Theorem 4001lem2 16591

Description: Lemma for 4001prm 16594. 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 12008	. . . . . 6 ⊢ 4 ∈ ℕ0
3		0nn0 12004	. . . . . 6 ⊢ 0 ∈ ℕ0
4	2, 3	deccl 12207	. . . . 5 ⊢ ;40 ∈ ℕ0
5	4, 3	deccl 12207	. . . 4 ⊢ ;;400 ∈ ℕ0
6		1nn 11740	. . . 4 ⊢ 1 ∈ ℕ
7	5, 6	decnncl 12212	. . 3 ⊢ ;;;4001 ∈ ℕ
8	1, 7	eqeltri 2830	. 2 ⊢ 𝑁 ∈ ℕ
9		2nn 11802	. 2 ⊢ 2 ∈ ℕ
10		9nn0 12013	. . . . 5 ⊢ 9 ∈ ℕ0
11	2, 10	deccl 12207	. . . 4 ⊢ ;49 ∈ ℕ0
12	11, 3	deccl 12207	. . 3 ⊢ ;;490 ∈ ℕ0
13	12	nn0zi 12101	. 2 ⊢ ;;490 ∈ ℤ
14		1nn0 12005	. . . . 5 ⊢ 1 ∈ ℕ0
15	14, 2	deccl 12207	. . . 4 ⊢ ;14 ∈ ℕ0
16	15, 3	deccl 12207	. . 3 ⊢ ;;140 ∈ ℕ0
17	16, 14	deccl 12207	. 2 ⊢ ;;;1401 ∈ ℕ0
18		2nn0 12006	. . . . 5 ⊢ 2 ∈ ℕ0
19		3nn0 12007	. . . . 5 ⊢ 3 ∈ ℕ0
20	18, 19	deccl 12207	. . . 4 ⊢ ;23 ∈ ℕ0
21	20, 14	deccl 12207	. . 3 ⊢ ;;231 ∈ ℕ0
22	21, 14	deccl 12207	. 2 ⊢ ;;;2311 ∈ ℕ0
23	18, 3	deccl 12207	. . . 4 ⊢ ;20 ∈ ℕ0
24	23, 3	deccl 12207	. . 3 ⊢ ;;200 ∈ ℕ0
25	23, 19	deccl 12207	. . . 4 ⊢ ;;203 ∈ ℕ0
26	25	nn0zi 12101	. . 3 ⊢ ;;203 ∈ ℤ
27	10, 3	deccl 12207	. . . 4 ⊢ ;90 ∈ ℕ0
28	27, 18	deccl 12207	. . 3 ⊢ ;;902 ∈ ℕ0
29	1	4001lem1 16590	. . 3 ⊢ ((2↑;;200) mod 𝑁) = (;;902 mod 𝑁)
30	24	nn0cni 12001	. . . 4 ⊢ ;;200 ∈ ℂ
31		2cn 11804	. . . 4 ⊢ 2 ∈ ℂ
32		eqid 2739	. . . . 5 ⊢ ;;200 = ;;200
33		eqid 2739	. . . . . 6 ⊢ ;20 = ;20
34		2t2e4 11893	. . . . . 6 ⊢ (2 · 2) = 4
35	31	mul02i 10920	. . . . . 6 ⊢ (0 · 2) = 0
36	18, 18, 3, 33, 34, 35	decmul1 12256	. . . . 5 ⊢ (;20 · 2) = ;40
37	18, 23, 3, 32, 36, 35	decmul1 12256	. . . 4 ⊢ (;;200 · 2) = ;;400
38	30, 31, 37	mulcomli 10741	. . 3 ⊢ (2 · ;;200) = ;;400
39		eqid 2739	. . . . 5 ⊢ ;;;1401 = ;;;1401
40		6nn0 12010	. . . . . . 7 ⊢ 6 ∈ ℕ0
41	14, 40	deccl 12207	. . . . . 6 ⊢ ;16 ∈ ℕ0
42		eqid 2739	. . . . . 6 ⊢ ;;400 = ;;400
43		eqid 2739	. . . . . . 7 ⊢ ;;140 = ;;140
44		eqid 2739	. . . . . . . 8 ⊢ ;14 = ;14
45		4p2e6 11882	. . . . . . . 8 ⊢ (4 + 2) = 6
46	14, 2, 18, 44, 45	decaddi 12252	. . . . . . 7 ⊢ (;14 + 2) = ;16
47		00id 10906	. . . . . . 7 ⊢ (0 + 0) = 0
48	15, 3, 18, 3, 43, 33, 46, 47	decadd 12246	. . . . . 6 ⊢ (;;140 + ;20) = ;;160
49		eqid 2739	. . . . . . 7 ⊢ ;40 = ;40
50	41	nn0cni 12001	. . . . . . . 8 ⊢ ;16 ∈ ℂ
51	50	addid1i 10918	. . . . . . 7 ⊢ (;16 + 0) = ;16
52		eqid 2739	. . . . . . . 8 ⊢ ;;203 = ;;203
53		ax-1cn 10686	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
54	53	addid1i 10918	. . . . . . . . 9 ⊢ (1 + 0) = 1
55	14	dec0h 12214	. . . . . . . . 9 ⊢ 1 = ;01
56	54, 55	eqtri 2762	. . . . . . . 8 ⊢ (1 + 0) = ;01
57	53	addid2i 10919	. . . . . . . . . 10 ⊢ (0 + 1) = 1
58	57, 14	eqeltri 2830	. . . . . . . . 9 ⊢ (0 + 1) ∈ ℕ0
59		4cn 11814	. . . . . . . . . 10 ⊢ 4 ∈ ℂ
60		4t2e8 11897	. . . . . . . . . 10 ⊢ (4 · 2) = 8
61	59, 31, 60	mulcomli 10741	. . . . . . . . 9 ⊢ (2 · 4) = 8
62	59	mul02i 10920	. . . . . . . . . . 11 ⊢ (0 · 4) = 0
63	62, 57	oveq12i 7195	. . . . . . . . . 10 ⊢ ((0 · 4) + (0 + 1)) = (0 + 1)
64	63, 57	eqtri 2762	. . . . . . . . 9 ⊢ ((0 · 4) + (0 + 1)) = 1
65	18, 3, 58, 33, 2, 61, 64	decrmanc 12249	. . . . . . . 8 ⊢ ((;20 · 4) + (0 + 1)) = ;81
66		2p1e3 11871	. . . . . . . . 9 ⊢ (2 + 1) = 3
67		3cn 11810	. . . . . . . . . 10 ⊢ 3 ∈ ℂ
68		4t3e12 12290	. . . . . . . . . 10 ⊢ (4 · 3) = ;12
69	59, 67, 68	mulcomli 10741	. . . . . . . . 9 ⊢ (3 · 4) = ;12
70	14, 18, 66, 69	decsuc 12223	. . . . . . . 8 ⊢ ((3 · 4) + 1) = ;13
71	23, 19, 3, 14, 52, 56, 2, 19, 14, 65, 70	decmac 12244	. . . . . . 7 ⊢ ((;;203 · 4) + (1 + 0)) = ;;813
72	25	nn0cni 12001	. . . . . . . . . 10 ⊢ ;;203 ∈ ℂ
73	72	mul01i 10921	. . . . . . . . 9 ⊢ (;;203 · 0) = 0
74	73	oveq1i 7193	. . . . . . . 8 ⊢ ((;;203 · 0) + 6) = (0 + 6)
75		6cn 11820	. . . . . . . . 9 ⊢ 6 ∈ ℂ
76	75	addid2i 10919	. . . . . . . 8 ⊢ (0 + 6) = 6
77	40	dec0h 12214	. . . . . . . 8 ⊢ 6 = ;06
78	74, 76, 77	3eqtri 2766	. . . . . . 7 ⊢ ((;;203 · 0) + 6) = ;06
79	2, 3, 14, 40, 49, 51, 25, 40, 3, 71, 78	decma2c 12245	. . . . . 6 ⊢ ((;;203 · ;40) + (;16 + 0)) = ;;;8136
80	73	oveq1i 7193	. . . . . . 7 ⊢ ((;;203 · 0) + 0) = (0 + 0)
81	3	dec0h 12214	. . . . . . 7 ⊢ 0 = ;00
82	80, 47, 81	3eqtri 2766	. . . . . 6 ⊢ ((;;203 · 0) + 0) = ;00
83	4, 3, 41, 3, 42, 48, 25, 3, 3, 79, 82	decma2c 12245	. . . . 5 ⊢ ((;;203 · ;;400) + (;;140 + ;20)) = ;;;;81360
84	31	mulid1i 10736	. . . . . . 7 ⊢ (2 · 1) = 2
85	53	mul02i 10920	. . . . . . 7 ⊢ (0 · 1) = 0
86	14, 18, 3, 33, 84, 85	decmul1 12256	. . . . . 6 ⊢ (;20 · 1) = ;20
87	67	mulid1i 10736	. . . . . . . 8 ⊢ (3 · 1) = 3
88	87	oveq1i 7193	. . . . . . 7 ⊢ ((3 · 1) + 1) = (3 + 1)
89		3p1e4 11874	. . . . . . 7 ⊢ (3 + 1) = 4
90	88, 89	eqtri 2762	. . . . . 6 ⊢ ((3 · 1) + 1) = 4
91	23, 19, 14, 52, 14, 86, 90	decrmanc 12249	. . . . 5 ⊢ ((;;203 · 1) + 1) = ;;204
92	5, 14, 16, 14, 1, 39, 25, 2, 23, 83, 91	decma2c 12245	. . . 4 ⊢ ((;;203 · 𝑁) + ;;;1401) = ;;;;;813604
93		eqid 2739	. . . . 5 ⊢ ;;902 = ;;902
94		8nn0 12012	. . . . . . 7 ⊢ 8 ∈ ℕ0
95	14, 94	deccl 12207	. . . . . 6 ⊢ ;18 ∈ ℕ0
96	95, 3	deccl 12207	. . . . 5 ⊢ ;;180 ∈ ℕ0
97		eqid 2739	. . . . . 6 ⊢ ;90 = ;90
98		eqid 2739	. . . . . 6 ⊢ ;;180 = ;;180
99	95	nn0cni 12001	. . . . . . . 8 ⊢ ;18 ∈ ℂ
100	99	addid1i 10918	. . . . . . 7 ⊢ (;18 + 0) = ;18
101		1p2e3 11872	. . . . . . . . 9 ⊢ (1 + 2) = 3
102	101, 19	eqeltri 2830	. . . . . . . 8 ⊢ (1 + 2) ∈ ℕ0
103		9t9e81 12321	. . . . . . . 8 ⊢ (9 · 9) = ;81
104		9cn 11829	. . . . . . . . . . 11 ⊢ 9 ∈ ℂ
105	104	mul02i 10920	. . . . . . . . . 10 ⊢ (0 · 9) = 0
106	105, 101	oveq12i 7195	. . . . . . . . 9 ⊢ ((0 · 9) + (1 + 2)) = (0 + 3)
107	67	addid2i 10919	. . . . . . . . 9 ⊢ (0 + 3) = 3
108	106, 107	eqtri 2762	. . . . . . . 8 ⊢ ((0 · 9) + (1 + 2)) = 3
109	10, 3, 102, 97, 10, 103, 108	decrmanc 12249	. . . . . . 7 ⊢ ((;90 · 9) + (1 + 2)) = ;;813
110		9t2e18 12314	. . . . . . . . 9 ⊢ (9 · 2) = ;18
111	104, 31, 110	mulcomli 10741	. . . . . . . 8 ⊢ (2 · 9) = ;18
112		1p1e2 11854	. . . . . . . 8 ⊢ (1 + 1) = 2
113		8p8e16 12278	. . . . . . . 8 ⊢ (8 + 8) = ;16
114	14, 94, 94, 111, 112, 40, 113	decaddci 12253	. . . . . . 7 ⊢ ((2 · 9) + 8) = ;26
115	27, 18, 14, 94, 93, 100, 10, 40, 18, 109, 114	decmac 12244	. . . . . 6 ⊢ ((;;902 · 9) + (;18 + 0)) = ;;;8136
116	28	nn0cni 12001	. . . . . . . . 9 ⊢ ;;902 ∈ ℂ
117	116	mul01i 10921	. . . . . . . 8 ⊢ (;;902 · 0) = 0
118	117	oveq1i 7193	. . . . . . 7 ⊢ ((;;902 · 0) + 0) = (0 + 0)
119	118, 47, 81	3eqtri 2766	. . . . . 6 ⊢ ((;;902 · 0) + 0) = ;00
120	10, 3, 95, 3, 97, 98, 28, 3, 3, 115, 119	decma2c 12245	. . . . 5 ⊢ ((;;902 · ;90) + ;;180) = ;;;;81360
121	18, 10, 3, 97, 110, 35	decmul1 12256	. . . . . 6 ⊢ (;90 · 2) = ;;180
122	18, 27, 18, 93, 121, 34	decmul1 12256	. . . . 5 ⊢ (;;902 · 2) = ;;;1804
123	28, 27, 18, 93, 2, 96, 120, 122	decmul2c 12258	. . . 4 ⊢ (;;902 · ;;902) = ;;;;;813604
124	92, 123	eqtr4i 2765	. . 3 ⊢ ((;;203 · 𝑁) + ;;;1401) = (;;902 · ;;902)
125	8, 9, 24, 26, 28, 17, 29, 38, 124	mod2xi 16518	. 2 ⊢ ((2↑;;400) mod 𝑁) = (;;;1401 mod 𝑁)
126	5	nn0cni 12001	. . 3 ⊢ ;;400 ∈ ℂ
127	18, 2, 3, 49, 60, 35	decmul1 12256	. . . 4 ⊢ (;40 · 2) = ;80
128	18, 4, 3, 42, 127, 35	decmul1 12256	. . 3 ⊢ (;;400 · 2) = ;;800
129	126, 31, 128	mulcomli 10741	. 2 ⊢ (2 · ;;400) = ;;800
130		eqid 2739	. . . 4 ⊢ ;;;2311 = ;;;2311
131	18, 94	deccl 12207	. . . . 5 ⊢ ;28 ∈ ℕ0
132		eqid 2739	. . . . . 6 ⊢ ;;231 = ;;231
133		eqid 2739	. . . . . 6 ⊢ ;49 = ;49
134		7nn0 12011	. . . . . . 7 ⊢ 7 ∈ ℕ0
135		7p1e8 11878	. . . . . . 7 ⊢ (7 + 1) = 8
136		eqid 2739	. . . . . . . 8 ⊢ ;23 = ;23
137		4p3e7 11883	. . . . . . . . 9 ⊢ (4 + 3) = 7
138	59, 67, 137	addcomli 10923	. . . . . . . 8 ⊢ (3 + 4) = 7
139	18, 19, 2, 136, 138	decaddi 12252	. . . . . . 7 ⊢ (;23 + 4) = ;27
140	18, 134, 135, 139	decsuc 12223	. . . . . 6 ⊢ ((;23 + 4) + 1) = ;28
141		9p1e10 12194	. . . . . . 7 ⊢ (9 + 1) = ;10
142	104, 53, 141	addcomli 10923	. . . . . 6 ⊢ (1 + 9) = ;10
143	20, 14, 2, 10, 132, 133, 140, 142	decaddc2 12248	. . . . 5 ⊢ (;;231 + ;49) = ;;280
144	131	nn0cni 12001	. . . . . . 7 ⊢ ;28 ∈ ℂ
145	144	addid1i 10918	. . . . . 6 ⊢ (;28 + 0) = ;28
146	31	addid1i 10918	. . . . . . . 8 ⊢ (2 + 0) = 2
147	146, 18	eqeltri 2830	. . . . . . 7 ⊢ (2 + 0) ∈ ℕ0
148		eqid 2739	. . . . . . 7 ⊢ ;;490 = ;;490
149		4t4e16 12291	. . . . . . . . 9 ⊢ (4 · 4) = ;16
150		6p3e9 11889	. . . . . . . . 9 ⊢ (6 + 3) = 9
151	14, 40, 19, 149, 150	decaddi 12252	. . . . . . . 8 ⊢ ((4 · 4) + 3) = ;19
152		9t4e36 12316	. . . . . . . 8 ⊢ (9 · 4) = ;36
153	2, 2, 10, 133, 40, 19, 151, 152	decmul1c 12257	. . . . . . 7 ⊢ (;49 · 4) = ;;196
154	62, 146	oveq12i 7195	. . . . . . . 8 ⊢ ((0 · 4) + (2 + 0)) = (0 + 2)
155	31	addid2i 10919	. . . . . . . 8 ⊢ (0 + 2) = 2
156	154, 155	eqtri 2762	. . . . . . 7 ⊢ ((0 · 4) + (2 + 0)) = 2
157	11, 3, 147, 148, 2, 153, 156	decrmanc 12249	. . . . . 6 ⊢ ((;;490 · 4) + (2 + 0)) = ;;;1962
158	12	nn0cni 12001	. . . . . . . . 9 ⊢ ;;490 ∈ ℂ
159	158	mul01i 10921	. . . . . . . 8 ⊢ (;;490 · 0) = 0
160	159	oveq1i 7193	. . . . . . 7 ⊢ ((;;490 · 0) + 8) = (0 + 8)
161		8cn 11826	. . . . . . . 8 ⊢ 8 ∈ ℂ
162	161	addid2i 10919	. . . . . . 7 ⊢ (0 + 8) = 8
163	94	dec0h 12214	. . . . . . 7 ⊢ 8 = ;08
164	160, 162, 163	3eqtri 2766	. . . . . 6 ⊢ ((;;490 · 0) + 8) = ;08
165	2, 3, 18, 94, 49, 145, 12, 94, 3, 157, 164	decma2c 12245	. . . . 5 ⊢ ((;;490 · ;40) + (;28 + 0)) = ;;;;19628
166	159	oveq1i 7193	. . . . . 6 ⊢ ((;;490 · 0) + 0) = (0 + 0)
167	166, 47, 81	3eqtri 2766	. . . . 5 ⊢ ((;;490 · 0) + 0) = ;00
168	4, 3, 131, 3, 42, 143, 12, 3, 3, 165, 167	decma2c 12245	. . . 4 ⊢ ((;;490 · ;;400) + (;;231 + ;49)) = ;;;;;196280
169	59	mulid1i 10736	. . . . . 6 ⊢ (4 · 1) = 4
170	104	mulid1i 10736	. . . . . 6 ⊢ (9 · 1) = 9
171	14, 2, 10, 133, 169, 170	decmul1 12256	. . . . 5 ⊢ (;49 · 1) = ;49
172	85	oveq1i 7193	. . . . . 6 ⊢ ((0 · 1) + 1) = (0 + 1)
173	172, 57	eqtri 2762	. . . . 5 ⊢ ((0 · 1) + 1) = 1
174	11, 3, 14, 148, 14, 171, 173	decrmanc 12249	. . . 4 ⊢ ((;;490 · 1) + 1) = ;;491
175	5, 14, 21, 14, 1, 130, 12, 14, 11, 168, 174	decma2c 12245	. . 3 ⊢ ((;;490 · 𝑁) + ;;;2311) = ;;;;;;1962801
176	15	nn0cni 12001	. . . . . . 7 ⊢ ;14 ∈ ℂ
177	176	addid1i 10918	. . . . . 6 ⊢ (;14 + 0) = ;14
178		5nn0 12009	. . . . . . . 8 ⊢ 5 ∈ ℕ0
179	178, 40	deccl 12207	. . . . . . 7 ⊢ ;56 ∈ ℕ0
180	179, 3	deccl 12207	. . . . . 6 ⊢ ;;560 ∈ ℕ0
181		eqid 2739	. . . . . . . 8 ⊢ ;;560 = ;;560
182	179	nn0cni 12001	. . . . . . . . 9 ⊢ ;56 ∈ ℂ
183	182	addid2i 10919	. . . . . . . 8 ⊢ (0 + ;56) = ;56
184	3, 14, 179, 3, 55, 181, 183, 54	decadd 12246	. . . . . . 7 ⊢ (1 + ;;560) = ;;561
185	182	addid1i 10918	. . . . . . . 8 ⊢ (;56 + 0) = ;56
186		5cn 11817	. . . . . . . . . . 11 ⊢ 5 ∈ ℂ
187	186	addid1i 10918	. . . . . . . . . 10 ⊢ (5 + 0) = 5
188	187, 178	eqeltri 2830	. . . . . . . . 9 ⊢ (5 + 0) ∈ ℕ0
189	53	mulid1i 10736	. . . . . . . . 9 ⊢ (1 · 1) = 1
190	169, 187	oveq12i 7195	. . . . . . . . . 10 ⊢ ((4 · 1) + (5 + 0)) = (4 + 5)
191		5p4e9 11887	. . . . . . . . . . 11 ⊢ (5 + 4) = 9
192	186, 59, 191	addcomli 10923	. . . . . . . . . 10 ⊢ (4 + 5) = 9
193	190, 192	eqtri 2762	. . . . . . . . 9 ⊢ ((4 · 1) + (5 + 0)) = 9
194	14, 2, 188, 44, 14, 189, 193	decrmanc 12249	. . . . . . . 8 ⊢ ((;14 · 1) + (5 + 0)) = ;19
195	85	oveq1i 7193	. . . . . . . . 9 ⊢ ((0 · 1) + 6) = (0 + 6)
196	195, 76, 77	3eqtri 2766	. . . . . . . 8 ⊢ ((0 · 1) + 6) = ;06
197	15, 3, 178, 40, 43, 185, 14, 40, 3, 194, 196	decmac 12244	. . . . . . 7 ⊢ ((;;140 · 1) + (;56 + 0)) = ;;196
198	189	oveq1i 7193	. . . . . . . 8 ⊢ ((1 · 1) + 1) = (1 + 1)
199	18	dec0h 12214	. . . . . . . 8 ⊢ 2 = ;02
200	198, 112, 199	3eqtri 2766	. . . . . . 7 ⊢ ((1 · 1) + 1) = ;02
201	16, 14, 179, 14, 39, 184, 14, 18, 3, 197, 200	decmac 12244	. . . . . 6 ⊢ ((;;;1401 · 1) + (1 + ;;560)) = ;;;1962
202	59	mulid2i 10737	. . . . . . . . . . . 12 ⊢ (1 · 4) = 4
203	202	oveq1i 7193	. . . . . . . . . . 11 ⊢ ((1 · 4) + 1) = (4 + 1)
204		4p1e5 11875	. . . . . . . . . . 11 ⊢ (4 + 1) = 5
205	203, 204	eqtri 2762	. . . . . . . . . 10 ⊢ ((1 · 4) + 1) = 5
206	2, 14, 2, 44, 40, 14, 205, 149	decmul1c 12257	. . . . . . . . 9 ⊢ (;14 · 4) = ;56
207	75	addid1i 10918	. . . . . . . . 9 ⊢ (6 + 0) = 6
208	178, 40, 3, 206, 207	decaddi 12252	. . . . . . . 8 ⊢ ((;14 · 4) + 0) = ;56
209		0cn 10724	. . . . . . . . 9 ⊢ 0 ∈ ℂ
210	59	mul01i 10921	. . . . . . . . . 10 ⊢ (4 · 0) = 0
211	210, 81	eqtri 2762	. . . . . . . . 9 ⊢ (4 · 0) = ;00
212	59, 209, 211	mulcomli 10741	. . . . . . . 8 ⊢ (0 · 4) = ;00
213	2, 15, 3, 43, 3, 3, 208, 212	decmul1c 12257	. . . . . . 7 ⊢ (;;140 · 4) = ;;560
214	202	oveq1i 7193	. . . . . . . 8 ⊢ ((1 · 4) + 4) = (4 + 4)
215		4p4e8 11884	. . . . . . . 8 ⊢ (4 + 4) = 8
216	214, 215	eqtri 2762	. . . . . . 7 ⊢ ((1 · 4) + 4) = 8
217	16, 14, 2, 39, 2, 213, 216	decrmanc 12249	. . . . . 6 ⊢ ((;;;1401 · 4) + 4) = ;;;5608
218	14, 2, 14, 2, 44, 177, 17, 94, 180, 201, 217	decma2c 12245	. . . . 5 ⊢ ((;;;1401 · ;14) + (;14 + 0)) = ;;;;19628
219	17	nn0cni 12001	. . . . . . . 8 ⊢ ;;;1401 ∈ ℂ
220	219	mul01i 10921	. . . . . . 7 ⊢ (;;;1401 · 0) = 0
221	220	oveq1i 7193	. . . . . 6 ⊢ ((;;;1401 · 0) + 0) = (0 + 0)
222	221, 47, 81	3eqtri 2766	. . . . 5 ⊢ ((;;;1401 · 0) + 0) = ;00
223	15, 3, 15, 3, 43, 43, 17, 3, 3, 218, 222	decma2c 12245	. . . 4 ⊢ ((;;;1401 · ;;140) + ;;140) = ;;;;;196280
224	219	mulid1i 10736	. . . 4 ⊢ (;;;1401 · 1) = ;;;1401
225	17, 16, 14, 39, 14, 16, 223, 224	decmul2c 12258	. . 3 ⊢ (;;;1401 · ;;;1401) = ;;;;;;1962801
226	175, 225	eqtr4i 2765	. 2 ⊢ ((;;490 · 𝑁) + ;;;2311) = (;;;1401 · ;;;1401)
227	8, 9, 5, 13, 17, 22, 125, 129, 226	mod2xi 16518	1 ⊢ ((2↑;;800) mod 𝑁) = (;;;2311 mod 𝑁)

Syntax hints:   = wceq 1542  (class class class)co 7183  0cc0 10628  1c1 10629   + caddc 10631   · cmul 10633  ℕcn 11729  2c2 11784  3c3 11785  4c4 11786  5c5 11787  6c6 11788  7c7 11789  8c8 11790  9c9 11791  ℕ₀cn0 11989  ;cdc 12192   mod cmo 13341  ↑cexp 13534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pow 5242  ax-pr 5306  ax-un 7492  ax-cnex 10684  ax-resscn 10685  ax-1cn 10686  ax-icn 10687  ax-addcl 10688  ax-addrcl 10689  ax-mulcl 10690  ax-mulrcl 10691  ax-mulcom 10692  ax-addass 10693  ax-mulass 10694  ax-distr 10695  ax-i2m1 10696  ax-1ne0 10697  ax-1rid 10698  ax-rnegex 10699  ax-rrecex 10700  ax-cnre 10701  ax-pre-lttri 10702  ax-pre-lttrn 10703  ax-pre-ltadd 10704  ax-pre-mulgt0 10705  ax-pre-sup 10706

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3402  df-sbc 3686  df-csb 3801  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4222  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-tp 4531  df-op 4533  df-uni 4807  df-iun 4893  df-br 5041  df-opab 5103  df-mpt 5121  df-tr 5147  df-id 5439  df-eprel 5444  df-po 5452  df-so 5453  df-fr 5493  df-we 5495  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-pred 6139  df-ord 6186  df-on 6187  df-lim 6188  df-suc 6189  df-iota 6308  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-riota 7140  df-ov 7186  df-oprab 7187  df-mpo 7188  df-om 7613  df-2nd 7728  df-wrecs 7989  df-recs 8050  df-rdg 8088  df-er 8333  df-en 8569  df-dom 8570  df-sdom 8571  df-sup 8992  df-inf 8993  df-pnf 10768  df-mnf 10769  df-xr 10770  df-ltxr 10771  df-le 10772  df-sub 10963  df-neg 10964  df-div 11389  df-nn 11730  df-2 11792  df-3 11793  df-4 11794  df-5 11795  df-6 11796  df-7 11797  df-8 11798  df-9 11799  df-n0 11990  df-z 12076  df-dec 12193  df-uz 12338  df-rp 12486  df-fl 13266  df-mod 13342  df-seq 13474  df-exp 13535

This theorem is referenced by:  4001lem3  16592  4001lem4  16593