4001lem2 - Metamath Proof Explorer

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Theorem 4001lem2 15687

Description: Lemma for 4001prm 15690. Calculate a power mod. In decimal, we calculate 2↑400 = (2↑200)↑2≡902↑2 = 203𝑁 + 1401 and 2↑800 = (2↑400)↑2≡1401↑2 = 490𝑁 + 2311 ≡2311. (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
4001lem2	⊢ ((2↑;;800) mod 𝑁) = (;;;2311 mod 𝑁)

**Proof of Theorem 4001lem2**
Step	Hyp	Ref	Expression
1		4001prm.1	. . 3 ⊢ 𝑁 = ;;;4001
2		4nn0 11188	. . . . . 6 ⊢ 4 ∈ ℕ₀
3		0nn0 11184	. . . . . 6 ⊢ 0 ∈ ℕ₀
4	2, 3	deccl 11388	. . . . 5 ⊢ ;40 ∈ ℕ₀
5	4, 3	deccl 11388	. . . 4 ⊢ ;;400 ∈ ℕ₀
6		1nn 10908	. . . 4 ⊢ 1 ∈ ℕ
7	5, 6	decnncl 11394	. . 3 ⊢ ;;;4001 ∈ ℕ
8	1, 7	eqeltri 2684	. 2 ⊢ 𝑁 ∈ ℕ
9		2nn 11062	. 2 ⊢ 2 ∈ ℕ
10		9nn0 11193	. . . . 5 ⊢ 9 ∈ ℕ₀
11	2, 10	deccl 11388	. . . 4 ⊢ ;49 ∈ ℕ₀
12	11, 3	deccl 11388	. . 3 ⊢ ;;490 ∈ ℕ₀
13	12	nn0zi 11279	. 2 ⊢ ;;490 ∈ ℤ
14		1nn0 11185	. . . . 5 ⊢ 1 ∈ ℕ₀
15	14, 2	deccl 11388	. . . 4 ⊢ ;14 ∈ ℕ₀
16	15, 3	deccl 11388	. . 3 ⊢ ;;140 ∈ ℕ₀
17	16, 14	deccl 11388	. 2 ⊢ ;;;1401 ∈ ℕ₀
18		2nn0 11186	. . . . 5 ⊢ 2 ∈ ℕ₀
19		3nn0 11187	. . . . 5 ⊢ 3 ∈ ℕ₀
20	18, 19	deccl 11388	. . . 4 ⊢ ;23 ∈ ℕ₀
21	20, 14	deccl 11388	. . 3 ⊢ ;;231 ∈ ℕ₀
22	21, 14	deccl 11388	. 2 ⊢ ;;;2311 ∈ ℕ₀
23	18, 3	deccl 11388	. . . 4 ⊢ ;20 ∈ ℕ₀
24	23, 3	deccl 11388	. . 3 ⊢ ;;200 ∈ ℕ₀
25	23, 19	deccl 11388	. . . 4 ⊢ ;;203 ∈ ℕ₀
26	25	nn0zi 11279	. . 3 ⊢ ;;203 ∈ ℤ
27	10, 3	deccl 11388	. . . 4 ⊢ ;90 ∈ ℕ₀
28	27, 18	deccl 11388	. . 3 ⊢ ;;902 ∈ ℕ₀
29	1	4001lem1 15686	. . 3 ⊢ ((2↑;;200) mod 𝑁) = (;;902 mod 𝑁)
30	24	nn0cni 11181	. . . 4 ⊢ ;;200 ∈ ℂ
31		2cn 10968	. . . 4 ⊢ 2 ∈ ℂ
32		eqid 2610	. . . . 5 ⊢ ;;200 = ;;200
33		eqid 2610	. . . . . 6 ⊢ ;20 = ;20
34		2t2e4 11054	. . . . . 6 ⊢ (2 · 2) = 4
35	31	mul02i 10104	. . . . . 6 ⊢ (0 · 2) = 0
36	18, 18, 3, 33, 3, 34, 35	decmul1 11461	. . . . 5 ⊢ (;20 · 2) = ;40
37	18, 23, 3, 32, 3, 36, 35	decmul1 11461	. . . 4 ⊢ (;;200 · 2) = ;;400
38	30, 31, 37	mulcomli 9926	. . 3 ⊢ (2 · ;;200) = ;;400
39		eqid 2610	. . . . 5 ⊢ ;;;1401 = ;;;1401
40		6nn0 11190	. . . . . . 7 ⊢ 6 ∈ ℕ₀
41	14, 40	deccl 11388	. . . . . 6 ⊢ ;16 ∈ ℕ₀
42		eqid 2610	. . . . . 6 ⊢ ;;400 = ;;400
43		eqid 2610	. . . . . . 7 ⊢ ;;140 = ;;140
44		eqid 2610	. . . . . . . 8 ⊢ ;14 = ;14
45		4p2e6 11039	. . . . . . . 8 ⊢ (4 + 2) = 6
46	14, 2, 18, 44, 45	decaddi 11455	. . . . . . 7 ⊢ (;14 + 2) = ;16
47		00id 10090	. . . . . . 7 ⊢ (0 + 0) = 0
48	15, 3, 18, 3, 43, 33, 46, 47	decadd 11446	. . . . . 6 ⊢ (;;140 + ;20) = ;;160
49		eqid 2610	. . . . . . 7 ⊢ ;40 = ;40
50	41	nn0cni 11181	. . . . . . . 8 ⊢ ;16 ∈ ℂ
51	50	addid1i 10102	. . . . . . 7 ⊢ (;16 + 0) = ;16
52		eqid 2610	. . . . . . . 8 ⊢ ;;203 = ;;203
53		ax-1cn 9873	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
54	53	addid1i 10102	. . . . . . . . 9 ⊢ (1 + 0) = 1
55	14	dec0h 11398	. . . . . . . . 9 ⊢ 1 = ;01
56	54, 55	eqtri 2632	. . . . . . . 8 ⊢ (1 + 0) = ;01
57	53	addid2i 10103	. . . . . . . . . 10 ⊢ (0 + 1) = 1
58	57, 14	eqeltri 2684	. . . . . . . . 9 ⊢ (0 + 1) ∈ ℕ₀
59		4cn 10975	. . . . . . . . . 10 ⊢ 4 ∈ ℂ
60		4t2e8 11058	. . . . . . . . . 10 ⊢ (4 · 2) = 8
61	59, 31, 60	mulcomli 9926	. . . . . . . . 9 ⊢ (2 · 4) = 8
62	59	mul02i 10104	. . . . . . . . . . 11 ⊢ (0 · 4) = 0
63	62, 57	oveq12i 6561	. . . . . . . . . 10 ⊢ ((0 · 4) + (0 + 1)) = (0 + 1)
64	63, 57	eqtri 2632	. . . . . . . . 9 ⊢ ((0 · 4) + (0 + 1)) = 1
65	18, 3, 58, 33, 2, 61, 64	decrmanc 11452	. . . . . . . 8 ⊢ ((;20 · 4) + (0 + 1)) = ;81
66		2p1e3 11028	. . . . . . . . 9 ⊢ (2 + 1) = 3
67		3cn 10972	. . . . . . . . . 10 ⊢ 3 ∈ ℂ
68		4t3e12 11508	. . . . . . . . . 10 ⊢ (4 · 3) = ;12
69	59, 67, 68	mulcomli 9926	. . . . . . . . 9 ⊢ (3 · 4) = ;12
70	14, 18, 66, 69	decsuc 11411	. . . . . . . 8 ⊢ ((3 · 4) + 1) = ;13
71	23, 19, 3, 14, 52, 56, 2, 19, 14, 65, 70	decmac 11442	. . . . . . 7 ⊢ ((;;203 · 4) + (1 + 0)) = ;;813
72	25	nn0cni 11181	. . . . . . . . . 10 ⊢ ;;203 ∈ ℂ
73	72	mul01i 10105	. . . . . . . . 9 ⊢ (;;203 · 0) = 0
74	73	oveq1i 6559	. . . . . . . 8 ⊢ ((;;203 · 0) + 6) = (0 + 6)
75		6cn 10979	. . . . . . . . 9 ⊢ 6 ∈ ℂ
76	75	addid2i 10103	. . . . . . . 8 ⊢ (0 + 6) = 6
77	40	dec0h 11398	. . . . . . . 8 ⊢ 6 = ;06
78	74, 76, 77	3eqtri 2636	. . . . . . 7 ⊢ ((;;203 · 0) + 6) = ;06
79	2, 3, 14, 40, 49, 51, 25, 40, 3, 71, 78	decma2c 11444	. . . . . 6 ⊢ ((;;203 · ;40) + (;16 + 0)) = ;;;8136
80	73	oveq1i 6559	. . . . . . 7 ⊢ ((;;203 · 0) + 0) = (0 + 0)
81	3	dec0h 11398	. . . . . . 7 ⊢ 0 = ;00
82	80, 47, 81	3eqtri 2636	. . . . . 6 ⊢ ((;;203 · 0) + 0) = ;00
83	4, 3, 41, 3, 42, 48, 25, 3, 3, 79, 82	decma2c 11444	. . . . 5 ⊢ ((;;203 · ;;400) + (;;140 + ;20)) = ;;;;81360
84	31	mulid1i 9921	. . . . . . 7 ⊢ (2 · 1) = 2
85	53	mul02i 10104	. . . . . . 7 ⊢ (0 · 1) = 0
86	14, 18, 3, 33, 3, 84, 85	decmul1 11461	. . . . . 6 ⊢ (;20 · 1) = ;20
87	67	mulid1i 9921	. . . . . . . 8 ⊢ (3 · 1) = 3
88	87	oveq1i 6559	. . . . . . 7 ⊢ ((3 · 1) + 1) = (3 + 1)
89		3p1e4 11030	. . . . . . 7 ⊢ (3 + 1) = 4
90	88, 89	eqtri 2632	. . . . . 6 ⊢ ((3 · 1) + 1) = 4
91	23, 19, 14, 52, 14, 86, 90	decrmanc 11452	. . . . 5 ⊢ ((;;203 · 1) + 1) = ;;204
92	5, 14, 16, 14, 1, 39, 25, 2, 23, 83, 91	decma2c 11444	. . . 4 ⊢ ((;;203 · 𝑁) + ;;;1401) = ;;;;;813604
93		eqid 2610	. . . . 5 ⊢ ;;902 = ;;902
94		8nn0 11192	. . . . . . 7 ⊢ 8 ∈ ℕ₀
95	14, 94	deccl 11388	. . . . . 6 ⊢ ;18 ∈ ℕ₀
96	95, 3	deccl 11388	. . . . 5 ⊢ ;;180 ∈ ℕ₀
97		eqid 2610	. . . . . 6 ⊢ ;90 = ;90
98		eqid 2610	. . . . . 6 ⊢ ;;180 = ;;180
99	95	nn0cni 11181	. . . . . . . 8 ⊢ ;18 ∈ ℂ
100	99	addid1i 10102	. . . . . . 7 ⊢ (;18 + 0) = ;18
101		1p2e3 11029	. . . . . . . . 9 ⊢ (1 + 2) = 3
102	101, 19	eqeltri 2684	. . . . . . . 8 ⊢ (1 + 2) ∈ ℕ₀
103		9t9e81 11546	. . . . . . . 8 ⊢ (9 · 9) = ;81
104		9cn 10985	. . . . . . . . . . 11 ⊢ 9 ∈ ℂ
105	104	mul02i 10104	. . . . . . . . . 10 ⊢ (0 · 9) = 0
106	105, 101	oveq12i 6561	. . . . . . . . 9 ⊢ ((0 · 9) + (1 + 2)) = (0 + 3)
107	67	addid2i 10103	. . . . . . . . 9 ⊢ (0 + 3) = 3
108	106, 107	eqtri 2632	. . . . . . . 8 ⊢ ((0 · 9) + (1 + 2)) = 3
109	10, 3, 102, 97, 10, 103, 108	decrmanc 11452	. . . . . . 7 ⊢ ((;90 · 9) + (1 + 2)) = ;;813
110		9t2e18 11539	. . . . . . . . 9 ⊢ (9 · 2) = ;18
111	104, 31, 110	mulcomli 9926	. . . . . . . 8 ⊢ (2 · 9) = ;18
112		1p1e2 11011	. . . . . . . 8 ⊢ (1 + 1) = 2
113		8p8e16 11494	. . . . . . . 8 ⊢ (8 + 8) = ;16
114	14, 94, 94, 111, 112, 40, 113	decaddci 11456	. . . . . . 7 ⊢ ((2 · 9) + 8) = ;26
115	27, 18, 14, 94, 93, 100, 10, 40, 18, 109, 114	decmac 11442	. . . . . 6 ⊢ ((;;902 · 9) + (;18 + 0)) = ;;;8136
116	28	nn0cni 11181	. . . . . . . . 9 ⊢ ;;902 ∈ ℂ
117	116	mul01i 10105	. . . . . . . 8 ⊢ (;;902 · 0) = 0
118	117	oveq1i 6559	. . . . . . 7 ⊢ ((;;902 · 0) + 0) = (0 + 0)
119	118, 47, 81	3eqtri 2636	. . . . . 6 ⊢ ((;;902 · 0) + 0) = ;00
120	10, 3, 95, 3, 97, 98, 28, 3, 3, 115, 119	decma2c 11444	. . . . 5 ⊢ ((;;902 · ;90) + ;;180) = ;;;;81360
121	18, 10, 3, 97, 3, 110, 35	decmul1 11461	. . . . . 6 ⊢ (;90 · 2) = ;;180
122	18, 27, 18, 93, 2, 121, 34	decmul1 11461	. . . . 5 ⊢ (;;902 · 2) = ;;;1804
123	28, 27, 18, 93, 2, 96, 120, 122	decmul2c 11465	. . . 4 ⊢ (;;902 · ;;902) = ;;;;;813604
124	92, 123	eqtr4i 2635	. . 3 ⊢ ((;;203 · 𝑁) + ;;;1401) = (;;902 · ;;902)
125	8, 9, 24, 26, 28, 17, 29, 38, 124	mod2xi 15611	. 2 ⊢ ((2↑;;400) mod 𝑁) = (;;;1401 mod 𝑁)
126	5	nn0cni 11181	. . 3 ⊢ ;;400 ∈ ℂ
127	18, 2, 3, 49, 3, 60, 35	decmul1 11461	. . . 4 ⊢ (;40 · 2) = ;80
128	18, 4, 3, 42, 3, 127, 35	decmul1 11461	. . 3 ⊢ (;;400 · 2) = ;;800
129	126, 31, 128	mulcomli 9926	. 2 ⊢ (2 · ;;400) = ;;800
130		eqid 2610	. . . 4 ⊢ ;;;2311 = ;;;2311
131	18, 94	deccl 11388	. . . . 5 ⊢ ;28 ∈ ℕ₀
132		eqid 2610	. . . . . 6 ⊢ ;;231 = ;;231
133		eqid 2610	. . . . . 6 ⊢ ;49 = ;49
134		7nn0 11191	. . . . . . 7 ⊢ 7 ∈ ℕ₀
135		7p1e8 11034	. . . . . . 7 ⊢ (7 + 1) = 8
136		eqid 2610	. . . . . . . 8 ⊢ ;23 = ;23
137		4p3e7 11040	. . . . . . . . 9 ⊢ (4 + 3) = 7
138	59, 67, 137	addcomli 10107	. . . . . . . 8 ⊢ (3 + 4) = 7
139	18, 19, 2, 136, 138	decaddi 11455	. . . . . . 7 ⊢ (;23 + 4) = ;27
140	18, 134, 135, 139	decsuc 11411	. . . . . 6 ⊢ ((;23 + 4) + 1) = ;28
141		9p1e10 11372	. . . . . . 7 ⊢ (9 + 1) = ;10
142	104, 53, 141	addcomli 10107	. . . . . 6 ⊢ (1 + 9) = ;10
143	20, 14, 2, 10, 132, 133, 140, 142	decaddc2 11451	. . . . 5 ⊢ (;;231 + ;49) = ;;280
144	131	nn0cni 11181	. . . . . . 7 ⊢ ;28 ∈ ℂ
145	144	addid1i 10102	. . . . . 6 ⊢ (;28 + 0) = ;28
146	31	addid1i 10102	. . . . . . . 8 ⊢ (2 + 0) = 2
147	146, 18	eqeltri 2684	. . . . . . 7 ⊢ (2 + 0) ∈ ℕ₀
148		eqid 2610	. . . . . . 7 ⊢ ;;490 = ;;490
149		4t4e16 11509	. . . . . . . . 9 ⊢ (4 · 4) = ;16
150		6p3e9 11047	. . . . . . . . 9 ⊢ (6 + 3) = 9
151	14, 40, 19, 149, 150	decaddi 11455	. . . . . . . 8 ⊢ ((4 · 4) + 3) = ;19
152		9t4e36 11541	. . . . . . . 8 ⊢ (9 · 4) = ;36
153	2, 2, 10, 133, 40, 19, 151, 152	decmul1c 11463	. . . . . . 7 ⊢ (;49 · 4) = ;;196
154	62, 146	oveq12i 6561	. . . . . . . 8 ⊢ ((0 · 4) + (2 + 0)) = (0 + 2)
155	31	addid2i 10103	. . . . . . . 8 ⊢ (0 + 2) = 2
156	154, 155	eqtri 2632	. . . . . . 7 ⊢ ((0 · 4) + (2 + 0)) = 2
157	11, 3, 147, 148, 2, 153, 156	decrmanc 11452	. . . . . 6 ⊢ ((;;490 · 4) + (2 + 0)) = ;;;1962
158	12	nn0cni 11181	. . . . . . . . 9 ⊢ ;;490 ∈ ℂ
159	158	mul01i 10105	. . . . . . . 8 ⊢ (;;490 · 0) = 0
160	159	oveq1i 6559	. . . . . . 7 ⊢ ((;;490 · 0) + 8) = (0 + 8)
161		8cn 10983	. . . . . . . 8 ⊢ 8 ∈ ℂ
162	161	addid2i 10103	. . . . . . 7 ⊢ (0 + 8) = 8
163	94	dec0h 11398	. . . . . . 7 ⊢ 8 = ;08
164	160, 162, 163	3eqtri 2636	. . . . . 6 ⊢ ((;;490 · 0) + 8) = ;08
165	2, 3, 18, 94, 49, 145, 12, 94, 3, 157, 164	decma2c 11444	. . . . 5 ⊢ ((;;490 · ;40) + (;28 + 0)) = ;;;;19628
166	159	oveq1i 6559	. . . . . 6 ⊢ ((;;490 · 0) + 0) = (0 + 0)
167	166, 47, 81	3eqtri 2636	. . . . 5 ⊢ ((;;490 · 0) + 0) = ;00
168	4, 3, 131, 3, 42, 143, 12, 3, 3, 165, 167	decma2c 11444	. . . 4 ⊢ ((;;490 · ;;400) + (;;231 + ;49)) = ;;;;;196280
169	59	mulid1i 9921	. . . . . 6 ⊢ (4 · 1) = 4
170	104	mulid1i 9921	. . . . . 6 ⊢ (9 · 1) = 9
171	14, 2, 10, 133, 10, 169, 170	decmul1 11461	. . . . 5 ⊢ (;49 · 1) = ;49
172	85	oveq1i 6559	. . . . . 6 ⊢ ((0 · 1) + 1) = (0 + 1)
173	172, 57	eqtri 2632	. . . . 5 ⊢ ((0 · 1) + 1) = 1
174	11, 3, 14, 148, 14, 171, 173	decrmanc 11452	. . . 4 ⊢ ((;;490 · 1) + 1) = ;;491
175	5, 14, 21, 14, 1, 130, 12, 14, 11, 168, 174	decma2c 11444	. . 3 ⊢ ((;;490 · 𝑁) + ;;;2311) = ;;;;;;1962801
176	15	nn0cni 11181	. . . . . . 7 ⊢ ;14 ∈ ℂ
177	176	addid1i 10102	. . . . . 6 ⊢ (;14 + 0) = ;14
178		5nn0 11189	. . . . . . . 8 ⊢ 5 ∈ ℕ₀
179	178, 40	deccl 11388	. . . . . . 7 ⊢ ;56 ∈ ℕ₀
180	179, 3	deccl 11388	. . . . . 6 ⊢ ;;560 ∈ ℕ₀
181		eqid 2610	. . . . . . . 8 ⊢ ;;560 = ;;560
182	179	nn0cni 11181	. . . . . . . . 9 ⊢ ;56 ∈ ℂ
183	182	addid2i 10103	. . . . . . . 8 ⊢ (0 + ;56) = ;56
184	3, 14, 179, 3, 55, 181, 183, 54	decadd 11446	. . . . . . 7 ⊢ (1 + ;;560) = ;;561
185	182	addid1i 10102	. . . . . . . 8 ⊢ (;56 + 0) = ;56
186		5cn 10977	. . . . . . . . . . 11 ⊢ 5 ∈ ℂ
187	186	addid1i 10102	. . . . . . . . . 10 ⊢ (5 + 0) = 5
188	187, 178	eqeltri 2684	. . . . . . . . 9 ⊢ (5 + 0) ∈ ℕ₀
189	53	mulid1i 9921	. . . . . . . . 9 ⊢ (1 · 1) = 1
190	169, 187	oveq12i 6561	. . . . . . . . . 10 ⊢ ((4 · 1) + (5 + 0)) = (4 + 5)
191		5p4e9 11044	. . . . . . . . . . 11 ⊢ (5 + 4) = 9
192	186, 59, 191	addcomli 10107	. . . . . . . . . 10 ⊢ (4 + 5) = 9
193	190, 192	eqtri 2632	. . . . . . . . 9 ⊢ ((4 · 1) + (5 + 0)) = 9
194	14, 2, 188, 44, 14, 189, 193	decrmanc 11452	. . . . . . . 8 ⊢ ((;14 · 1) + (5 + 0)) = ;19
195	85	oveq1i 6559	. . . . . . . . 9 ⊢ ((0 · 1) + 6) = (0 + 6)
196	195, 76, 77	3eqtri 2636	. . . . . . . 8 ⊢ ((0 · 1) + 6) = ;06
197	15, 3, 178, 40, 43, 185, 14, 40, 3, 194, 196	decmac 11442	. . . . . . 7 ⊢ ((;;140 · 1) + (;56 + 0)) = ;;196
198	189	oveq1i 6559	. . . . . . . 8 ⊢ ((1 · 1) + 1) = (1 + 1)
199	18	dec0h 11398	. . . . . . . 8 ⊢ 2 = ;02
200	198, 112, 199	3eqtri 2636	. . . . . . 7 ⊢ ((1 · 1) + 1) = ;02
201	16, 14, 179, 14, 39, 184, 14, 18, 3, 197, 200	decmac 11442	. . . . . 6 ⊢ ((;;;1401 · 1) + (1 + ;;560)) = ;;;1962
202	59	mulid2i 9922	. . . . . . . . . . . 12 ⊢ (1 · 4) = 4
203	202	oveq1i 6559	. . . . . . . . . . 11 ⊢ ((1 · 4) + 1) = (4 + 1)
204		4p1e5 11031	. . . . . . . . . . 11 ⊢ (4 + 1) = 5
205	203, 204	eqtri 2632	. . . . . . . . . 10 ⊢ ((1 · 4) + 1) = 5
206	2, 14, 2, 44, 40, 14, 205, 149	decmul1c 11463	. . . . . . . . 9 ⊢ (;14 · 4) = ;56
207	75	addid1i 10102	. . . . . . . . 9 ⊢ (6 + 0) = 6
208	178, 40, 3, 206, 207	decaddi 11455	. . . . . . . 8 ⊢ ((;14 · 4) + 0) = ;56
209		0cn 9911	. . . . . . . . 9 ⊢ 0 ∈ ℂ
210	59	mul01i 10105	. . . . . . . . . 10 ⊢ (4 · 0) = 0
211	210, 81	eqtri 2632	. . . . . . . . 9 ⊢ (4 · 0) = ;00
212	59, 209, 211	mulcomli 9926	. . . . . . . 8 ⊢ (0 · 4) = ;00
213	2, 15, 3, 43, 3, 3, 208, 212	decmul1c 11463	. . . . . . 7 ⊢ (;;140 · 4) = ;;560
214	202	oveq1i 6559	. . . . . . . 8 ⊢ ((1 · 4) + 4) = (4 + 4)
215		4p4e8 11041	. . . . . . . 8 ⊢ (4 + 4) = 8
216	214, 215	eqtri 2632	. . . . . . 7 ⊢ ((1 · 4) + 4) = 8
217	16, 14, 2, 39, 2, 213, 216	decrmanc 11452	. . . . . 6 ⊢ ((;;;1401 · 4) + 4) = ;;;5608
218	14, 2, 14, 2, 44, 177, 17, 94, 180, 201, 217	decma2c 11444	. . . . 5 ⊢ ((;;;1401 · ;14) + (;14 + 0)) = ;;;;19628
219	17	nn0cni 11181	. . . . . . . 8 ⊢ ;;;1401 ∈ ℂ
220	219	mul01i 10105	. . . . . . 7 ⊢ (;;;1401 · 0) = 0
221	220	oveq1i 6559	. . . . . 6 ⊢ ((;;;1401 · 0) + 0) = (0 + 0)
222	221, 47, 81	3eqtri 2636	. . . . 5 ⊢ ((;;;1401 · 0) + 0) = ;00
223	15, 3, 15, 3, 43, 43, 17, 3, 3, 218, 222	decma2c 11444	. . . 4 ⊢ ((;;;1401 · ;;140) + ;;140) = ;;;;;196280
224	219	mulid1i 9921	. . . 4 ⊢ (;;;1401 · 1) = ;;;1401
225	17, 16, 14, 39, 14, 16, 223, 224	decmul2c 11465	. . 3 ⊢ (;;;1401 · ;;;1401) = ;;;;;;1962801
226	175, 225	eqtr4i 2635	. 2 ⊢ ((;;490 · 𝑁) + ;;;2311) = (;;;1401 · ;;;1401)
227	8, 9, 5, 13, 17, 22, 125, 129, 226	mod2xi 15611	1 ⊢ ((2↑;;800) mod 𝑁) = (;;;2311 mod 𝑁)

Colors of variables: wff setvar class

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 ℕ₀cn0 11169 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: 4001lem3 15688 4001lem4 15689

W3C validator