Theorem 2503lem1 15682

Description: Lemma for 2503prm 15685. 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 11186	. . . . . 6 ⊢ 2 ∈ ℕ0
3		5nn0 11189	. . . . . 6 ⊢ 5 ∈ ℕ0
4	2, 3	deccl 11388	. . . . 5 ⊢ ;25 ∈ ℕ0
5		0nn0 11184	. . . . 5 ⊢ 0 ∈ ℕ0
6	4, 5	deccl 11388	. . . 4 ⊢ ;;250 ∈ ℕ0
7		3nn 11063	. . . 4 ⊢ 3 ∈ ℕ
8	6, 7	decnncl 11394	. . 3 ⊢ ;;;2503 ∈ ℕ
9	1, 8	eqeltri 2684	. 2 ⊢ 𝑁 ∈ ℕ
10		2nn 11062	. 2 ⊢ 2 ∈ ℕ
11		9nn0 11193	. 2 ⊢ 9 ∈ ℕ0
12		10nn0 11392	. . . 4 ⊢ ;10 ∈ ℕ0
13		4nn0 11188	. . . 4 ⊢ 4 ∈ ℕ0
14	12, 13	deccl 11388	. . 3 ⊢ ;;104 ∈ ℕ0
15	14	nn0zi 11279	. 2 ⊢ ;;104 ∈ ℤ
16		1nn0 11185	. . . 4 ⊢ 1 ∈ ℕ0
17	3, 16	deccl 11388	. . 3 ⊢ ;51 ∈ ℕ0
18	17, 2	deccl 11388	. 2 ⊢ ;;512 ∈ ℕ0
19		8nn0 11192	. . . . 5 ⊢ 8 ∈ ℕ0
20	16, 19	deccl 11388	. . . 4 ⊢ ;18 ∈ ℕ0
21		3nn0 11187	. . . 4 ⊢ 3 ∈ ℕ0
22	20, 21	deccl 11388	. . 3 ⊢ ;;183 ∈ ℕ0
23	22, 2	deccl 11388	. 2 ⊢ ;;;1832 ∈ ℕ0
24		8p1e9 11035	. . . 4 ⊢ (8 + 1) = 9
25		6nn0 11190	. . . . 5 ⊢ 6 ∈ ℕ0
26		2exp8 15634	. . . . 5 ⊢ (2↑8) = ;;256
27		eqid 2610	. . . . . 6 ⊢ ;25 = ;25
28	16	dec0h 11398	. . . . . 6 ⊢ 1 = ;01
29		2t2e4 11054	. . . . . . . 8 ⊢ (2 · 2) = 4
30		ax-1cn 9873	. . . . . . . . 9 ⊢ 1 ∈ ℂ
31	30	addid2i 10103	. . . . . . . 8 ⊢ (0 + 1) = 1
32	29, 31	oveq12i 6561	. . . . . . 7 ⊢ ((2 · 2) + (0 + 1)) = (4 + 1)
33		4p1e5 11031	. . . . . . 7 ⊢ (4 + 1) = 5
34	32, 33	eqtri 2632	. . . . . 6 ⊢ ((2 · 2) + (0 + 1)) = 5
35		5t2e10 11510	. . . . . . 7 ⊢ (5 · 2) = ;10
36	16, 5, 31, 35	decsuc 11411	. . . . . 6 ⊢ ((5 · 2) + 1) = ;11
37	2, 3, 5, 16, 27, 28, 2, 16, 16, 34, 36	decmac 11442	. . . . 5 ⊢ ((;25 · 2) + 1) = ;51
38		6t2e12 11517	. . . . 5 ⊢ (6 · 2) = ;12
39	2, 4, 25, 26, 2, 16, 37, 38	decmul1c 11463	. . . 4 ⊢ ((2↑8) · 2) = ;;512
40	2, 19, 24, 39	numexpp1 15620	. . 3 ⊢ (2↑9) = ;;512
41	40	oveq1i 6559	. 2 ⊢ ((2↑9) mod 𝑁) = (;;512 mod 𝑁)
42		9cn 10985	. . 3 ⊢ 9 ∈ ℂ
43		2cn 10968	. . 3 ⊢ 2 ∈ ℂ
44		9t2e18 11539	. . 3 ⊢ (9 · 2) = ;18
45	42, 43, 44	mulcomli 9926	. 2 ⊢ (2 · 9) = ;18
46		eqid 2610	. . . 4 ⊢ ;;;1832 = ;;;1832
47	21, 16	deccl 11388	. . . 4 ⊢ ;31 ∈ ℕ0
48	2, 16	deccl 11388	. . . . 5 ⊢ ;21 ∈ ℕ0
49		eqid 2610	. . . . 5 ⊢ ;;250 = ;;250
50		eqid 2610	. . . . . 6 ⊢ ;;183 = ;;183
51		eqid 2610	. . . . . 6 ⊢ ;31 = ;31
52		eqid 2610	. . . . . . 7 ⊢ ;18 = ;18
53		1p1e2 11011	. . . . . . 7 ⊢ (1 + 1) = 2
54		8p3e11 11488	. . . . . . 7 ⊢ (8 + 3) = ;11
55	16, 19, 21, 52, 53, 16, 54	decaddci 11456	. . . . . 6 ⊢ (;18 + 3) = ;21
56		3p1e4 11030	. . . . . 6 ⊢ (3 + 1) = 4
57	20, 21, 21, 16, 50, 51, 55, 56	decadd 11446	. . . . 5 ⊢ (;;183 + ;31) = ;;214
58	48	nn0cni 11181	. . . . . . 7 ⊢ ;21 ∈ ℂ
59	58	addid1i 10102	. . . . . 6 ⊢ (;21 + 0) = ;21
60	3, 2	deccl 11388	. . . . . 6 ⊢ ;52 ∈ ℕ0
61		eqid 2610	. . . . . . 7 ⊢ ;;104 = ;;104
62	60	nn0cni 11181	. . . . . . . 8 ⊢ ;52 ∈ ℂ
63		eqid 2610	. . . . . . . . 9 ⊢ ;52 = ;52
64		2p2e4 11021	. . . . . . . . 9 ⊢ (2 + 2) = 4
65	3, 2, 2, 63, 64	decaddi 11455	. . . . . . . 8 ⊢ (;52 + 2) = ;54
66	62, 43, 65	addcomli 10107	. . . . . . 7 ⊢ (2 + ;52) = ;54
67	2	dec0u 11396	. . . . . . . . 9 ⊢ (;10 · 2) = ;20
68		5p1e6 11032	. . . . . . . . 9 ⊢ (5 + 1) = 6
69	67, 68	oveq12i 6561	. . . . . . . 8 ⊢ ((;10 · 2) + (5 + 1)) = (;20 + 6)
70		eqid 2610	. . . . . . . . 9 ⊢ ;20 = ;20
71		6cn 10979	. . . . . . . . . 10 ⊢ 6 ∈ ℂ
72	71	addid2i 10103	. . . . . . . . 9 ⊢ (0 + 6) = 6
73	2, 5, 25, 70, 72	decaddi 11455	. . . . . . . 8 ⊢ (;20 + 6) = ;26
74	69, 73	eqtri 2632	. . . . . . 7 ⊢ ((;10 · 2) + (5 + 1)) = ;26
75		4t2e8 11058	. . . . . . . . 9 ⊢ (4 · 2) = 8
76	75	oveq1i 6559	. . . . . . . 8 ⊢ ((4 · 2) + 4) = (8 + 4)
77		8p4e12 11490	. . . . . . . 8 ⊢ (8 + 4) = ;12
78	76, 77	eqtri 2632	. . . . . . 7 ⊢ ((4 · 2) + 4) = ;12
79	12, 13, 3, 13, 61, 66, 2, 2, 16, 74, 78	decmac 11442	. . . . . 6 ⊢ ((;;104 · 2) + (2 + ;52)) = ;;262
80	3	dec0u 11396	. . . . . . . . 9 ⊢ (;10 · 5) = ;50
81	43	addid2i 10103	. . . . . . . . 9 ⊢ (0 + 2) = 2
82	80, 81	oveq12i 6561	. . . . . . . 8 ⊢ ((;10 · 5) + (0 + 2)) = (;50 + 2)
83		eqid 2610	. . . . . . . . 9 ⊢ ;50 = ;50
84	3, 5, 2, 83, 81	decaddi 11455	. . . . . . . 8 ⊢ (;50 + 2) = ;52
85	82, 84	eqtri 2632	. . . . . . 7 ⊢ ((;10 · 5) + (0 + 2)) = ;52
86		5cn 10977	. . . . . . . . 9 ⊢ 5 ∈ ℂ
87		4cn 10975	. . . . . . . . 9 ⊢ 4 ∈ ℂ
88		5t4e20 11513	. . . . . . . . 9 ⊢ (5 · 4) = ;20
89	86, 87, 88	mulcomli 9926	. . . . . . . 8 ⊢ (4 · 5) = ;20
90	2, 5, 31, 89	decsuc 11411	. . . . . . 7 ⊢ ((4 · 5) + 1) = ;21
91	12, 13, 5, 16, 61, 28, 3, 16, 2, 85, 90	decmac 11442	. . . . . 6 ⊢ ((;;104 · 5) + 1) = ;;521
92	2, 3, 2, 16, 27, 59, 14, 16, 60, 79, 91	decma2c 11444	. . . . 5 ⊢ ((;;104 · ;25) + (;21 + 0)) = ;;;2621
93	14	nn0cni 11181	. . . . . . . 8 ⊢ ;;104 ∈ ℂ
94	93	mul01i 10105	. . . . . . 7 ⊢ (;;104 · 0) = 0
95	94	oveq1i 6559	. . . . . 6 ⊢ ((;;104 · 0) + 4) = (0 + 4)
96	87	addid2i 10103	. . . . . 6 ⊢ (0 + 4) = 4
97	13	dec0h 11398	. . . . . 6 ⊢ 4 = ;04
98	95, 96, 97	3eqtri 2636	. . . . 5 ⊢ ((;;104 · 0) + 4) = ;04
99	4, 5, 48, 13, 49, 57, 14, 13, 5, 92, 98	decma2c 11444	. . . 4 ⊢ ((;;104 · ;;250) + (;;183 + ;31)) = ;;;;26214
100		eqid 2610	. . . . . 6 ⊢ ;10 = ;10
101		3cn 10972	. . . . . . . . 9 ⊢ 3 ∈ ℂ
102	101	mulid2i 9922	. . . . . . . 8 ⊢ (1 · 3) = 3
103		00id 10090	. . . . . . . 8 ⊢ (0 + 0) = 0
104	102, 103	oveq12i 6561	. . . . . . 7 ⊢ ((1 · 3) + (0 + 0)) = (3 + 0)
105	101	addid1i 10102	. . . . . . 7 ⊢ (3 + 0) = 3
106	104, 105	eqtri 2632	. . . . . 6 ⊢ ((1 · 3) + (0 + 0)) = 3
107	101	mul02i 10104	. . . . . . . 8 ⊢ (0 · 3) = 0
108	107	oveq1i 6559	. . . . . . 7 ⊢ ((0 · 3) + 1) = (0 + 1)
109	108, 31, 28	3eqtri 2636	. . . . . 6 ⊢ ((0 · 3) + 1) = ;01
110	16, 5, 5, 16, 100, 28, 21, 16, 5, 106, 109	decmac 11442	. . . . 5 ⊢ ((;10 · 3) + 1) = ;31
111		4t3e12 11508	. . . . . 6 ⊢ (4 · 3) = ;12
112	16, 2, 2, 111, 64	decaddi 11455	. . . . 5 ⊢ ((4 · 3) + 2) = ;14
113	12, 13, 2, 61, 21, 13, 16, 110, 112	decrmac 11453	. . . 4 ⊢ ((;;104 · 3) + 2) = ;;314
114	6, 21, 22, 2, 1, 46, 14, 13, 47, 99, 113	decma2c 11444	. . 3 ⊢ ((;;104 · 𝑁) + ;;;1832) = ;;;;;262144
115		eqid 2610	. . . 4 ⊢ ;;512 = ;;512
116	12, 2	deccl 11388	. . . 4 ⊢ ;;102 ∈ ℕ0
117		eqid 2610	. . . . 5 ⊢ ;51 = ;51
118		eqid 2610	. . . . 5 ⊢ ;;102 = ;;102
119	86, 30, 68	addcomli 10107	. . . . . . 7 ⊢ (1 + 5) = 6
120	16, 5, 3, 16, 100, 117, 119, 31	decadd 11446	. . . . . 6 ⊢ (;10 + ;51) = ;61
121		7nn0 11191	. . . . . . 7 ⊢ 7 ∈ ℕ0
122		6p1e7 11033	. . . . . . . 8 ⊢ (6 + 1) = 7
123	121	dec0h 11398	. . . . . . . 8 ⊢ 7 = ;07
124	122, 123	eqtri 2632	. . . . . . 7 ⊢ (6 + 1) = ;07
125	31	oveq2i 6560	. . . . . . . 8 ⊢ ((5 · 5) + (0 + 1)) = ((5 · 5) + 1)
126		5t5e25 11515	. . . . . . . . 9 ⊢ (5 · 5) = ;25
127	2, 3, 68, 126	decsuc 11411	. . . . . . . 8 ⊢ ((5 · 5) + 1) = ;26
128	125, 127	eqtri 2632	. . . . . . 7 ⊢ ((5 · 5) + (0 + 1)) = ;26
129	86	mulid2i 9922	. . . . . . . . 9 ⊢ (1 · 5) = 5
130	129	oveq1i 6559	. . . . . . . 8 ⊢ ((1 · 5) + 7) = (5 + 7)
131		7cn 10981	. . . . . . . . 9 ⊢ 7 ∈ ℂ
132		7p5e12 11483	. . . . . . . . 9 ⊢ (7 + 5) = ;12
133	131, 86, 132	addcomli 10107	. . . . . . . 8 ⊢ (5 + 7) = ;12
134	130, 133	eqtri 2632	. . . . . . 7 ⊢ ((1 · 5) + 7) = ;12
135	3, 16, 5, 121, 117, 124, 3, 2, 16, 128, 134	decmac 11442	. . . . . 6 ⊢ ((;51 · 5) + (6 + 1)) = ;;262
136	86, 43, 35	mulcomli 9926	. . . . . . 7 ⊢ (2 · 5) = ;10
137	16, 5, 31, 136	decsuc 11411	. . . . . 6 ⊢ ((2 · 5) + 1) = ;11
138	17, 2, 25, 16, 115, 120, 3, 16, 16, 135, 137	decmac 11442	. . . . 5 ⊢ ((;;512 · 5) + (;10 + ;51)) = ;;;2621
139	17	nn0cni 11181	. . . . . . 7 ⊢ ;51 ∈ ℂ
140	139	mulid1i 9921	. . . . . 6 ⊢ (;51 · 1) = ;51
141	43	mulid1i 9921	. . . . . . . 8 ⊢ (2 · 1) = 2
142	141	oveq1i 6559	. . . . . . 7 ⊢ ((2 · 1) + 2) = (2 + 2)
143	142, 64	eqtri 2632	. . . . . 6 ⊢ ((2 · 1) + 2) = 4
144	17, 2, 2, 115, 16, 140, 143	decrmanc 11452	. . . . 5 ⊢ ((;;512 · 1) + 2) = ;;514
145	3, 16, 12, 2, 117, 118, 18, 13, 17, 138, 144	decma2c 11444	. . . 4 ⊢ ((;;512 · ;51) + ;;102) = ;;;;26214
146	43	mulid2i 9922	. . . . . 6 ⊢ (1 · 2) = 2
147	2, 3, 16, 117, 2, 35, 146	decmul1 11461	. . . . 5 ⊢ (;51 · 2) = ;;102
148	2, 17, 2, 115, 13, 147, 29	decmul1 11461	. . . 4 ⊢ (;;512 · 2) = ;;;1024
149	18, 17, 2, 115, 13, 116, 145, 148	decmul2c 11465	. . 3 ⊢ (;;512 · ;;512) = ;;;;;262144
150	114, 149	eqtr4i 2635	. 2 ⊢ ((;;104 · 𝑁) + ;;;1832) = (;;512 · ;;512)
151	9, 10, 11, 15, 18, 23, 41, 45, 150	mod2xi 15611	1 ⊢ ((2↑;18) mod 𝑁) = (;;;1832 mod 𝑁)

Syntax hints:   = wceq 1475  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818   · cmul 9820  ℕcn 10897  2c2 10947  3c3 10948  4c4 10949  5c5 10950  6c6 10951  7c7 10952  8c8 10953  9c9 10954  ;cdc 11369   mod cmo 12530  ↑cexp 12722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-sup 8231  df-inf 8232  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-rp 11709  df-fl 12455  df-mod 12531  df-seq 12664  df-exp 12723

This theorem is referenced by:  2503lem2  15683  2503lem3  15684