4001lem3 - Metamath Proof Explorer

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Theorem 4001lem3 17229

Description: Lemma for 4001prm 17231. Calculate a power mod. In decimal, we calculate 2↑1000 = 2↑800 · 2↑200≡2311 · 902 = 521𝑁 + 1 and finally 2↑(𝑁 − 1) = (2↑1000)↑4≡1↑4 = 1. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.)

**Hypothesis**
Ref	Expression
4001prm.1	⊢ 𝑁 = ;;;4001

**Assertion**
Ref	Expression
4001lem3	⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)

**Proof of Theorem 4001lem3**
Step	Hyp	Ref	Expression
1		4001prm.1	. . 3 ⊢ 𝑁 = ;;;4001
2		4nn0 12542	. . . . . 6 ⊢ 4 ∈ ℕ₀
3		0nn0 12538	. . . . . 6 ⊢ 0 ∈ ℕ₀
4	2, 3	deccl 12746	. . . . 5 ⊢ ;40 ∈ ℕ₀
5	4, 3	deccl 12746	. . . 4 ⊢ ;;400 ∈ ℕ₀
6		1nn 12263	. . . 4 ⊢ 1 ∈ ℕ
7	5, 6	decnncl 12755	. . 3 ⊢ ;;;4001 ∈ ℕ
8	1, 7	eqeltri 2861	. 2 ⊢ 𝑁 ∈ ℕ
9		2nn 12333	. 2 ⊢ 2 ∈ ℕ
10		2nn0 12540	. . . . 5 ⊢ 2 ∈ ℕ₀
11	10, 3	deccl 12746	. . . 4 ⊢ ;20 ∈ ℕ₀
12	11, 3	deccl 12746	. . 3 ⊢ ;;200 ∈ ℕ₀
13	12, 3	deccl 12746	. 2 ⊢ ;;;2000 ∈ ℕ₀
14		0z 12621	. 2 ⊢ 0 ∈ ℤ
15		1nn0 12539	. 2 ⊢ 1 ∈ ℕ₀
16		10nn0 12753	. . . . 5 ⊢ ;10 ∈ ℕ₀
17	16, 3	deccl 12746	. . . 4 ⊢ ;;100 ∈ ℕ₀
18	17, 3	deccl 12746	. . 3 ⊢ ;;;1000 ∈ ℕ₀
19		8nn0 12546	. . . . . 6 ⊢ 8 ∈ ℕ₀
20	19, 3	deccl 12746	. . . . 5 ⊢ ;80 ∈ ℕ₀
21	20, 3	deccl 12746	. . . 4 ⊢ ;;800 ∈ ℕ₀
22		5nn0 12543	. . . . . . 7 ⊢ 5 ∈ ℕ₀
23	22, 10	deccl 12746	. . . . . 6 ⊢ ;52 ∈ ℕ₀
24	23, 15	deccl 12746	. . . . 5 ⊢ ;;521 ∈ ℕ₀
25	24	nn0zi 12638	. . . 4 ⊢ ;;521 ∈ ℤ
26		3nn0 12541	. . . . . . 7 ⊢ 3 ∈ ℕ₀
27	10, 26	deccl 12746	. . . . . 6 ⊢ ;23 ∈ ℕ₀
28	27, 15	deccl 12746	. . . . 5 ⊢ ;;231 ∈ ℕ₀
29	28, 15	deccl 12746	. . . 4 ⊢ ;;;2311 ∈ ℕ₀
30		9nn0 12547	. . . . . 6 ⊢ 9 ∈ ℕ₀
31	30, 3	deccl 12746	. . . . 5 ⊢ ;90 ∈ ℕ₀
32	31, 10	deccl 12746	. . . 4 ⊢ ;;902 ∈ ℕ₀
33	1	4001lem2 17228	. . . 4 ⊢ ((2↑;;800) mod 𝑁) = (;;;2311 mod 𝑁)
34	1	4001lem1 17227	. . . 4 ⊢ ((2↑;;200) mod 𝑁) = (;;902 mod 𝑁)
35		eqid 2765	. . . . 5 ⊢ ;;800 = ;;800
36		eqid 2765	. . . . 5 ⊢ ;;200 = ;;200
37		eqid 2765	. . . . . 6 ⊢ ;80 = ;80
38		eqid 2765	. . . . . 6 ⊢ ;20 = ;20
39		8p2e10 12816	. . . . . 6 ⊢ (8 + 2) = ;10
40		00id 11404	. . . . . 6 ⊢ (0 + 0) = 0
41	19, 3, 10, 3, 37, 38, 39, 40	decadd 12790	. . . . 5 ⊢ (;80 + ;20) = ;;100
42	20, 3, 11, 3, 35, 36, 41, 40	decadd 12790	. . . 4 ⊢ (;;800 + ;;200) = ;;;1000
43	15	dec0h 12758	. . . . . 6 ⊢ 1 = ;01
44		eqid 2765	. . . . . . 7 ⊢ ;;400 = ;;400
45	23	nn0cni 12535	. . . . . . . 8 ⊢ ;52 ∈ ℂ
46	45	addlidi 11417	. . . . . . 7 ⊢ (0 + ;52) = ;52
47		eqid 2765	. . . . . . . 8 ⊢ ;40 = ;40
48		5cn 12348	. . . . . . . . . 10 ⊢ 5 ∈ ℂ
49	48	addridi 11416	. . . . . . . . 9 ⊢ (5 + 0) = 5
50	22	dec0h 12758	. . . . . . . . 9 ⊢ 5 = ;05
51	49, 50	eqtri 2788	. . . . . . . 8 ⊢ (5 + 0) = ;05
52	40, 3	eqeltri 2861	. . . . . . . . 9 ⊢ (0 + 0) ∈ ℕ₀
53		eqid 2765	. . . . . . . . 9 ⊢ ;;521 = ;;521
54		eqid 2765	. . . . . . . . . 10 ⊢ ;52 = ;52
55		5t4e20 12838	. . . . . . . . . 10 ⊢ (5 · 4) = ;20
56		2t4e8 12429	. . . . . . . . . 10 ⊢ (2 · 4) = 8
57	2, 22, 10, 54, 55, 56	decmul1 12800	. . . . . . . . 9 ⊢ (;52 · 4) = ;;208
58		4cn 12345	. . . . . . . . . . . 12 ⊢ 4 ∈ ℂ
59	58	mullidi 11233	. . . . . . . . . . 11 ⊢ (1 · 4) = 4
60	59, 40	oveq12i 7431	. . . . . . . . . 10 ⊢ ((1 · 4) + (0 + 0)) = (4 + 0)
61	58	addridi 11416	. . . . . . . . . 10 ⊢ (4 + 0) = 4
62	60, 61	eqtri 2788	. . . . . . . . 9 ⊢ ((1 · 4) + (0 + 0)) = 4
63	23, 15, 52, 53, 2, 57, 62	decrmanc 12793	. . . . . . . 8 ⊢ ((;;521 · 4) + (0 + 0)) = ;;;2084
64	24	nn0cni 12535	. . . . . . . . . . 11 ⊢ ;;521 ∈ ℂ
65	64	mul01i 11419	. . . . . . . . . 10 ⊢ (;;521 · 0) = 0
66	65	oveq1i 7429	. . . . . . . . 9 ⊢ ((;;521 · 0) + 5) = (0 + 5)
67	48	addlidi 11417	. . . . . . . . 9 ⊢ (0 + 5) = 5
68	66, 67, 50	3eqtri 2792	. . . . . . . 8 ⊢ ((;;521 · 0) + 5) = ;05
69	2, 3, 3, 22, 47, 51, 24, 22, 3, 63, 68	decma2c 12789	. . . . . . 7 ⊢ ((;;521 · ;40) + (5 + 0)) = ;;;;20845
70	65	oveq1i 7429	. . . . . . . 8 ⊢ ((;;521 · 0) + 2) = (0 + 2)
71		2cn 12335	. . . . . . . . 9 ⊢ 2 ∈ ℂ
72	71	addlidi 11417	. . . . . . . 8 ⊢ (0 + 2) = 2
73	10	dec0h 12758	. . . . . . . 8 ⊢ 2 = ;02
74	70, 72, 73	3eqtri 2792	. . . . . . 7 ⊢ ((;;521 · 0) + 2) = ;02
75	4, 3, 22, 10, 44, 46, 24, 10, 3, 69, 74	decma2c 12789	. . . . . 6 ⊢ ((;;521 · ;;400) + (0 + ;52)) = ;;;;;208452
76	45	mulridi 11232	. . . . . . 7 ⊢ (;52 · 1) = ;52
77		ax-1cn 11177	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
78	77	mullidi 11233	. . . . . . . . 9 ⊢ (1 · 1) = 1
79	78	oveq1i 7429	. . . . . . . 8 ⊢ ((1 · 1) + 1) = (1 + 1)
80		1p1e2 12383	. . . . . . . 8 ⊢ (1 + 1) = 2
81	79, 80	eqtri 2788	. . . . . . 7 ⊢ ((1 · 1) + 1) = 2
82	23, 15, 15, 53, 15, 76, 81	decrmanc 12793	. . . . . 6 ⊢ ((;;521 · 1) + 1) = ;;522
83	5, 15, 3, 15, 1, 43, 24, 10, 23, 75, 82	decma2c 12789	. . . . 5 ⊢ ((;;521 · 𝑁) + 1) = ;;;;;;2084522
84		eqid 2765	. . . . . 6 ⊢ ;;902 = ;;902
85		6nn0 12544	. . . . . . . 8 ⊢ 6 ∈ ℕ₀
86	2, 85	deccl 12746	. . . . . . 7 ⊢ ;46 ∈ ℕ₀
87	86, 10	deccl 12746	. . . . . 6 ⊢ ;;462 ∈ ℕ₀
88		eqid 2765	. . . . . . 7 ⊢ ;90 = ;90
89		eqid 2765	. . . . . . 7 ⊢ ;;462 = ;;462
90		eqid 2765	. . . . . . . 8 ⊢ ;;;2311 = ;;;2311
91	86	nn0cni 12535	. . . . . . . . 9 ⊢ ;46 ∈ ℂ
92	91	addridi 11416	. . . . . . . 8 ⊢ (;46 + 0) = ;46
93		4p1e5 12405	. . . . . . . . . 10 ⊢ (4 + 1) = 5
94	93, 22	eqeltri 2861	. . . . . . . . 9 ⊢ (4 + 1) ∈ ℕ₀
95		eqid 2765	. . . . . . . . 9 ⊢ ;;231 = ;;231
96		eqid 2765	. . . . . . . . . 10 ⊢ ;23 = ;23
97		9cn 12360	. . . . . . . . . . . 12 ⊢ 9 ∈ ℂ
98		9t2e18 12858	. . . . . . . . . . . 12 ⊢ (9 · 2) = ;18
99	97, 71, 98	mulcomli 11237	. . . . . . . . . . 11 ⊢ (2 · 9) = ;18
100	15, 19, 10, 99, 80, 39	decaddci2 12798	. . . . . . . . . 10 ⊢ ((2 · 9) + 2) = ;20
101		7nn0 12545	. . . . . . . . . . 11 ⊢ 7 ∈ ℕ₀
102		7p1e8 12408	. . . . . . . . . . 11 ⊢ (7 + 1) = 8
103		3cn 12341	. . . . . . . . . . . 12 ⊢ 3 ∈ ℂ
104		9t3e27 12859	. . . . . . . . . . . 12 ⊢ (9 · 3) = ;27
105	97, 103, 104	mulcomli 11237	. . . . . . . . . . 11 ⊢ (3 · 9) = ;27
106	10, 101, 102, 105	decsuc 12767	. . . . . . . . . 10 ⊢ ((3 · 9) + 1) = ;28
107	10, 26, 15, 96, 30, 19, 10, 100, 106	decrmac 12794	. . . . . . . . 9 ⊢ ((;23 · 9) + 1) = ;;208
108	97	mullidi 11233	. . . . . . . . . . 11 ⊢ (1 · 9) = 9
109	108, 93	oveq12i 7431	. . . . . . . . . 10 ⊢ ((1 · 9) + (4 + 1)) = (9 + 5)
110		9p5e14 12826	. . . . . . . . . 10 ⊢ (9 + 5) = ;14
111	109, 110	eqtri 2788	. . . . . . . . 9 ⊢ ((1 · 9) + (4 + 1)) = ;14
112	27, 15, 94, 95, 30, 2, 15, 107, 111	decrmac 12794	. . . . . . . 8 ⊢ ((;;231 · 9) + (4 + 1)) = ;;;2084
113	108	oveq1i 7429	. . . . . . . . 9 ⊢ ((1 · 9) + 6) = (9 + 6)
114		9p6e15 12827	. . . . . . . . 9 ⊢ (9 + 6) = ;15
115	113, 114	eqtri 2788	. . . . . . . 8 ⊢ ((1 · 9) + 6) = ;15
116	28, 15, 2, 85, 90, 92, 30, 22, 15, 112, 115	decmac 12788	. . . . . . 7 ⊢ ((;;;2311 · 9) + (;46 + 0)) = ;;;;20845
117	29	nn0cni 12535	. . . . . . . . . 10 ⊢ ;;;2311 ∈ ℂ
118	117	mul01i 11419	. . . . . . . . 9 ⊢ (;;;2311 · 0) = 0
119	118	oveq1i 7429	. . . . . . . 8 ⊢ ((;;;2311 · 0) + 2) = (0 + 2)
120	119, 72, 73	3eqtri 2792	. . . . . . 7 ⊢ ((;;;2311 · 0) + 2) = ;02
121	30, 3, 86, 10, 88, 89, 29, 10, 3, 116, 120	decma2c 12789	. . . . . 6 ⊢ ((;;;2311 · ;90) + ;;462) = ;;;;;208452
122		2t2e4 12423	. . . . . . . . 9 ⊢ (2 · 2) = 4
123		3t2e6 12425	. . . . . . . . 9 ⊢ (3 · 2) = 6
124	10, 10, 26, 96, 122, 123	decmul1 12800	. . . . . . . 8 ⊢ (;23 · 2) = ;46
125	71	mullidi 11233	. . . . . . . 8 ⊢ (1 · 2) = 2
126	10, 27, 15, 95, 124, 125	decmul1 12800	. . . . . . 7 ⊢ (;;231 · 2) = ;;462
127	10, 28, 15, 90, 126, 125	decmul1 12800	. . . . . 6 ⊢ (;;;2311 · 2) = ;;;4622
128	29, 31, 10, 84, 10, 87, 121, 127	decmul2c 12802	. . . . 5 ⊢ (;;;2311 · ;;902) = ;;;;;;2084522
129	83, 128	eqtr4i 2791	. . . 4 ⊢ ((;;521 · 𝑁) + 1) = (;;;2311 · ;;902)
130	8, 9, 21, 25, 29, 15, 12, 32, 33, 34, 42, 129	modxai 17154	. . 3 ⊢ ((2↑;;;1000) mod 𝑁) = (1 mod 𝑁)
131	18	nn0cni 12535	. . . 4 ⊢ ;;;1000 ∈ ℂ
132		eqid 2765	. . . . 5 ⊢ ;;;1000 = ;;;1000
133		eqid 2765	. . . . . 6 ⊢ ;;100 = ;;100
134	10	dec0u 12757	. . . . . 6 ⊢ (;10 · 2) = ;20
135	71	mul02i 11418	. . . . . 6 ⊢ (0 · 2) = 0
136	10, 16, 3, 133, 134, 135	decmul1 12800	. . . . 5 ⊢ (;;100 · 2) = ;;200
137	10, 17, 3, 132, 136, 135	decmul1 12800	. . . 4 ⊢ (;;;1000 · 2) = ;;;2000
138	131, 71, 137	mulcomli 11237	. . 3 ⊢ (2 · ;;;1000) = ;;;2000
139	8	nncni 12262	. . . . . 6 ⊢ 𝑁 ∈ ℂ
140	139	mul02i 11418	. . . . 5 ⊢ (0 · 𝑁) = 0
141	140	oveq1i 7429	. . . 4 ⊢ ((0 · 𝑁) + 1) = (0 + 1)
142	77	addlidi 11417	. . . . 5 ⊢ (0 + 1) = 1
143	78, 142	eqtr4i 2791	. . . 4 ⊢ (1 · 1) = (0 + 1)
144	141, 143	eqtr4i 2791	. . 3 ⊢ ((0 · 𝑁) + 1) = (1 · 1)
145	8, 9, 18, 14, 15, 15, 130, 138, 144	mod2xi 17155	. 2 ⊢ ((2↑;;;2000) mod 𝑁) = (1 mod 𝑁)
146	13	nn0cni 12535	. . . 4 ⊢ ;;;2000 ∈ ℂ
147		eqid 2765	. . . . 5 ⊢ ;;;2000 = ;;;2000
148	10, 10, 3, 38, 122, 135	decmul1 12800	. . . . . 6 ⊢ (;20 · 2) = ;40
149	10, 11, 3, 36, 148, 135	decmul1 12800	. . . . 5 ⊢ (;;200 · 2) = ;;400
150	10, 12, 3, 147, 149, 135	decmul1 12800	. . . 4 ⊢ (;;;2000 · 2) = ;;;4000
151	146, 71, 150	mulcomli 11237	. . 3 ⊢ (2 · ;;;2000) = ;;;4000
152	5, 3	deccl 12746	. . . . 5 ⊢ ;;;4000 ∈ ℕ₀
153	152	nn0cni 12535	. . . 4 ⊢ ;;;4000 ∈ ℂ
154		eqid 2765	. . . . . 6 ⊢ ;;;4000 = ;;;4000
155	5, 3, 142, 154	decsuc 12767	. . . . 5 ⊢ (;;;4000 + 1) = ;;;4001
156	1, 155	eqtr4i 2791	. . . 4 ⊢ 𝑁 = (;;;4000 + 1)
157	153, 77, 156	mvrraddi 11493	. . 3 ⊢ (𝑁 − 1) = ;;;4000
158	151, 157	eqtr4i 2791	. 2 ⊢ (2 · ;;;2000) = (𝑁 − 1)
159	8, 9, 13, 14, 15, 15, 145, 158, 144	mod2xi 17155	1 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)

Colors of variables: wff setvar class

This proof depends on syntax axioms: = wceq 1570 (class class class)co 7419 0cc0 11119 1c1 11120 + caddc 11122 · cmul 11124 − cmin 11460 ℕcn 12252 2c2 12314 3c3 12315 4c4 12316 5c5 12317 6c6 12318 7c7 12319 8c8 12320 9c9 12321 ℕ₀cn0 12523 cdc 12731 mod cmo 13924 ↑cexp 14119

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11175 ax-resscn 11176 ax-1cn 11177 ax-icn 11178 ax-addcl 11179 ax-addrcl 11180 ax-mulcl 11181 ax-mulrcl 11182 ax-mulcom 11183 ax-addass 11184 ax-mulass 11185 ax-distr 11186 ax-i2m1 11187 ax-1ne0 11188 ax-1rid 11189 ax-rnegex 11190 ax-rrecex 11191 ax-cnre 11192 ax-pre-lttri 11193 ax-pre-lttrn 11194 ax-pre-ltadd 11195 ax-pre-mulgt0 11196 ax-pre-sup 11197

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 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7869 df-2nd 7993 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-er 8700 df-en 8950 df-dom 8951 df-sdom 8952 df-sup 9409 df-inf 9410 df-pnf 11264 df-mnf 11265 df-xr 11266 df-ltxr 11267 df-le 11268 df-sub 11462 df-neg 11463 df-div 11891 df-nn 12253 df-2 12322 df-3 12323 df-4 12324 df-5 12325 df-6 12326 df-7 12327 df-8 12328 df-9 12329 df-n0 12524 df-z 12611 df-dec 12732 df-uz 12883 df-rp 13037 df-fl 13847 df-mod 13925 df-seq 14060 df-exp 14120

This theorem is used by: 4001prm 17231

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