Theorem 1259lem3 17169

Description: Lemma for 1259prm 17172. 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 12497	. . . . . 6 ⊢ 1 ∈ ℕ0
3		2nn0 12498	. . . . . 6 ⊢ 2 ∈ ℕ0
4	2, 3	deccl 12703	. . . . 5 ⊢ ;12 ∈ ℕ0
5		5nn0 12501	. . . . 5 ⊢ 5 ∈ ℕ0
6	4, 5	deccl 12703	. . . 4 ⊢ ;;125 ∈ ℕ0
7		9nn 12316	. . . 4 ⊢ 9 ∈ ℕ
8	6, 7	decnncl 12712	. . 3 ⊢ ;;;1259 ∈ ℕ
9	1, 8	eqeltri 2858	. 2 ⊢ 𝑁 ∈ ℕ
10		2nn 12291	. 2 ⊢ 2 ∈ ℕ
11		3nn0 12499	. . 3 ⊢ 3 ∈ ℕ0
12		8nn0 12504	. . 3 ⊢ 8 ∈ ℕ0
13	11, 12	deccl 12703	. 2 ⊢ ;38 ∈ ℕ0
14		4z 12605	. 2 ⊢ 4 ∈ ℤ
15		7nn0 12503	. . 3 ⊢ 7 ∈ ℕ0
16	15, 2	deccl 12703	. 2 ⊢ ;71 ∈ ℕ0
17		4nn0 12500	. . . 4 ⊢ 4 ∈ ℕ0
18	11, 17	deccl 12703	. . 3 ⊢ ;34 ∈ ℕ0
19	2, 2	deccl 12703	. . . 4 ⊢ ;11 ∈ ℕ0
20	19	nn0zi 12596	. . 3 ⊢ ;11 ∈ ℤ
21	12, 15	deccl 12703	. . . 4 ⊢ ;87 ∈ ℕ0
22		0nn0 12496	. . . 4 ⊢ 0 ∈ ℕ0
23	21, 22	deccl 12703	. . 3 ⊢ ;;870 ∈ ℕ0
24		6nn0 12502	. . . 4 ⊢ 6 ∈ ℕ0
25	2, 24	deccl 12703	. . 3 ⊢ ;16 ∈ ℕ0
26	1	1259lem2 17168	. . 3 ⊢ ((2↑;34) mod 𝑁) = (;;870 mod 𝑁)
27		2exp4 17120	. . . 4 ⊢ (2↑4) = ;16
28	27	oveq1i 7406	. . 3 ⊢ ((2↑4) mod 𝑁) = (;16 mod 𝑁)
29		eqid 2762	. . . 4 ⊢ ;34 = ;34
30		4p4e8 12372	. . . 4 ⊢ (4 + 4) = 8
31	11, 17, 17, 29, 30	decaddi 12753	. . 3 ⊢ (;34 + 4) = ;38
32		9nn0 12505	. . . . 5 ⊢ 9 ∈ ℕ0
33		eqid 2762	. . . . 5 ⊢ ;71 = ;71
34		10nn0 12710	. . . . 5 ⊢ ;10 ∈ ℕ0
35		eqid 2762	. . . . . 6 ⊢ ;11 = ;11
36	34	nn0cni 12493	. . . . . . 7 ⊢ ;10 ∈ ℂ
37		7cn 12312	. . . . . . 7 ⊢ 7 ∈ ℂ
38		dec10p 12736	. . . . . . 7 ⊢ (;10 + 7) = ;17
39	36, 37, 38	addcomli 11375	. . . . . 6 ⊢ (7 + ;10) = ;17
40	2, 11	deccl 12703	. . . . . 6 ⊢ ;13 ∈ ℕ0
41	6	nn0cni 12493	. . . . . . . 8 ⊢ ;;125 ∈ ℂ
42	41	mullidi 11187	. . . . . . 7 ⊢ (1 · ;;125) = ;;125
43	2	dec0h 12715	. . . . . . . 8 ⊢ 1 = ;01
44		eqid 2762	. . . . . . . 8 ⊢ ;13 = ;13
45		0p1e1 12338	. . . . . . . 8 ⊢ (0 + 1) = 1
46		3cn 12299	. . . . . . . . 9 ⊢ 3 ∈ ℂ
47		ax-1cn 11131	. . . . . . . . 9 ⊢ 1 ∈ ℂ
48		3p1e4 12362	. . . . . . . . 9 ⊢ (3 + 1) = 4
49	46, 47, 48	addcomli 11375	. . . . . . . 8 ⊢ (1 + 3) = 4
50	22, 2, 2, 11, 43, 44, 45, 49	decadd 12747	. . . . . . 7 ⊢ (1 + ;13) = ;14
51		2p1e3 12359	. . . . . . . 8 ⊢ (2 + 1) = 3
52		eqid 2762	. . . . . . . 8 ⊢ ;12 = ;12
53	2, 3, 51, 52	decsuc 12724	. . . . . . 7 ⊢ (;12 + 1) = ;13
54		5p4e9 12375	. . . . . . 7 ⊢ (5 + 4) = 9
55	4, 5, 2, 17, 42, 50, 53, 54	decadd 12747	. . . . . 6 ⊢ ((1 · ;;125) + (1 + ;13)) = ;;139
56		5cn 12306	. . . . . . . 8 ⊢ 5 ∈ ℂ
57		7p5e12 12770	. . . . . . . 8 ⊢ (7 + 5) = ;12
58	37, 56, 57	addcomli 11375	. . . . . . 7 ⊢ (5 + 7) = ;12
59	4, 5, 15, 42, 53, 3, 58	decaddci 12754	. . . . . 6 ⊢ ((1 · ;;125) + 7) = ;;132
60	2, 2, 2, 15, 35, 39, 6, 3, 40, 55, 59	decmac 12745	. . . . 5 ⊢ ((;11 · ;;125) + (7 + ;10)) = ;;;1392
61		9p1e10 12690	. . . . . 6 ⊢ (9 + 1) = ;10
62		9cn 12318	. . . . . . 7 ⊢ 9 ∈ ℂ
63	19	nn0cni 12493	. . . . . . 7 ⊢ ;11 ∈ ℂ
64		9t11e99 12823	. . . . . . 7 ⊢ (9 · ;11) = ;99
65	62, 63, 64	mulcomli 11191	. . . . . 6 ⊢ (;11 · 9) = ;99
66	32, 61, 65	decsucc 12734	. . . . 5 ⊢ ((;11 · 9) + 1) = ;;100
67	6, 32, 15, 2, 1, 33, 19, 22, 34, 60, 66	decma2c 12746	. . . 4 ⊢ ((;11 · 𝑁) + ;71) = ;;;;13920
68		eqid 2762	. . . . 5 ⊢ ;16 = ;16
69	5, 3	deccl 12703	. . . . . 6 ⊢ ;52 ∈ ℕ0
70	69, 3	deccl 12703	. . . . 5 ⊢ ;;522 ∈ ℕ0
71		eqid 2762	. . . . . 6 ⊢ ;;870 = ;;870
72		eqid 2762	. . . . . 6 ⊢ ;;522 = ;;522
73		eqid 2762	. . . . . . 7 ⊢ ;87 = ;87
74	69	nn0cni 12493	. . . . . . . 8 ⊢ ;52 ∈ ℂ
75	74	addridi 11370	. . . . . . 7 ⊢ (;52 + 0) = ;52
76		8cn 12315	. . . . . . . . . 10 ⊢ 8 ∈ ℂ
77	76	mulridi 11186	. . . . . . . . 9 ⊢ (8 · 1) = 8
78	56	addridi 11370	. . . . . . . . 9 ⊢ (5 + 0) = 5
79	77, 78	oveq12i 7408	. . . . . . . 8 ⊢ ((8 · 1) + (5 + 0)) = (8 + 5)
80		8p5e13 12776	. . . . . . . 8 ⊢ (8 + 5) = ;13
81	79, 80	eqtri 2785	. . . . . . 7 ⊢ ((8 · 1) + (5 + 0)) = ;13
82	37	mulridi 11186	. . . . . . . . 9 ⊢ (7 · 1) = 7
83	82	oveq1i 7406	. . . . . . . 8 ⊢ ((7 · 1) + 2) = (7 + 2)
84		7p2e9 12378	. . . . . . . 8 ⊢ (7 + 2) = 9
85	32	dec0h 12715	. . . . . . . 8 ⊢ 9 = ;09
86	83, 84, 85	3eqtri 2789	. . . . . . 7 ⊢ ((7 · 1) + 2) = ;09
87	12, 15, 5, 3, 73, 75, 2, 32, 22, 81, 86	decmac 12745	. . . . . 6 ⊢ ((;87 · 1) + (;52 + 0)) = ;;139
88	47	mul02i 11372	. . . . . . . 8 ⊢ (0 · 1) = 0
89	88	oveq1i 7406	. . . . . . 7 ⊢ ((0 · 1) + 2) = (0 + 2)
90		2cn 12293	. . . . . . . 8 ⊢ 2 ∈ ℂ
91	90	addlidi 11371	. . . . . . 7 ⊢ (0 + 2) = 2
92	3	dec0h 12715	. . . . . . 7 ⊢ 2 = ;02
93	89, 91, 92	3eqtri 2789	. . . . . 6 ⊢ ((0 · 1) + 2) = ;02
94	21, 22, 69, 3, 71, 72, 2, 3, 22, 87, 93	decmac 12745	. . . . 5 ⊢ ((;;870 · 1) + ;;522) = ;;;1392
95		8t6e48 12812	. . . . . . . 8 ⊢ (8 · 6) = ;48
96		4p1e5 12363	. . . . . . . 8 ⊢ (4 + 1) = 5
97		8p4e12 12775	. . . . . . . 8 ⊢ (8 + 4) = ;12
98	17, 12, 17, 95, 96, 3, 97	decaddci 12754	. . . . . . 7 ⊢ ((8 · 6) + 4) = ;52
99		7t6e42 12806	. . . . . . 7 ⊢ (7 · 6) = ;42
100	24, 12, 15, 73, 3, 17, 98, 99	decmul1c 12758	. . . . . 6 ⊢ (;87 · 6) = ;;522
101		6cn 12309	. . . . . . 7 ⊢ 6 ∈ ℂ
102	101	mul02i 11372	. . . . . 6 ⊢ (0 · 6) = 0
103	24, 21, 22, 71, 100, 102	decmul1 12757	. . . . 5 ⊢ (;;870 · 6) = ;;;5220
104	23, 2, 24, 68, 22, 70, 94, 103	decmul2c 12759	. . . 4 ⊢ (;;870 · ;16) = ;;;;13920
105	67, 104	eqtr4i 2788	. . 3 ⊢ ((;11 · 𝑁) + ;71) = (;;870 · ;16)
106	9, 10, 18, 20, 23, 16, 17, 25, 26, 28, 31, 105	modxai 17104	. 2 ⊢ ((2↑;38) mod 𝑁) = (;71 mod 𝑁)
107		eqid 2762	. . 3 ⊢ ;38 = ;38
108		3t2e6 12383	. . . . . 6 ⊢ (3 · 2) = 6
109	46, 90, 108	mulcomli 11191	. . . . 5 ⊢ (2 · 3) = 6
110	109	oveq1i 7406	. . . 4 ⊢ ((2 · 3) + 1) = (6 + 1)
111		6p1e7 12365	. . . 4 ⊢ (6 + 1) = 7
112	110, 111	eqtri 2785	. . 3 ⊢ ((2 · 3) + 1) = 7
113		8t2e16 12808	. . . 4 ⊢ (8 · 2) = ;16
114	76, 90, 113	mulcomli 11191	. . 3 ⊢ (2 · 8) = ;16
115	3, 11, 12, 107, 24, 2, 112, 114	decmul2c 12759	. 2 ⊢ (2 · ;38) = ;76
116	5	dec0h 12715	. . . 4 ⊢ 5 = ;05
117		eqid 2762	. . . . 5 ⊢ ;;125 = ;;125
118		4cn 12303	. . . . . . 7 ⊢ 4 ∈ ℂ
119	118	addlidi 11371	. . . . . 6 ⊢ (0 + 4) = 4
120	17	dec0h 12715	. . . . . 6 ⊢ 4 = ;04
121	119, 120	eqtri 2785	. . . . 5 ⊢ (0 + 4) = ;04
122	91, 92	eqtri 2785	. . . . . 6 ⊢ (0 + 2) = ;02
123	118	mulridi 11186	. . . . . . . 8 ⊢ (4 · 1) = 4
124	123, 45	oveq12i 7408	. . . . . . 7 ⊢ ((4 · 1) + (0 + 1)) = (4 + 1)
125	124, 96	eqtri 2785	. . . . . 6 ⊢ ((4 · 1) + (0 + 1)) = 5
126		4t2e8 12386	. . . . . . . 8 ⊢ (4 · 2) = 8
127	126	oveq1i 7406	. . . . . . 7 ⊢ ((4 · 2) + 2) = (8 + 2)
128		8p2e10 12773	. . . . . . 7 ⊢ (8 + 2) = ;10
129	127, 128	eqtri 2785	. . . . . 6 ⊢ ((4 · 2) + 2) = ;10
130	2, 3, 22, 3, 52, 122, 17, 22, 2, 125, 129	decma2c 12746	. . . . 5 ⊢ ((4 · ;12) + (0 + 2)) = ;50
131		5t4e20 12795	. . . . . . 7 ⊢ (5 · 4) = ;20
132	56, 118, 131	mulcomli 11191	. . . . . 6 ⊢ (4 · 5) = ;20
133	3, 22, 17, 132, 119	decaddi 12753	. . . . 5 ⊢ ((4 · 5) + 4) = ;24
134	4, 5, 22, 17, 117, 121, 17, 17, 3, 130, 133	decma2c 12746	. . . 4 ⊢ ((4 · ;;125) + (0 + 4)) = ;;504
135		9t4e36 12817	. . . . . 6 ⊢ (9 · 4) = ;36
136	62, 118, 135	mulcomli 11191	. . . . 5 ⊢ (4 · 9) = ;36
137		6p5e11 12766	. . . . 5 ⊢ (6 + 5) = ;11
138	11, 24, 5, 136, 48, 2, 137	decaddci 12754	. . . 4 ⊢ ((4 · 9) + 5) = ;41
139	6, 32, 22, 5, 1, 116, 17, 2, 17, 134, 138	decma2c 12746	. . 3 ⊢ ((4 · 𝑁) + 5) = ;;;5041
140		7t7e49 12807	. . . . . 6 ⊢ (7 · 7) = ;49
141	17, 96, 140	decsucc 12734	. . . . 5 ⊢ ((7 · 7) + 1) = ;50
142	37	mullidi 11187	. . . . . . 7 ⊢ (1 · 7) = 7
143	142	oveq1i 7406	. . . . . 6 ⊢ ((1 · 7) + 7) = (7 + 7)
144		7p7e14 12772	. . . . . 6 ⊢ (7 + 7) = ;14
145	143, 144	eqtri 2785	. . . . 5 ⊢ ((1 · 7) + 7) = ;14
146	15, 2, 15, 33, 15, 17, 2, 141, 145	decrmac 12751	. . . 4 ⊢ ((;71 · 7) + 7) = ;;504
147	16	nn0cni 12493	. . . . 5 ⊢ ;71 ∈ ℂ
148	147	mulridi 11186	. . . 4 ⊢ (;71 · 1) = ;71
149	16, 15, 2, 33, 2, 15, 146, 148	decmul2c 12759	. . 3 ⊢ (;71 · ;71) = ;;;5041
150	139, 149	eqtr4i 2788	. 2 ⊢ ((4 · 𝑁) + 5) = (;71 · ;71)
151	9, 10, 13, 14, 16, 5, 106, 115, 150	mod2xi 17105	1 ⊢ ((2↑;76) mod 𝑁) = (5 mod 𝑁)

Syntax hints:   = wceq 1560  (class class class)co 7396  0cc0 11073  1c1 11074   + caddc 11076   · cmul 11078  ℕcn 12210  2c2 12272  3c3 12273  4c4 12274  5c5 12275  6c6 12276  7c7 12277  8c8 12278  9c9 12279  ;cdc 12688   mod cmo 13879  ↑cexp 14074

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718  ax-cnex 11129  ax-resscn 11130  ax-1cn 11131  ax-icn 11132  ax-addcl 11133  ax-addrcl 11134  ax-mulcl 11135  ax-mulrcl 11136  ax-mulcom 11137  ax-addass 11138  ax-mulass 11139  ax-distr 11140  ax-i2m1 11141  ax-1ne0 11142  ax-1rid 11143  ax-rnegex 11144  ax-rrecex 11145  ax-cnre 11146  ax-pre-lttri 11147  ax-pre-lttrn 11148  ax-pre-ltadd 11149  ax-pre-mulgt0 11150  ax-pre-sup 11151

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7847  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-er 8678  df-en 8928  df-dom 8929  df-sdom 8930  df-sup 9388  df-inf 9389  df-pnf 11218  df-mnf 11219  df-xr 11220  df-ltxr 11221  df-le 11222  df-sub 11416  df-neg 11417  df-div 11845  df-nn 12211  df-2 12280  df-3 12281  df-4 12282  df-5 12283  df-6 12284  df-7 12285  df-8 12286  df-9 12287  df-n0 12482  df-z 12569  df-dec 12689  df-uz 12840  df-rp 12994  df-fl 13802  df-mod 13880  df-seq 14015  df-exp 14075

This theorem is referenced by:  1259lem4  17170