Theorem 2503lem1 17175

Description: Lemma for 2503prm 17178. Calculate a power mod. In decimal, we calculate 2↑18 = 512↑2 = 104𝑁 + 1832≡1832. (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 = ;;;2503

Assertion

Ref	Expression
2503lem1	⊢ ((2↑;18) mod 𝑁) = (;;;1832 mod 𝑁)

Proof of Theorem 2503lem1

Step	Hyp	Ref	Expression
1		2503prm.1	. . 3 ⊢ 𝑁 = ;;;2503
2		2nn0 12500	. . . . . 6 ⊢ 2 ∈ ℕ0
3		5nn0 12503	. . . . . 6 ⊢ 5 ∈ ℕ0
4	2, 3	deccl 12705	. . . . 5 ⊢ ;25 ∈ ℕ0
5		0nn0 12498	. . . . 5 ⊢ 0 ∈ ℕ0
6	4, 5	deccl 12705	. . . 4 ⊢ ;;250 ∈ ℕ0
7		3nn 12299	. . . 4 ⊢ 3 ∈ ℕ
8	6, 7	decnncl 12714	. . 3 ⊢ ;;;2503 ∈ ℕ
9	1, 8	eqeltri 2860	. 2 ⊢ 𝑁 ∈ ℕ
10		2nn 12293	. 2 ⊢ 2 ∈ ℕ
11		9nn0 12507	. 2 ⊢ 9 ∈ ℕ0
12		10nn0 12712	. . . 4 ⊢ ;10 ∈ ℕ0
13		4nn0 12502	. . . 4 ⊢ 4 ∈ ℕ0
14	12, 13	deccl 12705	. . 3 ⊢ ;;104 ∈ ℕ0
15	14	nn0zi 12598	. 2 ⊢ ;;104 ∈ ℤ
16		1nn0 12499	. . . 4 ⊢ 1 ∈ ℕ0
17	3, 16	deccl 12705	. . 3 ⊢ ;51 ∈ ℕ0
18	17, 2	deccl 12705	. 2 ⊢ ;;512 ∈ ℕ0
19		8nn0 12506	. . . . 5 ⊢ 8 ∈ ℕ0
20	16, 19	deccl 12705	. . . 4 ⊢ ;18 ∈ ℕ0
21		3nn0 12501	. . . 4 ⊢ 3 ∈ ℕ0
22	20, 21	deccl 12705	. . 3 ⊢ ;;183 ∈ ℕ0
23	22, 2	deccl 12705	. 2 ⊢ ;;;1832 ∈ ℕ0
24		8p1e9 12369	. . . 4 ⊢ (8 + 1) = 9
25		6nn0 12504	. . . . 5 ⊢ 6 ∈ ℕ0
26		2exp8 17126	. . . . 5 ⊢ (2↑8) = ;;256
27		eqid 2764	. . . . . 6 ⊢ ;25 = ;25
28	16	dec0h 12717	. . . . . 6 ⊢ 1 = ;01
29		2t2e4 12383	. . . . . . . 8 ⊢ (2 · 2) = 4
30		ax-1cn 11133	. . . . . . . . 9 ⊢ 1 ∈ ℂ
31	30	addlidi 11373	. . . . . . . 8 ⊢ (0 + 1) = 1
32	29, 31	oveq12i 7410	. . . . . . 7 ⊢ ((2 · 2) + (0 + 1)) = (4 + 1)
33		4p1e5 12365	. . . . . . 7 ⊢ (4 + 1) = 5
34	32, 33	eqtri 2787	. . . . . 6 ⊢ ((2 · 2) + (0 + 1)) = 5
35		5t2e10 12795	. . . . . . 7 ⊢ (5 · 2) = ;10
36	16, 5, 31, 35	decsuc 12726	. . . . . 6 ⊢ ((5 · 2) + 1) = ;11
37	2, 3, 5, 16, 27, 28, 2, 16, 16, 34, 36	decmac 12747	. . . . 5 ⊢ ((;25 · 2) + 1) = ;51
38		6t2e12 12799	. . . . 5 ⊢ (6 · 2) = ;12
39	2, 4, 25, 26, 2, 16, 37, 38	decmul1c 12760	. . . 4 ⊢ ((2↑8) · 2) = ;;512
40	2, 19, 24, 39	numexpp1 17115	. . 3 ⊢ (2↑9) = ;;512
41	40	oveq1i 7408	. 2 ⊢ ((2↑9) mod 𝑁) = (;;512 mod 𝑁)
42		9cn 12320	. . 3 ⊢ 9 ∈ ℂ
43		2cn 12295	. . 3 ⊢ 2 ∈ ℂ
44		9t2e18 12817	. . 3 ⊢ (9 · 2) = ;18
45	42, 43, 44	mulcomli 11193	. 2 ⊢ (2 · 9) = ;18
46		eqid 2764	. . . 4 ⊢ ;;;1832 = ;;;1832
47	21, 16	deccl 12705	. . . 4 ⊢ ;31 ∈ ℕ0
48	2, 16	deccl 12705	. . . . 5 ⊢ ;21 ∈ ℕ0
49		eqid 2764	. . . . 5 ⊢ ;;250 = ;;250
50		eqid 2764	. . . . . 6 ⊢ ;;183 = ;;183
51		eqid 2764	. . . . . 6 ⊢ ;31 = ;31
52		eqid 2764	. . . . . . 7 ⊢ ;18 = ;18
53		1p1e2 12343	. . . . . . 7 ⊢ (1 + 1) = 2
54		8p3e11 12776	. . . . . . 7 ⊢ (8 + 3) = ;11
55	16, 19, 21, 52, 53, 16, 54	decaddci 12756	. . . . . 6 ⊢ (;18 + 3) = ;21
56		3p1e4 12364	. . . . . 6 ⊢ (3 + 1) = 4
57	20, 21, 21, 16, 50, 51, 55, 56	decadd 12749	. . . . 5 ⊢ (;;183 + ;31) = ;;214
58	48	nn0cni 12495	. . . . . . 7 ⊢ ;21 ∈ ℂ
59	58	addridi 11372	. . . . . 6 ⊢ (;21 + 0) = ;21
60	3, 2	deccl 12705	. . . . . 6 ⊢ ;52 ∈ ℕ0
61		eqid 2764	. . . . . . 7 ⊢ ;;104 = ;;104
62	60	nn0cni 12495	. . . . . . . 8 ⊢ ;52 ∈ ℂ
63		eqid 2764	. . . . . . . . 9 ⊢ ;52 = ;52
64		2p2e4 12354	. . . . . . . . 9 ⊢ (2 + 2) = 4
65	3, 2, 2, 63, 64	decaddi 12755	. . . . . . . 8 ⊢ (;52 + 2) = ;54
66	62, 43, 65	addcomli 11377	. . . . . . 7 ⊢ (2 + ;52) = ;54
67	2	dec0u 12716	. . . . . . . . 9 ⊢ (;10 · 2) = ;20
68		5p1e6 12366	. . . . . . . . 9 ⊢ (5 + 1) = 6
69	67, 68	oveq12i 7410	. . . . . . . 8 ⊢ ((;10 · 2) + (5 + 1)) = (;20 + 6)
70		eqid 2764	. . . . . . . . 9 ⊢ ;20 = ;20
71		6cn 12311	. . . . . . . . . 10 ⊢ 6 ∈ ℂ
72	71	addlidi 11373	. . . . . . . . 9 ⊢ (0 + 6) = 6
73	2, 5, 25, 70, 72	decaddi 12755	. . . . . . . 8 ⊢ (;20 + 6) = ;26
74	69, 73	eqtri 2787	. . . . . . 7 ⊢ ((;10 · 2) + (5 + 1)) = ;26
75		4t2e8 12388	. . . . . . . . 9 ⊢ (4 · 2) = 8
76	75	oveq1i 7408	. . . . . . . 8 ⊢ ((4 · 2) + 4) = (8 + 4)
77		8p4e12 12777	. . . . . . . 8 ⊢ (8 + 4) = ;12
78	76, 77	eqtri 2787	. . . . . . 7 ⊢ ((4 · 2) + 4) = ;12
79	12, 13, 3, 13, 61, 66, 2, 2, 16, 74, 78	decmac 12747	. . . . . 6 ⊢ ((;;104 · 2) + (2 + ;52)) = ;;262
80	3	dec0u 12716	. . . . . . . . 9 ⊢ (;10 · 5) = ;50
81	43	addlidi 11373	. . . . . . . . 9 ⊢ (0 + 2) = 2
82	80, 81	oveq12i 7410	. . . . . . . 8 ⊢ ((;10 · 5) + (0 + 2)) = (;50 + 2)
83		eqid 2764	. . . . . . . . 9 ⊢ ;50 = ;50
84	3, 5, 2, 83, 81	decaddi 12755	. . . . . . . 8 ⊢ (;50 + 2) = ;52
85	82, 84	eqtri 2787	. . . . . . 7 ⊢ ((;10 · 5) + (0 + 2)) = ;52
86		5cn 12308	. . . . . . . . 9 ⊢ 5 ∈ ℂ
87		4cn 12305	. . . . . . . . 9 ⊢ 4 ∈ ℂ
88		5t4e20 12797	. . . . . . . . 9 ⊢ (5 · 4) = ;20
89	86, 87, 88	mulcomli 11193	. . . . . . . 8 ⊢ (4 · 5) = ;20
90	2, 5, 31, 89	decsuc 12726	. . . . . . 7 ⊢ ((4 · 5) + 1) = ;21
91	12, 13, 5, 16, 61, 28, 3, 16, 2, 85, 90	decmac 12747	. . . . . 6 ⊢ ((;;104 · 5) + 1) = ;;521
92	2, 3, 2, 16, 27, 59, 14, 16, 60, 79, 91	decma2c 12748	. . . . 5 ⊢ ((;;104 · ;25) + (;21 + 0)) = ;;;2621
93	14	nn0cni 12495	. . . . . . . 8 ⊢ ;;104 ∈ ℂ
94	93	mul01i 11375	. . . . . . 7 ⊢ (;;104 · 0) = 0
95	94	oveq1i 7408	. . . . . 6 ⊢ ((;;104 · 0) + 4) = (0 + 4)
96	87	addlidi 11373	. . . . . 6 ⊢ (0 + 4) = 4
97	13	dec0h 12717	. . . . . 6 ⊢ 4 = ;04
98	95, 96, 97	3eqtri 2791	. . . . 5 ⊢ ((;;104 · 0) + 4) = ;04
99	4, 5, 48, 13, 49, 57, 14, 13, 5, 92, 98	decma2c 12748	. . . 4 ⊢ ((;;104 · ;;250) + (;;183 + ;31)) = ;;;;26214
100		eqid 2764	. . . . . 6 ⊢ ;10 = ;10
101		3cn 12301	. . . . . . . . 9 ⊢ 3 ∈ ℂ
102	101	mullidi 11189	. . . . . . . 8 ⊢ (1 · 3) = 3
103		00id 11360	. . . . . . . 8 ⊢ (0 + 0) = 0
104	102, 103	oveq12i 7410	. . . . . . 7 ⊢ ((1 · 3) + (0 + 0)) = (3 + 0)
105	101	addridi 11372	. . . . . . 7 ⊢ (3 + 0) = 3
106	104, 105	eqtri 2787	. . . . . 6 ⊢ ((1 · 3) + (0 + 0)) = 3
107	101	mul02i 11374	. . . . . . . 8 ⊢ (0 · 3) = 0
108	107	oveq1i 7408	. . . . . . 7 ⊢ ((0 · 3) + 1) = (0 + 1)
109	108, 31, 28	3eqtri 2791	. . . . . 6 ⊢ ((0 · 3) + 1) = ;01
110	16, 5, 5, 16, 100, 28, 21, 16, 5, 106, 109	decmac 12747	. . . . 5 ⊢ ((;10 · 3) + 1) = ;31
111		4t3e12 12793	. . . . . 6 ⊢ (4 · 3) = ;12
112	16, 2, 2, 111, 64	decaddi 12755	. . . . 5 ⊢ ((4 · 3) + 2) = ;14
113	12, 13, 2, 61, 21, 13, 16, 110, 112	decrmac 12753	. . . 4 ⊢ ((;;104 · 3) + 2) = ;;314
114	6, 21, 22, 2, 1, 46, 14, 13, 47, 99, 113	decma2c 12748	. . 3 ⊢ ((;;104 · 𝑁) + ;;;1832) = ;;;;;262144
115		eqid 2764	. . . 4 ⊢ ;;512 = ;;512
116	12, 2	deccl 12705	. . . 4 ⊢ ;;102 ∈ ℕ0
117		eqid 2764	. . . . 5 ⊢ ;51 = ;51
118		eqid 2764	. . . . 5 ⊢ ;;102 = ;;102
119	86, 30, 68	addcomli 11377	. . . . . . 7 ⊢ (1 + 5) = 6
120	16, 5, 3, 16, 100, 117, 119, 31	decadd 12749	. . . . . 6 ⊢ (;10 + ;51) = ;61
121		7nn0 12505	. . . . . . 7 ⊢ 7 ∈ ℕ0
122		6p1e7 12367	. . . . . . . 8 ⊢ (6 + 1) = 7
123	121	dec0h 12717	. . . . . . . 8 ⊢ 7 = ;07
124	122, 123	eqtri 2787	. . . . . . 7 ⊢ (6 + 1) = ;07
125	31	oveq2i 7409	. . . . . . . 8 ⊢ ((5 · 5) + (0 + 1)) = ((5 · 5) + 1)
126		5t5e25 12798	. . . . . . . . 9 ⊢ (5 · 5) = ;25
127	2, 3, 68, 126	decsuc 12726	. . . . . . . 8 ⊢ ((5 · 5) + 1) = ;26
128	125, 127	eqtri 2787	. . . . . . 7 ⊢ ((5 · 5) + (0 + 1)) = ;26
129	86	mullidi 11189	. . . . . . . . 9 ⊢ (1 · 5) = 5
130	129	oveq1i 7408	. . . . . . . 8 ⊢ ((1 · 5) + 7) = (5 + 7)
131		7cn 12314	. . . . . . . . 9 ⊢ 7 ∈ ℂ
132		7p5e12 12772	. . . . . . . . 9 ⊢ (7 + 5) = ;12
133	131, 86, 132	addcomli 11377	. . . . . . . 8 ⊢ (5 + 7) = ;12
134	130, 133	eqtri 2787	. . . . . . 7 ⊢ ((1 · 5) + 7) = ;12
135	3, 16, 5, 121, 117, 124, 3, 2, 16, 128, 134	decmac 12747	. . . . . 6 ⊢ ((;51 · 5) + (6 + 1)) = ;;262
136	86, 43, 35	mulcomli 11193	. . . . . . 7 ⊢ (2 · 5) = ;10
137	16, 5, 31, 136	decsuc 12726	. . . . . 6 ⊢ ((2 · 5) + 1) = ;11
138	17, 2, 25, 16, 115, 120, 3, 16, 16, 135, 137	decmac 12747	. . . . 5 ⊢ ((;;512 · 5) + (;10 + ;51)) = ;;;2621
139	17	nn0cni 12495	. . . . . . 7 ⊢ ;51 ∈ ℂ
140	139	mulridi 11188	. . . . . 6 ⊢ (;51 · 1) = ;51
141	43	mulridi 11188	. . . . . . . 8 ⊢ (2 · 1) = 2
142	141	oveq1i 7408	. . . . . . 7 ⊢ ((2 · 1) + 2) = (2 + 2)
143	142, 64	eqtri 2787	. . . . . 6 ⊢ ((2 · 1) + 2) = 4
144	17, 2, 2, 115, 16, 140, 143	decrmanc 12752	. . . . 5 ⊢ ((;;512 · 1) + 2) = ;;514
145	3, 16, 12, 2, 117, 118, 18, 13, 17, 138, 144	decma2c 12748	. . . 4 ⊢ ((;;512 · ;51) + ;;102) = ;;;;26214
146	43	mullidi 11189	. . . . . 6 ⊢ (1 · 2) = 2
147	2, 3, 16, 117, 35, 146	decmul1 12759	. . . . 5 ⊢ (;51 · 2) = ;;102
148	2, 17, 2, 115, 147, 29	decmul1 12759	. . . 4 ⊢ (;;512 · 2) = ;;;1024
149	18, 17, 2, 115, 13, 116, 145, 148	decmul2c 12761	. . 3 ⊢ (;;512 · ;;512) = ;;;;;262144
150	114, 149	eqtr4i 2790	. 2 ⊢ ((;;104 · 𝑁) + ;;;1832) = (;;512 · ;;512)
151	9, 10, 11, 15, 18, 23, 41, 45, 150	mod2xi 17107	1 ⊢ ((2↑;18) mod 𝑁) = (;;;1832 mod 𝑁)

Syntax hints:   = wceq 1562  (class class class)co 7398  0cc0 11075  1c1 11076   + caddc 11078   · cmul 11080  ℕcn 12212  2c2 12274  3c3 12275  4c4 12276  5c5 12277  6c6 12278  7c7 12279  8c8 12280  9c9 12281  ;cdc 12690   mod cmo 13881  ↑cexp 14076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-pre-sup 11153

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-riota 7355  df-ov 7401  df-oprab 7402  df-mpo 7403  df-om 7849  df-2nd 7973  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-rdg 8383  df-er 8680  df-en 8930  df-dom 8931  df-sdom 8932  df-sup 9390  df-inf 9391  df-pnf 11220  df-mnf 11221  df-xr 11222  df-ltxr 11223  df-le 11224  df-sub 11418  df-neg 11419  df-div 11847  df-nn 12213  df-2 12282  df-3 12283  df-4 12284  df-5 12285  df-6 12286  df-7 12287  df-8 12288  df-9 12289  df-n0 12484  df-z 12571  df-dec 12691  df-uz 12842  df-rp 12996  df-fl 13804  df-mod 13882  df-seq 14017  df-exp 14077

This theorem is referenced by:  2503lem2  17176  2503lem3  17177