Theorem 1259lem2 17303

Description: Lemma for 1259prm 17307. 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 12824	. . . . 5 ⊢ ;12 ∈ ℕ0
3		5nn0 12619	. . . . 5 ⊢ 5 ∈ ℕ0
4	2, 3	deccl 12822	. . . 4 ⊢ ;;125 ∈ ℕ0
5		9nn 12434	. . . 4 ⊢ 9 ∈ ℕ
6	4, 5	decnncl 12831	. . 3 ⊢ ;;;1259 ∈ ℕ
7	1, 6	eqeltri 2857	. 2 ⊢ 𝑁 ∈ ℕ
8		2nn 12409	. 2 ⊢ 2 ∈ ℕ
9		1nn0 12615	. . 3 ⊢ 1 ∈ ℕ0
10		7nn0 12621	. . 3 ⊢ 7 ∈ ℕ0
11	9, 10	deccl 12822	. 2 ⊢ ;17 ∈ ℕ0
12		4nn0 12618	. . . 4 ⊢ 4 ∈ ℕ0
13	9, 12	deccl 12822	. . 3 ⊢ ;14 ∈ ℕ0
14	13	nn0zi 12714	. 2 ⊢ ;14 ∈ ℤ
15		3nn0 12617	. . . 4 ⊢ 3 ∈ ℕ0
16	9, 15	deccl 12822	. . 3 ⊢ ;13 ∈ ℕ0
17		6nn0 12620	. . 3 ⊢ 6 ∈ ℕ0
18	16, 17	deccl 12822	. 2 ⊢ ;;136 ∈ ℕ0
19		8nn0 12622	. . . 4 ⊢ 8 ∈ ℕ0
20	19, 10	deccl 12822	. . 3 ⊢ ;87 ∈ ℕ0
21		0nn0 12614	. . 3 ⊢ 0 ∈ ℕ0
22	20, 21	deccl 12822	. 2 ⊢ ;;870 ∈ ℕ0
23	1	1259lem1 17302	. 2 ⊢ ((2↑;17) mod 𝑁) = (;;136 mod 𝑁)
24		2nn0 12616	. . 3 ⊢ 2 ∈ ℕ0
25		eqid 2761	. . 3 ⊢ ;17 = ;17
26		2cn 12411	. . . . . 6 ⊢ 2 ∈ ℂ
27	26	mulridi 11306	. . . . 5 ⊢ (2 · 1) = 2
28	27	oveq1i 7428	. . . 4 ⊢ ((2 · 1) + 1) = (2 + 1)
29		2p1e3 12477	. . . 4 ⊢ (2 + 1) = 3
30	28, 29	eqtri 2784	. . 3 ⊢ ((2 · 1) + 1) = 3
31		7cn 12430	. . . 4 ⊢ 7 ∈ ℂ
32		7t2e14 12921	. . . 4 ⊢ (7 · 2) = ;14
33	31, 26, 32	mulcomli 11311	. . 3 ⊢ (2 · 7) = ;14
34	24, 9, 10, 25, 12, 9, 30, 33	decmul2c 12878	. 2 ⊢ (2 · ;17) = ;34
35		9nn0 12623	. . . 4 ⊢ 9 ∈ ℕ0
36		eqid 2761	. . . 4 ⊢ ;;870 = ;;870
37		eqid 2761	. . . . 5 ⊢ ;;125 = ;;125
38		eqid 2761	. . . . . 6 ⊢ ;87 = ;87
39		eqid 2761	. . . . . 6 ⊢ ;12 = ;12
40		8p1e9 12485	. . . . . 6 ⊢ (8 + 1) = 9
41		7p2e9 12496	. . . . . 6 ⊢ (7 + 2) = 9
42	19, 10, 9, 24, 38, 39, 40, 41	decadd 12866	. . . . 5 ⊢ (;87 + ;12) = ;99
43		9p7e16 12904	. . . . . 6 ⊢ (9 + 7) = ;16
44		eqid 2761	. . . . . . 7 ⊢ ;14 = ;14
45		3cn 12417	. . . . . . . . 9 ⊢ 3 ∈ ℂ
46		ax-1cn 11251	. . . . . . . . 9 ⊢ 1 ∈ ℂ
47		3p1e4 12480	. . . . . . . . 9 ⊢ (3 + 1) = 4
48	45, 46, 47	addcomli 11495	. . . . . . . 8 ⊢ (1 + 3) = 4
49	12	dec0h 12834	. . . . . . . 8 ⊢ 4 = ;04
50	48, 49	eqtri 2784	. . . . . . 7 ⊢ (1 + 3) = ;04
51	46	mulridi 11306	. . . . . . . . 9 ⊢ (1 · 1) = 1
52		00id 11478	. . . . . . . . 9 ⊢ (0 + 0) = 0
53	51, 52	oveq12i 7430	. . . . . . . 8 ⊢ ((1 · 1) + (0 + 0)) = (1 + 0)
54	46	addridi 11490	. . . . . . . 8 ⊢ (1 + 0) = 1
55	53, 54	eqtri 2784	. . . . . . 7 ⊢ ((1 · 1) + (0 + 0)) = 1
56		4cn 12421	. . . . . . . . . 10 ⊢ 4 ∈ ℂ
57	56	mulridi 11306	. . . . . . . . 9 ⊢ (4 · 1) = 4
58	57	oveq1i 7428	. . . . . . . 8 ⊢ ((4 · 1) + 4) = (4 + 4)
59		4p4e8 12490	. . . . . . . 8 ⊢ (4 + 4) = 8
60	19	dec0h 12834	. . . . . . . 8 ⊢ 8 = ;08
61	58, 59, 60	3eqtri 2788	. . . . . . 7 ⊢ ((4 · 1) + 4) = ;08
62	9, 12, 21, 12, 44, 50, 9, 19, 21, 55, 61	decmac 12864	. . . . . 6 ⊢ ((;14 · 1) + (1 + 3)) = ;18
63	17	dec0h 12834	. . . . . . 7 ⊢ 6 = ;06
64	26	mullidi 11307	. . . . . . . . 9 ⊢ (1 · 2) = 2
65	46	addlidi 11491	. . . . . . . . 9 ⊢ (0 + 1) = 1
66	64, 65	oveq12i 7430	. . . . . . . 8 ⊢ ((1 · 2) + (0 + 1)) = (2 + 1)
67	66, 29	eqtri 2784	. . . . . . 7 ⊢ ((1 · 2) + (0 + 1)) = 3
68		4t2e8 12504	. . . . . . . . 9 ⊢ (4 · 2) = 8
69	68	oveq1i 7428	. . . . . . . 8 ⊢ ((4 · 2) + 6) = (8 + 6)
70		8p6e14 12896	. . . . . . . 8 ⊢ (8 + 6) = ;14
71	69, 70	eqtri 2784	. . . . . . 7 ⊢ ((4 · 2) + 6) = ;14
72	9, 12, 21, 17, 44, 63, 24, 12, 9, 67, 71	decmac 12864	. . . . . 6 ⊢ ((;14 · 2) + 6) = ;34
73	9, 24, 9, 17, 39, 43, 13, 12, 15, 62, 72	decma2c 12865	. . . . 5 ⊢ ((;14 · ;12) + (9 + 7)) = ;;184
74	35	dec0h 12834	. . . . . 6 ⊢ 9 = ;09
75		5cn 12424	. . . . . . . . 9 ⊢ 5 ∈ ℂ
76	75	mullidi 11307	. . . . . . . 8 ⊢ (1 · 5) = 5
77	26	addlidi 11491	. . . . . . . 8 ⊢ (0 + 2) = 2
78	76, 77	oveq12i 7430	. . . . . . 7 ⊢ ((1 · 5) + (0 + 2)) = (5 + 2)
79		5p2e7 12491	. . . . . . 7 ⊢ (5 + 2) = 7
80	78, 79	eqtri 2784	. . . . . 6 ⊢ ((1 · 5) + (0 + 2)) = 7
81		5t4e20 12914	. . . . . . . 8 ⊢ (5 · 4) = ;20
82	75, 56, 81	mulcomli 11311	. . . . . . 7 ⊢ (4 · 5) = ;20
83		9cn 12436	. . . . . . . 8 ⊢ 9 ∈ ℂ
84	83	addlidi 11491	. . . . . . 7 ⊢ (0 + 9) = 9
85	24, 21, 35, 82, 84	decaddi 12872	. . . . . 6 ⊢ ((4 · 5) + 9) = ;29
86	9, 12, 21, 35, 44, 74, 3, 35, 24, 80, 85	decmac 12864	. . . . 5 ⊢ ((;14 · 5) + 9) = ;79
87	2, 3, 35, 35, 37, 42, 13, 35, 10, 73, 86	decma2c 12865	. . . 4 ⊢ ((;14 · ;;125) + (;87 + ;12)) = ;;;1849
88	83	mullidi 11307	. . . . . . . . 9 ⊢ (1 · 9) = 9
89	88	oveq1i 7428	. . . . . . . 8 ⊢ ((1 · 9) + 3) = (9 + 3)
90		9p3e12 12900	. . . . . . . 8 ⊢ (9 + 3) = ;12
91	89, 90	eqtri 2784	. . . . . . 7 ⊢ ((1 · 9) + 3) = ;12
92		9t4e36 12936	. . . . . . . 8 ⊢ (9 · 4) = ;36
93	83, 56, 92	mulcomli 11311	. . . . . . 7 ⊢ (4 · 9) = ;36
94	35, 9, 12, 44, 17, 15, 91, 93	decmul1c 12877	. . . . . 6 ⊢ (;14 · 9) = ;;126
95	94	oveq1i 7428	. . . . 5 ⊢ ((;14 · 9) + 0) = (;;126 + 0)
96	2, 17	deccl 12822	. . . . . . 7 ⊢ ;;126 ∈ ℕ0
97	96	nn0cni 12611	. . . . . 6 ⊢ ;;126 ∈ ℂ
98	97	addridi 11490	. . . . 5 ⊢ (;;126 + 0) = ;;126
99	95, 98	eqtri 2784	. . . 4 ⊢ ((;14 · 9) + 0) = ;;126
100	4, 35, 20, 21, 1, 36, 13, 17, 2, 87, 99	decma2c 12865	. . 3 ⊢ ((;14 · 𝑁) + ;;870) = ;;;;18496
101		eqid 2761	. . . 4 ⊢ ;;136 = ;;136
102	19, 9	deccl 12822	. . . 4 ⊢ ;81 ∈ ℕ0
103		eqid 2761	. . . . 5 ⊢ ;13 = ;13
104		eqid 2761	. . . . 5 ⊢ ;81 = ;81
105	12, 21	deccl 12822	. . . . 5 ⊢ ;40 ∈ ℕ0
106		eqid 2761	. . . . . . 7 ⊢ ;40 = ;40
107	56	addlidi 11491	. . . . . . 7 ⊢ (0 + 4) = 4
108		8cn 12433	. . . . . . . 8 ⊢ 8 ∈ ℂ
109	108	addridi 11490	. . . . . . 7 ⊢ (8 + 0) = 8
110	21, 19, 12, 21, 60, 106, 107, 109	decadd 12866	. . . . . 6 ⊢ (8 + ;40) = ;48
111		4p1e5 12481	. . . . . . . 8 ⊢ (4 + 1) = 5
112	3	dec0h 12834	. . . . . . . 8 ⊢ 5 = ;05
113	111, 112	eqtri 2784	. . . . . . 7 ⊢ (4 + 1) = ;05
114	45	mulridi 11306	. . . . . . . . 9 ⊢ (3 · 1) = 3
115	114	oveq1i 7428	. . . . . . . 8 ⊢ ((3 · 1) + 5) = (3 + 5)
116		5p3e8 12492	. . . . . . . . 9 ⊢ (5 + 3) = 8
117	75, 45, 116	addcomli 11495	. . . . . . . 8 ⊢ (3 + 5) = 8
118	115, 117, 60	3eqtri 2788	. . . . . . 7 ⊢ ((3 · 1) + 5) = ;08
119	9, 15, 21, 3, 103, 113, 9, 19, 21, 55, 118	decmac 12864	. . . . . 6 ⊢ ((;13 · 1) + (4 + 1)) = ;18
120		6cn 12427	. . . . . . . . 9 ⊢ 6 ∈ ℂ
121	120	mulridi 11306	. . . . . . . 8 ⊢ (6 · 1) = 6
122	121	oveq1i 7428	. . . . . . 7 ⊢ ((6 · 1) + 8) = (6 + 8)
123	108, 120, 70	addcomli 11495	. . . . . . 7 ⊢ (6 + 8) = ;14
124	122, 123	eqtri 2784	. . . . . 6 ⊢ ((6 · 1) + 8) = ;14
125	16, 17, 12, 19, 101, 110, 9, 12, 9, 119, 124	decmac 12864	. . . . 5 ⊢ ((;;136 · 1) + (8 + ;40)) = ;;184
126	9	dec0h 12834	. . . . . 6 ⊢ 1 = ;01
127	65, 126	eqtri 2784	. . . . . . 7 ⊢ (0 + 1) = ;01
128	45	mullidi 11307	. . . . . . . . 9 ⊢ (1 · 3) = 3
129	128, 65	oveq12i 7430	. . . . . . . 8 ⊢ ((1 · 3) + (0 + 1)) = (3 + 1)
130	129, 47	eqtri 2784	. . . . . . 7 ⊢ ((1 · 3) + (0 + 1)) = 4
131		3t3e9 12503	. . . . . . . . 9 ⊢ (3 · 3) = 9
132	131	oveq1i 7428	. . . . . . . 8 ⊢ ((3 · 3) + 1) = (9 + 1)
133		9p1e10 12809	. . . . . . . 8 ⊢ (9 + 1) = ;10
134	132, 133	eqtri 2784	. . . . . . 7 ⊢ ((3 · 3) + 1) = ;10
135	9, 15, 21, 9, 103, 127, 15, 21, 9, 130, 134	decmac 12864	. . . . . 6 ⊢ ((;13 · 3) + (0 + 1)) = ;40
136		6t3e18 12917	. . . . . . 7 ⊢ (6 · 3) = ;18
137	9, 19, 9, 136, 40	decaddi 12872	. . . . . 6 ⊢ ((6 · 3) + 1) = ;19
138	16, 17, 21, 9, 101, 126, 15, 35, 9, 135, 137	decmac 12864	. . . . 5 ⊢ ((;;136 · 3) + 1) = ;;409
139	9, 15, 19, 9, 103, 104, 18, 35, 105, 125, 138	decma2c 12865	. . . 4 ⊢ ((;;136 · ;13) + ;81) = ;;;1849
140	15	dec0h 12834	. . . . . 6 ⊢ 3 = ;03
141	120	mullidi 11307	. . . . . . . 8 ⊢ (1 · 6) = 6
142	141, 77	oveq12i 7430	. . . . . . 7 ⊢ ((1 · 6) + (0 + 2)) = (6 + 2)
143		6p2e8 12494	. . . . . . 7 ⊢ (6 + 2) = 8
144	142, 143	eqtri 2784	. . . . . 6 ⊢ ((1 · 6) + (0 + 2)) = 8
145	120, 45, 136	mulcomli 11311	. . . . . . 7 ⊢ (3 · 6) = ;18
146		1p1e2 12459	. . . . . . 7 ⊢ (1 + 1) = 2
147		8p3e11 12893	. . . . . . 7 ⊢ (8 + 3) = ;11
148	9, 19, 15, 145, 146, 9, 147	decaddci 12873	. . . . . 6 ⊢ ((3 · 6) + 3) = ;21
149	9, 15, 21, 15, 103, 140, 17, 9, 24, 144, 148	decmac 12864	. . . . 5 ⊢ ((;13 · 6) + 3) = ;81
150		6t6e36 12920	. . . . 5 ⊢ (6 · 6) = ;36
151	17, 16, 17, 101, 17, 15, 149, 150	decmul1c 12877	. . . 4 ⊢ (;;136 · 6) = ;;816
152	18, 16, 17, 101, 17, 102, 139, 151	decmul2c 12878	. . 3 ⊢ (;;136 · ;;136) = ;;;;18496
153	100, 152	eqtr4i 2787	. 2 ⊢ ((;14 · 𝑁) + ;;870) = (;;136 · ;;136)
154	7, 8, 11, 14, 18, 22, 23, 34, 153	mod2xi 17240	1 ⊢ ((2↑;34) mod 𝑁) = (;;870 mod 𝑁)

This proof depends on syntax axioms:   = wceq 1570  (class class class)co 7418  0cc0 11193  1c1 11194   + caddc 11196   · cmul 11198  ℕcn 12328  2c2 12390  3c3 12391  4c4 12392  5c5 12393  6c6 12394  7c7 12395  8c8 12396  9c9 12397  ;cdc 12807   mod cmo 14002  ↑cexp 14197

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 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-cnex 11249  ax-resscn 11250  ax-1cn 11251  ax-icn 11252  ax-addcl 11253  ax-addrcl 11254  ax-mulcl 11255  ax-mulrcl 11256  ax-mulcom 11257  ax-addass 11258  ax-mulass 11259  ax-distr 11260  ax-i2m1 11261  ax-1ne0 11262  ax-1rid 11263  ax-rnegex 11264  ax-rrecex 11265  ax-cnre 11266  ax-pre-lttri 11267  ax-pre-lttrn 11268  ax-pre-ltadd 11269  ax-pre-mulgt0 11270  ax-pre-sup 11271

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  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 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-om 7876  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-er 8710  df-en 8967  df-dom 8968  df-sdom 8969  df-sup 9427  df-inf 9428  df-pnf 11338  df-mnf 11339  df-xr 11340  df-ltxr 11341  df-le 11342  df-sub 11536  df-neg 11537  df-div 11967  df-nn 12329  df-2 12398  df-3 12399  df-4 12400  df-5 12401  df-6 12402  df-7 12403  df-8 12404  df-9 12405  df-n0 12600  df-z 12687  df-dec 12808  df-uz 12959  df-rp 13114  df-fl 13925  df-mod 14003  df-seq 14138  df-exp 14198

This theorem is used by:  1259lem3  17304  1259lem5  17306