ; 2 0 = ; 2 0
71		6cn 11102	. . . . . . . . . 10 |- 6 e. CC
72	71	addid2i 10224	. . . . . . . . 9 |- ( 0 + 6 ) = 6
73	2, 5, 25, 70, 72	decaddi 11579	. . . . . . . 8 |- (; 2 0 + 6 ) = ; 2 6
74	69, 73	eqtri 2644	. . . . . . 7 |- ( (; 1 0 x. 2 ) + ( 5 + 1 ) ) = ; 2 6
75		4t2e8 11181	. . . . . . . . 9 |- ( 4 x. 2 ) = 8
76	75	oveq1i 6660	. . . . . . . 8 |- ( ( 4 x. 2 ) + 4 ) = ( 8 + 4 )
77		8p4e12 11614	. . . . . . . 8 |- ( 8 + 4 ) = ; 1 2
78	76, 77	eqtri 2644	. . . . . . 7 |- ( ( 4 x. 2 ) + 4 ) = ; 1 2
79	12, 13, 3, 13, 61, 66, 2, 2, 16, 74, 78	decmac 11566	. . . . . 6 |- ( (;; 1 0 4 x. 2 ) + ( 2 + ; 5 2 ) ) = ;; 2 6 2
80	3	dec0u 11520	. . . . . . . . 9 |- (; 1 0 x. 5 ) = ; 5 0
81	43	addid2i 10224	. . . . . . . . 9 |- ( 0 + 2 ) = 2
82	80, 81	oveq12i 6662	. . . . . . . 8 |- ( (; 1 0 x. 5 ) + ( 0 + 2 ) ) = (; 5 0 + 2 )
83		eqid 2622	. . . . . . . . 9 |- ; 5 0 = ; 5 0
84	3, 5, 2, 83, 81	decaddi 11579	. . . . . . . 8 |- (; 5 0 + 2 ) = ; 5 2
85	82, 84	eqtri 2644	. . . . . . 7 |- ( (; 1 0 x. 5 ) + ( 0 + 2 ) ) = ; 5 2
86		5cn 11100	. . . . . . . . 9 |- 5 e. CC
87		4cn 11098	. . . . . . . . 9 |- 4 e. CC
88		5t4e20 11637	. . . . . . . . 9 |- ( 5 x. 4 ) = ; 2 0
89	86, 87, 88	mulcomli 10047	. . . . . . . 8 |- ( 4 x. 5 ) = ; 2 0
90	2, 5, 31, 89	decsuc 11535	. . . . . . 7 |- ( ( 4 x. 5 ) + 1 ) = ; 2 1
91	12, 13, 5, 16, 61, 28, 3, 16, 2, 85, 90	decmac 11566	. . . . . 6 |- ( (;; 1 0 4 x. 5 ) + 1 ) = ;; 5 2 1
92	2, 3, 2, 16, 27, 59, 14, 16, 60, 79, 91	decma2c 11568	. . . . 5 |- ( (;; 1 0 4 x. ; 2 5 ) + (; 2 1 + 0 ) ) = ;;; 2 6 2 1
93	14	nn0cni 11304	. . . . . . . 8 |- ;; 1 0 4 e. CC
94	93	mul01i 10226	. . . . . . 7 |- (;; 1 0 4 x. 0 ) = 0
95	94	oveq1i 6660	. . . . . 6 |- ( (;; 1 0 4 x. 0 ) + 4 ) = ( 0 + 4 )
96	87	addid2i 10224	. . . . . 6 |- ( 0 + 4 ) = 4
97	13	dec0h 11522	. . . . . 6 |- 4 = ; 0 4
98	95, 96, 97	3eqtri 2648	. . . . 5 |- ( (;; 1 0 4 x. 0 ) + 4 ) = ; 0 4
99	4, 5, 48, 13, 49, 57, 14, 13, 5, 92, 98	decma2c 11568	. . . 4 |- ( (;; 1 0 4 x. ;; 2 5 0 ) + (;; 1 8 3 + ; 3 1 ) ) = ;;;; 2 6 2 1 4
100		eqid 2622	. . . . . 6 |- ; 1 0 = ; 1 0
101		3cn 11095	. . . . . . . . 9 |- 3 e. CC
102	101	mulid2i 10043	. . . . . . . 8 |- ( 1 x. 3 ) = 3
103		00id 10211	. . . . . . . 8 |- ( 0 + 0 ) = 0
104	102, 103	oveq12i 6662	. . . . . . 7 |- ( ( 1 x. 3 ) + ( 0 + 0 ) ) = ( 3 + 0 )
105	101	addid1i 10223	. . . . . . 7 |- ( 3 + 0 ) = 3
106	104, 105	eqtri 2644	. . . . . 6 |- ( ( 1 x. 3 ) + ( 0 + 0 ) ) = 3
107	101	mul02i 10225	. . . . . . . 8 |- ( 0 x. 3 ) = 0
108	107	oveq1i 6660	. . . . . . 7 |- ( ( 0 x. 3 ) + 1 ) = ( 0 + 1 )
109	108, 31, 28	3eqtri 2648	. . . . . 6 |- ( ( 0 x. 3 ) + 1 ) = ; 0 1
110	16, 5, 5, 16, 100, 28, 21, 16, 5, 106, 109	decmac 11566	. . . . 5 |- ( (; 1 0 x. 3 ) + 1 ) = ; 3 1
111		4t3e12 11632	. . . . . 6 |- ( 4 x. 3 ) = ; 1 2
112	16, 2, 2, 111, 64	decaddi 11579	. . . . 5 |- ( ( 4 x. 3 ) + 2 ) = ; 1 4
113	12, 13, 2, 61, 21, 13, 16, 110, 112	decrmac 11577	. . . 4 |- ( (;; 1 0 4 x. 3 ) + 2 ) = ;; 3 1 4
114	6, 21, 22, 2, 1, 46, 14, 13, 47, 99, 113	decma2c 11568	. . 3 |- ( (;; 1 0 4 x. N ) + ;;; 1 8 3 2 ) = ;;;;; 2 6 2 1 4 4
115		eqid 2622	. . . 4 |- ;; 5 1 2 = ;; 5 1 2
116	12, 2	deccl 11512	. . . 4 |- ;; 1 0 2 e. NN0
117		eqid 2622	. . . . 5 |- ; 5 1 = ; 5 1
118		eqid 2622	. . . . 5 |- ;; 1 0 2 = ;; 1 0 2
119	86, 30, 68	addcomli 10228	. . . . . . 7 |- ( 1 + 5 ) = 6
120	16, 5, 3, 16, 100, 117, 119, 31	decadd 11570	. . . . . 6 |- (; 1 0 + ; 5 1 ) = ; 6 1
121		7nn0 11314	. . . . . . 7 |- 7 e. NN0
122		6p1e7 11156	. . . . . . . 8 |- ( 6 + 1 ) = 7
123	121	dec0h 11522	. . . . . . . 8 |- 7 = ; 0 7
124	122, 123	eqtri 2644	. . . . . . 7 |- ( 6 + 1 ) = ; 0 7
125	31	oveq2i 6661	. . . . . . . 8 |- ( ( 5 x. 5 ) + ( 0 + 1 ) ) = ( ( 5 x. 5 ) + 1 )
126		5t5e25 11639	. . . . . . . . 9 |- ( 5 x. 5 ) = ; 2 5
127	2, 3, 68, 126	decsuc 11535	. . . . . . . 8 |- ( ( 5 x. 5 ) + 1 ) = ; 2 6
128	125, 127	eqtri 2644	. . . . . . 7 |- ( ( 5 x. 5 ) + ( 0 + 1 ) ) = ; 2 6
129	86	mulid2i 10043	. . . . . . . . 9 |- ( 1 x. 5 ) = 5
130	129	oveq1i 6660	. . . . . . . 8 |- ( ( 1 x. 5 ) + 7 ) = ( 5 + 7 )
131		7cn 11104	. . . . . . . . 9 |- 7 e. CC
132		7p5e12 11607	. . . . . . . . 9 |- ( 7 + 5 ) = ; 1 2
133	131, 86, 132	addcomli 10228	. . . . . . . 8 |- ( 5 + 7 ) = ; 1 2
134	130, 133	eqtri 2644	. . . . . . 7 |- ( ( 1 x. 5 ) + 7 ) = ; 1 2
135	3, 16, 5, 121, 117, 124, 3, 2, 16, 128, 134	decmac 11566	. . . . . 6 |- ( (; 5 1 x. 5 ) + ( 6 + 1 ) ) = ;; 2 6 2
136	86, 43, 35	mulcomli 10047	. . . . . . 7 |- ( 2 x. 5 ) = ; 1 0
137	16, 5, 31, 136	decsuc 11535	. . . . . 6 |- ( ( 2 x. 5 ) + 1 ) = ; 1 1
138	17, 2, 25, 16, 115, 120, 3, 16, 16, 135, 137	decmac 11566	. . . . 5 |- ( (;; 5 1 2 x. 5 ) + (; 1 0 + ; 5 1 ) ) = ;;; 2 6 2 1
139	17	nn0cni 11304	. . . . . . 7 |- ; 5 1 e. CC
140	139	mulid1i 10042	. . . . . 6 |- (; 5 1 x. 1 ) = ; 5 1
141	43	mulid1i 10042	. . . . . . . 8 |- ( 2 x. 1 ) = 2
142	141	oveq1i 6660	. . . . . . 7 |- ( ( 2 x. 1 ) + 2 ) = ( 2 + 2 )
143	142, 64	eqtri 2644	. . . . . 6 |- ( ( 2 x. 1 ) + 2 ) = 4
144	17, 2, 2, 115, 16, 140, 143	decrmanc 11576	. . . . 5 |- ( (;; 5 1 2 x. 1 ) + 2 ) = ;; 5 1 4
145	3, 16, 12, 2, 117, 118, 18, 13, 17, 138, 144	decma2c 11568	. . . 4 |- ( (;; 5 1 2 x. ; 5 1 ) + ;; 1 0 2 ) = ;;;; 2 6 2 1 4
146	43	mulid2i 10043	. . . . . 6 |- ( 1 x. 2 ) = 2
147	2, 3, 16, 117, 2, 35, 146	decmul1 11585	. . . . 5 |- (; 5 1 x. 2 ) = ;; 1 0 2
148	2, 17, 2, 115, 13, 147, 29	decmul1 11585	. . . 4 |- (;; 5 1 2 x. 2 ) = ;;; 1 0 2 4
149	18, 17, 2, 115, 13, 116, 145, 148	decmul2c 11589	. . 3 |- (;; 5 1 2 x. ;; 5 1 2 ) = ;;;;; 2 6 2 1 4 4
150	114, 149	eqtr4i 2647	. 2 |- ( (;; 1 0 4 x. N ) + ;;; 1 8 3 2 ) = (;; 5 1 2 x. ;; 5 1 2 )
151	9, 10, 11, 15, 18, 23, 41, 45, 150	mod2xi 15773	1 |- ( ( 2 ^; 1 8 ) mod N ) = (;;; 1 8 3 2 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:  2503lem2  15845  2503lem3  15846