Theorem 1259lem2 16831

Description: Lemma for 1259prm 16835. 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 12249	. . . . . 6 ⊢ 1 ∈ ℕ0
3		2nn0 12250	. . . . . 6 ⊢ 2 ∈ ℕ0
4	2, 3	deccl 12451	. . . . 5 ⊢ ;12 ∈ ℕ0
5		5nn0 12253	. . . . 5 ⊢ 5 ∈ ℕ0
6	4, 5	deccl 12451	. . . 4 ⊢ ;;125 ∈ ℕ0
7		9nn 12071	. . . 4 ⊢ 9 ∈ ℕ
8	6, 7	decnncl 12456	. . 3 ⊢ ;;;1259 ∈ ℕ
9	1, 8	eqeltri 2837	. 2 ⊢ 𝑁 ∈ ℕ
10		2nn 12046	. 2 ⊢ 2 ∈ ℕ
11		7nn0 12255	. . 3 ⊢ 7 ∈ ℕ0
12	2, 11	deccl 12451	. 2 ⊢ ;17 ∈ ℕ0
13		4nn0 12252	. . . 4 ⊢ 4 ∈ ℕ0
14	2, 13	deccl 12451	. . 3 ⊢ ;14 ∈ ℕ0
15	14	nn0zi 12345	. 2 ⊢ ;14 ∈ ℤ
16		3nn0 12251	. . . 4 ⊢ 3 ∈ ℕ0
17	2, 16	deccl 12451	. . 3 ⊢ ;13 ∈ ℕ0
18		6nn0 12254	. . 3 ⊢ 6 ∈ ℕ0
19	17, 18	deccl 12451	. 2 ⊢ ;;136 ∈ ℕ0
20		8nn0 12256	. . . 4 ⊢ 8 ∈ ℕ0
21	20, 11	deccl 12451	. . 3 ⊢ ;87 ∈ ℕ0
22		0nn0 12248	. . 3 ⊢ 0 ∈ ℕ0
23	21, 22	deccl 12451	. 2 ⊢ ;;870 ∈ ℕ0
24	1	1259lem1 16830	. 2 ⊢ ((2↑;17) mod 𝑁) = (;;136 mod 𝑁)
25		eqid 2740	. . 3 ⊢ ;17 = ;17
26		2cn 12048	. . . . . 6 ⊢ 2 ∈ ℂ
27	26	mulid1i 10980	. . . . 5 ⊢ (2 · 1) = 2
28	27	oveq1i 7281	. . . 4 ⊢ ((2 · 1) + 1) = (2 + 1)
29		2p1e3 12115	. . . 4 ⊢ (2 + 1) = 3
30	28, 29	eqtri 2768	. . 3 ⊢ ((2 · 1) + 1) = 3
31		7cn 12067	. . . 4 ⊢ 7 ∈ ℂ
32		7t2e14 12545	. . . 4 ⊢ (7 · 2) = ;14
33	31, 26, 32	mulcomli 10985	. . 3 ⊢ (2 · 7) = ;14
34	3, 2, 11, 25, 13, 2, 30, 33	decmul2c 12502	. 2 ⊢ (2 · ;17) = ;34
35		9nn0 12257	. . . 4 ⊢ 9 ∈ ℕ0
36		eqid 2740	. . . 4 ⊢ ;;870 = ;;870
37		eqid 2740	. . . . 5 ⊢ ;;125 = ;;125
38		eqid 2740	. . . . . 6 ⊢ ;87 = ;87
39		eqid 2740	. . . . . 6 ⊢ ;12 = ;12
40		8p1e9 12123	. . . . . 6 ⊢ (8 + 1) = 9
41		7p2e9 12134	. . . . . 6 ⊢ (7 + 2) = 9
42	20, 11, 2, 3, 38, 39, 40, 41	decadd 12490	. . . . 5 ⊢ (;87 + ;12) = ;99
43		9p7e16 12528	. . . . . 6 ⊢ (9 + 7) = ;16
44		eqid 2740	. . . . . . 7 ⊢ ;14 = ;14
45		3cn 12054	. . . . . . . . 9 ⊢ 3 ∈ ℂ
46		ax-1cn 10930	. . . . . . . . 9 ⊢ 1 ∈ ℂ
47		3p1e4 12118	. . . . . . . . 9 ⊢ (3 + 1) = 4
48	45, 46, 47	addcomli 11167	. . . . . . . 8 ⊢ (1 + 3) = 4
49	13	dec0h 12458	. . . . . . . 8 ⊢ 4 = ;04
50	48, 49	eqtri 2768	. . . . . . 7 ⊢ (1 + 3) = ;04
51	46	mulid1i 10980	. . . . . . . . 9 ⊢ (1 · 1) = 1
52		00id 11150	. . . . . . . . 9 ⊢ (0 + 0) = 0
53	51, 52	oveq12i 7283	. . . . . . . 8 ⊢ ((1 · 1) + (0 + 0)) = (1 + 0)
54	46	addid1i 11162	. . . . . . . 8 ⊢ (1 + 0) = 1
55	53, 54	eqtri 2768	. . . . . . 7 ⊢ ((1 · 1) + (0 + 0)) = 1
56		4cn 12058	. . . . . . . . . 10 ⊢ 4 ∈ ℂ
57	56	mulid1i 10980	. . . . . . . . 9 ⊢ (4 · 1) = 4
58	57	oveq1i 7281	. . . . . . . 8 ⊢ ((4 · 1) + 4) = (4 + 4)
59		4p4e8 12128	. . . . . . . 8 ⊢ (4 + 4) = 8
60	20	dec0h 12458	. . . . . . . 8 ⊢ 8 = ;08
61	58, 59, 60	3eqtri 2772	. . . . . . 7 ⊢ ((4 · 1) + 4) = ;08
62	2, 13, 22, 13, 44, 50, 2, 20, 22, 55, 61	decmac 12488	. . . . . 6 ⊢ ((;14 · 1) + (1 + 3)) = ;18
63	18	dec0h 12458	. . . . . . 7 ⊢ 6 = ;06
64	26	mulid2i 10981	. . . . . . . . 9 ⊢ (1 · 2) = 2
65	46	addid2i 11163	. . . . . . . . 9 ⊢ (0 + 1) = 1
66	64, 65	oveq12i 7283	. . . . . . . 8 ⊢ ((1 · 2) + (0 + 1)) = (2 + 1)
67	66, 29	eqtri 2768	. . . . . . 7 ⊢ ((1 · 2) + (0 + 1)) = 3
68		4t2e8 12141	. . . . . . . . 9 ⊢ (4 · 2) = 8
69	68	oveq1i 7281	. . . . . . . 8 ⊢ ((4 · 2) + 6) = (8 + 6)
70		8p6e14 12520	. . . . . . . 8 ⊢ (8 + 6) = ;14
71	69, 70	eqtri 2768	. . . . . . 7 ⊢ ((4 · 2) + 6) = ;14
72	2, 13, 22, 18, 44, 63, 3, 13, 2, 67, 71	decmac 12488	. . . . . 6 ⊢ ((;14 · 2) + 6) = ;34
73	2, 3, 2, 18, 39, 43, 14, 13, 16, 62, 72	decma2c 12489	. . . . 5 ⊢ ((;14 · ;12) + (9 + 7)) = ;;184
74	35	dec0h 12458	. . . . . 6 ⊢ 9 = ;09
75		5cn 12061	. . . . . . . . 9 ⊢ 5 ∈ ℂ
76	75	mulid2i 10981	. . . . . . . 8 ⊢ (1 · 5) = 5
77	26	addid2i 11163	. . . . . . . 8 ⊢ (0 + 2) = 2
78	76, 77	oveq12i 7283	. . . . . . 7 ⊢ ((1 · 5) + (0 + 2)) = (5 + 2)
79		5p2e7 12129	. . . . . . 7 ⊢ (5 + 2) = 7
80	78, 79	eqtri 2768	. . . . . 6 ⊢ ((1 · 5) + (0 + 2)) = 7
81		5t4e20 12538	. . . . . . . 8 ⊢ (5 · 4) = ;20
82	75, 56, 81	mulcomli 10985	. . . . . . 7 ⊢ (4 · 5) = ;20
83		9cn 12073	. . . . . . . 8 ⊢ 9 ∈ ℂ
84	83	addid2i 11163	. . . . . . 7 ⊢ (0 + 9) = 9
85	3, 22, 35, 82, 84	decaddi 12496	. . . . . 6 ⊢ ((4 · 5) + 9) = ;29
86	2, 13, 22, 35, 44, 74, 5, 35, 3, 80, 85	decmac 12488	. . . . 5 ⊢ ((;14 · 5) + 9) = ;79
87	4, 5, 35, 35, 37, 42, 14, 35, 11, 73, 86	decma2c 12489	. . . 4 ⊢ ((;14 · ;;125) + (;87 + ;12)) = ;;;1849
88	83	mulid2i 10981	. . . . . . . . 9 ⊢ (1 · 9) = 9
89	88	oveq1i 7281	. . . . . . . 8 ⊢ ((1 · 9) + 3) = (9 + 3)
90		9p3e12 12524	. . . . . . . 8 ⊢ (9 + 3) = ;12
91	89, 90	eqtri 2768	. . . . . . 7 ⊢ ((1 · 9) + 3) = ;12
92		9t4e36 12560	. . . . . . . 8 ⊢ (9 · 4) = ;36
93	83, 56, 92	mulcomli 10985	. . . . . . 7 ⊢ (4 · 9) = ;36
94	35, 2, 13, 44, 18, 16, 91, 93	decmul1c 12501	. . . . . 6 ⊢ (;14 · 9) = ;;126
95	94	oveq1i 7281	. . . . 5 ⊢ ((;14 · 9) + 0) = (;;126 + 0)
96	4, 18	deccl 12451	. . . . . . 7 ⊢ ;;126 ∈ ℕ0
97	96	nn0cni 12245	. . . . . 6 ⊢ ;;126 ∈ ℂ
98	97	addid1i 11162	. . . . 5 ⊢ (;;126 + 0) = ;;126
99	95, 98	eqtri 2768	. . . 4 ⊢ ((;14 · 9) + 0) = ;;126
100	6, 35, 21, 22, 1, 36, 14, 18, 4, 87, 99	decma2c 12489	. . 3 ⊢ ((;14 · 𝑁) + ;;870) = ;;;;18496
101		eqid 2740	. . . 4 ⊢ ;;136 = ;;136
102	20, 2	deccl 12451	. . . 4 ⊢ ;81 ∈ ℕ0
103		eqid 2740	. . . . 5 ⊢ ;13 = ;13
104		eqid 2740	. . . . 5 ⊢ ;81 = ;81
105	13, 22	deccl 12451	. . . . 5 ⊢ ;40 ∈ ℕ0
106		eqid 2740	. . . . . . 7 ⊢ ;40 = ;40
107	56	addid2i 11163	. . . . . . 7 ⊢ (0 + 4) = 4
108		8cn 12070	. . . . . . . 8 ⊢ 8 ∈ ℂ
109	108	addid1i 11162	. . . . . . 7 ⊢ (8 + 0) = 8
110	22, 20, 13, 22, 60, 106, 107, 109	decadd 12490	. . . . . 6 ⊢ (8 + ;40) = ;48
111		4p1e5 12119	. . . . . . . 8 ⊢ (4 + 1) = 5
112	5	dec0h 12458	. . . . . . . 8 ⊢ 5 = ;05
113	111, 112	eqtri 2768	. . . . . . 7 ⊢ (4 + 1) = ;05
114	45	mulid1i 10980	. . . . . . . . 9 ⊢ (3 · 1) = 3
115	114	oveq1i 7281	. . . . . . . 8 ⊢ ((3 · 1) + 5) = (3 + 5)
116		5p3e8 12130	. . . . . . . . 9 ⊢ (5 + 3) = 8
117	75, 45, 116	addcomli 11167	. . . . . . . 8 ⊢ (3 + 5) = 8
118	115, 117, 60	3eqtri 2772	. . . . . . 7 ⊢ ((3 · 1) + 5) = ;08
119	2, 16, 22, 5, 103, 113, 2, 20, 22, 55, 118	decmac 12488	. . . . . 6 ⊢ ((;13 · 1) + (4 + 1)) = ;18
120		6cn 12064	. . . . . . . . 9 ⊢ 6 ∈ ℂ
121	120	mulid1i 10980	. . . . . . . 8 ⊢ (6 · 1) = 6
122	121	oveq1i 7281	. . . . . . 7 ⊢ ((6 · 1) + 8) = (6 + 8)
123	108, 120, 70	addcomli 11167	. . . . . . 7 ⊢ (6 + 8) = ;14
124	122, 123	eqtri 2768	. . . . . 6 ⊢ ((6 · 1) + 8) = ;14
125	17, 18, 13, 20, 101, 110, 2, 13, 2, 119, 124	decmac 12488	. . . . 5 ⊢ ((;;136 · 1) + (8 + ;40)) = ;;184
126	2	dec0h 12458	. . . . . 6 ⊢ 1 = ;01
127	65, 126	eqtri 2768	. . . . . . 7 ⊢ (0 + 1) = ;01
128	45	mulid2i 10981	. . . . . . . . 9 ⊢ (1 · 3) = 3
129	128, 65	oveq12i 7283	. . . . . . . 8 ⊢ ((1 · 3) + (0 + 1)) = (3 + 1)
130	129, 47	eqtri 2768	. . . . . . 7 ⊢ ((1 · 3) + (0 + 1)) = 4
131		3t3e9 12140	. . . . . . . . 9 ⊢ (3 · 3) = 9
132	131	oveq1i 7281	. . . . . . . 8 ⊢ ((3 · 3) + 1) = (9 + 1)
133		9p1e10 12438	. . . . . . . 8 ⊢ (9 + 1) = ;10
134	132, 133	eqtri 2768	. . . . . . 7 ⊢ ((3 · 3) + 1) = ;10
135	2, 16, 22, 2, 103, 127, 16, 22, 2, 130, 134	decmac 12488	. . . . . 6 ⊢ ((;13 · 3) + (0 + 1)) = ;40
136		6t3e18 12541	. . . . . . 7 ⊢ (6 · 3) = ;18
137	2, 20, 2, 136, 40	decaddi 12496	. . . . . 6 ⊢ ((6 · 3) + 1) = ;19
138	17, 18, 22, 2, 101, 126, 16, 35, 2, 135, 137	decmac 12488	. . . . 5 ⊢ ((;;136 · 3) + 1) = ;;409
139	2, 16, 20, 2, 103, 104, 19, 35, 105, 125, 138	decma2c 12489	. . . 4 ⊢ ((;;136 · ;13) + ;81) = ;;;1849
140	16	dec0h 12458	. . . . . 6 ⊢ 3 = ;03
141	120	mulid2i 10981	. . . . . . . 8 ⊢ (1 · 6) = 6
142	141, 77	oveq12i 7283	. . . . . . 7 ⊢ ((1 · 6) + (0 + 2)) = (6 + 2)
143		6p2e8 12132	. . . . . . 7 ⊢ (6 + 2) = 8
144	142, 143	eqtri 2768	. . . . . 6 ⊢ ((1 · 6) + (0 + 2)) = 8
145	120, 45, 136	mulcomli 10985	. . . . . . 7 ⊢ (3 · 6) = ;18
146		1p1e2 12098	. . . . . . 7 ⊢ (1 + 1) = 2
147		8p3e11 12517	. . . . . . 7 ⊢ (8 + 3) = ;11
148	2, 20, 16, 145, 146, 2, 147	decaddci 12497	. . . . . 6 ⊢ ((3 · 6) + 3) = ;21
149	2, 16, 22, 16, 103, 140, 18, 2, 3, 144, 148	decmac 12488	. . . . 5 ⊢ ((;13 · 6) + 3) = ;81
150		6t6e36 12544	. . . . 5 ⊢ (6 · 6) = ;36
151	18, 17, 18, 101, 18, 16, 149, 150	decmul1c 12501	. . . 4 ⊢ (;;136 · 6) = ;;816
152	19, 17, 18, 101, 18, 102, 139, 151	decmul2c 12502	. . 3 ⊢ (;;136 · ;;136) = ;;;;18496
153	100, 152	eqtr4i 2771	. 2 ⊢ ((;14 · 𝑁) + ;;870) = (;;136 · ;;136)
154	9, 10, 12, 15, 19, 23, 24, 34, 153	mod2xi 16768	1 ⊢ ((2↑;34) mod 𝑁) = (;;870 mod 𝑁)

Syntax hints:   = wceq 1542  (class class class)co 7271  0cc0 10872  1c1 10873   + caddc 10875   · cmul 10877  ℕcn 11973  2c2 12028  3c3 12029  4c4 12030  5c5 12031  6c6 12032  7c7 12033  8c8 12034  9c9 12035  ;cdc 12436   mod cmo 13587  ↑cexp 13780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582  ax-cnex 10928  ax-resscn 10929  ax-1cn 10930  ax-icn 10931  ax-addcl 10932  ax-addrcl 10933  ax-mulcl 10934  ax-mulrcl 10935  ax-mulcom 10936  ax-addass 10937  ax-mulass 10938  ax-distr 10939  ax-i2m1 10940  ax-1ne0 10941  ax-1rid 10942  ax-rnegex 10943  ax-rrecex 10944  ax-cnre 10945  ax-pre-lttri 10946  ax-pre-lttrn 10947  ax-pre-ltadd 10948  ax-pre-mulgt0 10949  ax-pre-sup 10950

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-nel 3052  df-ral 3071  df-rex 3072  df-reu 3073  df-rmo 3074  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6201  df-ord 6268  df-on 6269  df-lim 6270  df-suc 6271  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-riota 7228  df-ov 7274  df-oprab 7275  df-mpo 7276  df-om 7707  df-2nd 7825  df-frecs 8088  df-wrecs 8119  df-recs 8193  df-rdg 8232  df-er 8481  df-en 8717  df-dom 8718  df-sdom 8719  df-sup 9179  df-inf 9180  df-pnf 11012  df-mnf 11013  df-xr 11014  df-ltxr 11015  df-le 11016  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-z 12320  df-dec 12437  df-uz 12582  df-rp 12730  df-fl 13510  df-mod 13588  df-seq 13720  df-exp 13781

This theorem is referenced by:  1259lem3  16832  1259lem5  16834