4001lem3 - Metamath Proof Explorer

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Theorem 4001lem3 17238

Description: Lemma for 4001prm 17240. 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 12550	. . . . . 6 ⊢ 4 ∈ ℕ₀
3		0nn0 12546	. . . . . 6 ⊢ 0 ∈ ℕ₀
4	2, 3	deccl 12754	. . . . 5 ⊢ ;40 ∈ ℕ₀
5	4, 3	deccl 12754	. . . 4 ⊢ ;;400 ∈ ℕ₀
6		1nn 12271	. . . 4 ⊢ 1 ∈ ℕ
7	5, 6	decnncl 12763	. . 3 ⊢ ;;;4001 ∈ ℕ
8	1, 7	eqeltri 2856	. 2 ⊢ 𝑁 ∈ ℕ
9		2nn 12341	. 2 ⊢ 2 ∈ ℕ
10		2nn0 12548	. . . . 5 ⊢ 2 ∈ ℕ₀
11	10, 3	deccl 12754	. . . 4 ⊢ ;20 ∈ ℕ₀
12	11, 3	deccl 12754	. . 3 ⊢ ;;200 ∈ ℕ₀
13	12, 3	deccl 12754	. 2 ⊢ ;;;2000 ∈ ℕ₀
14		0z 12629	. 2 ⊢ 0 ∈ ℤ
15		1nn0 12547	. 2 ⊢ 1 ∈ ℕ₀
16		10nn0 12761	. . . . 5 ⊢ ;10 ∈ ℕ₀
17	16, 3	deccl 12754	. . . 4 ⊢ ;;100 ∈ ℕ₀
18	17, 3	deccl 12754	. . 3 ⊢ ;;;1000 ∈ ℕ₀
19		8nn0 12554	. . . . . 6 ⊢ 8 ∈ ℕ₀
20	19, 3	deccl 12754	. . . . 5 ⊢ ;80 ∈ ℕ₀
21	20, 3	deccl 12754	. . . 4 ⊢ ;;800 ∈ ℕ₀
22		5nn0 12551	. . . . . . 7 ⊢ 5 ∈ ℕ₀
23	22, 10	deccl 12754	. . . . . 6 ⊢ ;52 ∈ ℕ₀
24	23, 15	deccl 12754	. . . . 5 ⊢ ;;521 ∈ ℕ₀
25	24	nn0zi 12646	. . . 4 ⊢ ;;521 ∈ ℤ
26		3nn0 12549	. . . . . . 7 ⊢ 3 ∈ ℕ₀
27	10, 26	deccl 12754	. . . . . 6 ⊢ ;23 ∈ ℕ₀
28	27, 15	deccl 12754	. . . . 5 ⊢ ;;231 ∈ ℕ₀
29	28, 15	deccl 12754	. . . 4 ⊢ ;;;2311 ∈ ℕ₀
30		9nn0 12555	. . . . . 6 ⊢ 9 ∈ ℕ₀
31	30, 3	deccl 12754	. . . . 5 ⊢ ;90 ∈ ℕ₀
32	31, 10	deccl 12754	. . . 4 ⊢ ;;902 ∈ ℕ₀
33	1	4001lem2 17237	. . . 4 ⊢ ((2↑;;800) mod 𝑁) = (;;;2311 mod 𝑁)
34	1	4001lem1 17236	. . . 4 ⊢ ((2↑;;200) mod 𝑁) = (;;902 mod 𝑁)
35		eqid 2760	. . . . 5 ⊢ ;;800 = ;;800
36		eqid 2760	. . . . 5 ⊢ ;;200 = ;;200
37		eqid 2760	. . . . . 6 ⊢ ;80 = ;80
38		eqid 2760	. . . . . 6 ⊢ ;20 = ;20
39		8p2e10 12824	. . . . . 6 ⊢ (8 + 2) = ;10
40		00id 11412	. . . . . 6 ⊢ (0 + 0) = 0
41	19, 3, 10, 3, 37, 38, 39, 40	decadd 12798	. . . . 5 ⊢ (;80 + ;20) = ;;100
42	20, 3, 11, 3, 35, 36, 41, 40	decadd 12798	. . . 4 ⊢ (;;800 + ;;200) = ;;;1000
43	15	dec0h 12766	. . . . . 6 ⊢ 1 = ;01
44		eqid 2760	. . . . . . 7 ⊢ ;;400 = ;;400
45	23	nn0cni 12543	. . . . . . . 8 ⊢ ;52 ∈ ℂ
46	45	addlidi 11425	. . . . . . 7 ⊢ (0 + ;52) = ;52
47		eqid 2760	. . . . . . . 8 ⊢ ;40 = ;40
48		5cn 12356	. . . . . . . . . 10 ⊢ 5 ∈ ℂ
49	48	addridi 11424	. . . . . . . . 9 ⊢ (5 + 0) = 5
50	22	dec0h 12766	. . . . . . . . 9 ⊢ 5 = ;05
51	49, 50	eqtri 2783	. . . . . . . 8 ⊢ (5 + 0) = ;05
52	40, 3	eqeltri 2856	. . . . . . . . 9 ⊢ (0 + 0) ∈ ℕ₀
53		eqid 2760	. . . . . . . . 9 ⊢ ;;521 = ;;521
54		eqid 2760	. . . . . . . . . 10 ⊢ ;52 = ;52
55		5t4e20 12846	. . . . . . . . . 10 ⊢ (5 · 4) = ;20
56		2t4e8 12437	. . . . . . . . . 10 ⊢ (2 · 4) = 8
57	2, 22, 10, 54, 55, 56	decmul1 12808	. . . . . . . . 9 ⊢ (;52 · 4) = ;;208
58		4cn 12353	. . . . . . . . . . . 12 ⊢ 4 ∈ ℂ
59	58	mullidi 11241	. . . . . . . . . . 11 ⊢ (1 · 4) = 4
60	59, 40	oveq12i 7426	. . . . . . . . . 10 ⊢ ((1 · 4) + (0 + 0)) = (4 + 0)
61	58	addridi 11424	. . . . . . . . . 10 ⊢ (4 + 0) = 4
62	60, 61	eqtri 2783	. . . . . . . . 9 ⊢ ((1 · 4) + (0 + 0)) = 4
63	23, 15, 52, 53, 2, 57, 62	decrmanc 12801	. . . . . . . 8 ⊢ ((;;521 · 4) + (0 + 0)) = ;;;2084
64	24	nn0cni 12543	. . . . . . . . . . 11 ⊢ ;;521 ∈ ℂ
65	64	mul01i 11427	. . . . . . . . . 10 ⊢ (;;521 · 0) = 0
66	65	oveq1i 7424	. . . . . . . . 9 ⊢ ((;;521 · 0) + 5) = (0 + 5)
67	48	addlidi 11425	. . . . . . . . 9 ⊢ (0 + 5) = 5
68	66, 67, 50	3eqtri 2787	. . . . . . . 8 ⊢ ((;;521 · 0) + 5) = ;05
69	2, 3, 3, 22, 47, 51, 24, 22, 3, 63, 68	decma2c 12797	. . . . . . 7 ⊢ ((;;521 · ;40) + (5 + 0)) = ;;;;20845
70	65	oveq1i 7424	. . . . . . . 8 ⊢ ((;;521 · 0) + 2) = (0 + 2)
71		2cn 12343	. . . . . . . . 9 ⊢ 2 ∈ ℂ
72	71	addlidi 11425	. . . . . . . 8 ⊢ (0 + 2) = 2
73	10	dec0h 12766	. . . . . . . 8 ⊢ 2 = ;02
74	70, 72, 73	3eqtri 2787	. . . . . . 7 ⊢ ((;;521 · 0) + 2) = ;02
75	4, 3, 22, 10, 44, 46, 24, 10, 3, 69, 74	decma2c 12797	. . . . . 6 ⊢ ((;;521 · ;;400) + (0 + ;52)) = ;;;;;208452
76	45	mulridi 11240	. . . . . . 7 ⊢ (;52 · 1) = ;52
77		ax-1cn 11185	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
78	77	mullidi 11241	. . . . . . . . 9 ⊢ (1 · 1) = 1
79	78	oveq1i 7424	. . . . . . . 8 ⊢ ((1 · 1) + 1) = (1 + 1)
80		1p1e2 12391	. . . . . . . 8 ⊢ (1 + 1) = 2
81	79, 80	eqtri 2783	. . . . . . 7 ⊢ ((1 · 1) + 1) = 2
82	23, 15, 15, 53, 15, 76, 81	decrmanc 12801	. . . . . 6 ⊢ ((;;521 · 1) + 1) = ;;522
83	5, 15, 3, 15, 1, 43, 24, 10, 23, 75, 82	decma2c 12797	. . . . 5 ⊢ ((;;521 · 𝑁) + 1) = ;;;;;;2084522
84		eqid 2760	. . . . . 6 ⊢ ;;902 = ;;902
85		6nn0 12552	. . . . . . . 8 ⊢ 6 ∈ ℕ₀
86	2, 85	deccl 12754	. . . . . . 7 ⊢ ;46 ∈ ℕ₀
87	86, 10	deccl 12754	. . . . . 6 ⊢ ;;462 ∈ ℕ₀
88		eqid 2760	. . . . . . 7 ⊢ ;90 = ;90
89		eqid 2760	. . . . . . 7 ⊢ ;;462 = ;;462
90		eqid 2760	. . . . . . . 8 ⊢ ;;;2311 = ;;;2311
91	86	nn0cni 12543	. . . . . . . . 9 ⊢ ;46 ∈ ℂ
92	91	addridi 11424	. . . . . . . 8 ⊢ (;46 + 0) = ;46
93		4p1e5 12413	. . . . . . . . . 10 ⊢ (4 + 1) = 5
94	93, 22	eqeltri 2856	. . . . . . . . 9 ⊢ (4 + 1) ∈ ℕ₀
95		eqid 2760	. . . . . . . . 9 ⊢ ;;231 = ;;231
96		eqid 2760	. . . . . . . . . 10 ⊢ ;23 = ;23
97		9cn 12368	. . . . . . . . . . . 12 ⊢ 9 ∈ ℂ
98		9t2e18 12866	. . . . . . . . . . . 12 ⊢ (9 · 2) = ;18
99	97, 71, 98	mulcomli 11245	. . . . . . . . . . 11 ⊢ (2 · 9) = ;18
100	15, 19, 10, 99, 80, 39	decaddci2 12806	. . . . . . . . . 10 ⊢ ((2 · 9) + 2) = ;20
101		7nn0 12553	. . . . . . . . . . 11 ⊢ 7 ∈ ℕ₀
102		7p1e8 12416	. . . . . . . . . . 11 ⊢ (7 + 1) = 8
103		3cn 12349	. . . . . . . . . . . 12 ⊢ 3 ∈ ℂ
104		9t3e27 12867	. . . . . . . . . . . 12 ⊢ (9 · 3) = ;27
105	97, 103, 104	mulcomli 11245	. . . . . . . . . . 11 ⊢ (3 · 9) = ;27
106	10, 101, 102, 105	decsuc 12775	. . . . . . . . . 10 ⊢ ((3 · 9) + 1) = ;28
107	10, 26, 15, 96, 30, 19, 10, 100, 106	decrmac 12802	. . . . . . . . 9 ⊢ ((;23 · 9) + 1) = ;;208
108	97	mullidi 11241	. . . . . . . . . . 11 ⊢ (1 · 9) = 9
109	108, 93	oveq12i 7426	. . . . . . . . . 10 ⊢ ((1 · 9) + (4 + 1)) = (9 + 5)
110		9p5e14 12834	. . . . . . . . . 10 ⊢ (9 + 5) = ;14
111	109, 110	eqtri 2783	. . . . . . . . 9 ⊢ ((1 · 9) + (4 + 1)) = ;14
112	27, 15, 94, 95, 30, 2, 15, 107, 111	decrmac 12802	. . . . . . . 8 ⊢ ((;;231 · 9) + (4 + 1)) = ;;;2084
113	108	oveq1i 7424	. . . . . . . . 9 ⊢ ((1 · 9) + 6) = (9 + 6)
114		9p6e15 12835	. . . . . . . . 9 ⊢ (9 + 6) = ;15
115	113, 114	eqtri 2783	. . . . . . . 8 ⊢ ((1 · 9) + 6) = ;15
116	28, 15, 2, 85, 90, 92, 30, 22, 15, 112, 115	decmac 12796	. . . . . . 7 ⊢ ((;;;2311 · 9) + (;46 + 0)) = ;;;;20845
117	29	nn0cni 12543	. . . . . . . . . 10 ⊢ ;;;2311 ∈ ℂ
118	117	mul01i 11427	. . . . . . . . 9 ⊢ (;;;2311 · 0) = 0
119	118	oveq1i 7424	. . . . . . . 8 ⊢ ((;;;2311 · 0) + 2) = (0 + 2)
120	119, 72, 73	3eqtri 2787	. . . . . . 7 ⊢ ((;;;2311 · 0) + 2) = ;02
121	30, 3, 86, 10, 88, 89, 29, 10, 3, 116, 120	decma2c 12797	. . . . . 6 ⊢ ((;;;2311 · ;90) + ;;462) = ;;;;;208452
122		2t2e4 12431	. . . . . . . . 9 ⊢ (2 · 2) = 4
123		3t2e6 12433	. . . . . . . . 9 ⊢ (3 · 2) = 6
124	10, 10, 26, 96, 122, 123	decmul1 12808	. . . . . . . 8 ⊢ (;23 · 2) = ;46
125	71	mullidi 11241	. . . . . . . 8 ⊢ (1 · 2) = 2
126	10, 27, 15, 95, 124, 125	decmul1 12808	. . . . . . 7 ⊢ (;;231 · 2) = ;;462
127	10, 28, 15, 90, 126, 125	decmul1 12808	. . . . . 6 ⊢ (;;;2311 · 2) = ;;;4622
128	29, 31, 10, 84, 10, 87, 121, 127	decmul2c 12810	. . . . 5 ⊢ (;;;2311 · ;;902) = ;;;;;;2084522
129	83, 128	eqtr4i 2786	. . . 4 ⊢ ((;;521 · 𝑁) + 1) = (;;;2311 · ;;902)
130	8, 9, 21, 25, 29, 15, 12, 32, 33, 34, 42, 129	modxai 17163	. . 3 ⊢ ((2↑;;;1000) mod 𝑁) = (1 mod 𝑁)
131	18	nn0cni 12543	. . . 4 ⊢ ;;;1000 ∈ ℂ
132		eqid 2760	. . . . 5 ⊢ ;;;1000 = ;;;1000
133		eqid 2760	. . . . . 6 ⊢ ;;100 = ;;100
134	10	dec0u 12765	. . . . . 6 ⊢ (;10 · 2) = ;20
135	71	mul02i 11426	. . . . . 6 ⊢ (0 · 2) = 0
136	10, 16, 3, 133, 134, 135	decmul1 12808	. . . . 5 ⊢ (;;100 · 2) = ;;200
137	10, 17, 3, 132, 136, 135	decmul1 12808	. . . 4 ⊢ (;;;1000 · 2) = ;;;2000
138	131, 71, 137	mulcomli 11245	. . 3 ⊢ (2 · ;;;1000) = ;;;2000
139	8	nncni 12270	. . . . . 6 ⊢ 𝑁 ∈ ℂ
140	139	mul02i 11426	. . . . 5 ⊢ (0 · 𝑁) = 0
141	140	oveq1i 7424	. . . 4 ⊢ ((0 · 𝑁) + 1) = (0 + 1)
142	77	addlidi 11425	. . . . 5 ⊢ (0 + 1) = 1
143	78, 142	eqtr4i 2786	. . . 4 ⊢ (1 · 1) = (0 + 1)
144	141, 143	eqtr4i 2786	. . 3 ⊢ ((0 · 𝑁) + 1) = (1 · 1)
145	8, 9, 18, 14, 15, 15, 130, 138, 144	mod2xi 17164	. 2 ⊢ ((2↑;;;2000) mod 𝑁) = (1 mod 𝑁)
146	13	nn0cni 12543	. . . 4 ⊢ ;;;2000 ∈ ℂ
147		eqid 2760	. . . . 5 ⊢ ;;;2000 = ;;;2000
148	10, 10, 3, 38, 122, 135	decmul1 12808	. . . . . 6 ⊢ (;20 · 2) = ;40
149	10, 11, 3, 36, 148, 135	decmul1 12808	. . . . 5 ⊢ (;;200 · 2) = ;;400
150	10, 12, 3, 147, 149, 135	decmul1 12808	. . . 4 ⊢ (;;;2000 · 2) = ;;;4000
151	146, 71, 150	mulcomli 11245	. . 3 ⊢ (2 · ;;;2000) = ;;;4000
152	5, 3	deccl 12754	. . . . 5 ⊢ ;;;4000 ∈ ℕ₀
153	152	nn0cni 12543	. . . 4 ⊢ ;;;4000 ∈ ℂ
154		eqid 2760	. . . . . 6 ⊢ ;;;4000 = ;;;4000
155	5, 3, 142, 154	decsuc 12775	. . . . 5 ⊢ (;;;4000 + 1) = ;;;4001
156	1, 155	eqtr4i 2786	. . . 4 ⊢ 𝑁 = (;;;4000 + 1)
157	153, 77, 156	mvrraddi 11501	. . 3 ⊢ (𝑁 − 1) = ;;;4000
158	151, 157	eqtr4i 2786	. 2 ⊢ (2 · ;;;2000) = (𝑁 − 1)
159	8, 9, 13, 14, 15, 15, 145, 158, 144	mod2xi 17164	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 7414 0cc0 11127 1c1 11128 + caddc 11130 · cmul 11132 − cmin 11468 ℕcn 12260 2c2 12322 3c3 12323 4c4 12324 5c5 12325 6c6 12326 7c7 12327 8c8 12328 9c9 12329 ℕ₀cn0 12531 cdc 12739 mod cmo 13933 ↑cexp 14128

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 2732 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205

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 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 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 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7864 df-2nd 7988 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-er 8699 df-en 8956 df-dom 8957 df-sdom 8958 df-sup 9415 df-inf 9416 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-div 11899 df-nn 12261 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12532 df-z 12619 df-dec 12740 df-uz 12891 df-rp 13046 df-fl 13856 df-mod 13934 df-seq 14069 df-exp 14129

This theorem is used by: 4001prm 17240

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