Theorem 4001lem1 17319

Description: Lemma for 4001prm 17323. Calculate a power mod. In decimal, we calculate 2↑12 = 4096 = 𝑁 + 95, 2↑24 = (2↑12)↑2≡95↑2 = 2𝑁 + 1023, 2↑25 = 2↑24 · 2≡1023 · 2 = 2046, 2↑50 = (2↑25)↑2≡2046↑2 = 1046𝑁 + 1070, 2↑100 = (2↑50)↑2≡1070↑2 = 286𝑁 + 614 and 2↑200 = (2↑100)↑2≡614↑2 = 94𝑁 + 902 ≡902. (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
4001lem1	⊢ ((2↑;;200) mod 𝑁) = (;;902 mod 𝑁)

Proof of Theorem 4001lem1

Step	Hyp	Ref	Expression
1		4001prm.1	. . 3 ⊢ 𝑁 = ;;;4001
2		4nn0 12625	. . . . . 6 ⊢ 4 ∈ ℕ0
3		0nn0 12621	. . . . . 6 ⊢ 0 ∈ ℕ0
4	2, 3	deccl 12829	. . . . 5 ⊢ ;40 ∈ ℕ0
5	4, 3	deccl 12829	. . . 4 ⊢ ;;400 ∈ ℕ0
6		1nn 12346	. . . 4 ⊢ 1 ∈ ℕ
7	5, 6	decnncl 12838	. . 3 ⊢ ;;;4001 ∈ ℕ
8	1, 7	eqeltri 2857	. 2 ⊢ 𝑁 ∈ ℕ
9		2nn 12416	. 2 ⊢ 2 ∈ ℕ
10		10nn0 12836	. . 3 ⊢ ;10 ∈ ℕ0
11	10, 3	deccl 12829	. 2 ⊢ ;;100 ∈ ℕ0
12		9nn0 12630	. . . 4 ⊢ 9 ∈ ℕ0
13	12, 2	deccl 12829	. . 3 ⊢ ;94 ∈ ℕ0
14	13	nn0zi 12721	. 2 ⊢ ;94 ∈ ℤ
15		6nn0 12627	. . . 4 ⊢ 6 ∈ ℕ0
16		1nn0 12622	. . . 4 ⊢ 1 ∈ ℕ0
17	15, 16	deccl 12829	. . 3 ⊢ ;61 ∈ ℕ0
18	17, 2	deccl 12829	. 2 ⊢ ;;614 ∈ ℕ0
19	12, 3	deccl 12829	. . 3 ⊢ ;90 ∈ ℕ0
20		2nn0 12623	. . 3 ⊢ 2 ∈ ℕ0
21	19, 20	deccl 12829	. 2 ⊢ ;;902 ∈ ℕ0
22		5nn0 12626	. . . 4 ⊢ 5 ∈ ℕ0
23	22, 3	deccl 12829	. . 3 ⊢ ;50 ∈ ℕ0
24		8nn0 12629	. . . . . 6 ⊢ 8 ∈ ℕ0
25	20, 24	deccl 12829	. . . . 5 ⊢ ;28 ∈ ℕ0
26	25, 15	deccl 12829	. . . 4 ⊢ ;;286 ∈ ℕ0
27	26	nn0zi 12721	. . 3 ⊢ ;;286 ∈ ℤ
28		7nn0 12628	. . . . 5 ⊢ 7 ∈ ℕ0
29	10, 28	deccl 12829	. . . 4 ⊢ ;;107 ∈ ℕ0
30	29, 3	deccl 12829	. . 3 ⊢ ;;;1070 ∈ ℕ0
31	20, 22	deccl 12829	. . . 4 ⊢ ;25 ∈ ℕ0
32	10, 2	deccl 12829	. . . . . 6 ⊢ ;;104 ∈ ℕ0
33	32, 15	deccl 12829	. . . . 5 ⊢ ;;;1046 ∈ ℕ0
34	33	nn0zi 12721	. . . 4 ⊢ ;;;1046 ∈ ℤ
35	20, 3	deccl 12829	. . . . . 6 ⊢ ;20 ∈ ℕ0
36	35, 2	deccl 12829	. . . . 5 ⊢ ;;204 ∈ ℕ0
37	36, 15	deccl 12829	. . . 4 ⊢ ;;;2046 ∈ ℕ0
38	20, 2	deccl 12829	. . . . 5 ⊢ ;24 ∈ ℕ0
39		0z 12704	. . . . 5 ⊢ 0 ∈ ℤ
40	10, 20	deccl 12829	. . . . . 6 ⊢ ;;102 ∈ ℕ0
41		3nn0 12624	. . . . . 6 ⊢ 3 ∈ ℕ0
42	40, 41	deccl 12829	. . . . 5 ⊢ ;;;1023 ∈ ℕ0
43	16, 20	deccl 12829	. . . . . 6 ⊢ ;12 ∈ ℕ0
44		2z 12728	. . . . . 6 ⊢ 2 ∈ ℤ
45	12, 22	deccl 12829	. . . . . 6 ⊢ ;95 ∈ ℕ0
46		1z 12726	. . . . . . 7 ⊢ 1 ∈ ℤ
47	15, 2	deccl 12829	. . . . . . 7 ⊢ ;64 ∈ ℕ0
48		2exp6 17264	. . . . . . . 8 ⊢ (2↑6) = ;64
49	48	oveq1i 7430	. . . . . . 7 ⊢ ((2↑6) mod 𝑁) = (;64 mod 𝑁)
50		6cn 12434	. . . . . . . 8 ⊢ 6 ∈ ℂ
51		2cn 12418	. . . . . . . 8 ⊢ 2 ∈ ℂ
52		6t2e12 12923	. . . . . . . 8 ⊢ (6 · 2) = ;12
53	50, 51, 52	mulcomli 11318	. . . . . . 7 ⊢ (2 · 6) = ;12
54		eqid 2761	. . . . . . . . 9 ⊢ ;95 = ;95
55		eqid 2761	. . . . . . . . . 10 ⊢ ;;400 = ;;400
56		9cn 12443	. . . . . . . . . . . 12 ⊢ 9 ∈ ℂ
57	56	addridi 11497	. . . . . . . . . . 11 ⊢ (9 + 0) = 9
58	12	dec0h 12841	. . . . . . . . . . 11 ⊢ 9 = ;09
59	57, 58	eqtri 2784	. . . . . . . . . 10 ⊢ (9 + 0) = ;09
60		eqid 2761	. . . . . . . . . . 11 ⊢ ;40 = ;40
61		00id 11485	. . . . . . . . . . . 12 ⊢ (0 + 0) = 0
62	3	dec0h 12841	. . . . . . . . . . . 12 ⊢ 0 = ;00
63	61, 62	eqtri 2784	. . . . . . . . . . 11 ⊢ (0 + 0) = ;00
64		4cn 12428	. . . . . . . . . . . . . 14 ⊢ 4 ∈ ℂ
65	64	mullidi 11314	. . . . . . . . . . . . 13 ⊢ (1 · 4) = 4
66	65, 61	oveq12i 7432	. . . . . . . . . . . 12 ⊢ ((1 · 4) + (0 + 0)) = (4 + 0)
67	64	addridi 11497	. . . . . . . . . . . 12 ⊢ (4 + 0) = 4
68	66, 67	eqtri 2784	. . . . . . . . . . 11 ⊢ ((1 · 4) + (0 + 0)) = 4
69		ax-1cn 11258	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℂ
70	69	mul01i 11500	. . . . . . . . . . . . 13 ⊢ (1 · 0) = 0
71	70	oveq1i 7430	. . . . . . . . . . . 12 ⊢ ((1 · 0) + 0) = (0 + 0)
72	71, 61, 62	3eqtri 2788	. . . . . . . . . . 11 ⊢ ((1 · 0) + 0) = ;00
73	2, 3, 3, 3, 60, 63, 16, 3, 3, 68, 72	decma2c 12872	. . . . . . . . . 10 ⊢ ((1 · ;40) + (0 + 0)) = ;40
74	70	oveq1i 7430	. . . . . . . . . . 11 ⊢ ((1 · 0) + 9) = (0 + 9)
75	56	addlidi 11498	. . . . . . . . . . 11 ⊢ (0 + 9) = 9
76	74, 75, 58	3eqtri 2788	. . . . . . . . . 10 ⊢ ((1 · 0) + 9) = ;09
77	4, 3, 3, 12, 55, 59, 16, 12, 3, 73, 76	decma2c 12872	. . . . . . . . 9 ⊢ ((1 · ;;400) + (9 + 0)) = ;;409
78	69	mulridi 11313	. . . . . . . . . . 11 ⊢ (1 · 1) = 1
79	78	oveq1i 7430	. . . . . . . . . 10 ⊢ ((1 · 1) + 5) = (1 + 5)
80		5cn 12431	. . . . . . . . . . 11 ⊢ 5 ∈ ℂ
81		5p1e6 12489	. . . . . . . . . . 11 ⊢ (5 + 1) = 6
82	80, 69, 81	addcomli 11502	. . . . . . . . . 10 ⊢ (1 + 5) = 6
83	15	dec0h 12841	. . . . . . . . . 10 ⊢ 6 = ;06
84	79, 82, 83	3eqtri 2788	. . . . . . . . 9 ⊢ ((1 · 1) + 5) = ;06
85	5, 16, 12, 22, 1, 54, 16, 15, 3, 77, 84	decma2c 12872	. . . . . . . 8 ⊢ ((1 · 𝑁) + ;95) = ;;;4096
86		eqid 2761	. . . . . . . . 9 ⊢ ;64 = ;64
87		eqid 2761	. . . . . . . . . 10 ⊢ ;25 = ;25
88		2p2e4 12477	. . . . . . . . . . . 12 ⊢ (2 + 2) = 4
89	88	oveq2i 7431	. . . . . . . . . . 11 ⊢ ((6 · 6) + (2 + 2)) = ((6 · 6) + 4)
90		6t6e36 12927	. . . . . . . . . . . 12 ⊢ (6 · 6) = ;36
91		3p1e4 12487	. . . . . . . . . . . 12 ⊢ (3 + 1) = 4
92		6p4e10 12891	. . . . . . . . . . . 12 ⊢ (6 + 4) = ;10
93	41, 15, 2, 90, 91, 92	decaddci2 12881	. . . . . . . . . . 11 ⊢ ((6 · 6) + 4) = ;40
94	89, 93	eqtri 2784	. . . . . . . . . 10 ⊢ ((6 · 6) + (2 + 2)) = ;40
95		6t4e24 12925	. . . . . . . . . . . 12 ⊢ (6 · 4) = ;24
96	50, 64, 95	mulcomli 11318	. . . . . . . . . . 11 ⊢ (4 · 6) = ;24
97		5p4e9 12500	. . . . . . . . . . . 12 ⊢ (5 + 4) = 9
98	80, 64, 97	addcomli 11502	. . . . . . . . . . 11 ⊢ (4 + 5) = 9
99	20, 2, 22, 96, 98	decaddi 12879	. . . . . . . . . 10 ⊢ ((4 · 6) + 5) = ;29
100	15, 2, 20, 22, 86, 87, 15, 12, 20, 94, 99	decmac 12871	. . . . . . . . 9 ⊢ ((;64 · 6) + ;25) = ;;409
101		4p1e5 12488	. . . . . . . . . . 11 ⊢ (4 + 1) = 5
102	20, 2, 101, 95	decsuc 12850	. . . . . . . . . 10 ⊢ ((6 · 4) + 1) = ;25
103		4t4e16 12918	. . . . . . . . . 10 ⊢ (4 · 4) = ;16
104	2, 15, 2, 86, 15, 16, 102, 103	decmul1c 12884	. . . . . . . . 9 ⊢ (;64 · 4) = ;;256
105	47, 15, 2, 86, 15, 31, 100, 104	decmul2c 12885	. . . . . . . 8 ⊢ (;64 · ;64) = ;;;4096
106	85, 105	eqtr4i 2787	. . . . . . 7 ⊢ ((1 · 𝑁) + ;95) = (;64 · ;64)
107	8, 9, 15, 46, 47, 45, 49, 53, 106	mod2xi 17247	. . . . . 6 ⊢ ((2↑;12) mod 𝑁) = (;95 mod 𝑁)
108		eqid 2761	. . . . . . 7 ⊢ ;12 = ;12
109	51	mulridi 11313	. . . . . . . . 9 ⊢ (2 · 1) = 2
110	109	oveq1i 7430	. . . . . . . 8 ⊢ ((2 · 1) + 0) = (2 + 0)
111	51	addridi 11497	. . . . . . . 8 ⊢ (2 + 0) = 2
112	110, 111	eqtri 2784	. . . . . . 7 ⊢ ((2 · 1) + 0) = 2
113		2t2e4 12506	. . . . . . . 8 ⊢ (2 · 2) = 4
114	2	dec0h 12841	. . . . . . . 8 ⊢ 4 = ;04
115	113, 114	eqtri 2784	. . . . . . 7 ⊢ (2 · 2) = ;04
116	20, 16, 20, 108, 2, 3, 112, 115	decmul2c 12885	. . . . . 6 ⊢ (2 · ;12) = ;24
117		eqid 2761	. . . . . . . 8 ⊢ ;;;1023 = ;;;1023
118	40	nn0cni 12618	. . . . . . . . . 10 ⊢ ;;102 ∈ ℂ
119	118	addridi 11497	. . . . . . . . 9 ⊢ (;;102 + 0) = ;;102
120		dec10p 12862	. . . . . . . . . 10 ⊢ (;10 + 0) = ;10
121		2t4e8 12512	. . . . . . . . . . . 12 ⊢ (2 · 4) = 8
122	69	addridi 11497	. . . . . . . . . . . 12 ⊢ (1 + 0) = 1
123	121, 122	oveq12i 7432	. . . . . . . . . . 11 ⊢ ((2 · 4) + (1 + 0)) = (8 + 1)
124		8p1e9 12492	. . . . . . . . . . 11 ⊢ (8 + 1) = 9
125	123, 124	eqtri 2784	. . . . . . . . . 10 ⊢ ((2 · 4) + (1 + 0)) = 9
126	51	mul01i 11500	. . . . . . . . . . . 12 ⊢ (2 · 0) = 0
127	126	oveq1i 7430	. . . . . . . . . . 11 ⊢ ((2 · 0) + 0) = (0 + 0)
128	127, 61, 62	3eqtri 2788	. . . . . . . . . 10 ⊢ ((2 · 0) + 0) = ;00
129	2, 3, 16, 3, 60, 120, 20, 3, 3, 125, 128	decma2c 12872	. . . . . . . . 9 ⊢ ((2 · ;40) + (;10 + 0)) = ;90
130	126	oveq1i 7430	. . . . . . . . . 10 ⊢ ((2 · 0) + 2) = (0 + 2)
131	51	addlidi 11498	. . . . . . . . . 10 ⊢ (0 + 2) = 2
132	20	dec0h 12841	. . . . . . . . . 10 ⊢ 2 = ;02
133	130, 131, 132	3eqtri 2788	. . . . . . . . 9 ⊢ ((2 · 0) + 2) = ;02
134	4, 3, 10, 20, 55, 119, 20, 20, 3, 129, 133	decma2c 12872	. . . . . . . 8 ⊢ ((2 · ;;400) + (;;102 + 0)) = ;;902
135	109	oveq1i 7430	. . . . . . . . 9 ⊢ ((2 · 1) + 3) = (2 + 3)
136		3cn 12424	. . . . . . . . . 10 ⊢ 3 ∈ ℂ
137		3p2e5 12493	. . . . . . . . . 10 ⊢ (3 + 2) = 5
138	136, 51, 137	addcomli 11502	. . . . . . . . 9 ⊢ (2 + 3) = 5
139	22	dec0h 12841	. . . . . . . . 9 ⊢ 5 = ;05
140	135, 138, 139	3eqtri 2788	. . . . . . . 8 ⊢ ((2 · 1) + 3) = ;05
141	5, 16, 40, 41, 1, 117, 20, 22, 3, 134, 140	decma2c 12872	. . . . . . 7 ⊢ ((2 · 𝑁) + ;;;1023) = ;;;9025
142	2, 28	deccl 12829	. . . . . . . 8 ⊢ ;47 ∈ ℕ0
143		eqid 2761	. . . . . . . . 9 ⊢ ;47 = ;47
144	98	oveq2i 7431	. . . . . . . . . 10 ⊢ ((9 · 9) + (4 + 5)) = ((9 · 9) + 9)
145		9t9e81 12948	. . . . . . . . . . 11 ⊢ (9 · 9) = ;81
146		9p1e10 12816	. . . . . . . . . . . 12 ⊢ (9 + 1) = ;10
147	56, 69, 146	addcomli 11502	. . . . . . . . . . 11 ⊢ (1 + 9) = ;10
148	24, 16, 12, 145, 124, 147	decaddci2 12881	. . . . . . . . . 10 ⊢ ((9 · 9) + 9) = ;90
149	144, 148	eqtri 2784	. . . . . . . . 9 ⊢ ((9 · 9) + (4 + 5)) = ;90
150		9t5e45 12944	. . . . . . . . . . 11 ⊢ (9 · 5) = ;45
151	56, 80, 150	mulcomli 11318	. . . . . . . . . 10 ⊢ (5 · 9) = ;45
152		7cn 12437	. . . . . . . . . . 11 ⊢ 7 ∈ ℂ
153		7p5e12 12896	. . . . . . . . . . 11 ⊢ (7 + 5) = ;12
154	152, 80, 153	addcomli 11502	. . . . . . . . . 10 ⊢ (5 + 7) = ;12
155	2, 22, 28, 151, 101, 20, 154	decaddci 12880	. . . . . . . . 9 ⊢ ((5 · 9) + 7) = ;52
156	12, 22, 2, 28, 54, 143, 12, 20, 22, 149, 155	decmac 12871	. . . . . . . 8 ⊢ ((;95 · 9) + ;47) = ;;902
157		5p2e7 12498	. . . . . . . . . 10 ⊢ (5 + 2) = 7
158	2, 22, 20, 150, 157	decaddi 12879	. . . . . . . . 9 ⊢ ((9 · 5) + 2) = ;47
159		5t5e25 12922	. . . . . . . . 9 ⊢ (5 · 5) = ;25
160	22, 12, 22, 54, 22, 20, 158, 159	decmul1c 12884	. . . . . . . 8 ⊢ (;95 · 5) = ;;475
161	45, 12, 22, 54, 22, 142, 156, 160	decmul2c 12885	. . . . . . 7 ⊢ (;95 · ;95) = ;;;9025
162	141, 161	eqtr4i 2787	. . . . . 6 ⊢ ((2 · 𝑁) + ;;;1023) = (;95 · ;95)
163	8, 9, 43, 44, 45, 42, 107, 116, 162	mod2xi 17247	. . . . 5 ⊢ ((2↑;24) mod 𝑁) = (;;;1023 mod 𝑁)
164		eqid 2761	. . . . . 6 ⊢ ;24 = ;24
165	20, 2, 101, 164	decsuc 12850	. . . . 5 ⊢ (;24 + 1) = ;25
166	37	nn0cni 12618	. . . . . . 7 ⊢ ;;;2046 ∈ ℂ
167	166	addlidi 11498	. . . . . 6 ⊢ (0 + ;;;2046) = ;;;2046
168	8	nncni 12345	. . . . . . . 8 ⊢ 𝑁 ∈ ℂ
169	168	mul02i 11499	. . . . . . 7 ⊢ (0 · 𝑁) = 0
170	169	oveq1i 7430	. . . . . 6 ⊢ ((0 · 𝑁) + ;;;2046) = (0 + ;;;2046)
171		eqid 2761	. . . . . . . 8 ⊢ ;;102 = ;;102
172	20	dec0u 12840	. . . . . . . 8 ⊢ (;10 · 2) = ;20
173	20, 10, 20, 171, 172, 113	decmul1 12883	. . . . . . 7 ⊢ (;;102 · 2) = ;;204
174		3t2e6 12508	. . . . . . 7 ⊢ (3 · 2) = 6
175	20, 40, 41, 117, 173, 174	decmul1 12883	. . . . . 6 ⊢ (;;;1023 · 2) = ;;;2046
176	167, 170, 175	3eqtr4i 2794	. . . . 5 ⊢ ((0 · 𝑁) + ;;;2046) = (;;;1023 · 2)
177	8, 9, 38, 39, 42, 37, 163, 165, 176	modxp1i 17248	. . . 4 ⊢ ((2↑;25) mod 𝑁) = (;;;2046 mod 𝑁)
178	113	oveq1i 7430	. . . . . 6 ⊢ ((2 · 2) + 1) = (4 + 1)
179	178, 101	eqtri 2784	. . . . 5 ⊢ ((2 · 2) + 1) = 5
180		5t2e10 12919	. . . . . 6 ⊢ (5 · 2) = ;10
181	80, 51, 180	mulcomli 11318	. . . . 5 ⊢ (2 · 5) = ;10
182	20, 20, 22, 87, 3, 16, 179, 181	decmul2c 12885	. . . 4 ⊢ (2 · ;25) = ;50
183		eqid 2761	. . . . . 6 ⊢ ;;;1070 = ;;;1070
184	20, 16	deccl 12829	. . . . . . 7 ⊢ ;21 ∈ ℕ0
185		eqid 2761	. . . . . . . 8 ⊢ ;;107 = ;;107
186		eqid 2761	. . . . . . . 8 ⊢ ;;104 = ;;104
187		0p1e1 12463	. . . . . . . . 9 ⊢ (0 + 1) = 1
188		10p10e20 12914	. . . . . . . . 9 ⊢ (;10 + ;10) = ;20
189	20, 3, 187, 188	decsuc 12850	. . . . . . . 8 ⊢ ((;10 + ;10) + 1) = ;21
190		7p4e11 12895	. . . . . . . 8 ⊢ (7 + 4) = ;11
191	10, 28, 10, 2, 185, 186, 189, 16, 190	decaddc 12874	. . . . . . 7 ⊢ (;;107 + ;;104) = ;;211
192	184	nn0cni 12618	. . . . . . . . 9 ⊢ ;21 ∈ ℂ
193	192	addridi 11497	. . . . . . . 8 ⊢ (;21 + 0) = ;21
194	111, 20	eqeltri 2857	. . . . . . . . 9 ⊢ (2 + 0) ∈ ℕ0
195		eqid 2761	. . . . . . . . 9 ⊢ ;;;1046 = ;;;1046
196		dfdec10 12817	. . . . . . . . . . 11 ⊢ ;41 = ((;10 · 4) + 1)
197	196	eqcomi 2770	. . . . . . . . . 10 ⊢ ((;10 · 4) + 1) = ;41
198		6p2e8 12501	. . . . . . . . . . 11 ⊢ (6 + 2) = 8
199	16, 15, 20, 103, 198	decaddi 12879	. . . . . . . . . 10 ⊢ ((4 · 4) + 2) = ;18
200	10, 2, 20, 186, 2, 24, 16, 197, 199	decrmac 12877	. . . . . . . . 9 ⊢ ((;;104 · 4) + 2) = ;;418
201	95, 111	oveq12i 7432	. . . . . . . . . 10 ⊢ ((6 · 4) + (2 + 0)) = (;24 + 2)
202		4p2e6 12495	. . . . . . . . . . 11 ⊢ (4 + 2) = 6
203	20, 2, 20, 164, 202	decaddi 12879	. . . . . . . . . 10 ⊢ (;24 + 2) = ;26
204	201, 203	eqtri 2784	. . . . . . . . 9 ⊢ ((6 · 4) + (2 + 0)) = ;26
205	32, 15, 194, 195, 2, 15, 20, 200, 204	decrmac 12877	. . . . . . . 8 ⊢ ((;;;1046 · 4) + (2 + 0)) = ;;;4186
206	33	nn0cni 12618	. . . . . . . . . . 11 ⊢ ;;;1046 ∈ ℂ
207	206	mul01i 11500	. . . . . . . . . 10 ⊢ (;;;1046 · 0) = 0
208	207	oveq1i 7430	. . . . . . . . 9 ⊢ ((;;;1046 · 0) + 1) = (0 + 1)
209	16	dec0h 12841	. . . . . . . . 9 ⊢ 1 = ;01
210	208, 187, 209	3eqtri 2788	. . . . . . . 8 ⊢ ((;;;1046 · 0) + 1) = ;01
211	2, 3, 20, 16, 60, 193, 33, 16, 3, 205, 210	decma2c 12872	. . . . . . 7 ⊢ ((;;;1046 · ;40) + (;21 + 0)) = ;;;;41861
212	4, 3, 184, 16, 55, 191, 33, 16, 3, 211, 210	decma2c 12872	. . . . . 6 ⊢ ((;;;1046 · ;;400) + (;;107 + ;;104)) = ;;;;;418611
213	206	mulridi 11313	. . . . . . . 8 ⊢ (;;;1046 · 1) = ;;;1046
214	213	oveq1i 7430	. . . . . . 7 ⊢ ((;;;1046 · 1) + 0) = (;;;1046 + 0)
215	206	addridi 11497	. . . . . . 7 ⊢ (;;;1046 + 0) = ;;;1046
216	214, 215	eqtri 2784	. . . . . 6 ⊢ ((;;;1046 · 1) + 0) = ;;;1046
217	5, 16, 29, 3, 1, 183, 33, 15, 32, 212, 216	decma2c 12872	. . . . 5 ⊢ ((;;;1046 · 𝑁) + ;;;1070) = ;;;;;;4186116
218		eqid 2761	. . . . . 6 ⊢ ;;;2046 = ;;;2046
219	43, 20	deccl 12829	. . . . . . 7 ⊢ ;;122 ∈ ℕ0
220	219, 28	deccl 12829	. . . . . 6 ⊢ ;;;1227 ∈ ℕ0
221		eqid 2761	. . . . . . 7 ⊢ ;;204 = ;;204
222		eqid 2761	. . . . . . 7 ⊢ ;;;1227 = ;;;1227
223	24, 16	deccl 12829	. . . . . . . 8 ⊢ ;81 ∈ ℕ0
224	223, 12	deccl 12829	. . . . . . 7 ⊢ ;;819 ∈ ℕ0
225		eqid 2761	. . . . . . . 8 ⊢ ;20 = ;20
226		eqid 2761	. . . . . . . . 9 ⊢ ;;122 = ;;122
227		eqid 2761	. . . . . . . . 9 ⊢ ;;819 = ;;819
228		eqid 2761	. . . . . . . . . . 11 ⊢ ;81 = ;81
229		8cn 12440	. . . . . . . . . . . 12 ⊢ 8 ∈ ℂ
230	229, 69, 124	addcomli 11502	. . . . . . . . . . 11 ⊢ (1 + 8) = 9
231		2p1e3 12484	. . . . . . . . . . 11 ⊢ (2 + 1) = 3
232	16, 20, 24, 16, 108, 228, 230, 231	decadd 12873	. . . . . . . . . 10 ⊢ (;12 + ;81) = ;93
233	12, 41, 91, 232	decsuc 12850	. . . . . . . . 9 ⊢ ((;12 + ;81) + 1) = ;94
234		9p2e11 12906	. . . . . . . . . 10 ⊢ (9 + 2) = ;11
235	56, 51, 234	addcomli 11502	. . . . . . . . 9 ⊢ (2 + 9) = ;11
236	43, 20, 223, 12, 226, 227, 233, 16, 235	decaddc 12874	. . . . . . . 8 ⊢ (;;122 + ;;819) = ;;941
237	13	nn0cni 12618	. . . . . . . . . 10 ⊢ ;94 ∈ ℂ
238	237	addridi 11497	. . . . . . . . 9 ⊢ (;94 + 0) = ;94
239	122, 16	eqeltri 2857	. . . . . . . . . . 11 ⊢ (1 + 0) ∈ ℕ0
240	51	mul02i 11499	. . . . . . . . . . . . 13 ⊢ (0 · 2) = 0
241	240, 122	oveq12i 7432	. . . . . . . . . . . 12 ⊢ ((0 · 2) + (1 + 0)) = (0 + 1)
242	241, 187	eqtri 2784	. . . . . . . . . . 11 ⊢ ((0 · 2) + (1 + 0)) = 1
243	20, 3, 239, 225, 20, 113, 242	decrmanc 12876	. . . . . . . . . 10 ⊢ ((;20 · 2) + (1 + 0)) = ;41
244		4t2e8 12511	. . . . . . . . . . . 12 ⊢ (4 · 2) = 8
245	244	oveq1i 7430	. . . . . . . . . . 11 ⊢ ((4 · 2) + 0) = (8 + 0)
246	229	addridi 11497	. . . . . . . . . . 11 ⊢ (8 + 0) = 8
247	24	dec0h 12841	. . . . . . . . . . 11 ⊢ 8 = ;08
248	245, 246, 247	3eqtri 2788	. . . . . . . . . 10 ⊢ ((4 · 2) + 0) = ;08
249	35, 2, 16, 3, 221, 146, 20, 24, 3, 243, 248	decmac 12871	. . . . . . . . 9 ⊢ ((;;204 · 2) + (9 + 1)) = ;;418
250	64, 51, 202	addcomli 11502	. . . . . . . . . 10 ⊢ (2 + 4) = 6
251	16, 20, 2, 52, 250	decaddi 12879	. . . . . . . . 9 ⊢ ((6 · 2) + 4) = ;16
252	36, 15, 12, 2, 218, 238, 20, 15, 16, 249, 251	decmac 12871	. . . . . . . 8 ⊢ ((;;;2046 · 2) + (;94 + 0)) = ;;;4186
253	166	mul01i 11500	. . . . . . . . . 10 ⊢ (;;;2046 · 0) = 0
254	253	oveq1i 7430	. . . . . . . . 9 ⊢ ((;;;2046 · 0) + 1) = (0 + 1)
255	254, 187, 209	3eqtri 2788	. . . . . . . 8 ⊢ ((;;;2046 · 0) + 1) = ;01
256	20, 3, 13, 16, 225, 236, 37, 16, 3, 252, 255	decma2c 12872	. . . . . . 7 ⊢ ((;;;2046 · ;20) + (;;122 + ;;819)) = ;;;;41861
257	41	dec0h 12841	. . . . . . . . 9 ⊢ 3 = ;03
258	187, 16	eqeltri 2857	. . . . . . . . . 10 ⊢ (0 + 1) ∈ ℕ0
259	64	mul02i 11499	. . . . . . . . . . . 12 ⊢ (0 · 4) = 0
260	259, 187	oveq12i 7432	. . . . . . . . . . 11 ⊢ ((0 · 4) + (0 + 1)) = (0 + 1)
261	260, 187	eqtri 2784	. . . . . . . . . 10 ⊢ ((0 · 4) + (0 + 1)) = 1
262	20, 3, 258, 225, 2, 121, 261	decrmanc 12876	. . . . . . . . 9 ⊢ ((;20 · 4) + (0 + 1)) = ;81
263		6p3e9 12502	. . . . . . . . . 10 ⊢ (6 + 3) = 9
264	16, 15, 41, 103, 263	decaddi 12879	. . . . . . . . 9 ⊢ ((4 · 4) + 3) = ;19
265	35, 2, 3, 41, 221, 257, 2, 12, 16, 262, 264	decmac 12871	. . . . . . . 8 ⊢ ((;;204 · 4) + 3) = ;;819
266	152, 64, 190	addcomli 11502	. . . . . . . . 9 ⊢ (4 + 7) = ;11
267	20, 2, 28, 95, 231, 16, 266	decaddci 12880	. . . . . . . 8 ⊢ ((6 · 4) + 7) = ;31
268	36, 15, 28, 218, 2, 16, 41, 265, 267	decrmac 12877	. . . . . . 7 ⊢ ((;;;2046 · 4) + 7) = ;;;8191
269	35, 2, 219, 28, 221, 222, 37, 16, 224, 256, 268	decma2c 12872	. . . . . 6 ⊢ ((;;;2046 · ;;204) + ;;;1227) = ;;;;;418611
270	50	mul02i 11499	. . . . . . . . . . 11 ⊢ (0 · 6) = 0
271	270	oveq1i 7430	. . . . . . . . . 10 ⊢ ((0 · 6) + 2) = (0 + 2)
272	271, 131	eqtri 2784	. . . . . . . . 9 ⊢ ((0 · 6) + 2) = 2
273	20, 3, 20, 225, 15, 53, 272	decrmanc 12876	. . . . . . . 8 ⊢ ((;20 · 6) + 2) = ;;122
274		4p3e7 12496	. . . . . . . . 9 ⊢ (4 + 3) = 7
275	20, 2, 41, 96, 274	decaddi 12879	. . . . . . . 8 ⊢ ((4 · 6) + 3) = ;27
276	35, 2, 41, 221, 15, 28, 20, 273, 275	decrmac 12877	. . . . . . 7 ⊢ ((;;204 · 6) + 3) = ;;;1227
277	15, 36, 15, 218, 15, 41, 276, 90	decmul1c 12884	. . . . . 6 ⊢ (;;;2046 · 6) = ;;;;12276
278	37, 36, 15, 218, 15, 220, 269, 277	decmul2c 12885	. . . . 5 ⊢ (;;;2046 · ;;;2046) = ;;;;;;4186116
279	217, 278	eqtr4i 2787	. . . 4 ⊢ ((;;;1046 · 𝑁) + ;;;1070) = (;;;2046 · ;;;2046)
280	8, 9, 31, 34, 37, 30, 177, 182, 279	mod2xi 17247	. . 3 ⊢ ((2↑;50) mod 𝑁) = (;;;1070 mod 𝑁)
281	23	nn0cni 12618	. . . 4 ⊢ ;50 ∈ ℂ
282		eqid 2761	. . . . 5 ⊢ ;50 = ;50
283	20, 22, 3, 282, 180, 240	decmul1 12883	. . . 4 ⊢ (;50 · 2) = ;;100
284	281, 51, 283	mulcomli 11318	. . 3 ⊢ (2 · ;50) = ;;100
285		eqid 2761	. . . . 5 ⊢ ;;614 = ;;614
286	20, 12	deccl 12829	. . . . 5 ⊢ ;29 ∈ ℕ0
287		eqid 2761	. . . . . . 7 ⊢ ;61 = ;61
288		eqid 2761	. . . . . . 7 ⊢ ;29 = ;29
289	198	oveq1i 7430	. . . . . . . 8 ⊢ ((6 + 2) + 1) = (8 + 1)
290	289, 124	eqtri 2784	. . . . . . 7 ⊢ ((6 + 2) + 1) = 9
291	15, 16, 20, 12, 287, 288, 290, 147	decaddc2 12875	. . . . . 6 ⊢ (;61 + ;29) = ;90
292	61, 3	eqeltri 2857	. . . . . . . 8 ⊢ (0 + 0) ∈ ℕ0
293		eqid 2761	. . . . . . . 8 ⊢ ;;286 = ;;286
294		eqid 2761	. . . . . . . . 9 ⊢ ;28 = ;28
295	121	oveq1i 7430	. . . . . . . . . 10 ⊢ ((2 · 4) + 3) = (8 + 3)
296		8p3e11 12900	. . . . . . . . . 10 ⊢ (8 + 3) = ;11
297	295, 296	eqtri 2784	. . . . . . . . 9 ⊢ ((2 · 4) + 3) = ;11
298		8t4e32 12936	. . . . . . . . . 10 ⊢ (8 · 4) = ;32
299	41, 20, 20, 298, 88	decaddi 12879	. . . . . . . . 9 ⊢ ((8 · 4) + 2) = ;34
300	20, 24, 20, 294, 2, 2, 41, 297, 299	decrmac 12877	. . . . . . . 8 ⊢ ((;28 · 4) + 2) = ;;114
301	95, 61	oveq12i 7432	. . . . . . . . 9 ⊢ ((6 · 4) + (0 + 0)) = (;24 + 0)
302	38	nn0cni 12618	. . . . . . . . . 10 ⊢ ;24 ∈ ℂ
303	302	addridi 11497	. . . . . . . . 9 ⊢ (;24 + 0) = ;24
304	301, 303	eqtri 2784	. . . . . . . 8 ⊢ ((6 · 4) + (0 + 0)) = ;24
305	25, 15, 292, 293, 2, 2, 20, 300, 304	decrmac 12877	. . . . . . 7 ⊢ ((;;286 · 4) + (0 + 0)) = ;;;1144
306	26	nn0cni 12618	. . . . . . . . . 10 ⊢ ;;286 ∈ ℂ
307	306	mul01i 11500	. . . . . . . . 9 ⊢ (;;286 · 0) = 0
308	307	oveq1i 7430	. . . . . . . 8 ⊢ ((;;286 · 0) + 9) = (0 + 9)
309	308, 75, 58	3eqtri 2788	. . . . . . 7 ⊢ ((;;286 · 0) + 9) = ;09
310	2, 3, 3, 12, 60, 59, 26, 12, 3, 305, 309	decma2c 12872	. . . . . 6 ⊢ ((;;286 · ;40) + (9 + 0)) = ;;;;11449
311	307	oveq1i 7430	. . . . . . 7 ⊢ ((;;286 · 0) + 0) = (0 + 0)
312	311, 61, 62	3eqtri 2788	. . . . . 6 ⊢ ((;;286 · 0) + 0) = ;00
313	4, 3, 12, 3, 55, 291, 26, 3, 3, 310, 312	decma2c 12872	. . . . 5 ⊢ ((;;286 · ;;400) + (;61 + ;29)) = ;;;;;114490
314	229	mulridi 11313	. . . . . . . 8 ⊢ (8 · 1) = 8
315	16, 20, 24, 294, 109, 314	decmul1 12883	. . . . . . 7 ⊢ (;28 · 1) = ;28
316	20, 24, 124, 315	decsuc 12850	. . . . . 6 ⊢ ((;28 · 1) + 1) = ;29
317	50	mulridi 11313	. . . . . . . 8 ⊢ (6 · 1) = 6
318	317	oveq1i 7430	. . . . . . 7 ⊢ ((6 · 1) + 4) = (6 + 4)
319	318, 92	eqtri 2784	. . . . . 6 ⊢ ((6 · 1) + 4) = ;10
320	25, 15, 2, 293, 16, 3, 16, 316, 319	decrmac 12877	. . . . 5 ⊢ ((;;286 · 1) + 4) = ;;290
321	5, 16, 17, 2, 1, 285, 26, 3, 286, 313, 320	decma2c 12872	. . . 4 ⊢ ((;;286 · 𝑁) + ;;614) = ;;;;;;1144900
322	16, 16	deccl 12829	. . . . . . . . 9 ⊢ ;11 ∈ ℕ0
323	322, 2	deccl 12829	. . . . . . . 8 ⊢ ;;114 ∈ ℕ0
324	323, 2	deccl 12829	. . . . . . 7 ⊢ ;;;1144 ∈ ℕ0
325	324, 12	deccl 12829	. . . . . 6 ⊢ ;;;;11449 ∈ ℕ0
326	28, 2	deccl 12829	. . . . . . . 8 ⊢ ;74 ∈ ℕ0
327	326, 12	deccl 12829	. . . . . . 7 ⊢ ;;749 ∈ ℕ0
328		eqid 2761	. . . . . . . 8 ⊢ ;10 = ;10
329		eqid 2761	. . . . . . . 8 ⊢ ;;749 = ;;749
330	326	nn0cni 12618	. . . . . . . . . 10 ⊢ ;74 ∈ ℂ
331	330	addridi 11497	. . . . . . . . 9 ⊢ (;74 + 0) = ;74
332	152	addridi 11497	. . . . . . . . . . 11 ⊢ (7 + 0) = 7
333	332, 28	eqeltri 2857	. . . . . . . . . 10 ⊢ (7 + 0) ∈ ℕ0
334	10	nn0cni 12618	. . . . . . . . . . . 12 ⊢ ;10 ∈ ℂ
335	334	mulridi 11313	. . . . . . . . . . 11 ⊢ (;10 · 1) = ;10
336	16, 3, 187, 335	decsuc 12850	. . . . . . . . . 10 ⊢ ((;10 · 1) + 1) = ;11
337	152	mulridi 11313	. . . . . . . . . . . 12 ⊢ (7 · 1) = 7
338	337, 332	oveq12i 7432	. . . . . . . . . . 11 ⊢ ((7 · 1) + (7 + 0)) = (7 + 7)
339		7p7e14 12898	. . . . . . . . . . 11 ⊢ (7 + 7) = ;14
340	338, 339	eqtri 2784	. . . . . . . . . 10 ⊢ ((7 · 1) + (7 + 0)) = ;14
341	10, 28, 333, 185, 16, 2, 16, 336, 340	decrmac 12877	. . . . . . . . 9 ⊢ ((;;107 · 1) + (7 + 0)) = ;;114
342	69	mul02i 11499	. . . . . . . . . . 11 ⊢ (0 · 1) = 0
343	342	oveq1i 7430	. . . . . . . . . 10 ⊢ ((0 · 1) + 4) = (0 + 4)
344	64	addlidi 11498	. . . . . . . . . 10 ⊢ (0 + 4) = 4
345	343, 344, 114	3eqtri 2788	. . . . . . . . 9 ⊢ ((0 · 1) + 4) = ;04
346	29, 3, 28, 2, 183, 331, 16, 2, 3, 341, 345	decmac 12871	. . . . . . . 8 ⊢ ((;;;1070 · 1) + (;74 + 0)) = ;;;1144
347	30	nn0cni 12618	. . . . . . . . . . 11 ⊢ ;;;1070 ∈ ℂ
348	347	mul01i 11500	. . . . . . . . . 10 ⊢ (;;;1070 · 0) = 0
349	348	oveq1i 7430	. . . . . . . . 9 ⊢ ((;;;1070 · 0) + 9) = (0 + 9)
350	349, 75, 58	3eqtri 2788	. . . . . . . 8 ⊢ ((;;;1070 · 0) + 9) = ;09
351	16, 3, 326, 12, 328, 329, 30, 12, 3, 346, 350	decma2c 12872	. . . . . . 7 ⊢ ((;;;1070 · ;10) + ;;749) = ;;;;11449
352		dfdec10 12817	. . . . . . . . . 10 ⊢ ;74 = ((;10 · 7) + 4)
353	352	eqcomi 2770	. . . . . . . . 9 ⊢ ((;10 · 7) + 4) = ;74
354		7t7e49 12933	. . . . . . . . 9 ⊢ (7 · 7) = ;49
355	28, 10, 28, 185, 12, 2, 353, 354	decmul1c 12884	. . . . . . . 8 ⊢ (;;107 · 7) = ;;749
356	152	mul02i 11499	. . . . . . . 8 ⊢ (0 · 7) = 0
357	28, 29, 3, 183, 355, 356	decmul1 12883	. . . . . . 7 ⊢ (;;;1070 · 7) = ;;;7490
358	30, 10, 28, 185, 3, 327, 351, 357	decmul2c 12885	. . . . . 6 ⊢ (;;;1070 · ;;107) = ;;;;;114490
359	325, 3, 3, 358, 61	decaddi 12879	. . . . 5 ⊢ ((;;;1070 · ;;107) + 0) = ;;;;;114490
360	348, 62	eqtri 2784	. . . . 5 ⊢ (;;;1070 · 0) = ;00
361	30, 29, 3, 183, 3, 3, 359, 360	decmul2c 12885	. . . 4 ⊢ (;;;1070 · ;;;1070) = ;;;;;;1144900
362	321, 361	eqtr4i 2787	. . 3 ⊢ ((;;286 · 𝑁) + ;;614) = (;;;1070 · ;;;1070)
363	8, 9, 23, 27, 30, 18, 280, 284, 362	mod2xi 17247	. 2 ⊢ ((2↑;;100) mod 𝑁) = (;;614 mod 𝑁)
364	11	nn0cni 12618	. . 3 ⊢ ;;100 ∈ ℂ
365		eqid 2761	. . . 4 ⊢ ;;100 = ;;100
366	20, 10, 3, 365, 172, 240	decmul1 12883	. . 3 ⊢ (;;100 · 2) = ;;200
367	364, 51, 366	mulcomli 11318	. 2 ⊢ (2 · ;;100) = ;;200
368		eqid 2761	. . . 4 ⊢ ;;902 = ;;902
369		eqid 2761	. . . . . 6 ⊢ ;90 = ;90
370	12, 3, 12, 369, 75	decaddi 12879	. . . . 5 ⊢ (;90 + 9) = ;99
371		eqid 2761	. . . . . . 7 ⊢ ;94 = ;94
372		6p1e7 12490	. . . . . . . 8 ⊢ (6 + 1) = 7
373		9t4e36 12943	. . . . . . . 8 ⊢ (9 · 4) = ;36
374	41, 15, 372, 373	decsuc 12850	. . . . . . 7 ⊢ ((9 · 4) + 1) = ;37
375	103, 61	oveq12i 7432	. . . . . . . 8 ⊢ ((4 · 4) + (0 + 0)) = (;16 + 0)
376	16, 15	deccl 12829	. . . . . . . . . 10 ⊢ ;16 ∈ ℕ0
377	376	nn0cni 12618	. . . . . . . . 9 ⊢ ;16 ∈ ℂ
378	377	addridi 11497	. . . . . . . 8 ⊢ (;16 + 0) = ;16
379	375, 378	eqtri 2784	. . . . . . 7 ⊢ ((4 · 4) + (0 + 0)) = ;16
380	12, 2, 292, 371, 2, 15, 16, 374, 379	decrmac 12877	. . . . . 6 ⊢ ((;94 · 4) + (0 + 0)) = ;;376
381	237	mul01i 11500	. . . . . . . 8 ⊢ (;94 · 0) = 0
382	381	oveq1i 7430	. . . . . . 7 ⊢ ((;94 · 0) + 9) = (0 + 9)
383	382, 75, 58	3eqtri 2788	. . . . . 6 ⊢ ((;94 · 0) + 9) = ;09
384	2, 3, 3, 12, 60, 59, 13, 12, 3, 380, 383	decma2c 12872	. . . . 5 ⊢ ((;94 · ;40) + (9 + 0)) = ;;;3769
385	4, 3, 12, 12, 55, 370, 13, 12, 3, 384, 383	decma2c 12872	. . . 4 ⊢ ((;94 · ;;400) + (;90 + 9)) = ;;;;37699
386	56	mulridi 11313	. . . . 5 ⊢ (9 · 1) = 9
387	64	mulridi 11313	. . . . . . 7 ⊢ (4 · 1) = 4
388	387	oveq1i 7430	. . . . . 6 ⊢ ((4 · 1) + 2) = (4 + 2)
389	388, 202	eqtri 2784	. . . . 5 ⊢ ((4 · 1) + 2) = 6
390	12, 2, 20, 371, 16, 386, 389	decrmanc 12876	. . . 4 ⊢ ((;94 · 1) + 2) = ;96
391	5, 16, 19, 20, 1, 368, 13, 15, 12, 385, 390	decma2c 12872	. . 3 ⊢ ((;94 · 𝑁) + ;;902) = ;;;;;376996
392	38, 22	deccl 12829	. . . 4 ⊢ ;;245 ∈ ℕ0
393		eqid 2761	. . . . 5 ⊢ ;;245 = ;;245
394	50, 51, 198	addcomli 11502	. . . . . . 7 ⊢ (2 + 6) = 8
395	20, 2, 15, 16, 164, 287, 394, 101	decadd 12873	. . . . . 6 ⊢ (;24 + ;61) = ;85
396		8p2e10 12899	. . . . . . 7 ⊢ (8 + 2) = ;10
397	41, 15, 372, 90	decsuc 12850	. . . . . . 7 ⊢ ((6 · 6) + 1) = ;37
398	50	mullidi 11314	. . . . . . . . 9 ⊢ (1 · 6) = 6
399	398	oveq1i 7430	. . . . . . . 8 ⊢ ((1 · 6) + 0) = (6 + 0)
400	50	addridi 11497	. . . . . . . 8 ⊢ (6 + 0) = 6
401	399, 400	eqtri 2784	. . . . . . 7 ⊢ ((1 · 6) + 0) = 6
402	15, 16, 16, 3, 287, 396, 15, 397, 401	decma 12870	. . . . . 6 ⊢ ((;61 · 6) + (8 + 2)) = ;;376
403	17, 2, 24, 22, 285, 395, 15, 12, 20, 402, 99	decmac 12871	. . . . 5 ⊢ ((;;614 · 6) + (;24 + ;61)) = ;;;3769
404	16, 15, 16, 287, 317, 78	decmul1 12883	. . . . . 6 ⊢ (;61 · 1) = ;61
405	387	oveq1i 7430	. . . . . . 7 ⊢ ((4 · 1) + 5) = (4 + 5)
406	405, 98	eqtri 2784	. . . . . 6 ⊢ ((4 · 1) + 5) = 9
407	17, 2, 22, 285, 16, 404, 406	decrmanc 12876	. . . . 5 ⊢ ((;;614 · 1) + 5) = ;;619
408	15, 16, 38, 22, 287, 393, 18, 12, 17, 403, 407	decma2c 12872	. . . 4 ⊢ ((;;614 · ;61) + ;;245) = ;;;;37699
409	65	oveq1i 7430	. . . . . . 7 ⊢ ((1 · 4) + 1) = (4 + 1)
410	409, 101	eqtri 2784	. . . . . 6 ⊢ ((1 · 4) + 1) = 5
411	15, 16, 16, 287, 2, 95, 410	decrmanc 12876	. . . . 5 ⊢ ((;61 · 4) + 1) = ;;245
412	2, 17, 2, 285, 15, 16, 411, 103	decmul1c 12884	. . . 4 ⊢ (;;614 · 4) = ;;;2456
413	18, 17, 2, 285, 15, 392, 408, 412	decmul2c 12885	. . 3 ⊢ (;;614 · ;;614) = ;;;;;376996
414	391, 413	eqtr4i 2787	. 2 ⊢ ((;94 · 𝑁) + ;;902) = (;;614 · ;;614)
415	8, 9, 11, 14, 18, 21, 363, 367, 414	mod2xi 17247	1 ⊢ ((2↑;;200) mod 𝑁) = (;;902 mod 𝑁)

This proof depends on syntax axioms:   = wceq 1570  (class class class)co 7420  0cc0 11200  1c1 11201   + caddc 11203   · cmul 11205  ℕcn 12335  2c2 12397  3c3 12398  4c4 12399  5c5 12400  6c6 12401  7c7 12402  8c8 12403  9c9 12404  ℕ₀cn0 12606  ;cdc 12814   mod cmo 14009  ↑cexp 14204

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 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277  ax-pre-sup 11278

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  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 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-er 8717  df-en 8974  df-dom 8975  df-sdom 8976  df-sup 9434  df-inf 9435  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-div 11974  df-nn 12336  df-2 12405  df-3 12406  df-4 12407  df-5 12408  df-6 12409  df-7 12410  df-8 12411  df-9 12412  df-n0 12607  df-z 12694  df-dec 12815  df-uz 12966  df-rp 13121  df-fl 13932  df-mod 14010  df-seq 14145  df-exp 14205

This theorem is used by:  4001lem2  17320  4001lem3  17321