Theorem 1259lem1 13262

Description: Lemma for 1259prm 13267. Calculate a power mod. In decimal, we calculate 2↑16 = 52𝑁 + 68≡68 and 2↑17≡68 · 2 = 136 in this lemma. (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 = ;;;1259

Assertion

Ref	Expression
1259lem1	⊢ ((2↑;17) mod 𝑁) = (;;136 mod 𝑁)

Proof of Theorem 1259lem1

Step	Hyp	Ref	Expression
1		1259prm.1	. . 3 ⊢ 𝑁 = ;;;1259
2		1nn0 9583	. . . . . 6 ⊢ 1 ∈ ℕ0
3		2nn0 9584	. . . . . 6 ⊢ 2 ∈ ℕ0
4	2, 3	deccl 9795	. . . . 5 ⊢ ;12 ∈ ℕ0
5		5nn0 9587	. . . . 5 ⊢ 5 ∈ ℕ0
6	4, 5	deccl 9795	. . . 4 ⊢ ;;125 ∈ ℕ0
7		9nn 9477	. . . 4 ⊢ 9 ∈ ℕ
8	6, 7	decnncl 9804	. . 3 ⊢ ;;;1259 ∈ ℕ
9	1, 8	eqeltri 2311	. 2 ⊢ 𝑁 ∈ ℕ
10		2nn 9470	. 2 ⊢ 2 ∈ ℕ
11		6nn0 9588	. . 3 ⊢ 6 ∈ ℕ0
12	2, 11	deccl 9795	. 2 ⊢ ;16 ∈ ℕ0
13		0z 9659	. 2 ⊢ 0 ∈ ℤ
14		8nn0 9590	. . 3 ⊢ 8 ∈ ℕ0
15	11, 14	deccl 9795	. 2 ⊢ ;68 ∈ ℕ0
16		3nn0 9585	. . . 4 ⊢ 3 ∈ ℕ0
17	2, 16	deccl 9795	. . 3 ⊢ ;13 ∈ ℕ0
18	17, 11	deccl 9795	. 2 ⊢ ;;136 ∈ ℕ0
19	5, 3	deccl 9795	. . . 4 ⊢ ;52 ∈ ℕ0
20	19	nn0zi 9670	. . 3 ⊢ ;52 ∈ ℤ
21	3, 14	nn0expcli 11015	. . 3 ⊢ (2↑8) ∈ ℕ0
22		eqid 2238	. . 3 ⊢ ((2↑8) mod 𝑁) = ((2↑8) mod 𝑁)
23	14	nn0cni 9579	. . . 4 ⊢ 8 ∈ ℂ
24		2cn 9377	. . . 4 ⊢ 2 ∈ ℂ
25		8t2e16 9900	. . . 4 ⊢ (8 · 2) = ;16
26	23, 24, 25	mulcomli 8333	. . 3 ⊢ (2 · 8) = ;16
27		9nn0 9591	. . . . 5 ⊢ 9 ∈ ℕ0
28		eqid 2238	. . . . 5 ⊢ ;68 = ;68
29		4nn0 9586	. . . . . 6 ⊢ 4 ∈ ℕ0
30		7nn0 9589	. . . . . 6 ⊢ 7 ∈ ℕ0
31	29, 30	deccl 9795	. . . . 5 ⊢ ;47 ∈ ℕ0
32		eqid 2238	. . . . . 6 ⊢ ;;125 = ;;125
33		0nn0 9582	. . . . . . 7 ⊢ 0 ∈ ℕ0
34	11	dec0h 9807	. . . . . . 7 ⊢ 6 = ;06
35		eqid 2238	. . . . . . 7 ⊢ ;47 = ;47
36		4cn 9384	. . . . . . . . . 10 ⊢ 4 ∈ ℂ
37	36	addlidi 8470	. . . . . . . . 9 ⊢ (0 + 4) = 4
38	37	oveq1i 6095	. . . . . . . 8 ⊢ ((0 + 4) + 1) = (4 + 1)
39		4p1e5 9443	. . . . . . . 8 ⊢ (4 + 1) = 5
40	38, 39	eqtri 2259	. . . . . . 7 ⊢ ((0 + 4) + 1) = 5
41		7cn 9390	. . . . . . . 8 ⊢ 7 ∈ ℂ
42		6cn 9388	. . . . . . . 8 ⊢ 6 ∈ ℂ
43		7p6e13 9863	. . . . . . . 8 ⊢ (7 + 6) = ;13
44	41, 42, 43	addcomli 8472	. . . . . . 7 ⊢ (6 + 7) = ;13
45	33, 11, 29, 30, 34, 35, 40, 16, 44	decaddc 9840	. . . . . 6 ⊢ (6 + ;47) = ;53
46	3, 11	deccl 9795	. . . . . 6 ⊢ ;26 ∈ ℕ0
47		eqid 2238	. . . . . . 7 ⊢ ;12 = ;12
48	5	dec0h 9807	. . . . . . . 8 ⊢ 5 = ;05
49		eqid 2238	. . . . . . . 8 ⊢ ;26 = ;26
50	24	addlidi 8470	. . . . . . . . . 10 ⊢ (0 + 2) = 2
51	50	oveq1i 6095	. . . . . . . . 9 ⊢ ((0 + 2) + 1) = (2 + 1)
52		2p1e3 9440	. . . . . . . . 9 ⊢ (2 + 1) = 3
53	51, 52	eqtri 2259	. . . . . . . 8 ⊢ ((0 + 2) + 1) = 3
54		5cn 9386	. . . . . . . . 9 ⊢ 5 ∈ ℂ
55		6p5e11 9858	. . . . . . . . 9 ⊢ (6 + 5) = ;11
56	42, 54, 55	addcomli 8472	. . . . . . . 8 ⊢ (5 + 6) = ;11
57	33, 5, 3, 11, 48, 49, 53, 2, 56	decaddc 9840	. . . . . . 7 ⊢ (5 + ;26) = ;31
58		10nn0 9802	. . . . . . 7 ⊢ ;10 ∈ ℕ0
59		eqid 2238	. . . . . . . 8 ⊢ ;52 = ;52
60	58	nn0cni 9579	. . . . . . . . 9 ⊢ ;10 ∈ ℂ
61		3cn 9381	. . . . . . . . 9 ⊢ 3 ∈ ℂ
62		dec10p 9828	. . . . . . . . 9 ⊢ (;10 + 3) = ;13
63	60, 61, 62	addcomli 8472	. . . . . . . 8 ⊢ (3 + ;10) = ;13
64	54	mulridi 8328	. . . . . . . . . 10 ⊢ (5 · 1) = 5
65		1p0e1 9422	. . . . . . . . . 10 ⊢ (1 + 0) = 1
66	64, 65	oveq12i 6097	. . . . . . . . 9 ⊢ ((5 · 1) + (1 + 0)) = (5 + 1)
67		5p1e6 9444	. . . . . . . . 9 ⊢ (5 + 1) = 6
68	66, 67	eqtri 2259	. . . . . . . 8 ⊢ ((5 · 1) + (1 + 0)) = 6
69	24	mulridi 8328	. . . . . . . . . 10 ⊢ (2 · 1) = 2
70	69	oveq1i 6095	. . . . . . . . 9 ⊢ ((2 · 1) + 3) = (2 + 3)
71		3p2e5 9448	. . . . . . . . . 10 ⊢ (3 + 2) = 5
72	61, 24, 71	addcomli 8472	. . . . . . . . 9 ⊢ (2 + 3) = 5
73	70, 72, 48	3eqtri 2263	. . . . . . . 8 ⊢ ((2 · 1) + 3) = ;05
74	5, 3, 2, 16, 59, 63, 2, 5, 33, 68, 73	decmac 9837	. . . . . . 7 ⊢ ((;52 · 1) + (3 + ;10)) = ;65
75	2	dec0h 9807	. . . . . . . 8 ⊢ 1 = ;01
76		5t2e10 9885	. . . . . . . . . 10 ⊢ (5 · 2) = ;10
77		00id 8468	. . . . . . . . . 10 ⊢ (0 + 0) = 0
78	76, 77	oveq12i 6097	. . . . . . . . 9 ⊢ ((5 · 2) + (0 + 0)) = (;10 + 0)
79		dec10p 9828	. . . . . . . . 9 ⊢ (;10 + 0) = ;10
80	78, 79	eqtri 2259	. . . . . . . 8 ⊢ ((5 · 2) + (0 + 0)) = ;10
81		2t2e4 9461	. . . . . . . . . 10 ⊢ (2 · 2) = 4
82	81	oveq1i 6095	. . . . . . . . 9 ⊢ ((2 · 2) + 1) = (4 + 1)
83	82, 39, 48	3eqtri 2263	. . . . . . . 8 ⊢ ((2 · 2) + 1) = ;05
84	5, 3, 33, 2, 59, 75, 3, 5, 33, 80, 83	decmac 9837	. . . . . . 7 ⊢ ((;52 · 2) + 1) = ;;105
85	2, 3, 16, 2, 47, 57, 19, 5, 58, 74, 84	decma2c 9838	. . . . . 6 ⊢ ((;52 · ;12) + (5 + ;26)) = ;;655
86		5t5e25 9888	. . . . . . . 8 ⊢ (5 · 5) = ;25
87	3, 5, 67, 86	decsuc 9816	. . . . . . 7 ⊢ ((5 · 5) + 1) = ;26
88	54, 24, 76	mulcomli 8333	. . . . . . . 8 ⊢ (2 · 5) = ;10
89	61	addlidi 8470	. . . . . . . 8 ⊢ (0 + 3) = 3
90	2, 33, 16, 88, 89	decaddi 9845	. . . . . . 7 ⊢ ((2 · 5) + 3) = ;13
91	5, 3, 16, 59, 5, 16, 2, 87, 90	decrmac 9843	. . . . . 6 ⊢ ((;52 · 5) + 3) = ;;263
92	4, 5, 5, 16, 32, 45, 19, 16, 46, 85, 91	decma2c 9838	. . . . 5 ⊢ ((;52 · ;;125) + (6 + ;47)) = ;;;6553
93		9cn 9394	. . . . . . . 8 ⊢ 9 ∈ ℂ
94		9t5e45 9910	. . . . . . . 8 ⊢ (9 · 5) = ;45
95	93, 54, 94	mulcomli 8333	. . . . . . 7 ⊢ (5 · 9) = ;45
96		5p2e7 9453	. . . . . . 7 ⊢ (5 + 2) = 7
97	29, 5, 3, 95, 96	decaddi 9845	. . . . . 6 ⊢ ((5 · 9) + 2) = ;47
98		9t2e18 9907	. . . . . . . 8 ⊢ (9 · 2) = ;18
99	93, 24, 98	mulcomli 8333	. . . . . . 7 ⊢ (2 · 9) = ;18
100		1p1e2 9423	. . . . . . 7 ⊢ (1 + 1) = 2
101		8p8e16 9871	. . . . . . 7 ⊢ (8 + 8) = ;16
102	2, 14, 14, 99, 100, 11, 101	decaddci 9846	. . . . . 6 ⊢ ((2 · 9) + 8) = ;26
103	5, 3, 14, 59, 27, 11, 3, 97, 102	decrmac 9843	. . . . 5 ⊢ ((;52 · 9) + 8) = ;;476
104	6, 27, 11, 14, 1, 28, 19, 11, 31, 92, 103	decma2c 9838	. . . 4 ⊢ ((;52 · 𝑁) + ;68) = ;;;;65536
105		2exp16 13237	. . . 4 ⊢ (2↑;16) = ;;;;65536
106		eqid 2238	. . . . 5 ⊢ (2↑8) = (2↑8)
107		eqid 2238	. . . . 5 ⊢ ((2↑8) · (2↑8)) = ((2↑8) · (2↑8))
108	3, 14, 26, 106, 107	numexp2x 13225	. . . 4 ⊢ (2↑;16) = ((2↑8) · (2↑8))
109	104, 105, 108	3eqtr2i 2265	. . 3 ⊢ ((;52 · 𝑁) + ;68) = ((2↑8) · (2↑8))
110	9, 10, 14, 20, 21, 15, 22, 26, 109	mod2xi 13216	. 2 ⊢ ((2↑;16) mod 𝑁) = (;68 mod 𝑁)
111		6p1e7 9445	. . 3 ⊢ (6 + 1) = 7
112		eqid 2238	. . 3 ⊢ ;16 = ;16
113	2, 11, 111, 112	decsuc 9816	. 2 ⊢ (;16 + 1) = ;17
114	18	nn0cni 9579	. . . 4 ⊢ ;;136 ∈ ℂ
115	114	addlidi 8470	. . 3 ⊢ (0 + ;;136) = ;;136
116	9	nncni 9316	. . . . 5 ⊢ 𝑁 ∈ ℂ
117	116	mul02i 8718	. . . 4 ⊢ (0 · 𝑁) = 0
118	117	oveq1i 6095	. . 3 ⊢ ((0 · 𝑁) + ;;136) = (0 + ;;136)
119		6t2e12 9889	. . . . 5 ⊢ (6 · 2) = ;12
120	2, 3, 52, 119	decsuc 9816	. . . 4 ⊢ ((6 · 2) + 1) = ;13
121	3, 11, 14, 28, 11, 2, 120, 25	decmul1c 9850	. . 3 ⊢ (;68 · 2) = ;;136
122	115, 118, 121	3eqtr4i 2269	. 2 ⊢ ((0 · 𝑁) + ;;136) = (;68 · 2)
123	9, 10, 12, 13, 15, 18, 110, 113, 122	modxp1i 13217	1 ⊢ ((2↑;17) mod 𝑁) = (;;136 mod 𝑁)

This proof depends on syntax axioms:   = wceq 1402  (class class class)co 6085  0cc0 8179  1c1 8180   + caddc 8182   · cmul 8184  ℕcn 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:  1259lem2  13263  1259lem4  13265