Theorem 1259lem3 13273

Description: Lemma for 1259prm 13276. 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 9590	. . . . . 6 ⊢ 1 ∈ ℕ0
3		2nn0 9591	. . . . . 6 ⊢ 2 ∈ ℕ0
4	2, 3	deccl 9802	. . . . 5 ⊢ ;12 ∈ ℕ0
5		5nn0 9594	. . . . 5 ⊢ 5 ∈ ℕ0
6	4, 5	deccl 9802	. . . 4 ⊢ ;;125 ∈ ℕ0
7		9nn 9484	. . . 4 ⊢ 9 ∈ ℕ
8	6, 7	decnncl 9811	. . 3 ⊢ ;;;1259 ∈ ℕ
9	1, 8	eqeltri 2311	. 2 ⊢ 𝑁 ∈ ℕ
10		2nn 9477	. 2 ⊢ 2 ∈ ℕ
11		3nn0 9592	. . 3 ⊢ 3 ∈ ℕ0
12		8nn0 9597	. . 3 ⊢ 8 ∈ ℕ0
13	11, 12	deccl 9802	. 2 ⊢ ;38 ∈ ℕ0
14		4z 9685	. 2 ⊢ 4 ∈ ℤ
15		7nn0 9596	. . 3 ⊢ 7 ∈ ℕ0
16	15, 2	deccl 9802	. 2 ⊢ ;71 ∈ ℕ0
17		4nn0 9593	. . . 4 ⊢ 4 ∈ ℕ0
18	11, 17	deccl 9802	. . 3 ⊢ ;34 ∈ ℕ0
19	2, 2	deccl 9802	. . . 4 ⊢ ;11 ∈ ℕ0
20	19	nn0zi 9677	. . 3 ⊢ ;11 ∈ ℤ
21	12, 15	deccl 9802	. . . 4 ⊢ ;87 ∈ ℕ0
22		0nn0 9589	. . . 4 ⊢ 0 ∈ ℕ0
23	21, 22	deccl 9802	. . 3 ⊢ ;;870 ∈ ℕ0
24		6nn0 9595	. . . 4 ⊢ 6 ∈ ℕ0
25	2, 24	deccl 9802	. . 3 ⊢ ;16 ∈ ℕ0
26	1	1259lem2 13272	. . 3 ⊢ ((2↑;34) mod 𝑁) = (;;870 mod 𝑁)
27		2exp4 13240	. . . 4 ⊢ (2↑4) = ;16
28	27	oveq1i 6098	. . 3 ⊢ ((2↑4) mod 𝑁) = (;16 mod 𝑁)
29		eqid 2238	. . . 4 ⊢ ;34 = ;34
30		4p4e8 9459	. . . 4 ⊢ (4 + 4) = 8
31	11, 17, 17, 29, 30	decaddi 9852	. . 3 ⊢ (;34 + 4) = ;38
32		9nn0 9598	. . . . 5 ⊢ 9 ∈ ℕ0
33		eqid 2238	. . . . 5 ⊢ ;71 = ;71
34		10nn0 9809	. . . . 5 ⊢ ;10 ∈ ℕ0
35		eqid 2238	. . . . . 6 ⊢ ;11 = ;11
36	34	nn0cni 9586	. . . . . . 7 ⊢ ;10 ∈ ℂ
37		7cn 9397	. . . . . . 7 ⊢ 7 ∈ ℂ
38		dec10p 9835	. . . . . . 7 ⊢ (;10 + 7) = ;17
39	36, 37, 38	addcomli 8478	. . . . . 6 ⊢ (7 + ;10) = ;17
40	2, 11	deccl 9802	. . . . . 6 ⊢ ;13 ∈ ℕ0
41	6	nn0cni 9586	. . . . . . . 8 ⊢ ;;125 ∈ ℂ
42	41	mullidi 8335	. . . . . . 7 ⊢ (1 · ;;125) = ;;125
43	2	dec0h 9814	. . . . . . . 8 ⊢ 1 = ;01
44		eqid 2238	. . . . . . . 8 ⊢ ;13 = ;13
45		0p1e1 9427	. . . . . . . 8 ⊢ (0 + 1) = 1
46		3cn 9388	. . . . . . . . 9 ⊢ 3 ∈ ℂ
47		ax-1cn 8278	. . . . . . . . 9 ⊢ 1 ∈ ℂ
48		3p1e4 9449	. . . . . . . . 9 ⊢ (3 + 1) = 4
49	46, 47, 48	addcomli 8478	. . . . . . . 8 ⊢ (1 + 3) = 4
50	22, 2, 2, 11, 43, 44, 45, 49	decadd 9846	. . . . . . 7 ⊢ (1 + ;13) = ;14
51		2p1e3 9447	. . . . . . . 8 ⊢ (2 + 1) = 3
52		eqid 2238	. . . . . . . 8 ⊢ ;12 = ;12
53	2, 3, 51, 52	decsuc 9823	. . . . . . 7 ⊢ (;12 + 1) = ;13
54		5p4e9 9462	. . . . . . 7 ⊢ (5 + 4) = 9
55	4, 5, 2, 17, 42, 50, 53, 54	decadd 9846	. . . . . 6 ⊢ ((1 · ;;125) + (1 + ;13)) = ;;139
56		5cn 9393	. . . . . . . 8 ⊢ 5 ∈ ℂ
57		7p5e12 9869	. . . . . . . 8 ⊢ (7 + 5) = ;12
58	37, 56, 57	addcomli 8478	. . . . . . 7 ⊢ (5 + 7) = ;12
59	4, 5, 15, 42, 53, 3, 58	decaddci 9853	. . . . . 6 ⊢ ((1 · ;;125) + 7) = ;;132
60	2, 2, 2, 15, 35, 39, 6, 3, 40, 55, 59	decmac 9844	. . . . 5 ⊢ ((;11 · ;;125) + (7 + ;10)) = ;;;1392
61		9p1e10 9790	. . . . . 6 ⊢ (9 + 1) = ;10
62		9cn 9401	. . . . . . 7 ⊢ 9 ∈ ℂ
63	19	nn0cni 9586	. . . . . . 7 ⊢ ;11 ∈ ℂ
64		9t11e99 9922	. . . . . . 7 ⊢ (9 · ;11) = ;99
65	62, 63, 64	mulcomli 8339	. . . . . 6 ⊢ (;11 · 9) = ;99
66	32, 61, 65	decsucc 9833	. . . . 5 ⊢ ((;11 · 9) + 1) = ;;100
67	6, 32, 15, 2, 1, 33, 19, 22, 34, 60, 66	decma2c 9845	. . . 4 ⊢ ((;11 · 𝑁) + ;71) = ;;;;13920
68		eqid 2238	. . . . 5 ⊢ ;16 = ;16
69	5, 3	deccl 9802	. . . . . 6 ⊢ ;52 ∈ ℕ0
70	69, 3	deccl 9802	. . . . 5 ⊢ ;;522 ∈ ℕ0
71		eqid 2238	. . . . . 6 ⊢ ;;870 = ;;870
72		eqid 2238	. . . . . 6 ⊢ ;;522 = ;;522
73		eqid 2238	. . . . . . 7 ⊢ ;87 = ;87
74	69	nn0cni 9586	. . . . . . . 8 ⊢ ;52 ∈ ℂ
75	74	addridi 8475	. . . . . . 7 ⊢ (;52 + 0) = ;52
76		8cn 9399	. . . . . . . . . 10 ⊢ 8 ∈ ℂ
77	76	mulridi 8334	. . . . . . . . 9 ⊢ (8 · 1) = 8
78	56	addridi 8475	. . . . . . . . 9 ⊢ (5 + 0) = 5
79	77, 78	oveq12i 6100	. . . . . . . 8 ⊢ ((8 · 1) + (5 + 0)) = (8 + 5)
80		8p5e13 9875	. . . . . . . 8 ⊢ (8 + 5) = ;13
81	79, 80	eqtri 2259	. . . . . . 7 ⊢ ((8 · 1) + (5 + 0)) = ;13
82	37	mulridi 8334	. . . . . . . . 9 ⊢ (7 · 1) = 7
83	82	oveq1i 6098	. . . . . . . 8 ⊢ ((7 · 1) + 2) = (7 + 2)
84		7p2e9 9465	. . . . . . . 8 ⊢ (7 + 2) = 9
85	32	dec0h 9814	. . . . . . . 8 ⊢ 9 = ;09
86	83, 84, 85	3eqtri 2263	. . . . . . 7 ⊢ ((7 · 1) + 2) = ;09
87	12, 15, 5, 3, 73, 75, 2, 32, 22, 81, 86	decmac 9844	. . . . . 6 ⊢ ((;87 · 1) + (;52 + 0)) = ;;139
88	47	mul02i 8724	. . . . . . . 8 ⊢ (0 · 1) = 0
89	88	oveq1i 6098	. . . . . . 7 ⊢ ((0 · 1) + 2) = (0 + 2)
90		2cn 9384	. . . . . . . 8 ⊢ 2 ∈ ℂ
91	90	addlidi 8476	. . . . . . 7 ⊢ (0 + 2) = 2
92	3	dec0h 9814	. . . . . . 7 ⊢ 2 = ;02
93	89, 91, 92	3eqtri 2263	. . . . . 6 ⊢ ((0 · 1) + 2) = ;02
94	21, 22, 69, 3, 71, 72, 2, 3, 22, 87, 93	decmac 9844	. . . . 5 ⊢ ((;;870 · 1) + ;;522) = ;;;1392
95		8t6e48 9911	. . . . . . . 8 ⊢ (8 · 6) = ;48
96		4p1e5 9450	. . . . . . . 8 ⊢ (4 + 1) = 5
97		8p4e12 9874	. . . . . . . 8 ⊢ (8 + 4) = ;12
98	17, 12, 17, 95, 96, 3, 97	decaddci 9853	. . . . . . 7 ⊢ ((8 · 6) + 4) = ;52
99		7t6e42 9905	. . . . . . 7 ⊢ (7 · 6) = ;42
100	24, 12, 15, 73, 3, 17, 98, 99	decmul1c 9857	. . . . . 6 ⊢ (;87 · 6) = ;;522
101		6cn 9395	. . . . . . 7 ⊢ 6 ∈ ℂ
102	101	mul02i 8724	. . . . . 6 ⊢ (0 · 6) = 0
103	24, 21, 22, 71, 22, 100, 102	decmul1 9856	. . . . 5 ⊢ (;;870 · 6) = ;;;5220
104	23, 2, 24, 68, 22, 70, 94, 103	decmul2c 9858	. . . 4 ⊢ (;;870 · ;16) = ;;;;13920
105	67, 104	eqtr4i 2262	. . 3 ⊢ ((;11 · 𝑁) + ;71) = (;;870 · ;16)
106	9, 10, 18, 20, 23, 16, 17, 25, 26, 28, 31, 105	modxai 13224	. 2 ⊢ ((2↑;38) mod 𝑁) = (;71 mod 𝑁)
107		eqid 2238	. . 3 ⊢ ;38 = ;38
108		2t3e6 9471	. . . . 5 ⊢ (2 · 3) = 6
109	108	oveq1i 6098	. . . 4 ⊢ ((2 · 3) + 1) = (6 + 1)
110		6p1e7 9452	. . . 4 ⊢ (6 + 1) = 7
111	109, 110	eqtri 2259	. . 3 ⊢ ((2 · 3) + 1) = 7
112		8t2e16 9907	. . . 4 ⊢ (8 · 2) = ;16
113	76, 90, 112	mulcomli 8339	. . 3 ⊢ (2 · 8) = ;16
114	3, 11, 12, 107, 24, 2, 111, 113	decmul2c 9858	. 2 ⊢ (2 · ;38) = ;76
115	5	dec0h 9814	. . . 4 ⊢ 5 = ;05
116		eqid 2238	. . . . 5 ⊢ ;;125 = ;;125
117		4cn 9391	. . . . . . 7 ⊢ 4 ∈ ℂ
118	117	addlidi 8476	. . . . . 6 ⊢ (0 + 4) = 4
119	17	dec0h 9814	. . . . . 6 ⊢ 4 = ;04
120	118, 119	eqtri 2259	. . . . 5 ⊢ (0 + 4) = ;04
121	91, 92	eqtri 2259	. . . . . 6 ⊢ (0 + 2) = ;02
122	117	mulridi 8334	. . . . . . . 8 ⊢ (4 · 1) = 4
123	122, 45	oveq12i 6100	. . . . . . 7 ⊢ ((4 · 1) + (0 + 1)) = (4 + 1)
124	123, 96	eqtri 2259	. . . . . 6 ⊢ ((4 · 1) + (0 + 1)) = 5
125		4t2e8 9473	. . . . . . . 8 ⊢ (4 · 2) = 8
126	125	oveq1i 6098	. . . . . . 7 ⊢ ((4 · 2) + 2) = (8 + 2)
127		8p2e10 9872	. . . . . . 7 ⊢ (8 + 2) = ;10
128	126, 127	eqtri 2259	. . . . . 6 ⊢ ((4 · 2) + 2) = ;10
129	2, 3, 22, 3, 52, 121, 17, 22, 2, 124, 128	decma2c 9845	. . . . 5 ⊢ ((4 · ;12) + (0 + 2)) = ;50
130		5t4e20 9894	. . . . . . 7 ⊢ (5 · 4) = ;20
131	56, 117, 130	mulcomli 8339	. . . . . 6 ⊢ (4 · 5) = ;20
132	3, 22, 17, 131, 118	decaddi 9852	. . . . 5 ⊢ ((4 · 5) + 4) = ;24
133	4, 5, 22, 17, 116, 120, 17, 17, 3, 129, 132	decma2c 9845	. . . 4 ⊢ ((4 · ;;125) + (0 + 4)) = ;;504
134		9t4e36 9916	. . . . . 6 ⊢ (9 · 4) = ;36
135	62, 117, 134	mulcomli 8339	. . . . 5 ⊢ (4 · 9) = ;36
136		6p5e11 9865	. . . . 5 ⊢ (6 + 5) = ;11
137	11, 24, 5, 135, 48, 2, 136	decaddci 9853	. . . 4 ⊢ ((4 · 9) + 5) = ;41
138	6, 32, 22, 5, 1, 115, 17, 2, 17, 133, 137	decma2c 9845	. . 3 ⊢ ((4 · 𝑁) + 5) = ;;;5041
139		7t7e49 9906	. . . . . 6 ⊢ (7 · 7) = ;49
140	17, 96, 139	decsucc 9833	. . . . 5 ⊢ ((7 · 7) + 1) = ;50
141	37	mullidi 8335	. . . . . . 7 ⊢ (1 · 7) = 7
142	141	oveq1i 6098	. . . . . 6 ⊢ ((1 · 7) + 7) = (7 + 7)
143		7p7e14 9871	. . . . . 6 ⊢ (7 + 7) = ;14
144	142, 143	eqtri 2259	. . . . 5 ⊢ ((1 · 7) + 7) = ;14
145	15, 2, 15, 33, 15, 17, 2, 140, 144	decrmac 9850	. . . 4 ⊢ ((;71 · 7) + 7) = ;;504
146	16	nn0cni 9586	. . . . 5 ⊢ ;71 ∈ ℂ
147	146	mulridi 8334	. . . 4 ⊢ (;71 · 1) = ;71
148	16, 15, 2, 33, 2, 15, 145, 147	decmul2c 9858	. . 3 ⊢ (;71 · ;71) = ;;;5041
149	138, 148	eqtr4i 2262	. 2 ⊢ ((4 · 𝑁) + 5) = (;71 · ;71)
150	9, 10, 13, 14, 16, 5, 106, 114, 149	mod2xi 13225	1 ⊢ ((2↑;76) mod 𝑁) = (5 mod 𝑁)

This proof depends on syntax axioms:   = wceq 1402  (class class class)co 6088  0cc0 8185  1c1 8186   + caddc 8188   · cmul 8190  ℕcn 9313  2c2 9364  3c3 9365  4c4 9366  5c5 9367  6c6 9368  7c7 9369  8c8 9370  9c9 9371  ;cdc 9788   mod cmo 10780  ↑cexp 10996

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303  ax-arch 8304

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-5 9375  df-6 9376  df-7 9377  df-8 9378  df-9 9379  df-n0 9575  df-z 9656  df-dec 9789  df-uz 9938  df-q 10036  df-rp 10072  df-fl 10722  df-mod 10781  df-seqfrec 10906  df-exp 10997

This theorem is used by:  1259lem4  13274