Theorem 1259lem2 15677

Description: Lemma for 1259prm 15681. 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		1nn0 11185	. . . . . 6 ⊢ 1 ∈ ℕ0
3		2nn0 11186	. . . . . 6 ⊢ 2 ∈ ℕ0
4	2, 3	deccl 11388	. . . . 5 ⊢ ;12 ∈ ℕ0
5		5nn0 11189	. . . . 5 ⊢ 5 ∈ ℕ0
6	4, 5	deccl 11388	. . . 4 ⊢ ;;125 ∈ ℕ0
7		9nn 11069	. . . 4 ⊢ 9 ∈ ℕ
8	6, 7	decnncl 11394	. . 3 ⊢ ;;;1259 ∈ ℕ
9	1, 8	eqeltri 2684	. 2 ⊢ 𝑁 ∈ ℕ
10		2nn 11062	. 2 ⊢ 2 ∈ ℕ
11		7nn0 11191	. . 3 ⊢ 7 ∈ ℕ0
12	2, 11	deccl 11388	. 2 ⊢ ;17 ∈ ℕ0
13		4nn0 11188	. . . 4 ⊢ 4 ∈ ℕ0
14	2, 13	deccl 11388	. . 3 ⊢ ;14 ∈ ℕ0
15	14	nn0zi 11279	. 2 ⊢ ;14 ∈ ℤ
16		3nn0 11187	. . . 4 ⊢ 3 ∈ ℕ0
17	2, 16	deccl 11388	. . 3 ⊢ ;13 ∈ ℕ0
18		6nn0 11190	. . 3 ⊢ 6 ∈ ℕ0
19	17, 18	deccl 11388	. 2 ⊢ ;;136 ∈ ℕ0
20		8nn0 11192	. . . 4 ⊢ 8 ∈ ℕ0
21	20, 11	deccl 11388	. . 3 ⊢ ;87 ∈ ℕ0
22		0nn0 11184	. . 3 ⊢ 0 ∈ ℕ0
23	21, 22	deccl 11388	. 2 ⊢ ;;870 ∈ ℕ0
24	1	1259lem1 15676	. 2 ⊢ ((2↑;17) mod 𝑁) = (;;136 mod 𝑁)
25		eqid 2610	. . 3 ⊢ ;17 = ;17
26		2cn 10968	. . . . . 6 ⊢ 2 ∈ ℂ
27	26	mulid1i 9921	. . . . 5 ⊢ (2 · 1) = 2
28	27	oveq1i 6559	. . . 4 ⊢ ((2 · 1) + 1) = (2 + 1)
29		2p1e3 11028	. . . 4 ⊢ (2 + 1) = 3
30	28, 29	eqtri 2632	. . 3 ⊢ ((2 · 1) + 1) = 3
31		7cn 10981	. . . 4 ⊢ 7 ∈ ℂ
32		7t2e14 11524	. . . 4 ⊢ (7 · 2) = ;14
33	31, 26, 32	mulcomli 9926	. . 3 ⊢ (2 · 7) = ;14
34	3, 2, 11, 25, 13, 2, 30, 33	decmul2c 11465	. 2 ⊢ (2 · ;17) = ;34
35		9nn0 11193	. . . 4 ⊢ 9 ∈ ℕ0
36		eqid 2610	. . . 4 ⊢ ;;870 = ;;870
37		eqid 2610	. . . . 5 ⊢ ;;125 = ;;125
38		eqid 2610	. . . . . 6 ⊢ ;87 = ;87
39		eqid 2610	. . . . . 6 ⊢ ;12 = ;12
40		8p1e9 11035	. . . . . 6 ⊢ (8 + 1) = 9
41		7p2e9 11049	. . . . . 6 ⊢ (7 + 2) = 9
42	20, 11, 2, 3, 38, 39, 40, 41	decadd 11446	. . . . 5 ⊢ (;87 + ;12) = ;99
43		9p7e16 11501	. . . . . 6 ⊢ (9 + 7) = ;16
44		eqid 2610	. . . . . . 7 ⊢ ;14 = ;14
45		3cn 10972	. . . . . . . . 9 ⊢ 3 ∈ ℂ
46		ax-1cn 9873	. . . . . . . . 9 ⊢ 1 ∈ ℂ
47		3p1e4 11030	. . . . . . . . 9 ⊢ (3 + 1) = 4
48	45, 46, 47	addcomli 10107	. . . . . . . 8 ⊢ (1 + 3) = 4
49	13	dec0h 11398	. . . . . . . 8 ⊢ 4 = ;04
50	48, 49	eqtri 2632	. . . . . . 7 ⊢ (1 + 3) = ;04
51	46	mulid1i 9921	. . . . . . . . 9 ⊢ (1 · 1) = 1
52		00id 10090	. . . . . . . . 9 ⊢ (0 + 0) = 0
53	51, 52	oveq12i 6561	. . . . . . . 8 ⊢ ((1 · 1) + (0 + 0)) = (1 + 0)
54	46	addid1i 10102	. . . . . . . 8 ⊢ (1 + 0) = 1
55	53, 54	eqtri 2632	. . . . . . 7 ⊢ ((1 · 1) + (0 + 0)) = 1
56		4cn 10975	. . . . . . . . . 10 ⊢ 4 ∈ ℂ
57	56	mulid1i 9921	. . . . . . . . 9 ⊢ (4 · 1) = 4
58	57	oveq1i 6559	. . . . . . . 8 ⊢ ((4 · 1) + 4) = (4 + 4)
59		4p4e8 11041	. . . . . . . 8 ⊢ (4 + 4) = 8
60	20	dec0h 11398	. . . . . . . 8 ⊢ 8 = ;08
61	58, 59, 60	3eqtri 2636	. . . . . . 7 ⊢ ((4 · 1) + 4) = ;08
62	2, 13, 22, 13, 44, 50, 2, 20, 22, 55, 61	decmac 11442	. . . . . 6 ⊢ ((;14 · 1) + (1 + 3)) = ;18
63	18	dec0h 11398	. . . . . . 7 ⊢ 6 = ;06
64	26	mulid2i 9922	. . . . . . . . 9 ⊢ (1 · 2) = 2
65	46	addid2i 10103	. . . . . . . . 9 ⊢ (0 + 1) = 1
66	64, 65	oveq12i 6561	. . . . . . . 8 ⊢ ((1 · 2) + (0 + 1)) = (2 + 1)
67	66, 29	eqtri 2632	. . . . . . 7 ⊢ ((1 · 2) + (0 + 1)) = 3
68		4t2e8 11058	. . . . . . . . 9 ⊢ (4 · 2) = 8
69	68	oveq1i 6559	. . . . . . . 8 ⊢ ((4 · 2) + 6) = (8 + 6)
70		8p6e14 11492	. . . . . . . 8 ⊢ (8 + 6) = ;14
71	69, 70	eqtri 2632	. . . . . . 7 ⊢ ((4 · 2) + 6) = ;14
72	2, 13, 22, 18, 44, 63, 3, 13, 2, 67, 71	decmac 11442	. . . . . 6 ⊢ ((;14 · 2) + 6) = ;34
73	2, 3, 2, 18, 39, 43, 14, 13, 16, 62, 72	decma2c 11444	. . . . 5 ⊢ ((;14 · ;12) + (9 + 7)) = ;;184
74	35	dec0h 11398	. . . . . 6 ⊢ 9 = ;09
75		5cn 10977	. . . . . . . . 9 ⊢ 5 ∈ ℂ
76	75	mulid2i 9922	. . . . . . . 8 ⊢ (1 · 5) = 5
77	26	addid2i 10103	. . . . . . . 8 ⊢ (0 + 2) = 2
78	76, 77	oveq12i 6561	. . . . . . 7 ⊢ ((1 · 5) + (0 + 2)) = (5 + 2)
79		5p2e7 11042	. . . . . . 7 ⊢ (5 + 2) = 7
80	78, 79	eqtri 2632	. . . . . 6 ⊢ ((1 · 5) + (0 + 2)) = 7
81		5t4e20 11513	. . . . . . . 8 ⊢ (5 · 4) = ;20
82	75, 56, 81	mulcomli 9926	. . . . . . 7 ⊢ (4 · 5) = ;20
83		9cn 10985	. . . . . . . 8 ⊢ 9 ∈ ℂ
84	83	addid2i 10103	. . . . . . 7 ⊢ (0 + 9) = 9
85	3, 22, 35, 82, 84	decaddi 11455	. . . . . 6 ⊢ ((4 · 5) + 9) = ;29
86	2, 13, 22, 35, 44, 74, 5, 35, 3, 80, 85	decmac 11442	. . . . 5 ⊢ ((;14 · 5) + 9) = ;79
87	4, 5, 35, 35, 37, 42, 14, 35, 11, 73, 86	decma2c 11444	. . . 4 ⊢ ((;14 · ;;125) + (;87 + ;12)) = ;;;1849
88	83	mulid2i 9922	. . . . . . . . 9 ⊢ (1 · 9) = 9
89	88	oveq1i 6559	. . . . . . . 8 ⊢ ((1 · 9) + 3) = (9 + 3)
90		9p3e12 11497	. . . . . . . 8 ⊢ (9 + 3) = ;12
91	89, 90	eqtri 2632	. . . . . . 7 ⊢ ((1 · 9) + 3) = ;12
92		9t4e36 11541	. . . . . . . 8 ⊢ (9 · 4) = ;36
93	83, 56, 92	mulcomli 9926	. . . . . . 7 ⊢ (4 · 9) = ;36
94	35, 2, 13, 44, 18, 16, 91, 93	decmul1c 11463	. . . . . 6 ⊢ (;14 · 9) = ;;126
95	94	oveq1i 6559	. . . . 5 ⊢ ((;14 · 9) + 0) = (;;126 + 0)
96	4, 18	deccl 11388	. . . . . . 7 ⊢ ;;126 ∈ ℕ0
97	96	nn0cni 11181	. . . . . 6 ⊢ ;;126 ∈ ℂ
98	97	addid1i 10102	. . . . 5 ⊢ (;;126 + 0) = ;;126
99	95, 98	eqtri 2632	. . . 4 ⊢ ((;14 · 9) + 0) = ;;126
100	6, 35, 21, 22, 1, 36, 14, 18, 4, 87, 99	decma2c 11444	. . 3 ⊢ ((;14 · 𝑁) + ;;870) = ;;;;18496
101		eqid 2610	. . . 4 ⊢ ;;136 = ;;136
102	20, 2	deccl 11388	. . . 4 ⊢ ;81 ∈ ℕ0
103		eqid 2610	. . . . 5 ⊢ ;13 = ;13
104		eqid 2610	. . . . 5 ⊢ ;81 = ;81
105	13, 22	deccl 11388	. . . . 5 ⊢ ;40 ∈ ℕ0
106		eqid 2610	. . . . . . 7 ⊢ ;40 = ;40
107	56	addid2i 10103	. . . . . . 7 ⊢ (0 + 4) = 4
108		8cn 10983	. . . . . . . 8 ⊢ 8 ∈ ℂ
109	108	addid1i 10102	. . . . . . 7 ⊢ (8 + 0) = 8
110	22, 20, 13, 22, 60, 106, 107, 109	decadd 11446	. . . . . 6 ⊢ (8 + ;40) = ;48
111		4p1e5 11031	. . . . . . . 8 ⊢ (4 + 1) = 5
112	5	dec0h 11398	. . . . . . . 8 ⊢ 5 = ;05
113	111, 112	eqtri 2632	. . . . . . 7 ⊢ (4 + 1) = ;05
114	45	mulid1i 9921	. . . . . . . . 9 ⊢ (3 · 1) = 3
115	114	oveq1i 6559	. . . . . . . 8 ⊢ ((3 · 1) + 5) = (3 + 5)
116		5p3e8 11043	. . . . . . . . 9 ⊢ (5 + 3) = 8
117	75, 45, 116	addcomli 10107	. . . . . . . 8 ⊢ (3 + 5) = 8
118	115, 117, 60	3eqtri 2636	. . . . . . 7 ⊢ ((3 · 1) + 5) = ;08
119	2, 16, 22, 5, 103, 113, 2, 20, 22, 55, 118	decmac 11442	. . . . . 6 ⊢ ((;13 · 1) + (4 + 1)) = ;18
120		6cn 10979	. . . . . . . . 9 ⊢ 6 ∈ ℂ
121	120	mulid1i 9921	. . . . . . . 8 ⊢ (6 · 1) = 6
122	121	oveq1i 6559	. . . . . . 7 ⊢ ((6 · 1) + 8) = (6 + 8)
123	108, 120, 70	addcomli 10107	. . . . . . 7 ⊢ (6 + 8) = ;14
124	122, 123	eqtri 2632	. . . . . 6 ⊢ ((6 · 1) + 8) = ;14
125	17, 18, 13, 20, 101, 110, 2, 13, 2, 119, 124	decmac 11442	. . . . 5 ⊢ ((;;136 · 1) + (8 + ;40)) = ;;184
126	2	dec0h 11398	. . . . . 6 ⊢ 1 = ;01
127	65, 126	eqtri 2632	. . . . . . 7 ⊢ (0 + 1) = ;01
128	45	mulid2i 9922	. . . . . . . . 9 ⊢ (1 · 3) = 3
129	128, 65	oveq12i 6561	. . . . . . . 8 ⊢ ((1 · 3) + (0 + 1)) = (3 + 1)
130	129, 47	eqtri 2632	. . . . . . 7 ⊢ ((1 · 3) + (0 + 1)) = 4
131		3t3e9 11057	. . . . . . . . 9 ⊢ (3 · 3) = 9
132	131	oveq1i 6559	. . . . . . . 8 ⊢ ((3 · 3) + 1) = (9 + 1)
133		9p1e10 11372	. . . . . . . 8 ⊢ (9 + 1) = ;10
134	132, 133	eqtri 2632	. . . . . . 7 ⊢ ((3 · 3) + 1) = ;10
135	2, 16, 22, 2, 103, 127, 16, 22, 2, 130, 134	decmac 11442	. . . . . 6 ⊢ ((;13 · 3) + (0 + 1)) = ;40
136		6t3e18 11518	. . . . . . 7 ⊢ (6 · 3) = ;18
137	2, 20, 2, 136, 40	decaddi 11455	. . . . . 6 ⊢ ((6 · 3) + 1) = ;19
138	17, 18, 22, 2, 101, 126, 16, 35, 2, 135, 137	decmac 11442	. . . . 5 ⊢ ((;;136 · 3) + 1) = ;;409
139	2, 16, 20, 2, 103, 104, 19, 35, 105, 125, 138	decma2c 11444	. . . 4 ⊢ ((;;136 · ;13) + ;81) = ;;;1849
140	16	dec0h 11398	. . . . . 6 ⊢ 3 = ;03
141	120	mulid2i 9922	. . . . . . . 8 ⊢ (1 · 6) = 6
142	141, 77	oveq12i 6561	. . . . . . 7 ⊢ ((1 · 6) + (0 + 2)) = (6 + 2)
143		6p2e8 11046	. . . . . . 7 ⊢ (6 + 2) = 8
144	142, 143	eqtri 2632	. . . . . 6 ⊢ ((1 · 6) + (0 + 2)) = 8
145	120, 45, 136	mulcomli 9926	. . . . . . 7 ⊢ (3 · 6) = ;18
146		1p1e2 11011	. . . . . . 7 ⊢ (1 + 1) = 2
147		8p3e11 11488	. . . . . . 7 ⊢ (8 + 3) = ;11
148	2, 20, 16, 145, 146, 2, 147	decaddci 11456	. . . . . 6 ⊢ ((3 · 6) + 3) = ;21
149	2, 16, 22, 16, 103, 140, 18, 2, 3, 144, 148	decmac 11442	. . . . 5 ⊢ ((;13 · 6) + 3) = ;81
150		6t6e36 11522	. . . . 5 ⊢ (6 · 6) = ;36
151	18, 17, 18, 101, 18, 16, 149, 150	decmul1c 11463	. . . 4 ⊢ (;;136 · 6) = ;;816
152	19, 17, 18, 101, 18, 102, 139, 151	decmul2c 11465	. . 3 ⊢ (;;136 · ;;136) = ;;;;18496
153	100, 152	eqtr4i 2635	. 2 ⊢ ((;14 · 𝑁) + ;;870) = (;;136 · ;;136)
154	9, 10, 12, 15, 19, 23, 24, 34, 153	mod2xi 15611	1 ⊢ ((2↑;34) mod 𝑁) = (;;870 mod 𝑁)

Syntax hints:   = wceq 1475  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818   · cmul 9820  ℕcn 10897  2c2 10947  3c3 10948  4c4 10949  5c5 10950  6c6 10951  7c7 10952  8c8 10953  9c9 10954  ;cdc 11369   mod cmo 12530  ↑cexp 12722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-sup 8231  df-inf 8232  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-rp 11709  df-fl 12455  df-mod 12531  df-seq 12664  df-exp 12723

This theorem is referenced by:  1259lem3  15678  1259lem5  15680