Theorem 1259lem2 13272

Description: Lemma for 1259prm 13276. 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		12nn0 9804	. . . . 5 ⊢ ;12 ∈ ℕ0
3		5nn0 9594	. . . . 5 ⊢ 5 ∈ ℕ0
4	2, 3	deccl 9802	. . . 4 ⊢ ;;125 ∈ ℕ0
5		9nn 9484	. . . 4 ⊢ 9 ∈ ℕ
6	4, 5	decnncl 9811	. . 3 ⊢ ;;;1259 ∈ ℕ
7	1, 6	eqeltri 2311	. 2 ⊢ 𝑁 ∈ ℕ
8		2nn 9477	. 2 ⊢ 2 ∈ ℕ
9		1nn0 9590	. . 3 ⊢ 1 ∈ ℕ0
10		7nn0 9596	. . 3 ⊢ 7 ∈ ℕ0
11	9, 10	deccl 9802	. 2 ⊢ ;17 ∈ ℕ0
12		4nn0 9593	. . . 4 ⊢ 4 ∈ ℕ0
13	9, 12	deccl 9802	. . 3 ⊢ ;14 ∈ ℕ0
14	13	nn0zi 9677	. 2 ⊢ ;14 ∈ ℤ
15		3nn0 9592	. . . 4 ⊢ 3 ∈ ℕ0
16	9, 15	deccl 9802	. . 3 ⊢ ;13 ∈ ℕ0
17		6nn0 9595	. . 3 ⊢ 6 ∈ ℕ0
18	16, 17	deccl 9802	. 2 ⊢ ;;136 ∈ ℕ0
19		8nn0 9597	. . . 4 ⊢ 8 ∈ ℕ0
20	19, 10	deccl 9802	. . 3 ⊢ ;87 ∈ ℕ0
21		0nn0 9589	. . 3 ⊢ 0 ∈ ℕ0
22	20, 21	deccl 9802	. 2 ⊢ ;;870 ∈ ℕ0
23	1	1259lem1 13271	. 2 ⊢ ((2↑;17) mod 𝑁) = (;;136 mod 𝑁)
24		2nn0 9591	. . 3 ⊢ 2 ∈ ℕ0
25		eqid 2238	. . 3 ⊢ ;17 = ;17
26		2cn 9384	. . . . . 6 ⊢ 2 ∈ ℂ
27	26	mulridi 8334	. . . . 5 ⊢ (2 · 1) = 2
28	27	oveq1i 6098	. . . 4 ⊢ ((2 · 1) + 1) = (2 + 1)
29		2p1e3 9447	. . . 4 ⊢ (2 + 1) = 3
30	28, 29	eqtri 2259	. . 3 ⊢ ((2 · 1) + 1) = 3
31		7cn 9397	. . . 4 ⊢ 7 ∈ ℂ
32		7t2e14 9901	. . . 4 ⊢ (7 · 2) = ;14
33	31, 26, 32	mulcomli 8339	. . 3 ⊢ (2 · 7) = ;14
34	24, 9, 10, 25, 12, 9, 30, 33	decmul2c 9858	. 2 ⊢ (2 · ;17) = ;34
35		9nn0 9598	. . . 4 ⊢ 9 ∈ ℕ0
36		eqid 2238	. . . 4 ⊢ ;;870 = ;;870
37		eqid 2238	. . . . 5 ⊢ ;;125 = ;;125
38		eqid 2238	. . . . . 6 ⊢ ;87 = ;87
39		eqid 2238	. . . . . 6 ⊢ ;12 = ;12
40		8p1e9 9454	. . . . . 6 ⊢ (8 + 1) = 9
41		7p2e9 9465	. . . . . 6 ⊢ (7 + 2) = 9
42	19, 10, 9, 24, 38, 39, 40, 41	decadd 9846	. . . . 5 ⊢ (;87 + ;12) = ;99
43		9p7e16 9884	. . . . . 6 ⊢ (9 + 7) = ;16
44		eqid 2238	. . . . . . 7 ⊢ ;14 = ;14
45		3cn 9388	. . . . . . . . 9 ⊢ 3 ∈ ℂ
46		ax-1cn 8278	. . . . . . . . 9 ⊢ 1 ∈ ℂ
47		3p1e4 9449	. . . . . . . . 9 ⊢ (3 + 1) = 4
48	45, 46, 47	addcomli 8478	. . . . . . . 8 ⊢ (1 + 3) = 4
49	12	dec0h 9814	. . . . . . . 8 ⊢ 4 = ;04
50	48, 49	eqtri 2259	. . . . . . 7 ⊢ (1 + 3) = ;04
51	46	mulridi 8334	. . . . . . . . 9 ⊢ (1 · 1) = 1
52		00id 8474	. . . . . . . . 9 ⊢ (0 + 0) = 0
53	51, 52	oveq12i 6100	. . . . . . . 8 ⊢ ((1 · 1) + (0 + 0)) = (1 + 0)
54	46	addridi 8475	. . . . . . . 8 ⊢ (1 + 0) = 1
55	53, 54	eqtri 2259	. . . . . . 7 ⊢ ((1 · 1) + (0 + 0)) = 1
56		4cn 9391	. . . . . . . . . 10 ⊢ 4 ∈ ℂ
57	56	mulridi 8334	. . . . . . . . 9 ⊢ (4 · 1) = 4
58	57	oveq1i 6098	. . . . . . . 8 ⊢ ((4 · 1) + 4) = (4 + 4)
59		4p4e8 9459	. . . . . . . 8 ⊢ (4 + 4) = 8
60	19	dec0h 9814	. . . . . . . 8 ⊢ 8 = ;08
61	58, 59, 60	3eqtri 2263	. . . . . . 7 ⊢ ((4 · 1) + 4) = ;08
62	9, 12, 21, 12, 44, 50, 9, 19, 21, 55, 61	decmac 9844	. . . . . 6 ⊢ ((;14 · 1) + (1 + 3)) = ;18
63	17	dec0h 9814	. . . . . . 7 ⊢ 6 = ;06
64	26	mullidi 8335	. . . . . . . . 9 ⊢ (1 · 2) = 2
65	46	addlidi 8476	. . . . . . . . 9 ⊢ (0 + 1) = 1
66	64, 65	oveq12i 6100	. . . . . . . 8 ⊢ ((1 · 2) + (0 + 1)) = (2 + 1)
67	66, 29	eqtri 2259	. . . . . . 7 ⊢ ((1 · 2) + (0 + 1)) = 3
68		4t2e8 9473	. . . . . . . . 9 ⊢ (4 · 2) = 8
69	68	oveq1i 6098	. . . . . . . 8 ⊢ ((4 · 2) + 6) = (8 + 6)
70		8p6e14 9876	. . . . . . . 8 ⊢ (8 + 6) = ;14
71	69, 70	eqtri 2259	. . . . . . 7 ⊢ ((4 · 2) + 6) = ;14
72	9, 12, 21, 17, 44, 63, 24, 12, 9, 67, 71	decmac 9844	. . . . . 6 ⊢ ((;14 · 2) + 6) = ;34
73	9, 24, 9, 17, 39, 43, 13, 12, 15, 62, 72	decma2c 9845	. . . . 5 ⊢ ((;14 · ;12) + (9 + 7)) = ;;184
74	35	dec0h 9814	. . . . . 6 ⊢ 9 = ;09
75		5cn 9393	. . . . . . . . 9 ⊢ 5 ∈ ℂ
76	75	mullidi 8335	. . . . . . . 8 ⊢ (1 · 5) = 5
77	26	addlidi 8476	. . . . . . . 8 ⊢ (0 + 2) = 2
78	76, 77	oveq12i 6100	. . . . . . 7 ⊢ ((1 · 5) + (0 + 2)) = (5 + 2)
79		5p2e7 9460	. . . . . . 7 ⊢ (5 + 2) = 7
80	78, 79	eqtri 2259	. . . . . 6 ⊢ ((1 · 5) + (0 + 2)) = 7
81		5t4e20 9894	. . . . . . . 8 ⊢ (5 · 4) = ;20
82	75, 56, 81	mulcomli 8339	. . . . . . 7 ⊢ (4 · 5) = ;20
83		9cn 9401	. . . . . . . 8 ⊢ 9 ∈ ℂ
84	83	addlidi 8476	. . . . . . 7 ⊢ (0 + 9) = 9
85	24, 21, 35, 82, 84	decaddi 9852	. . . . . 6 ⊢ ((4 · 5) + 9) = ;29
86	9, 12, 21, 35, 44, 74, 3, 35, 24, 80, 85	decmac 9844	. . . . 5 ⊢ ((;14 · 5) + 9) = ;79
87	2, 3, 35, 35, 37, 42, 13, 35, 10, 73, 86	decma2c 9845	. . . 4 ⊢ ((;14 · ;;125) + (;87 + ;12)) = ;;;1849
88	83	mullidi 8335	. . . . . . . . 9 ⊢ (1 · 9) = 9
89	88	oveq1i 6098	. . . . . . . 8 ⊢ ((1 · 9) + 3) = (9 + 3)
90		9p3e12 9880	. . . . . . . 8 ⊢ (9 + 3) = ;12
91	89, 90	eqtri 2259	. . . . . . 7 ⊢ ((1 · 9) + 3) = ;12
92		9t4e36 9916	. . . . . . . 8 ⊢ (9 · 4) = ;36
93	83, 56, 92	mulcomli 8339	. . . . . . 7 ⊢ (4 · 9) = ;36
94	35, 9, 12, 44, 17, 15, 91, 93	decmul1c 9857	. . . . . 6 ⊢ (;14 · 9) = ;;126
95	94	oveq1i 6098	. . . . 5 ⊢ ((;14 · 9) + 0) = (;;126 + 0)
96	2, 17	deccl 9802	. . . . . . 7 ⊢ ;;126 ∈ ℕ0
97	96	nn0cni 9586	. . . . . 6 ⊢ ;;126 ∈ ℂ
98	97	addridi 8475	. . . . 5 ⊢ (;;126 + 0) = ;;126
99	95, 98	eqtri 2259	. . . 4 ⊢ ((;14 · 9) + 0) = ;;126
100	4, 35, 20, 21, 1, 36, 13, 17, 2, 87, 99	decma2c 9845	. . 3 ⊢ ((;14 · 𝑁) + ;;870) = ;;;;18496
101		eqid 2238	. . . 4 ⊢ ;;136 = ;;136
102	19, 9	deccl 9802	. . . 4 ⊢ ;81 ∈ ℕ0
103		eqid 2238	. . . . 5 ⊢ ;13 = ;13
104		eqid 2238	. . . . 5 ⊢ ;81 = ;81
105	12, 21	deccl 9802	. . . . 5 ⊢ ;40 ∈ ℕ0
106		eqid 2238	. . . . . . 7 ⊢ ;40 = ;40
107	56	addlidi 8476	. . . . . . 7 ⊢ (0 + 4) = 4
108		8cn 9399	. . . . . . . 8 ⊢ 8 ∈ ℂ
109	108	addridi 8475	. . . . . . 7 ⊢ (8 + 0) = 8
110	21, 19, 12, 21, 60, 106, 107, 109	decadd 9846	. . . . . 6 ⊢ (8 + ;40) = ;48
111		4p1e5 9450	. . . . . . . 8 ⊢ (4 + 1) = 5
112	3	dec0h 9814	. . . . . . . 8 ⊢ 5 = ;05
113	111, 112	eqtri 2259	. . . . . . 7 ⊢ (4 + 1) = ;05
114	45	mulridi 8334	. . . . . . . . 9 ⊢ (3 · 1) = 3
115	114	oveq1i 6098	. . . . . . . 8 ⊢ ((3 · 1) + 5) = (3 + 5)
116		5p3e8 9461	. . . . . . . . 9 ⊢ (5 + 3) = 8
117	75, 45, 116	addcomli 8478	. . . . . . . 8 ⊢ (3 + 5) = 8
118	115, 117, 60	3eqtri 2263	. . . . . . 7 ⊢ ((3 · 1) + 5) = ;08
119	9, 15, 21, 3, 103, 113, 9, 19, 21, 55, 118	decmac 9844	. . . . . 6 ⊢ ((;13 · 1) + (4 + 1)) = ;18
120		6cn 9395	. . . . . . . . 9 ⊢ 6 ∈ ℂ
121	120	mulridi 8334	. . . . . . . 8 ⊢ (6 · 1) = 6
122	121	oveq1i 6098	. . . . . . 7 ⊢ ((6 · 1) + 8) = (6 + 8)
123	108, 120, 70	addcomli 8478	. . . . . . 7 ⊢ (6 + 8) = ;14
124	122, 123	eqtri 2259	. . . . . 6 ⊢ ((6 · 1) + 8) = ;14
125	16, 17, 12, 19, 101, 110, 9, 12, 9, 119, 124	decmac 9844	. . . . 5 ⊢ ((;;136 · 1) + (8 + ;40)) = ;;184
126	9	dec0h 9814	. . . . . 6 ⊢ 1 = ;01
127	65, 126	eqtri 2259	. . . . . . 7 ⊢ (0 + 1) = ;01
128	45	mullidi 8335	. . . . . . . . 9 ⊢ (1 · 3) = 3
129	128, 65	oveq12i 6100	. . . . . . . 8 ⊢ ((1 · 3) + (0 + 1)) = (3 + 1)
130	129, 47	eqtri 2259	. . . . . . 7 ⊢ ((1 · 3) + (0 + 1)) = 4
131		3t3e9 9472	. . . . . . . . 9 ⊢ (3 · 3) = 9
132	131	oveq1i 6098	. . . . . . . 8 ⊢ ((3 · 3) + 1) = (9 + 1)
133		9p1e10 9790	. . . . . . . 8 ⊢ (9 + 1) = ;10
134	132, 133	eqtri 2259	. . . . . . 7 ⊢ ((3 · 3) + 1) = ;10
135	9, 15, 21, 9, 103, 127, 15, 21, 9, 130, 134	decmac 9844	. . . . . 6 ⊢ ((;13 · 3) + (0 + 1)) = ;40
136		6t3e18 9897	. . . . . . 7 ⊢ (6 · 3) = ;18
137	9, 19, 9, 136, 40	decaddi 9852	. . . . . 6 ⊢ ((6 · 3) + 1) = ;19
138	16, 17, 21, 9, 101, 126, 15, 35, 9, 135, 137	decmac 9844	. . . . 5 ⊢ ((;;136 · 3) + 1) = ;;409
139	9, 15, 19, 9, 103, 104, 18, 35, 105, 125, 138	decma2c 9845	. . . 4 ⊢ ((;;136 · ;13) + ;81) = ;;;1849
140	15	dec0h 9814	. . . . . 6 ⊢ 3 = ;03
141	120	mullidi 8335	. . . . . . . 8 ⊢ (1 · 6) = 6
142	141, 77	oveq12i 6100	. . . . . . 7 ⊢ ((1 · 6) + (0 + 2)) = (6 + 2)
143		6p2e8 9463	. . . . . . 7 ⊢ (6 + 2) = 8
144	142, 143	eqtri 2259	. . . . . 6 ⊢ ((1 · 6) + (0 + 2)) = 8
145	120, 45, 136	mulcomli 8339	. . . . . . 7 ⊢ (3 · 6) = ;18
146		1p1e2 9430	. . . . . . 7 ⊢ (1 + 1) = 2
147		8p3e11 9873	. . . . . . 7 ⊢ (8 + 3) = ;11
148	9, 19, 15, 145, 146, 9, 147	decaddci 9853	. . . . . 6 ⊢ ((3 · 6) + 3) = ;21
149	9, 15, 21, 15, 103, 140, 17, 9, 24, 144, 148	decmac 9844	. . . . 5 ⊢ ((;13 · 6) + 3) = ;81
150		6t6e36 9900	. . . . 5 ⊢ (6 · 6) = ;36
151	17, 16, 17, 101, 17, 15, 149, 150	decmul1c 9857	. . . 4 ⊢ (;;136 · 6) = ;;816
152	18, 16, 17, 101, 17, 102, 139, 151	decmul2c 9858	. . 3 ⊢ (;;136 · ;;136) = ;;;;18496
153	100, 152	eqtr4i 2262	. 2 ⊢ ((;14 · 𝑁) + ;;870) = (;;136 · ;;136)
154	7, 8, 11, 14, 18, 22, 23, 34, 153	mod2xi 13225	1 ⊢ ((2↑;34) mod 𝑁) = (;;870 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:  1259lem3  13273  1259lem5  13275