Theorem 1259lem3 16466

Description: Lemma for 1259prm 16469. Calculate a power mod. In decimal, we calculate 2↑38 = 2↑34 · 2↑4≡870 · 16 = 11𝑁 + 71 and 2↑76 = (2↑34)↑2≡71↑2 = 4𝑁 + 5≡5. (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
1259lem3	⊢ ((2↑;76) mod 𝑁) = (5 mod 𝑁)

Proof of Theorem 1259lem3

Step	Hyp	Ref	Expression
1		1259prm.1	. . 3 ⊢ 𝑁 = ;;;1259
2		1nn0 11914	. . . . . 6 ⊢ 1 ∈ ℕ0
3		2nn0 11915	. . . . . 6 ⊢ 2 ∈ ℕ0
4	2, 3	deccl 12114	. . . . 5 ⊢ ;12 ∈ ℕ0
5		5nn0 11918	. . . . 5 ⊢ 5 ∈ ℕ0
6	4, 5	deccl 12114	. . . 4 ⊢ ;;125 ∈ ℕ0
7		9nn 11736	. . . 4 ⊢ 9 ∈ ℕ
8	6, 7	decnncl 12119	. . 3 ⊢ ;;;1259 ∈ ℕ
9	1, 8	eqeltri 2909	. 2 ⊢ 𝑁 ∈ ℕ
10		2nn 11711	. 2 ⊢ 2 ∈ ℕ
11		3nn0 11916	. . 3 ⊢ 3 ∈ ℕ0
12		8nn0 11921	. . 3 ⊢ 8 ∈ ℕ0
13	11, 12	deccl 12114	. 2 ⊢ ;38 ∈ ℕ0
14		4z 12017	. 2 ⊢ 4 ∈ ℤ
15		7nn0 11920	. . 3 ⊢ 7 ∈ ℕ0
16	15, 2	deccl 12114	. 2 ⊢ ;71 ∈ ℕ0
17		4nn0 11917	. . . 4 ⊢ 4 ∈ ℕ0
18	11, 17	deccl 12114	. . 3 ⊢ ;34 ∈ ℕ0
19	2, 2	deccl 12114	. . . 4 ⊢ ;11 ∈ ℕ0
20	19	nn0zi 12008	. . 3 ⊢ ;11 ∈ ℤ
21	12, 15	deccl 12114	. . . 4 ⊢ ;87 ∈ ℕ0
22		0nn0 11913	. . . 4 ⊢ 0 ∈ ℕ0
23	21, 22	deccl 12114	. . 3 ⊢ ;;870 ∈ ℕ0
24		6nn0 11919	. . . 4 ⊢ 6 ∈ ℕ0
25	2, 24	deccl 12114	. . 3 ⊢ ;16 ∈ ℕ0
26	1	1259lem2 16465	. . 3 ⊢ ((2↑;34) mod 𝑁) = (;;870 mod 𝑁)
27		2exp4 16421	. . . 4 ⊢ (2↑4) = ;16
28	27	oveq1i 7166	. . 3 ⊢ ((2↑4) mod 𝑁) = (;16 mod 𝑁)
29		eqid 2821	. . . 4 ⊢ ;34 = ;34
30		4p4e8 11793	. . . 4 ⊢ (4 + 4) = 8
31	11, 17, 17, 29, 30	decaddi 12159	. . 3 ⊢ (;34 + 4) = ;38
32		9nn0 11922	. . . . 5 ⊢ 9 ∈ ℕ0
33		eqid 2821	. . . . 5 ⊢ ;71 = ;71
34		10nn0 12117	. . . . 5 ⊢ ;10 ∈ ℕ0
35		eqid 2821	. . . . . 6 ⊢ ;11 = ;11
36	34	nn0cni 11910	. . . . . . 7 ⊢ ;10 ∈ ℂ
37		7cn 11732	. . . . . . 7 ⊢ 7 ∈ ℂ
38		dec10p 12142	. . . . . . 7 ⊢ (;10 + 7) = ;17
39	36, 37, 38	addcomli 10832	. . . . . 6 ⊢ (7 + ;10) = ;17
40	2, 11	deccl 12114	. . . . . 6 ⊢ ;13 ∈ ℕ0
41	6	nn0cni 11910	. . . . . . . 8 ⊢ ;;125 ∈ ℂ
42	41	mulid2i 10646	. . . . . . 7 ⊢ (1 · ;;125) = ;;125
43	2	dec0h 12121	. . . . . . . 8 ⊢ 1 = ;01
44		eqid 2821	. . . . . . . 8 ⊢ ;13 = ;13
45		0p1e1 11760	. . . . . . . 8 ⊢ (0 + 1) = 1
46		3cn 11719	. . . . . . . . 9 ⊢ 3 ∈ ℂ
47		ax-1cn 10595	. . . . . . . . 9 ⊢ 1 ∈ ℂ
48		3p1e4 11783	. . . . . . . . 9 ⊢ (3 + 1) = 4
49	46, 47, 48	addcomli 10832	. . . . . . . 8 ⊢ (1 + 3) = 4
50	22, 2, 2, 11, 43, 44, 45, 49	decadd 12153	. . . . . . 7 ⊢ (1 + ;13) = ;14
51		2p1e3 11780	. . . . . . . 8 ⊢ (2 + 1) = 3
52		eqid 2821	. . . . . . . 8 ⊢ ;12 = ;12
53	2, 3, 51, 52	decsuc 12130	. . . . . . 7 ⊢ (;12 + 1) = ;13
54		5p4e9 11796	. . . . . . 7 ⊢ (5 + 4) = 9
55	4, 5, 2, 17, 42, 50, 53, 54	decadd 12153	. . . . . 6 ⊢ ((1 · ;;125) + (1 + ;13)) = ;;139
56		5cn 11726	. . . . . . . 8 ⊢ 5 ∈ ℂ
57		7p5e12 12176	. . . . . . . 8 ⊢ (7 + 5) = ;12
58	37, 56, 57	addcomli 10832	. . . . . . 7 ⊢ (5 + 7) = ;12
59	4, 5, 15, 42, 53, 3, 58	decaddci 12160	. . . . . 6 ⊢ ((1 · ;;125) + 7) = ;;132
60	2, 2, 2, 15, 35, 39, 6, 3, 40, 55, 59	decmac 12151	. . . . 5 ⊢ ((;11 · ;;125) + (7 + ;10)) = ;;;1392
61		9p1e10 12101	. . . . . 6 ⊢ (9 + 1) = ;10
62		9cn 11738	. . . . . . 7 ⊢ 9 ∈ ℂ
63	19	nn0cni 11910	. . . . . . 7 ⊢ ;11 ∈ ℂ
64		9t11e99 12229	. . . . . . 7 ⊢ (9 · ;11) = ;99
65	62, 63, 64	mulcomli 10650	. . . . . 6 ⊢ (;11 · 9) = ;99
66	32, 61, 65	decsucc 12140	. . . . 5 ⊢ ((;11 · 9) + 1) = ;;100
67	6, 32, 15, 2, 1, 33, 19, 22, 34, 60, 66	decma2c 12152	. . . 4 ⊢ ((;11 · 𝑁) + ;71) = ;;;;13920
68		eqid 2821	. . . . 5 ⊢ ;16 = ;16
69	5, 3	deccl 12114	. . . . . 6 ⊢ ;52 ∈ ℕ0
70	69, 3	deccl 12114	. . . . 5 ⊢ ;;522 ∈ ℕ0
71		eqid 2821	. . . . . 6 ⊢ ;;870 = ;;870
72		eqid 2821	. . . . . 6 ⊢ ;;522 = ;;522
73		eqid 2821	. . . . . . 7 ⊢ ;87 = ;87
74	69	nn0cni 11910	. . . . . . . 8 ⊢ ;52 ∈ ℂ
75	74	addid1i 10827	. . . . . . 7 ⊢ (;52 + 0) = ;52
76		8cn 11735	. . . . . . . . . 10 ⊢ 8 ∈ ℂ
77	76	mulid1i 10645	. . . . . . . . 9 ⊢ (8 · 1) = 8
78	56	addid1i 10827	. . . . . . . . 9 ⊢ (5 + 0) = 5
79	77, 78	oveq12i 7168	. . . . . . . 8 ⊢ ((8 · 1) + (5 + 0)) = (8 + 5)
80		8p5e13 12182	. . . . . . . 8 ⊢ (8 + 5) = ;13
81	79, 80	eqtri 2844	. . . . . . 7 ⊢ ((8 · 1) + (5 + 0)) = ;13
82	37	mulid1i 10645	. . . . . . . . 9 ⊢ (7 · 1) = 7
83	82	oveq1i 7166	. . . . . . . 8 ⊢ ((7 · 1) + 2) = (7 + 2)
84		7p2e9 11799	. . . . . . . 8 ⊢ (7 + 2) = 9
85	32	dec0h 12121	. . . . . . . 8 ⊢ 9 = ;09
86	83, 84, 85	3eqtri 2848	. . . . . . 7 ⊢ ((7 · 1) + 2) = ;09
87	12, 15, 5, 3, 73, 75, 2, 32, 22, 81, 86	decmac 12151	. . . . . 6 ⊢ ((;87 · 1) + (;52 + 0)) = ;;139
88	47	mul02i 10829	. . . . . . . 8 ⊢ (0 · 1) = 0
89	88	oveq1i 7166	. . . . . . 7 ⊢ ((0 · 1) + 2) = (0 + 2)
90		2cn 11713	. . . . . . . 8 ⊢ 2 ∈ ℂ
91	90	addid2i 10828	. . . . . . 7 ⊢ (0 + 2) = 2
92	3	dec0h 12121	. . . . . . 7 ⊢ 2 = ;02
93	89, 91, 92	3eqtri 2848	. . . . . 6 ⊢ ((0 · 1) + 2) = ;02
94	21, 22, 69, 3, 71, 72, 2, 3, 22, 87, 93	decmac 12151	. . . . 5 ⊢ ((;;870 · 1) + ;;522) = ;;;1392
95		8t6e48 12218	. . . . . . . 8 ⊢ (8 · 6) = ;48
96		4p1e5 11784	. . . . . . . 8 ⊢ (4 + 1) = 5
97		8p4e12 12181	. . . . . . . 8 ⊢ (8 + 4) = ;12
98	17, 12, 17, 95, 96, 3, 97	decaddci 12160	. . . . . . 7 ⊢ ((8 · 6) + 4) = ;52
99		7t6e42 12212	. . . . . . 7 ⊢ (7 · 6) = ;42
100	24, 12, 15, 73, 3, 17, 98, 99	decmul1c 12164	. . . . . 6 ⊢ (;87 · 6) = ;;522
101		6cn 11729	. . . . . . 7 ⊢ 6 ∈ ℂ
102	101	mul02i 10829	. . . . . 6 ⊢ (0 · 6) = 0
103	24, 21, 22, 71, 100, 102	decmul1 12163	. . . . 5 ⊢ (;;870 · 6) = ;;;5220
104	23, 2, 24, 68, 22, 70, 94, 103	decmul2c 12165	. . . 4 ⊢ (;;870 · ;16) = ;;;;13920
105	67, 104	eqtr4i 2847	. . 3 ⊢ ((;11 · 𝑁) + ;71) = (;;870 · ;16)
106	9, 10, 18, 20, 23, 16, 17, 25, 26, 28, 31, 105	modxai 16404	. 2 ⊢ ((2↑;38) mod 𝑁) = (;71 mod 𝑁)
107		eqid 2821	. . 3 ⊢ ;38 = ;38
108		3t2e6 11804	. . . . . 6 ⊢ (3 · 2) = 6
109	46, 90, 108	mulcomli 10650	. . . . 5 ⊢ (2 · 3) = 6
110	109	oveq1i 7166	. . . 4 ⊢ ((2 · 3) + 1) = (6 + 1)
111		6p1e7 11786	. . . 4 ⊢ (6 + 1) = 7
112	110, 111	eqtri 2844	. . 3 ⊢ ((2 · 3) + 1) = 7
113		8t2e16 12214	. . . 4 ⊢ (8 · 2) = ;16
114	76, 90, 113	mulcomli 10650	. . 3 ⊢ (2 · 8) = ;16
115	3, 11, 12, 107, 24, 2, 112, 114	decmul2c 12165	. 2 ⊢ (2 · ;38) = ;76
116	5	dec0h 12121	. . . 4 ⊢ 5 = ;05
117		eqid 2821	. . . . 5 ⊢ ;;125 = ;;125
118		4cn 11723	. . . . . . 7 ⊢ 4 ∈ ℂ
119	118	addid2i 10828	. . . . . 6 ⊢ (0 + 4) = 4
120	17	dec0h 12121	. . . . . 6 ⊢ 4 = ;04
121	119, 120	eqtri 2844	. . . . 5 ⊢ (0 + 4) = ;04
122	91, 92	eqtri 2844	. . . . . 6 ⊢ (0 + 2) = ;02
123	118	mulid1i 10645	. . . . . . . 8 ⊢ (4 · 1) = 4
124	123, 45	oveq12i 7168	. . . . . . 7 ⊢ ((4 · 1) + (0 + 1)) = (4 + 1)
125	124, 96	eqtri 2844	. . . . . 6 ⊢ ((4 · 1) + (0 + 1)) = 5
126		4t2e8 11806	. . . . . . . 8 ⊢ (4 · 2) = 8
127	126	oveq1i 7166	. . . . . . 7 ⊢ ((4 · 2) + 2) = (8 + 2)
128		8p2e10 12179	. . . . . . 7 ⊢ (8 + 2) = ;10
129	127, 128	eqtri 2844	. . . . . 6 ⊢ ((4 · 2) + 2) = ;10
130	2, 3, 22, 3, 52, 122, 17, 22, 2, 125, 129	decma2c 12152	. . . . 5 ⊢ ((4 · ;12) + (0 + 2)) = ;50
131		5t4e20 12201	. . . . . . 7 ⊢ (5 · 4) = ;20
132	56, 118, 131	mulcomli 10650	. . . . . 6 ⊢ (4 · 5) = ;20
133	3, 22, 17, 132, 119	decaddi 12159	. . . . 5 ⊢ ((4 · 5) + 4) = ;24
134	4, 5, 22, 17, 117, 121, 17, 17, 3, 130, 133	decma2c 12152	. . . 4 ⊢ ((4 · ;;125) + (0 + 4)) = ;;504
135		9t4e36 12223	. . . . . 6 ⊢ (9 · 4) = ;36
136	62, 118, 135	mulcomli 10650	. . . . 5 ⊢ (4 · 9) = ;36
137		6p5e11 12172	. . . . 5 ⊢ (6 + 5) = ;11
138	11, 24, 5, 136, 48, 2, 137	decaddci 12160	. . . 4 ⊢ ((4 · 9) + 5) = ;41
139	6, 32, 22, 5, 1, 116, 17, 2, 17, 134, 138	decma2c 12152	. . 3 ⊢ ((4 · 𝑁) + 5) = ;;;5041
140		7t7e49 12213	. . . . . 6 ⊢ (7 · 7) = ;49
141	17, 96, 140	decsucc 12140	. . . . 5 ⊢ ((7 · 7) + 1) = ;50
142	37	mulid2i 10646	. . . . . . 7 ⊢ (1 · 7) = 7
143	142	oveq1i 7166	. . . . . 6 ⊢ ((1 · 7) + 7) = (7 + 7)
144		7p7e14 12178	. . . . . 6 ⊢ (7 + 7) = ;14
145	143, 144	eqtri 2844	. . . . 5 ⊢ ((1 · 7) + 7) = ;14
146	15, 2, 15, 33, 15, 17, 2, 141, 145	decrmac 12157	. . . 4 ⊢ ((;71 · 7) + 7) = ;;504
147	16	nn0cni 11910	. . . . 5 ⊢ ;71 ∈ ℂ
148	147	mulid1i 10645	. . . 4 ⊢ (;71 · 1) = ;71
149	16, 15, 2, 33, 2, 15, 146, 148	decmul2c 12165	. . 3 ⊢ (;71 · ;71) = ;;;5041
150	139, 149	eqtr4i 2847	. 2 ⊢ ((4 · 𝑁) + 5) = (;71 · ;71)
151	9, 10, 13, 14, 16, 5, 106, 115, 150	mod2xi 16405	1 ⊢ ((2↑;76) mod 𝑁) = (5 mod 𝑁)

Syntax hints:   = wceq 1537  (class class class)co 7156  0cc0 10537  1c1 10538   + caddc 10540   · cmul 10542  ℕcn 11638  2c2 11693  3c3 11694  4c4 11695  5c5 11696  6c6 11697  7c7 11698  8c8 11699  9c9 11700  ;cdc 12099   mod cmo 13238  ↑cexp 13430

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614  ax-pre-sup 10615

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-er 8289  df-en 8510  df-dom 8511  df-sdom 8512  df-sup 8906  df-inf 8907  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-div 11298  df-nn 11639  df-2 11701  df-3 11702  df-4 11703  df-5 11704  df-6 11705  df-7 11706  df-8 11707  df-9 11708  df-n0 11899  df-z 11983  df-dec 12100  df-uz 12245  df-rp 12391  df-fl 13163  df-mod 13239  df-seq 13371  df-exp 13431

This theorem is referenced by:  1259lem4  16467