Theorem 1259lem2 17071

Description: Lemma for 1259prm 17075. 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 12429	. . . . . 6 ⊢ 1 ∈ ℕ0
3		2nn0 12430	. . . . . 6 ⊢ 2 ∈ ℕ0
4	2, 3	deccl 12634	. . . . 5 ⊢ ;12 ∈ ℕ0
5		5nn0 12433	. . . . 5 ⊢ 5 ∈ ℕ0
6	4, 5	deccl 12634	. . . 4 ⊢ ;;125 ∈ ℕ0
7		9nn 12255	. . . 4 ⊢ 9 ∈ ℕ
8	6, 7	decnncl 12639	. . 3 ⊢ ;;;1259 ∈ ℕ
9	1, 8	eqeltri 2833	. 2 ⊢ 𝑁 ∈ ℕ
10		2nn 12230	. 2 ⊢ 2 ∈ ℕ
11		7nn0 12435	. . 3 ⊢ 7 ∈ ℕ0
12	2, 11	deccl 12634	. 2 ⊢ ;17 ∈ ℕ0
13		4nn0 12432	. . . 4 ⊢ 4 ∈ ℕ0
14	2, 13	deccl 12634	. . 3 ⊢ ;14 ∈ ℕ0
15	14	nn0zi 12528	. 2 ⊢ ;14 ∈ ℤ
16		3nn0 12431	. . . 4 ⊢ 3 ∈ ℕ0
17	2, 16	deccl 12634	. . 3 ⊢ ;13 ∈ ℕ0
18		6nn0 12434	. . 3 ⊢ 6 ∈ ℕ0
19	17, 18	deccl 12634	. 2 ⊢ ;;136 ∈ ℕ0
20		8nn0 12436	. . . 4 ⊢ 8 ∈ ℕ0
21	20, 11	deccl 12634	. . 3 ⊢ ;87 ∈ ℕ0
22		0nn0 12428	. . 3 ⊢ 0 ∈ ℕ0
23	21, 22	deccl 12634	. 2 ⊢ ;;870 ∈ ℕ0
24	1	1259lem1 17070	. 2 ⊢ ((2↑;17) mod 𝑁) = (;;136 mod 𝑁)
25		eqid 2737	. . 3 ⊢ ;17 = ;17
26		2cn 12232	. . . . . 6 ⊢ 2 ∈ ℂ
27	26	mulridi 11148	. . . . 5 ⊢ (2 · 1) = 2
28	27	oveq1i 7378	. . . 4 ⊢ ((2 · 1) + 1) = (2 + 1)
29		2p1e3 12294	. . . 4 ⊢ (2 + 1) = 3
30	28, 29	eqtri 2760	. . 3 ⊢ ((2 · 1) + 1) = 3
31		7cn 12251	. . . 4 ⊢ 7 ∈ ℂ
32		7t2e14 12728	. . . 4 ⊢ (7 · 2) = ;14
33	31, 26, 32	mulcomli 11153	. . 3 ⊢ (2 · 7) = ;14
34	3, 2, 11, 25, 13, 2, 30, 33	decmul2c 12685	. 2 ⊢ (2 · ;17) = ;34
35		9nn0 12437	. . . 4 ⊢ 9 ∈ ℕ0
36		eqid 2737	. . . 4 ⊢ ;;870 = ;;870
37		eqid 2737	. . . . 5 ⊢ ;;125 = ;;125
38		eqid 2737	. . . . . 6 ⊢ ;87 = ;87
39		eqid 2737	. . . . . 6 ⊢ ;12 = ;12
40		8p1e9 12302	. . . . . 6 ⊢ (8 + 1) = 9
41		7p2e9 12313	. . . . . 6 ⊢ (7 + 2) = 9
42	20, 11, 2, 3, 38, 39, 40, 41	decadd 12673	. . . . 5 ⊢ (;87 + ;12) = ;99
43		9p7e16 12711	. . . . . 6 ⊢ (9 + 7) = ;16
44		eqid 2737	. . . . . . 7 ⊢ ;14 = ;14
45		3cn 12238	. . . . . . . . 9 ⊢ 3 ∈ ℂ
46		ax-1cn 11096	. . . . . . . . 9 ⊢ 1 ∈ ℂ
47		3p1e4 12297	. . . . . . . . 9 ⊢ (3 + 1) = 4
48	45, 46, 47	addcomli 11337	. . . . . . . 8 ⊢ (1 + 3) = 4
49	13	dec0h 12641	. . . . . . . 8 ⊢ 4 = ;04
50	48, 49	eqtri 2760	. . . . . . 7 ⊢ (1 + 3) = ;04
51	46	mulridi 11148	. . . . . . . . 9 ⊢ (1 · 1) = 1
52		00id 11320	. . . . . . . . 9 ⊢ (0 + 0) = 0
53	51, 52	oveq12i 7380	. . . . . . . 8 ⊢ ((1 · 1) + (0 + 0)) = (1 + 0)
54	46	addridi 11332	. . . . . . . 8 ⊢ (1 + 0) = 1
55	53, 54	eqtri 2760	. . . . . . 7 ⊢ ((1 · 1) + (0 + 0)) = 1
56		4cn 12242	. . . . . . . . . 10 ⊢ 4 ∈ ℂ
57	56	mulridi 11148	. . . . . . . . 9 ⊢ (4 · 1) = 4
58	57	oveq1i 7378	. . . . . . . 8 ⊢ ((4 · 1) + 4) = (4 + 4)
59		4p4e8 12307	. . . . . . . 8 ⊢ (4 + 4) = 8
60	20	dec0h 12641	. . . . . . . 8 ⊢ 8 = ;08
61	58, 59, 60	3eqtri 2764	. . . . . . 7 ⊢ ((4 · 1) + 4) = ;08
62	2, 13, 22, 13, 44, 50, 2, 20, 22, 55, 61	decmac 12671	. . . . . 6 ⊢ ((;14 · 1) + (1 + 3)) = ;18
63	18	dec0h 12641	. . . . . . 7 ⊢ 6 = ;06
64	26	mullidi 11149	. . . . . . . . 9 ⊢ (1 · 2) = 2
65	46	addlidi 11333	. . . . . . . . 9 ⊢ (0 + 1) = 1
66	64, 65	oveq12i 7380	. . . . . . . 8 ⊢ ((1 · 2) + (0 + 1)) = (2 + 1)
67	66, 29	eqtri 2760	. . . . . . 7 ⊢ ((1 · 2) + (0 + 1)) = 3
68		4t2e8 12320	. . . . . . . . 9 ⊢ (4 · 2) = 8
69	68	oveq1i 7378	. . . . . . . 8 ⊢ ((4 · 2) + 6) = (8 + 6)
70		8p6e14 12703	. . . . . . . 8 ⊢ (8 + 6) = ;14
71	69, 70	eqtri 2760	. . . . . . 7 ⊢ ((4 · 2) + 6) = ;14
72	2, 13, 22, 18, 44, 63, 3, 13, 2, 67, 71	decmac 12671	. . . . . 6 ⊢ ((;14 · 2) + 6) = ;34
73	2, 3, 2, 18, 39, 43, 14, 13, 16, 62, 72	decma2c 12672	. . . . 5 ⊢ ((;14 · ;12) + (9 + 7)) = ;;184
74	35	dec0h 12641	. . . . . 6 ⊢ 9 = ;09
75		5cn 12245	. . . . . . . . 9 ⊢ 5 ∈ ℂ
76	75	mullidi 11149	. . . . . . . 8 ⊢ (1 · 5) = 5
77	26	addlidi 11333	. . . . . . . 8 ⊢ (0 + 2) = 2
78	76, 77	oveq12i 7380	. . . . . . 7 ⊢ ((1 · 5) + (0 + 2)) = (5 + 2)
79		5p2e7 12308	. . . . . . 7 ⊢ (5 + 2) = 7
80	78, 79	eqtri 2760	. . . . . 6 ⊢ ((1 · 5) + (0 + 2)) = 7
81		5t4e20 12721	. . . . . . . 8 ⊢ (5 · 4) = ;20
82	75, 56, 81	mulcomli 11153	. . . . . . 7 ⊢ (4 · 5) = ;20
83		9cn 12257	. . . . . . . 8 ⊢ 9 ∈ ℂ
84	83	addlidi 11333	. . . . . . 7 ⊢ (0 + 9) = 9
85	3, 22, 35, 82, 84	decaddi 12679	. . . . . 6 ⊢ ((4 · 5) + 9) = ;29
86	2, 13, 22, 35, 44, 74, 5, 35, 3, 80, 85	decmac 12671	. . . . 5 ⊢ ((;14 · 5) + 9) = ;79
87	4, 5, 35, 35, 37, 42, 14, 35, 11, 73, 86	decma2c 12672	. . . 4 ⊢ ((;14 · ;;125) + (;87 + ;12)) = ;;;1849
88	83	mullidi 11149	. . . . . . . . 9 ⊢ (1 · 9) = 9
89	88	oveq1i 7378	. . . . . . . 8 ⊢ ((1 · 9) + 3) = (9 + 3)
90		9p3e12 12707	. . . . . . . 8 ⊢ (9 + 3) = ;12
91	89, 90	eqtri 2760	. . . . . . 7 ⊢ ((1 · 9) + 3) = ;12
92		9t4e36 12743	. . . . . . . 8 ⊢ (9 · 4) = ;36
93	83, 56, 92	mulcomli 11153	. . . . . . 7 ⊢ (4 · 9) = ;36
94	35, 2, 13, 44, 18, 16, 91, 93	decmul1c 12684	. . . . . 6 ⊢ (;14 · 9) = ;;126
95	94	oveq1i 7378	. . . . 5 ⊢ ((;14 · 9) + 0) = (;;126 + 0)
96	4, 18	deccl 12634	. . . . . . 7 ⊢ ;;126 ∈ ℕ0
97	96	nn0cni 12425	. . . . . 6 ⊢ ;;126 ∈ ℂ
98	97	addridi 11332	. . . . 5 ⊢ (;;126 + 0) = ;;126
99	95, 98	eqtri 2760	. . . 4 ⊢ ((;14 · 9) + 0) = ;;126
100	6, 35, 21, 22, 1, 36, 14, 18, 4, 87, 99	decma2c 12672	. . 3 ⊢ ((;14 · 𝑁) + ;;870) = ;;;;18496
101		eqid 2737	. . . 4 ⊢ ;;136 = ;;136
102	20, 2	deccl 12634	. . . 4 ⊢ ;81 ∈ ℕ0
103		eqid 2737	. . . . 5 ⊢ ;13 = ;13
104		eqid 2737	. . . . 5 ⊢ ;81 = ;81
105	13, 22	deccl 12634	. . . . 5 ⊢ ;40 ∈ ℕ0
106		eqid 2737	. . . . . . 7 ⊢ ;40 = ;40
107	56	addlidi 11333	. . . . . . 7 ⊢ (0 + 4) = 4
108		8cn 12254	. . . . . . . 8 ⊢ 8 ∈ ℂ
109	108	addridi 11332	. . . . . . 7 ⊢ (8 + 0) = 8
110	22, 20, 13, 22, 60, 106, 107, 109	decadd 12673	. . . . . 6 ⊢ (8 + ;40) = ;48
111		4p1e5 12298	. . . . . . . 8 ⊢ (4 + 1) = 5
112	5	dec0h 12641	. . . . . . . 8 ⊢ 5 = ;05
113	111, 112	eqtri 2760	. . . . . . 7 ⊢ (4 + 1) = ;05
114	45	mulridi 11148	. . . . . . . . 9 ⊢ (3 · 1) = 3
115	114	oveq1i 7378	. . . . . . . 8 ⊢ ((3 · 1) + 5) = (3 + 5)
116		5p3e8 12309	. . . . . . . . 9 ⊢ (5 + 3) = 8
117	75, 45, 116	addcomli 11337	. . . . . . . 8 ⊢ (3 + 5) = 8
118	115, 117, 60	3eqtri 2764	. . . . . . 7 ⊢ ((3 · 1) + 5) = ;08
119	2, 16, 22, 5, 103, 113, 2, 20, 22, 55, 118	decmac 12671	. . . . . 6 ⊢ ((;13 · 1) + (4 + 1)) = ;18
120		6cn 12248	. . . . . . . . 9 ⊢ 6 ∈ ℂ
121	120	mulridi 11148	. . . . . . . 8 ⊢ (6 · 1) = 6
122	121	oveq1i 7378	. . . . . . 7 ⊢ ((6 · 1) + 8) = (6 + 8)
123	108, 120, 70	addcomli 11337	. . . . . . 7 ⊢ (6 + 8) = ;14
124	122, 123	eqtri 2760	. . . . . 6 ⊢ ((6 · 1) + 8) = ;14
125	17, 18, 13, 20, 101, 110, 2, 13, 2, 119, 124	decmac 12671	. . . . 5 ⊢ ((;;136 · 1) + (8 + ;40)) = ;;184
126	2	dec0h 12641	. . . . . 6 ⊢ 1 = ;01
127	65, 126	eqtri 2760	. . . . . . 7 ⊢ (0 + 1) = ;01
128	45	mullidi 11149	. . . . . . . . 9 ⊢ (1 · 3) = 3
129	128, 65	oveq12i 7380	. . . . . . . 8 ⊢ ((1 · 3) + (0 + 1)) = (3 + 1)
130	129, 47	eqtri 2760	. . . . . . 7 ⊢ ((1 · 3) + (0 + 1)) = 4
131		3t3e9 12319	. . . . . . . . 9 ⊢ (3 · 3) = 9
132	131	oveq1i 7378	. . . . . . . 8 ⊢ ((3 · 3) + 1) = (9 + 1)
133		9p1e10 12621	. . . . . . . 8 ⊢ (9 + 1) = ;10
134	132, 133	eqtri 2760	. . . . . . 7 ⊢ ((3 · 3) + 1) = ;10
135	2, 16, 22, 2, 103, 127, 16, 22, 2, 130, 134	decmac 12671	. . . . . 6 ⊢ ((;13 · 3) + (0 + 1)) = ;40
136		6t3e18 12724	. . . . . . 7 ⊢ (6 · 3) = ;18
137	2, 20, 2, 136, 40	decaddi 12679	. . . . . 6 ⊢ ((6 · 3) + 1) = ;19
138	17, 18, 22, 2, 101, 126, 16, 35, 2, 135, 137	decmac 12671	. . . . 5 ⊢ ((;;136 · 3) + 1) = ;;409
139	2, 16, 20, 2, 103, 104, 19, 35, 105, 125, 138	decma2c 12672	. . . 4 ⊢ ((;;136 · ;13) + ;81) = ;;;1849
140	16	dec0h 12641	. . . . . 6 ⊢ 3 = ;03
141	120	mullidi 11149	. . . . . . . 8 ⊢ (1 · 6) = 6
142	141, 77	oveq12i 7380	. . . . . . 7 ⊢ ((1 · 6) + (0 + 2)) = (6 + 2)
143		6p2e8 12311	. . . . . . 7 ⊢ (6 + 2) = 8
144	142, 143	eqtri 2760	. . . . . 6 ⊢ ((1 · 6) + (0 + 2)) = 8
145	120, 45, 136	mulcomli 11153	. . . . . . 7 ⊢ (3 · 6) = ;18
146		1p1e2 12277	. . . . . . 7 ⊢ (1 + 1) = 2
147		8p3e11 12700	. . . . . . 7 ⊢ (8 + 3) = ;11
148	2, 20, 16, 145, 146, 2, 147	decaddci 12680	. . . . . 6 ⊢ ((3 · 6) + 3) = ;21
149	2, 16, 22, 16, 103, 140, 18, 2, 3, 144, 148	decmac 12671	. . . . 5 ⊢ ((;13 · 6) + 3) = ;81
150		6t6e36 12727	. . . . 5 ⊢ (6 · 6) = ;36
151	18, 17, 18, 101, 18, 16, 149, 150	decmul1c 12684	. . . 4 ⊢ (;;136 · 6) = ;;816
152	19, 17, 18, 101, 18, 102, 139, 151	decmul2c 12685	. . 3 ⊢ (;;136 · ;;136) = ;;;;18496
153	100, 152	eqtr4i 2763	. 2 ⊢ ((;14 · 𝑁) + ;;870) = (;;136 · ;;136)
154	9, 10, 12, 15, 19, 23, 24, 34, 153	mod2xi 17009	1 ⊢ ((2↑;34) mod 𝑁) = (;;870 mod 𝑁)

Syntax hints:   = wceq 1542  (class class class)co 7368  0cc0 11038  1c1 11039   + caddc 11041   · cmul 11043  ℕcn 12157  2c2 12212  3c3 12213  4c4 12214  5c5 12215  6c6 12216  7c7 12217  8c8 12218  9c9 12219  ;cdc 12619   mod cmo 13801  ↑cexp 13996

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-er 8645  df-en 8896  df-dom 8897  df-sdom 8898  df-sup 9357  df-inf 9358  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-div 11807  df-nn 12158  df-2 12220  df-3 12221  df-4 12222  df-5 12223  df-6 12224  df-7 12225  df-8 12226  df-9 12227  df-n0 12414  df-z 12501  df-dec 12620  df-uz 12764  df-rp 12918  df-fl 13724  df-mod 13802  df-seq 13937  df-exp 13997

This theorem is referenced by:  1259lem3  17072  1259lem5  17074