Theorem 1259lem3 17225

Description: Lemma for 1259prm 17228. 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 12544	. . . . . 6 ⊢ 1 ∈ ℕ0
3		2nn0 12545	. . . . . 6 ⊢ 2 ∈ ℕ0
4	2, 3	deccl 12751	. . . . 5 ⊢ ;12 ∈ ℕ0
5		5nn0 12548	. . . . 5 ⊢ 5 ∈ ℕ0
6	4, 5	deccl 12751	. . . 4 ⊢ ;;125 ∈ ℕ0
7		9nn 12363	. . . 4 ⊢ 9 ∈ ℕ
8	6, 7	decnncl 12760	. . 3 ⊢ ;;;1259 ∈ ℕ
9	1, 8	eqeltri 2856	. 2 ⊢ 𝑁 ∈ ℕ
10		2nn 12338	. 2 ⊢ 2 ∈ ℕ
11		3nn0 12546	. . 3 ⊢ 3 ∈ ℕ0
12		8nn0 12551	. . 3 ⊢ 8 ∈ ℕ0
13	11, 12	deccl 12751	. 2 ⊢ ;38 ∈ ℕ0
14		4z 12652	. 2 ⊢ 4 ∈ ℤ
15		7nn0 12550	. . 3 ⊢ 7 ∈ ℕ0
16	15, 2	deccl 12751	. 2 ⊢ ;71 ∈ ℕ0
17		4nn0 12547	. . . 4 ⊢ 4 ∈ ℕ0
18	11, 17	deccl 12751	. . 3 ⊢ ;34 ∈ ℕ0
19	2, 2	deccl 12751	. . . 4 ⊢ ;11 ∈ ℕ0
20	19	nn0zi 12643	. . 3 ⊢ ;11 ∈ ℤ
21	12, 15	deccl 12751	. . . 4 ⊢ ;87 ∈ ℕ0
22		0nn0 12543	. . . 4 ⊢ 0 ∈ ℕ0
23	21, 22	deccl 12751	. . 3 ⊢ ;;870 ∈ ℕ0
24		6nn0 12549	. . . 4 ⊢ 6 ∈ ℕ0
25	2, 24	deccl 12751	. . 3 ⊢ ;16 ∈ ℕ0
26	1	1259lem2 17224	. . 3 ⊢ ((2↑;34) mod 𝑁) = (;;870 mod 𝑁)
27		2exp4 17176	. . . 4 ⊢ (2↑4) = ;16
28	27	oveq1i 7423	. . 3 ⊢ ((2↑4) mod 𝑁) = (;16 mod 𝑁)
29		eqid 2760	. . . 4 ⊢ ;34 = ;34
30		4p4e8 12419	. . . 4 ⊢ (4 + 4) = 8
31	11, 17, 17, 29, 30	decaddi 12801	. . 3 ⊢ (;34 + 4) = ;38
32		9nn0 12552	. . . . 5 ⊢ 9 ∈ ℕ0
33		eqid 2760	. . . . 5 ⊢ ;71 = ;71
34		10nn0 12758	. . . . 5 ⊢ ;10 ∈ ℕ0
35		eqid 2760	. . . . . 6 ⊢ ;11 = ;11
36	34	nn0cni 12540	. . . . . . 7 ⊢ ;10 ∈ ℂ
37		7cn 12359	. . . . . . 7 ⊢ 7 ∈ ℂ
38		dec10p 12784	. . . . . . 7 ⊢ (;10 + 7) = ;17
39	36, 37, 38	addcomli 11426	. . . . . 6 ⊢ (7 + ;10) = ;17
40	2, 11	deccl 12751	. . . . . 6 ⊢ ;13 ∈ ℕ0
41	6	nn0cni 12540	. . . . . . . 8 ⊢ ;;125 ∈ ℂ
42	41	mullidi 11238	. . . . . . 7 ⊢ (1 · ;;125) = ;;125
43	2	dec0h 12763	. . . . . . . 8 ⊢ 1 = ;01
44		eqid 2760	. . . . . . . 8 ⊢ ;13 = ;13
45		0p1e1 12385	. . . . . . . 8 ⊢ (0 + 1) = 1
46		3cn 12346	. . . . . . . . 9 ⊢ 3 ∈ ℂ
47		ax-1cn 11182	. . . . . . . . 9 ⊢ 1 ∈ ℂ
48		3p1e4 12409	. . . . . . . . 9 ⊢ (3 + 1) = 4
49	46, 47, 48	addcomli 11426	. . . . . . . 8 ⊢ (1 + 3) = 4
50	22, 2, 2, 11, 43, 44, 45, 49	decadd 12795	. . . . . . 7 ⊢ (1 + ;13) = ;14
51		2p1e3 12406	. . . . . . . 8 ⊢ (2 + 1) = 3
52		eqid 2760	. . . . . . . 8 ⊢ ;12 = ;12
53	2, 3, 51, 52	decsuc 12772	. . . . . . 7 ⊢ (;12 + 1) = ;13
54		5p4e9 12422	. . . . . . 7 ⊢ (5 + 4) = 9
55	4, 5, 2, 17, 42, 50, 53, 54	decadd 12795	. . . . . 6 ⊢ ((1 · ;;125) + (1 + ;13)) = ;;139
56		5cn 12353	. . . . . . . 8 ⊢ 5 ∈ ℂ
57		7p5e12 12818	. . . . . . . 8 ⊢ (7 + 5) = ;12
58	37, 56, 57	addcomli 11426	. . . . . . 7 ⊢ (5 + 7) = ;12
59	4, 5, 15, 42, 53, 3, 58	decaddci 12802	. . . . . 6 ⊢ ((1 · ;;125) + 7) = ;;132
60	2, 2, 2, 15, 35, 39, 6, 3, 40, 55, 59	decmac 12793	. . . . 5 ⊢ ((;11 · ;;125) + (7 + ;10)) = ;;;1392
61		9p1e10 12738	. . . . . 6 ⊢ (9 + 1) = ;10
62		9cn 12365	. . . . . . 7 ⊢ 9 ∈ ℂ
63	19	nn0cni 12540	. . . . . . 7 ⊢ ;11 ∈ ℂ
64		9t11e99 12871	. . . . . . 7 ⊢ (9 · ;11) = ;99
65	62, 63, 64	mulcomli 11242	. . . . . 6 ⊢ (;11 · 9) = ;99
66	32, 61, 65	decsucc 12782	. . . . 5 ⊢ ((;11 · 9) + 1) = ;;100
67	6, 32, 15, 2, 1, 33, 19, 22, 34, 60, 66	decma2c 12794	. . . 4 ⊢ ((;11 · 𝑁) + ;71) = ;;;;13920
68		eqid 2760	. . . . 5 ⊢ ;16 = ;16
69	5, 3	deccl 12751	. . . . . 6 ⊢ ;52 ∈ ℕ0
70	69, 3	deccl 12751	. . . . 5 ⊢ ;;522 ∈ ℕ0
71		eqid 2760	. . . . . 6 ⊢ ;;870 = ;;870
72		eqid 2760	. . . . . 6 ⊢ ;;522 = ;;522
73		eqid 2760	. . . . . . 7 ⊢ ;87 = ;87
74	69	nn0cni 12540	. . . . . . . 8 ⊢ ;52 ∈ ℂ
75	74	addridi 11421	. . . . . . 7 ⊢ (;52 + 0) = ;52
76		8cn 12362	. . . . . . . . . 10 ⊢ 8 ∈ ℂ
77	76	mulridi 11237	. . . . . . . . 9 ⊢ (8 · 1) = 8
78	56	addridi 11421	. . . . . . . . 9 ⊢ (5 + 0) = 5
79	77, 78	oveq12i 7425	. . . . . . . 8 ⊢ ((8 · 1) + (5 + 0)) = (8 + 5)
80		8p5e13 12824	. . . . . . . 8 ⊢ (8 + 5) = ;13
81	79, 80	eqtri 2783	. . . . . . 7 ⊢ ((8 · 1) + (5 + 0)) = ;13
82	37	mulridi 11237	. . . . . . . . 9 ⊢ (7 · 1) = 7
83	82	oveq1i 7423	. . . . . . . 8 ⊢ ((7 · 1) + 2) = (7 + 2)
84		7p2e9 12425	. . . . . . . 8 ⊢ (7 + 2) = 9
85	32	dec0h 12763	. . . . . . . 8 ⊢ 9 = ;09
86	83, 84, 85	3eqtri 2787	. . . . . . 7 ⊢ ((7 · 1) + 2) = ;09
87	12, 15, 5, 3, 73, 75, 2, 32, 22, 81, 86	decmac 12793	. . . . . 6 ⊢ ((;87 · 1) + (;52 + 0)) = ;;139
88	47	mul02i 11423	. . . . . . . 8 ⊢ (0 · 1) = 0
89	88	oveq1i 7423	. . . . . . 7 ⊢ ((0 · 1) + 2) = (0 + 2)
90		2cn 12340	. . . . . . . 8 ⊢ 2 ∈ ℂ
91	90	addlidi 11422	. . . . . . 7 ⊢ (0 + 2) = 2
92	3	dec0h 12763	. . . . . . 7 ⊢ 2 = ;02
93	89, 91, 92	3eqtri 2787	. . . . . 6 ⊢ ((0 · 1) + 2) = ;02
94	21, 22, 69, 3, 71, 72, 2, 3, 22, 87, 93	decmac 12793	. . . . 5 ⊢ ((;;870 · 1) + ;;522) = ;;;1392
95		8t6e48 12860	. . . . . . . 8 ⊢ (8 · 6) = ;48
96		4p1e5 12410	. . . . . . . 8 ⊢ (4 + 1) = 5
97		8p4e12 12823	. . . . . . . 8 ⊢ (8 + 4) = ;12
98	17, 12, 17, 95, 96, 3, 97	decaddci 12802	. . . . . . 7 ⊢ ((8 · 6) + 4) = ;52
99		7t6e42 12854	. . . . . . 7 ⊢ (7 · 6) = ;42
100	24, 12, 15, 73, 3, 17, 98, 99	decmul1c 12806	. . . . . 6 ⊢ (;87 · 6) = ;;522
101		6cn 12356	. . . . . . 7 ⊢ 6 ∈ ℂ
102	101	mul02i 11423	. . . . . 6 ⊢ (0 · 6) = 0
103	24, 21, 22, 71, 100, 102	decmul1 12805	. . . . 5 ⊢ (;;870 · 6) = ;;;5220
104	23, 2, 24, 68, 22, 70, 94, 103	decmul2c 12807	. . . 4 ⊢ (;;870 · ;16) = ;;;;13920
105	67, 104	eqtr4i 2786	. . 3 ⊢ ((;11 · 𝑁) + ;71) = (;;870 · ;16)
106	9, 10, 18, 20, 23, 16, 17, 25, 26, 28, 31, 105	modxai 17160	. 2 ⊢ ((2↑;38) mod 𝑁) = (;71 mod 𝑁)
107		eqid 2760	. . 3 ⊢ ;38 = ;38
108		2t3e6 12431	. . . . 5 ⊢ (2 · 3) = 6
109	108	oveq1i 7423	. . . 4 ⊢ ((2 · 3) + 1) = (6 + 1)
110		6p1e7 12412	. . . 4 ⊢ (6 + 1) = 7
111	109, 110	eqtri 2783	. . 3 ⊢ ((2 · 3) + 1) = 7
112		8t2e16 12856	. . . 4 ⊢ (8 · 2) = ;16
113	76, 90, 112	mulcomli 11242	. . 3 ⊢ (2 · 8) = ;16
114	3, 11, 12, 107, 24, 2, 111, 113	decmul2c 12807	. 2 ⊢ (2 · ;38) = ;76
115	5	dec0h 12763	. . . 4 ⊢ 5 = ;05
116		eqid 2760	. . . . 5 ⊢ ;;125 = ;;125
117		4cn 12350	. . . . . . 7 ⊢ 4 ∈ ℂ
118	117	addlidi 11422	. . . . . 6 ⊢ (0 + 4) = 4
119	17	dec0h 12763	. . . . . 6 ⊢ 4 = ;04
120	118, 119	eqtri 2783	. . . . 5 ⊢ (0 + 4) = ;04
121	91, 92	eqtri 2783	. . . . . 6 ⊢ (0 + 2) = ;02
122	117	mulridi 11237	. . . . . . . 8 ⊢ (4 · 1) = 4
123	122, 45	oveq12i 7425	. . . . . . 7 ⊢ ((4 · 1) + (0 + 1)) = (4 + 1)
124	123, 96	eqtri 2783	. . . . . 6 ⊢ ((4 · 1) + (0 + 1)) = 5
125		4t2e8 12433	. . . . . . . 8 ⊢ (4 · 2) = 8
126	125	oveq1i 7423	. . . . . . 7 ⊢ ((4 · 2) + 2) = (8 + 2)
127		8p2e10 12821	. . . . . . 7 ⊢ (8 + 2) = ;10
128	126, 127	eqtri 2783	. . . . . 6 ⊢ ((4 · 2) + 2) = ;10
129	2, 3, 22, 3, 52, 121, 17, 22, 2, 124, 128	decma2c 12794	. . . . 5 ⊢ ((4 · ;12) + (0 + 2)) = ;50
130		5t4e20 12843	. . . . . . 7 ⊢ (5 · 4) = ;20
131	56, 117, 130	mulcomli 11242	. . . . . 6 ⊢ (4 · 5) = ;20
132	3, 22, 17, 131, 118	decaddi 12801	. . . . 5 ⊢ ((4 · 5) + 4) = ;24
133	4, 5, 22, 17, 116, 120, 17, 17, 3, 129, 132	decma2c 12794	. . . 4 ⊢ ((4 · ;;125) + (0 + 4)) = ;;504
134		9t4e36 12865	. . . . . 6 ⊢ (9 · 4) = ;36
135	62, 117, 134	mulcomli 11242	. . . . 5 ⊢ (4 · 9) = ;36
136		6p5e11 12814	. . . . 5 ⊢ (6 + 5) = ;11
137	11, 24, 5, 135, 48, 2, 136	decaddci 12802	. . . 4 ⊢ ((4 · 9) + 5) = ;41
138	6, 32, 22, 5, 1, 115, 17, 2, 17, 133, 137	decma2c 12794	. . 3 ⊢ ((4 · 𝑁) + 5) = ;;;5041
139		7t7e49 12855	. . . . . 6 ⊢ (7 · 7) = ;49
140	17, 96, 139	decsucc 12782	. . . . 5 ⊢ ((7 · 7) + 1) = ;50
141	37	mullidi 11238	. . . . . . 7 ⊢ (1 · 7) = 7
142	141	oveq1i 7423	. . . . . 6 ⊢ ((1 · 7) + 7) = (7 + 7)
143		7p7e14 12820	. . . . . 6 ⊢ (7 + 7) = ;14
144	142, 143	eqtri 2783	. . . . 5 ⊢ ((1 · 7) + 7) = ;14
145	15, 2, 15, 33, 15, 17, 2, 140, 144	decrmac 12799	. . . 4 ⊢ ((;71 · 7) + 7) = ;;504
146	16	nn0cni 12540	. . . . 5 ⊢ ;71 ∈ ℂ
147	146	mulridi 11237	. . . 4 ⊢ (;71 · 1) = ;71
148	16, 15, 2, 33, 2, 15, 145, 147	decmul2c 12807	. . 3 ⊢ (;71 · ;71) = ;;;5041
149	138, 148	eqtr4i 2786	. 2 ⊢ ((4 · 𝑁) + 5) = (;71 · ;71)
150	9, 10, 13, 14, 16, 5, 106, 114, 149	mod2xi 17161	1 ⊢ ((2↑;76) mod 𝑁) = (5 mod 𝑁)

This proof depends on syntax axioms:   = wceq 1570  (class class class)co 7413  0cc0 11124  1c1 11125   + caddc 11127   · cmul 11129  ℕcn 12257  2c2 12319  3c3 12320  4c4 12321  5c5 12322  6c6 12323  7c7 12324  8c8 12325  9c9 12326  ;cdc 12736   mod cmo 13930  ↑cexp 14125

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201  ax-pre-sup 11202

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7863  df-2nd 7987  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-er 8696  df-en 8953  df-dom 8954  df-sdom 8955  df-sup 9412  df-inf 9413  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11467  df-neg 11468  df-div 11896  df-nn 12258  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12529  df-z 12616  df-dec 12737  df-uz 12888  df-rp 13043  df-fl 13853  df-mod 13931  df-seq 14066  df-exp 14126

This theorem is used by:  1259lem4  17226