1259lem2 - Metamath Proof Explorer

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Theorem 1259lem2 17095

Description: Lemma for 1259prm 17099. 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 12513	. . . . . 6 ⊢ 1 ∈ ℕ₀
3		2nn0 12514	. . . . . 6 ⊢ 2 ∈ ℕ₀
4	2, 3	deccl 12717	. . . . 5 ⊢ ;12 ∈ ℕ₀
5		5nn0 12517	. . . . 5 ⊢ 5 ∈ ℕ₀
6	4, 5	deccl 12717	. . . 4 ⊢ ;;125 ∈ ℕ₀
7		9nn 12335	. . . 4 ⊢ 9 ∈ ℕ
8	6, 7	decnncl 12722	. . 3 ⊢ ;;;1259 ∈ ℕ
9	1, 8	eqeltri 2825	. 2 ⊢ 𝑁 ∈ ℕ
10		2nn 12310	. 2 ⊢ 2 ∈ ℕ
11		7nn0 12519	. . 3 ⊢ 7 ∈ ℕ₀
12	2, 11	deccl 12717	. 2 ⊢ ;17 ∈ ℕ₀
13		4nn0 12516	. . . 4 ⊢ 4 ∈ ℕ₀
14	2, 13	deccl 12717	. . 3 ⊢ ;14 ∈ ℕ₀
15	14	nn0zi 12612	. 2 ⊢ ;14 ∈ ℤ
16		3nn0 12515	. . . 4 ⊢ 3 ∈ ℕ₀
17	2, 16	deccl 12717	. . 3 ⊢ ;13 ∈ ℕ₀
18		6nn0 12518	. . 3 ⊢ 6 ∈ ℕ₀
19	17, 18	deccl 12717	. 2 ⊢ ;;136 ∈ ℕ₀
20		8nn0 12520	. . . 4 ⊢ 8 ∈ ℕ₀
21	20, 11	deccl 12717	. . 3 ⊢ ;87 ∈ ℕ₀
22		0nn0 12512	. . 3 ⊢ 0 ∈ ℕ₀
23	21, 22	deccl 12717	. 2 ⊢ ;;870 ∈ ℕ₀
24	1	1259lem1 17094	. 2 ⊢ ((2↑;17) mod 𝑁) = (;;136 mod 𝑁)
25		eqid 2728	. . 3 ⊢ ;17 = ;17
26		2cn 12312	. . . . . 6 ⊢ 2 ∈ ℂ
27	26	mulridi 11243	. . . . 5 ⊢ (2 · 1) = 2
28	27	oveq1i 7425	. . . 4 ⊢ ((2 · 1) + 1) = (2 + 1)
29		2p1e3 12379	. . . 4 ⊢ (2 + 1) = 3
30	28, 29	eqtri 2756	. . 3 ⊢ ((2 · 1) + 1) = 3
31		7cn 12331	. . . 4 ⊢ 7 ∈ ℂ
32		7t2e14 12811	. . . 4 ⊢ (7 · 2) = ;14
33	31, 26, 32	mulcomli 11248	. . 3 ⊢ (2 · 7) = ;14
34	3, 2, 11, 25, 13, 2, 30, 33	decmul2c 12768	. 2 ⊢ (2 · ;17) = ;34
35		9nn0 12521	. . . 4 ⊢ 9 ∈ ℕ₀
36		eqid 2728	. . . 4 ⊢ ;;870 = ;;870
37		eqid 2728	. . . . 5 ⊢ ;;125 = ;;125
38		eqid 2728	. . . . . 6 ⊢ ;87 = ;87
39		eqid 2728	. . . . . 6 ⊢ ;12 = ;12
40		8p1e9 12387	. . . . . 6 ⊢ (8 + 1) = 9
41		7p2e9 12398	. . . . . 6 ⊢ (7 + 2) = 9
42	20, 11, 2, 3, 38, 39, 40, 41	decadd 12756	. . . . 5 ⊢ (;87 + ;12) = ;99
43		9p7e16 12794	. . . . . 6 ⊢ (9 + 7) = ;16
44		eqid 2728	. . . . . . 7 ⊢ ;14 = ;14
45		3cn 12318	. . . . . . . . 9 ⊢ 3 ∈ ℂ
46		ax-1cn 11191	. . . . . . . . 9 ⊢ 1 ∈ ℂ
47		3p1e4 12382	. . . . . . . . 9 ⊢ (3 + 1) = 4
48	45, 46, 47	addcomli 11431	. . . . . . . 8 ⊢ (1 + 3) = 4
49	13	dec0h 12724	. . . . . . . 8 ⊢ 4 = ;04
50	48, 49	eqtri 2756	. . . . . . 7 ⊢ (1 + 3) = ;04
51	46	mulridi 11243	. . . . . . . . 9 ⊢ (1 · 1) = 1
52		00id 11414	. . . . . . . . 9 ⊢ (0 + 0) = 0
53	51, 52	oveq12i 7427	. . . . . . . 8 ⊢ ((1 · 1) + (0 + 0)) = (1 + 0)
54	46	addridi 11426	. . . . . . . 8 ⊢ (1 + 0) = 1
55	53, 54	eqtri 2756	. . . . . . 7 ⊢ ((1 · 1) + (0 + 0)) = 1
56		4cn 12322	. . . . . . . . . 10 ⊢ 4 ∈ ℂ
57	56	mulridi 11243	. . . . . . . . 9 ⊢ (4 · 1) = 4
58	57	oveq1i 7425	. . . . . . . 8 ⊢ ((4 · 1) + 4) = (4 + 4)
59		4p4e8 12392	. . . . . . . 8 ⊢ (4 + 4) = 8
60	20	dec0h 12724	. . . . . . . 8 ⊢ 8 = ;08
61	58, 59, 60	3eqtri 2760	. . . . . . 7 ⊢ ((4 · 1) + 4) = ;08
62	2, 13, 22, 13, 44, 50, 2, 20, 22, 55, 61	decmac 12754	. . . . . 6 ⊢ ((;14 · 1) + (1 + 3)) = ;18
63	18	dec0h 12724	. . . . . . 7 ⊢ 6 = ;06
64	26	mullidi 11244	. . . . . . . . 9 ⊢ (1 · 2) = 2
65	46	addlidi 11427	. . . . . . . . 9 ⊢ (0 + 1) = 1
66	64, 65	oveq12i 7427	. . . . . . . 8 ⊢ ((1 · 2) + (0 + 1)) = (2 + 1)
67	66, 29	eqtri 2756	. . . . . . 7 ⊢ ((1 · 2) + (0 + 1)) = 3
68		4t2e8 12405	. . . . . . . . 9 ⊢ (4 · 2) = 8
69	68	oveq1i 7425	. . . . . . . 8 ⊢ ((4 · 2) + 6) = (8 + 6)
70		8p6e14 12786	. . . . . . . 8 ⊢ (8 + 6) = ;14
71	69, 70	eqtri 2756	. . . . . . 7 ⊢ ((4 · 2) + 6) = ;14
72	2, 13, 22, 18, 44, 63, 3, 13, 2, 67, 71	decmac 12754	. . . . . 6 ⊢ ((;14 · 2) + 6) = ;34
73	2, 3, 2, 18, 39, 43, 14, 13, 16, 62, 72	decma2c 12755	. . . . 5 ⊢ ((;14 · ;12) + (9 + 7)) = ;;184
74	35	dec0h 12724	. . . . . 6 ⊢ 9 = ;09
75		5cn 12325	. . . . . . . . 9 ⊢ 5 ∈ ℂ
76	75	mullidi 11244	. . . . . . . 8 ⊢ (1 · 5) = 5
77	26	addlidi 11427	. . . . . . . 8 ⊢ (0 + 2) = 2
78	76, 77	oveq12i 7427	. . . . . . 7 ⊢ ((1 · 5) + (0 + 2)) = (5 + 2)
79		5p2e7 12393	. . . . . . 7 ⊢ (5 + 2) = 7
80	78, 79	eqtri 2756	. . . . . 6 ⊢ ((1 · 5) + (0 + 2)) = 7
81		5t4e20 12804	. . . . . . . 8 ⊢ (5 · 4) = ;20
82	75, 56, 81	mulcomli 11248	. . . . . . 7 ⊢ (4 · 5) = ;20
83		9cn 12337	. . . . . . . 8 ⊢ 9 ∈ ℂ
84	83	addlidi 11427	. . . . . . 7 ⊢ (0 + 9) = 9
85	3, 22, 35, 82, 84	decaddi 12762	. . . . . 6 ⊢ ((4 · 5) + 9) = ;29
86	2, 13, 22, 35, 44, 74, 5, 35, 3, 80, 85	decmac 12754	. . . . 5 ⊢ ((;14 · 5) + 9) = ;79
87	4, 5, 35, 35, 37, 42, 14, 35, 11, 73, 86	decma2c 12755	. . . 4 ⊢ ((;14 · ;;125) + (;87 + ;12)) = ;;;1849
88	83	mullidi 11244	. . . . . . . . 9 ⊢ (1 · 9) = 9
89	88	oveq1i 7425	. . . . . . . 8 ⊢ ((1 · 9) + 3) = (9 + 3)
90		9p3e12 12790	. . . . . . . 8 ⊢ (9 + 3) = ;12
91	89, 90	eqtri 2756	. . . . . . 7 ⊢ ((1 · 9) + 3) = ;12
92		9t4e36 12826	. . . . . . . 8 ⊢ (9 · 4) = ;36
93	83, 56, 92	mulcomli 11248	. . . . . . 7 ⊢ (4 · 9) = ;36
94	35, 2, 13, 44, 18, 16, 91, 93	decmul1c 12767	. . . . . 6 ⊢ (;14 · 9) = ;;126
95	94	oveq1i 7425	. . . . 5 ⊢ ((;14 · 9) + 0) = (;;126 + 0)
96	4, 18	deccl 12717	. . . . . . 7 ⊢ ;;126 ∈ ℕ₀
97	96	nn0cni 12509	. . . . . 6 ⊢ ;;126 ∈ ℂ
98	97	addridi 11426	. . . . 5 ⊢ (;;126 + 0) = ;;126
99	95, 98	eqtri 2756	. . . 4 ⊢ ((;14 · 9) + 0) = ;;126
100	6, 35, 21, 22, 1, 36, 14, 18, 4, 87, 99	decma2c 12755	. . 3 ⊢ ((;14 · 𝑁) + ;;870) = ;;;;18496
101		eqid 2728	. . . 4 ⊢ ;;136 = ;;136
102	20, 2	deccl 12717	. . . 4 ⊢ ;81 ∈ ℕ₀
103		eqid 2728	. . . . 5 ⊢ ;13 = ;13
104		eqid 2728	. . . . 5 ⊢ ;81 = ;81
105	13, 22	deccl 12717	. . . . 5 ⊢ ;40 ∈ ℕ₀
106		eqid 2728	. . . . . . 7 ⊢ ;40 = ;40
107	56	addlidi 11427	. . . . . . 7 ⊢ (0 + 4) = 4
108		8cn 12334	. . . . . . . 8 ⊢ 8 ∈ ℂ
109	108	addridi 11426	. . . . . . 7 ⊢ (8 + 0) = 8
110	22, 20, 13, 22, 60, 106, 107, 109	decadd 12756	. . . . . 6 ⊢ (8 + ;40) = ;48
111		4p1e5 12383	. . . . . . . 8 ⊢ (4 + 1) = 5
112	5	dec0h 12724	. . . . . . . 8 ⊢ 5 = ;05
113	111, 112	eqtri 2756	. . . . . . 7 ⊢ (4 + 1) = ;05
114	45	mulridi 11243	. . . . . . . . 9 ⊢ (3 · 1) = 3
115	114	oveq1i 7425	. . . . . . . 8 ⊢ ((3 · 1) + 5) = (3 + 5)
116		5p3e8 12394	. . . . . . . . 9 ⊢ (5 + 3) = 8
117	75, 45, 116	addcomli 11431	. . . . . . . 8 ⊢ (3 + 5) = 8
118	115, 117, 60	3eqtri 2760	. . . . . . 7 ⊢ ((3 · 1) + 5) = ;08
119	2, 16, 22, 5, 103, 113, 2, 20, 22, 55, 118	decmac 12754	. . . . . 6 ⊢ ((;13 · 1) + (4 + 1)) = ;18
120		6cn 12328	. . . . . . . . 9 ⊢ 6 ∈ ℂ
121	120	mulridi 11243	. . . . . . . 8 ⊢ (6 · 1) = 6
122	121	oveq1i 7425	. . . . . . 7 ⊢ ((6 · 1) + 8) = (6 + 8)
123	108, 120, 70	addcomli 11431	. . . . . . 7 ⊢ (6 + 8) = ;14
124	122, 123	eqtri 2756	. . . . . 6 ⊢ ((6 · 1) + 8) = ;14
125	17, 18, 13, 20, 101, 110, 2, 13, 2, 119, 124	decmac 12754	. . . . 5 ⊢ ((;;136 · 1) + (8 + ;40)) = ;;184
126	2	dec0h 12724	. . . . . 6 ⊢ 1 = ;01
127	65, 126	eqtri 2756	. . . . . . 7 ⊢ (0 + 1) = ;01
128	45	mullidi 11244	. . . . . . . . 9 ⊢ (1 · 3) = 3
129	128, 65	oveq12i 7427	. . . . . . . 8 ⊢ ((1 · 3) + (0 + 1)) = (3 + 1)
130	129, 47	eqtri 2756	. . . . . . 7 ⊢ ((1 · 3) + (0 + 1)) = 4
131		3t3e9 12404	. . . . . . . . 9 ⊢ (3 · 3) = 9
132	131	oveq1i 7425	. . . . . . . 8 ⊢ ((3 · 3) + 1) = (9 + 1)
133		9p1e10 12704	. . . . . . . 8 ⊢ (9 + 1) = ;10
134	132, 133	eqtri 2756	. . . . . . 7 ⊢ ((3 · 3) + 1) = ;10
135	2, 16, 22, 2, 103, 127, 16, 22, 2, 130, 134	decmac 12754	. . . . . 6 ⊢ ((;13 · 3) + (0 + 1)) = ;40
136		6t3e18 12807	. . . . . . 7 ⊢ (6 · 3) = ;18
137	2, 20, 2, 136, 40	decaddi 12762	. . . . . 6 ⊢ ((6 · 3) + 1) = ;19
138	17, 18, 22, 2, 101, 126, 16, 35, 2, 135, 137	decmac 12754	. . . . 5 ⊢ ((;;136 · 3) + 1) = ;;409
139	2, 16, 20, 2, 103, 104, 19, 35, 105, 125, 138	decma2c 12755	. . . 4 ⊢ ((;;136 · ;13) + ;81) = ;;;1849
140	16	dec0h 12724	. . . . . 6 ⊢ 3 = ;03
141	120	mullidi 11244	. . . . . . . 8 ⊢ (1 · 6) = 6
142	141, 77	oveq12i 7427	. . . . . . 7 ⊢ ((1 · 6) + (0 + 2)) = (6 + 2)
143		6p2e8 12396	. . . . . . 7 ⊢ (6 + 2) = 8
144	142, 143	eqtri 2756	. . . . . 6 ⊢ ((1 · 6) + (0 + 2)) = 8
145	120, 45, 136	mulcomli 11248	. . . . . . 7 ⊢ (3 · 6) = ;18
146		1p1e2 12362	. . . . . . 7 ⊢ (1 + 1) = 2
147		8p3e11 12783	. . . . . . 7 ⊢ (8 + 3) = ;11
148	2, 20, 16, 145, 146, 2, 147	decaddci 12763	. . . . . 6 ⊢ ((3 · 6) + 3) = ;21
149	2, 16, 22, 16, 103, 140, 18, 2, 3, 144, 148	decmac 12754	. . . . 5 ⊢ ((;13 · 6) + 3) = ;81
150		6t6e36 12810	. . . . 5 ⊢ (6 · 6) = ;36
151	18, 17, 18, 101, 18, 16, 149, 150	decmul1c 12767	. . . 4 ⊢ (;;136 · 6) = ;;816
152	19, 17, 18, 101, 18, 102, 139, 151	decmul2c 12768	. . 3 ⊢ (;;136 · ;;136) = ;;;;18496
153	100, 152	eqtr4i 2759	. 2 ⊢ ((;14 · 𝑁) + ;;870) = (;;136 · ;;136)
154	9, 10, 12, 15, 19, 23, 24, 34, 153	mod2xi 17032	1 ⊢ ((2↑;34) mod 𝑁) = (;;870 mod 𝑁)

Colors of variables: wff setvar class

Syntax hints: = wceq 1534 (class class class)co 7415 0cc0 11133 1c1 11134 + caddc 11136 · cmul 11138 ℕcn 12237 2c2 12292 3c3 12293 4c4 12294 5c5 12295 6c6 12296 7c7 12297 8c8 12298 9c9 12299 cdc 12702 mod cmo 13861 ↑cexp 14053

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-pre-sup 11211

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7866 df-2nd 7989 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-er 8719 df-en 8959 df-dom 8960 df-sdom 8961 df-sup 9460 df-inf 9461 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-div 11897 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-z 12584 df-dec 12703 df-uz 12848 df-rp 13002 df-fl 13784 df-mod 13862 df-seq 13994 df-exp 14054

This theorem is referenced by: 1259lem3 17096 1259lem5 17098

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