Theorem 1259lem2 16311

Description: Lemma for 1259prm 16315. Calculate a power mod. In decimal, we calculate 2↑34 = (2↑17)↑2≡136↑2≡14𝑁 + 870. (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 = ;;;1259

Assertion

Ref	Expression
1259lem2	⊢ ((2↑;34) mod 𝑁) = (;;870 mod 𝑁)

Proof of Theorem 1259lem2

Step	Hyp	Ref	Expression
1		1259prm.1	. . 3 ⊢ 𝑁 = ;;;1259
2		1nn0 11718	. . . . . 6 ⊢ 1 ∈ ℕ0
3		2nn0 11719	. . . . . 6 ⊢ 2 ∈ ℕ0
4	2, 3	deccl 11919	. . . . 5 ⊢ ;12 ∈ ℕ0
5		5nn0 11722	. . . . 5 ⊢ 5 ∈ ℕ0
6	4, 5	deccl 11919	. . . 4 ⊢ ;;125 ∈ ℕ0
7		9nn 11537	. . . 4 ⊢ 9 ∈ ℕ
8	6, 7	decnncl 11925	. . 3 ⊢ ;;;1259 ∈ ℕ
9	1, 8	eqeltri 2856	. 2 ⊢ 𝑁 ∈ ℕ
10		2nn 11506	. 2 ⊢ 2 ∈ ℕ
11		7nn0 11724	. . 3 ⊢ 7 ∈ ℕ0
12	2, 11	deccl 11919	. 2 ⊢ ;17 ∈ ℕ0
13		4nn0 11721	. . . 4 ⊢ 4 ∈ ℕ0
14	2, 13	deccl 11919	. . 3 ⊢ ;14 ∈ ℕ0
15	14	nn0zi 11813	. 2 ⊢ ;14 ∈ ℤ
16		3nn0 11720	. . . 4 ⊢ 3 ∈ ℕ0
17	2, 16	deccl 11919	. . 3 ⊢ ;13 ∈ ℕ0
18		6nn0 11723	. . 3 ⊢ 6 ∈ ℕ0
19	17, 18	deccl 11919	. 2 ⊢ ;;136 ∈ ℕ0
20		8nn0 11725	. . . 4 ⊢ 8 ∈ ℕ0
21	20, 11	deccl 11919	. . 3 ⊢ ;87 ∈ ℕ0
22		0nn0 11717	. . 3 ⊢ 0 ∈ ℕ0
23	21, 22	deccl 11919	. 2 ⊢ ;;870 ∈ ℕ0
24	1	1259lem1 16310	. 2 ⊢ ((2↑;17) mod 𝑁) = (;;136 mod 𝑁)
25		eqid 2772	. . 3 ⊢ ;17 = ;17
26		2cn 11508	. . . . . 6 ⊢ 2 ∈ ℂ
27	26	mulid1i 10436	. . . . 5 ⊢ (2 · 1) = 2
28	27	oveq1i 6980	. . . 4 ⊢ ((2 · 1) + 1) = (2 + 1)
29		2p1e3 11582	. . . 4 ⊢ (2 + 1) = 3
30	28, 29	eqtri 2796	. . 3 ⊢ ((2 · 1) + 1) = 3
31		7cn 11531	. . . 4 ⊢ 7 ∈ ℂ
32		7t2e14 12015	. . . 4 ⊢ (7 · 2) = ;14
33	31, 26, 32	mulcomli 10441	. . 3 ⊢ (2 · 7) = ;14
34	3, 2, 11, 25, 13, 2, 30, 33	decmul2c 11972	. 2 ⊢ (2 · ;17) = ;34
35		9nn0 11726	. . . 4 ⊢ 9 ∈ ℕ0
36		eqid 2772	. . . 4 ⊢ ;;870 = ;;870
37		eqid 2772	. . . . 5 ⊢ ;;125 = ;;125
38		eqid 2772	. . . . . 6 ⊢ ;87 = ;87
39		eqid 2772	. . . . . 6 ⊢ ;12 = ;12
40		8p1e9 11590	. . . . . 6 ⊢ (8 + 1) = 9
41		7p2e9 11601	. . . . . 6 ⊢ (7 + 2) = 9
42	20, 11, 2, 3, 38, 39, 40, 41	decadd 11959	. . . . 5 ⊢ (;87 + ;12) = ;99
43		9p7e16 11998	. . . . . 6 ⊢ (9 + 7) = ;16
44		eqid 2772	. . . . . . 7 ⊢ ;14 = ;14
45		3cn 11514	. . . . . . . . 9 ⊢ 3 ∈ ℂ
46		ax-1cn 10385	. . . . . . . . 9 ⊢ 1 ∈ ℂ
47		3p1e4 11585	. . . . . . . . 9 ⊢ (3 + 1) = 4
48	45, 46, 47	addcomli 10624	. . . . . . . 8 ⊢ (1 + 3) = 4
49	13	dec0h 11927	. . . . . . . 8 ⊢ 4 = ;04
50	48, 49	eqtri 2796	. . . . . . 7 ⊢ (1 + 3) = ;04
51	46	mulid1i 10436	. . . . . . . . 9 ⊢ (1 · 1) = 1
52		00id 10607	. . . . . . . . 9 ⊢ (0 + 0) = 0
53	51, 52	oveq12i 6982	. . . . . . . 8 ⊢ ((1 · 1) + (0 + 0)) = (1 + 0)
54	46	addid1i 10619	. . . . . . . 8 ⊢ (1 + 0) = 1
55	53, 54	eqtri 2796	. . . . . . 7 ⊢ ((1 · 1) + (0 + 0)) = 1
56		4cn 11519	. . . . . . . . . 10 ⊢ 4 ∈ ℂ
57	56	mulid1i 10436	. . . . . . . . 9 ⊢ (4 · 1) = 4
58	57	oveq1i 6980	. . . . . . . 8 ⊢ ((4 · 1) + 4) = (4 + 4)
59		4p4e8 11595	. . . . . . . 8 ⊢ (4 + 4) = 8
60	20	dec0h 11927	. . . . . . . 8 ⊢ 8 = ;08
61	58, 59, 60	3eqtri 2800	. . . . . . 7 ⊢ ((4 · 1) + 4) = ;08
62	2, 13, 22, 13, 44, 50, 2, 20, 22, 55, 61	decmac 11957	. . . . . 6 ⊢ ((;14 · 1) + (1 + 3)) = ;18
63	18	dec0h 11927	. . . . . . 7 ⊢ 6 = ;06
64	26	mulid2i 10437	. . . . . . . . 9 ⊢ (1 · 2) = 2
65	46	addid2i 10620	. . . . . . . . 9 ⊢ (0 + 1) = 1
66	64, 65	oveq12i 6982	. . . . . . . 8 ⊢ ((1 · 2) + (0 + 1)) = (2 + 1)
67	66, 29	eqtri 2796	. . . . . . 7 ⊢ ((1 · 2) + (0 + 1)) = 3
68		4t2e8 11608	. . . . . . . . 9 ⊢ (4 · 2) = 8
69	68	oveq1i 6980	. . . . . . . 8 ⊢ ((4 · 2) + 6) = (8 + 6)
70		8p6e14 11990	. . . . . . . 8 ⊢ (8 + 6) = ;14
71	69, 70	eqtri 2796	. . . . . . 7 ⊢ ((4 · 2) + 6) = ;14
72	2, 13, 22, 18, 44, 63, 3, 13, 2, 67, 71	decmac 11957	. . . . . 6 ⊢ ((;14 · 2) + 6) = ;34
73	2, 3, 2, 18, 39, 43, 14, 13, 16, 62, 72	decma2c 11958	. . . . 5 ⊢ ((;14 · ;12) + (9 + 7)) = ;;184
74	35	dec0h 11927	. . . . . 6 ⊢ 9 = ;09
75		5cn 11523	. . . . . . . . 9 ⊢ 5 ∈ ℂ
76	75	mulid2i 10437	. . . . . . . 8 ⊢ (1 · 5) = 5
77	26	addid2i 10620	. . . . . . . 8 ⊢ (0 + 2) = 2
78	76, 77	oveq12i 6982	. . . . . . 7 ⊢ ((1 · 5) + (0 + 2)) = (5 + 2)
79		5p2e7 11596	. . . . . . 7 ⊢ (5 + 2) = 7
80	78, 79	eqtri 2796	. . . . . 6 ⊢ ((1 · 5) + (0 + 2)) = 7
81		5t4e20 12008	. . . . . . . 8 ⊢ (5 · 4) = ;20
82	75, 56, 81	mulcomli 10441	. . . . . . 7 ⊢ (4 · 5) = ;20
83		9cn 11539	. . . . . . . 8 ⊢ 9 ∈ ℂ
84	83	addid2i 10620	. . . . . . 7 ⊢ (0 + 9) = 9
85	3, 22, 35, 82, 84	decaddi 11965	. . . . . 6 ⊢ ((4 · 5) + 9) = ;29
86	2, 13, 22, 35, 44, 74, 5, 35, 3, 80, 85	decmac 11957	. . . . 5 ⊢ ((;14 · 5) + 9) = ;79
87	4, 5, 35, 35, 37, 42, 14, 35, 11, 73, 86	decma2c 11958	. . . 4 ⊢ ((;14 · ;;125) + (;87 + ;12)) = ;;;1849
88	83	mulid2i 10437	. . . . . . . . 9 ⊢ (1 · 9) = 9
89	88	oveq1i 6980	. . . . . . . 8 ⊢ ((1 · 9) + 3) = (9 + 3)
90		9p3e12 11994	. . . . . . . 8 ⊢ (9 + 3) = ;12
91	89, 90	eqtri 2796	. . . . . . 7 ⊢ ((1 · 9) + 3) = ;12
92		9t4e36 12030	. . . . . . . 8 ⊢ (9 · 4) = ;36
93	83, 56, 92	mulcomli 10441	. . . . . . 7 ⊢ (4 · 9) = ;36
94	35, 2, 13, 44, 18, 16, 91, 93	decmul1c 11971	. . . . . 6 ⊢ (;14 · 9) = ;;126
95	94	oveq1i 6980	. . . . 5 ⊢ ((;14 · 9) + 0) = (;;126 + 0)
96	4, 18	deccl 11919	. . . . . . 7 ⊢ ;;126 ∈ ℕ0
97	96	nn0cni 11713	. . . . . 6 ⊢ ;;126 ∈ ℂ
98	97	addid1i 10619	. . . . 5 ⊢ (;;126 + 0) = ;;126
99	95, 98	eqtri 2796	. . . 4 ⊢ ((;14 · 9) + 0) = ;;126
100	6, 35, 21, 22, 1, 36, 14, 18, 4, 87, 99	decma2c 11958	. . 3 ⊢ ((;14 · 𝑁) + ;;870) = ;;;;18496
101		eqid 2772	. . . 4 ⊢ ;;136 = ;;136
102	20, 2	deccl 11919	. . . 4 ⊢ ;81 ∈ ℕ0
103		eqid 2772	. . . . 5 ⊢ ;13 = ;13
104		eqid 2772	. . . . 5 ⊢ ;81 = ;81
105	13, 22	deccl 11919	. . . . 5 ⊢ ;40 ∈ ℕ0
106		eqid 2772	. . . . . . 7 ⊢ ;40 = ;40
107	56	addid2i 10620	. . . . . . 7 ⊢ (0 + 4) = 4
108		8cn 11535	. . . . . . . 8 ⊢ 8 ∈ ℂ
109	108	addid1i 10619	. . . . . . 7 ⊢ (8 + 0) = 8
110	22, 20, 13, 22, 60, 106, 107, 109	decadd 11959	. . . . . 6 ⊢ (8 + ;40) = ;48
111		4p1e5 11586	. . . . . . . 8 ⊢ (4 + 1) = 5
112	5	dec0h 11927	. . . . . . . 8 ⊢ 5 = ;05
113	111, 112	eqtri 2796	. . . . . . 7 ⊢ (4 + 1) = ;05
114	45	mulid1i 10436	. . . . . . . . 9 ⊢ (3 · 1) = 3
115	114	oveq1i 6980	. . . . . . . 8 ⊢ ((3 · 1) + 5) = (3 + 5)
116		5p3e8 11597	. . . . . . . . 9 ⊢ (5 + 3) = 8
117	75, 45, 116	addcomli 10624	. . . . . . . 8 ⊢ (3 + 5) = 8
118	115, 117, 60	3eqtri 2800	. . . . . . 7 ⊢ ((3 · 1) + 5) = ;08
119	2, 16, 22, 5, 103, 113, 2, 20, 22, 55, 118	decmac 11957	. . . . . 6 ⊢ ((;13 · 1) + (4 + 1)) = ;18
120		6cn 11527	. . . . . . . . 9 ⊢ 6 ∈ ℂ
121	120	mulid1i 10436	. . . . . . . 8 ⊢ (6 · 1) = 6
122	121	oveq1i 6980	. . . . . . 7 ⊢ ((6 · 1) + 8) = (6 + 8)
123	108, 120, 70	addcomli 10624	. . . . . . 7 ⊢ (6 + 8) = ;14
124	122, 123	eqtri 2796	. . . . . 6 ⊢ ((6 · 1) + 8) = ;14
125	17, 18, 13, 20, 101, 110, 2, 13, 2, 119, 124	decmac 11957	. . . . 5 ⊢ ((;;136 · 1) + (8 + ;40)) = ;;184
126	2	dec0h 11927	. . . . . 6 ⊢ 1 = ;01
127	65, 126	eqtri 2796	. . . . . . 7 ⊢ (0 + 1) = ;01
128	45	mulid2i 10437	. . . . . . . . 9 ⊢ (1 · 3) = 3
129	128, 65	oveq12i 6982	. . . . . . . 8 ⊢ ((1 · 3) + (0 + 1)) = (3 + 1)
130	129, 47	eqtri 2796	. . . . . . 7 ⊢ ((1 · 3) + (0 + 1)) = 4
131		3t3e9 11607	. . . . . . . . 9 ⊢ (3 · 3) = 9
132	131	oveq1i 6980	. . . . . . . 8 ⊢ ((3 · 3) + 1) = (9 + 1)
133		9p1e10 11906	. . . . . . . 8 ⊢ (9 + 1) = ;10
134	132, 133	eqtri 2796	. . . . . . 7 ⊢ ((3 · 3) + 1) = ;10
135	2, 16, 22, 2, 103, 127, 16, 22, 2, 130, 134	decmac 11957	. . . . . 6 ⊢ ((;13 · 3) + (0 + 1)) = ;40
136		6t3e18 12011	. . . . . . 7 ⊢ (6 · 3) = ;18
137	2, 20, 2, 136, 40	decaddi 11965	. . . . . 6 ⊢ ((6 · 3) + 1) = ;19
138	17, 18, 22, 2, 101, 126, 16, 35, 2, 135, 137	decmac 11957	. . . . 5 ⊢ ((;;136 · 3) + 1) = ;;409
139	2, 16, 20, 2, 103, 104, 19, 35, 105, 125, 138	decma2c 11958	. . . 4 ⊢ ((;;136 · ;13) + ;81) = ;;;1849
140	16	dec0h 11927	. . . . . 6 ⊢ 3 = ;03
141	120	mulid2i 10437	. . . . . . . 8 ⊢ (1 · 6) = 6
142	141, 77	oveq12i 6982	. . . . . . 7 ⊢ ((1 · 6) + (0 + 2)) = (6 + 2)
143		6p2e8 11599	. . . . . . 7 ⊢ (6 + 2) = 8
144	142, 143	eqtri 2796	. . . . . 6 ⊢ ((1 · 6) + (0 + 2)) = 8
145	120, 45, 136	mulcomli 10441	. . . . . . 7 ⊢ (3 · 6) = ;18
146		1p1e2 11565	. . . . . . 7 ⊢ (1 + 1) = 2
147		8p3e11 11987	. . . . . . 7 ⊢ (8 + 3) = ;11
148	2, 20, 16, 145, 146, 2, 147	decaddci 11966	. . . . . 6 ⊢ ((3 · 6) + 3) = ;21
149	2, 16, 22, 16, 103, 140, 18, 2, 3, 144, 148	decmac 11957	. . . . 5 ⊢ ((;13 · 6) + 3) = ;81
150		6t6e36 12014	. . . . 5 ⊢ (6 · 6) = ;36
151	18, 17, 18, 101, 18, 16, 149, 150	decmul1c 11971	. . . 4 ⊢ (;;136 · 6) = ;;816
152	19, 17, 18, 101, 18, 102, 139, 151	decmul2c 11972	. . 3 ⊢ (;;136 · ;;136) = ;;;;18496
153	100, 152	eqtr4i 2799	. 2 ⊢ ((;14 · 𝑁) + ;;870) = (;;136 · ;;136)
154	9, 10, 12, 15, 19, 23, 24, 34, 153	mod2xi 16251	1 ⊢ ((2↑;34) mod 𝑁) = (;;870 mod 𝑁)

Syntax hints:   = wceq 1507  (class class class)co 6970  0cc0 10327  1c1 10328   + caddc 10330   · cmul 10332  ℕcn 11431  2c2 11488  3c3 11489  4c4 11490  5c5 11491  6c6 11492  7c7 11493  8c8 11494  9c9 11495  ;cdc 11904   mod cmo 13045  ↑cexp 13237

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273  ax-cnex 10383  ax-resscn 10384  ax-1cn 10385  ax-icn 10386  ax-addcl 10387  ax-addrcl 10388  ax-mulcl 10389  ax-mulrcl 10390  ax-mulcom 10391  ax-addass 10392  ax-mulass 10393  ax-distr 10394  ax-i2m1 10395  ax-1ne0 10396  ax-1rid 10397  ax-rnegex 10398  ax-rrecex 10399  ax-cnre 10400  ax-pre-lttri 10401  ax-pre-lttrn 10402  ax-pre-ltadd 10403  ax-pre-mulgt0 10404  ax-pre-sup 10405

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-nel 3068  df-ral 3087  df-rex 3088  df-reu 3089  df-rmo 3090  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-pss 3841  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4707  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-tr 5025  df-id 5305  df-eprel 5310  df-po 5319  df-so 5320  df-fr 5359  df-we 5361  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-pred 5980  df-ord 6026  df-on 6027  df-lim 6028  df-suc 6029  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-riota 6931  df-ov 6973  df-oprab 6974  df-mpo 6975  df-om 7391  df-2nd 7495  df-wrecs 7743  df-recs 7805  df-rdg 7843  df-er 8081  df-en 8299  df-dom 8300  df-sdom 8301  df-sup 8693  df-inf 8694  df-pnf 10468  df-mnf 10469  df-xr 10470  df-ltxr 10471  df-le 10472  df-sub 10664  df-neg 10665  df-div 11091  df-nn 11432  df-2 11496  df-3 11497  df-4 11498  df-5 11499  df-6 11500  df-7 11501  df-8 11502  df-9 11503  df-n0 11701  df-z 11787  df-dec 11905  df-uz 12052  df-rp 12198  df-fl 12970  df-mod 13046  df-seq 13178  df-exp 13238

This theorem is referenced by:  1259lem3  16312  1259lem5  16314