Theorem 1259lem4 17312

Description: Lemma for 1259prm 17314. Calculate a power mod. In decimal, we calculate 2↑306 = (2↑76)↑4 · 4≡5↑4 · 4 = 2𝑁 − 18, 2↑612 = (2↑306)↑2≡18↑2 = 324, 2↑629 = 2↑612 · 2↑17≡324 · 136 = 35𝑁 − 1 and finally 2↑(𝑁 − 1) = (2↑629)↑2≡1↑2 = 1. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.)

Hypothesis

Ref	Expression
1259prm.1	⊢ 𝑁 = ;;;1259

Assertion

Ref	Expression
1259lem4	⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)

Proof of Theorem 1259lem4

Step	Hyp	Ref	Expression
1		2nn 12416	. 2 ⊢ 2 ∈ ℕ
2		6nn0 12627	. . . 4 ⊢ 6 ∈ ℕ0
3		2nn0 12623	. . . 4 ⊢ 2 ∈ ℕ0
4	2, 3	deccl 12829	. . 3 ⊢ ;62 ∈ ℕ0
5		9nn0 12630	. . 3 ⊢ 9 ∈ ℕ0
6	4, 5	deccl 12829	. 2 ⊢ ;;629 ∈ ℕ0
7		0z 12704	. 2 ⊢ 0 ∈ ℤ
8		1nn 12346	. 2 ⊢ 1 ∈ ℕ
9		1nn0 12622	. 2 ⊢ 1 ∈ ℕ0
10		12nn0 12831	. . . . . . 7 ⊢ ;12 ∈ ℕ0
11		5nn0 12626	. . . . . . 7 ⊢ 5 ∈ ℕ0
12	10, 11	deccl 12829	. . . . . 6 ⊢ ;;125 ∈ ℕ0
13		8nn0 12629	. . . . . 6 ⊢ 8 ∈ ℕ0
14	12, 13	deccl 12829	. . . . 5 ⊢ ;;;1258 ∈ ℕ0
15	14	nn0cni 12618	. . . 4 ⊢ ;;;1258 ∈ ℂ
16		ax-1cn 11258	. . . 4 ⊢ 1 ∈ ℂ
17		1259prm.1	. . . . 5 ⊢ 𝑁 = ;;;1259
18		8p1e9 12492	. . . . . 6 ⊢ (8 + 1) = 9
19		eqid 2761	. . . . . 6 ⊢ ;;;1258 = ;;;1258
20	12, 13, 18, 19	decsuc 12850	. . . . 5 ⊢ (;;;1258 + 1) = ;;;1259
21	17, 20	eqtr4i 2787	. . . 4 ⊢ 𝑁 = (;;;1258 + 1)
22	15, 16, 21	mvrraddi 11574	. . 3 ⊢ (𝑁 − 1) = ;;;1258
23	22, 14	eqeltri 2857	. 2 ⊢ (𝑁 − 1) ∈ ℕ0
24		9nn 12441	. . . . 5 ⊢ 9 ∈ ℕ
25	12, 24	decnncl 12838	. . . 4 ⊢ ;;;1259 ∈ ℕ
26	17, 25	eqeltri 2857	. . 3 ⊢ 𝑁 ∈ ℕ
27	2, 9	deccl 12829	. . . 4 ⊢ ;61 ∈ ℕ0
28	27, 3	deccl 12829	. . 3 ⊢ ;;612 ∈ ℕ0
29		3nn0 12624	. . . . 5 ⊢ 3 ∈ ℕ0
30		4nn0 12625	. . . . 5 ⊢ 4 ∈ ℕ0
31	29, 30	deccl 12829	. . . 4 ⊢ ;34 ∈ ℕ0
32	31	nn0zi 12721	. . 3 ⊢ ;34 ∈ ℤ
33	29, 3	deccl 12829	. . . 4 ⊢ ;32 ∈ ℕ0
34	33, 30	deccl 12829	. . 3 ⊢ ;;324 ∈ ℕ0
35		7nn0 12628	. . . 4 ⊢ 7 ∈ ℕ0
36	9, 35	deccl 12829	. . 3 ⊢ ;17 ∈ ℕ0
37	9, 29	deccl 12829	. . . 4 ⊢ ;13 ∈ ℕ0
38	37, 2	deccl 12829	. . 3 ⊢ ;;136 ∈ ℕ0
39		0nn0 12621	. . . . . 6 ⊢ 0 ∈ ℕ0
40	29, 39	deccl 12829	. . . . 5 ⊢ ;30 ∈ ℕ0
41	40, 2	deccl 12829	. . . 4 ⊢ ;;306 ∈ ℕ0
42		8nn 12438	. . . . 5 ⊢ 8 ∈ ℕ
43	9, 42	decnncl 12838	. . . 4 ⊢ ;18 ∈ ℕ
44	10, 30	deccl 12829	. . . . 5 ⊢ ;;124 ∈ ℕ0
45	44, 9	deccl 12829	. . . 4 ⊢ ;;;1241 ∈ ℕ0
46	9, 11	deccl 12829	. . . . . 6 ⊢ ;15 ∈ ℕ0
47	46, 29	deccl 12829	. . . . 5 ⊢ ;;153 ∈ ℕ0
48		1z 12726	. . . . 5 ⊢ 1 ∈ ℤ
49	11, 39	deccl 12829	. . . . 5 ⊢ ;50 ∈ ℕ0
50	46, 3	deccl 12829	. . . . . 6 ⊢ ;;152 ∈ ℕ0
51		25nn0 12833	. . . . . 6 ⊢ ;25 ∈ ℕ0
52	35, 2	deccl 12829	. . . . . . 7 ⊢ ;76 ∈ ℕ0
53	17	1259lem3 17311	. . . . . . 7 ⊢ ((2↑;76) mod 𝑁) = (5 mod 𝑁)
54		eqid 2761	. . . . . . . 8 ⊢ ;76 = ;76
55		4p1e5 12488	. . . . . . . . 9 ⊢ (4 + 1) = 5
56		7cn 12437	. . . . . . . . . 10 ⊢ 7 ∈ ℂ
57		2cn 12418	. . . . . . . . . 10 ⊢ 2 ∈ ℂ
58		7t2e14 12928	. . . . . . . . . 10 ⊢ (7 · 2) = ;14
59	56, 57, 58	mulcomli 11318	. . . . . . . . 9 ⊢ (2 · 7) = ;14
60	9, 30, 55, 59	decsuc 12850	. . . . . . . 8 ⊢ ((2 · 7) + 1) = ;15
61		6cn 12434	. . . . . . . . 9 ⊢ 6 ∈ ℂ
62		6t2e12 12923	. . . . . . . . 9 ⊢ (6 · 2) = ;12
63	61, 57, 62	mulcomli 11318	. . . . . . . 8 ⊢ (2 · 6) = ;12
64	3, 35, 2, 54, 3, 9, 60, 63	decmul2c 12885	. . . . . . 7 ⊢ (2 · ;76) = ;;152
65	51	nn0cni 12618	. . . . . . . . 9 ⊢ ;25 ∈ ℂ
66	65	addlidi 11498	. . . . . . . 8 ⊢ (0 + ;25) = ;25
67	26	nncni 12345	. . . . . . . . . 10 ⊢ 𝑁 ∈ ℂ
68	67	mul02i 11499	. . . . . . . . 9 ⊢ (0 · 𝑁) = 0
69	68	oveq1i 7430	. . . . . . . 8 ⊢ ((0 · 𝑁) + ;25) = (0 + ;25)
70		5t5e25 12922	. . . . . . . 8 ⊢ (5 · 5) = ;25
71	66, 69, 70	3eqtr4i 2794	. . . . . . 7 ⊢ ((0 · 𝑁) + ;25) = (5 · 5)
72	26, 1, 52, 7, 11, 51, 53, 64, 71	mod2xi 17247	. . . . . 6 ⊢ ((2↑;;152) mod 𝑁) = (;25 mod 𝑁)
73		2p1e3 12484	. . . . . . 7 ⊢ (2 + 1) = 3
74		eqid 2761	. . . . . . 7 ⊢ ;;152 = ;;152
75	46, 3, 73, 74	decsuc 12850	. . . . . 6 ⊢ (;;152 + 1) = ;;153
76	49	nn0cni 12618	. . . . . . . 8 ⊢ ;50 ∈ ℂ
77	76	addlidi 11498	. . . . . . 7 ⊢ (0 + ;50) = ;50
78	68	oveq1i 7430	. . . . . . 7 ⊢ ((0 · 𝑁) + ;50) = (0 + ;50)
79		eqid 2761	. . . . . . . 8 ⊢ ;25 = ;25
80		2t2e4 12506	. . . . . . . . . 10 ⊢ (2 · 2) = 4
81	80	oveq1i 7430	. . . . . . . . 9 ⊢ ((2 · 2) + 1) = (4 + 1)
82	81, 55	eqtri 2784	. . . . . . . 8 ⊢ ((2 · 2) + 1) = 5
83		5t2e10 12919	. . . . . . . 8 ⊢ (5 · 2) = ;10
84	3, 3, 11, 79, 39, 9, 82, 83	decmul1c 12884	. . . . . . 7 ⊢ (;25 · 2) = ;50
85	77, 78, 84	3eqtr4i 2794	. . . . . 6 ⊢ ((0 · 𝑁) + ;50) = (;25 · 2)
86	26, 1, 50, 7, 51, 49, 72, 75, 85	modxp1i 17248	. . . . 5 ⊢ ((2↑;;153) mod 𝑁) = (;50 mod 𝑁)
87		eqid 2761	. . . . . 6 ⊢ ;;153 = ;;153
88		eqid 2761	. . . . . . . . 9 ⊢ ;15 = ;15
89	57	mulridi 11313	. . . . . . . . . . 11 ⊢ (2 · 1) = 2
90	89	oveq1i 7430	. . . . . . . . . 10 ⊢ ((2 · 1) + 1) = (2 + 1)
91	90, 73	eqtri 2784	. . . . . . . . 9 ⊢ ((2 · 1) + 1) = 3
92		5cn 12431	. . . . . . . . . 10 ⊢ 5 ∈ ℂ
93	92, 57, 83	mulcomli 11318	. . . . . . . . 9 ⊢ (2 · 5) = ;10
94	3, 9, 11, 88, 39, 9, 91, 93	decmul2c 12885	. . . . . . . 8 ⊢ (2 · ;15) = ;30
95	94	oveq1i 7430	. . . . . . 7 ⊢ ((2 · ;15) + 0) = (;30 + 0)
96	40	nn0cni 12618	. . . . . . . 8 ⊢ ;30 ∈ ℂ
97	96	addridi 11497	. . . . . . 7 ⊢ (;30 + 0) = ;30
98	95, 97	eqtri 2784	. . . . . 6 ⊢ ((2 · ;15) + 0) = ;30
99		2t3e6 12509	. . . . . . 7 ⊢ (2 · 3) = 6
100	2	dec0h 12841	. . . . . . 7 ⊢ 6 = ;06
101	99, 100	eqtri 2784	. . . . . 6 ⊢ (2 · 3) = ;06
102	3, 46, 29, 87, 2, 39, 98, 101	decmul2c 12885	. . . . 5 ⊢ (2 · ;;153) = ;;306
103	67	mullidi 11314	. . . . . . . 8 ⊢ (1 · 𝑁) = 𝑁
104	103, 17	eqtri 2784	. . . . . . 7 ⊢ (1 · 𝑁) = ;;;1259
105		eqid 2761	. . . . . . 7 ⊢ ;;;1241 = ;;;1241
106	3, 30	deccl 12829	. . . . . . . 8 ⊢ ;24 ∈ ℕ0
107		eqid 2761	. . . . . . . . 9 ⊢ ;24 = ;24
108	3, 30, 55, 107	decsuc 12850	. . . . . . . 8 ⊢ (;24 + 1) = ;25
109		eqid 2761	. . . . . . . . 9 ⊢ ;;125 = ;;125
110		eqid 2761	. . . . . . . . 9 ⊢ ;;124 = ;;124
111		eqid 2761	. . . . . . . . . 10 ⊢ ;12 = ;12
112		1p1e2 12466	. . . . . . . . . 10 ⊢ (1 + 1) = 2
113		2p2e4 12477	. . . . . . . . . 10 ⊢ (2 + 2) = 4
114	9, 3, 9, 3, 111, 111, 112, 113	decadd 12873	. . . . . . . . 9 ⊢ (;12 + ;12) = ;24
115		5p4e9 12500	. . . . . . . . 9 ⊢ (5 + 4) = 9
116	10, 11, 10, 30, 109, 110, 114, 115	decadd 12873	. . . . . . . 8 ⊢ (;;125 + ;;124) = ;;249
117	106, 108, 116	decsucc 12860	. . . . . . 7 ⊢ ((;;125 + ;;124) + 1) = ;;250
118		9p1e10 12816	. . . . . . 7 ⊢ (9 + 1) = ;10
119	12, 5, 44, 9, 104, 105, 117, 118	decaddc2 12875	. . . . . 6 ⊢ ((1 · 𝑁) + ;;;1241) = ;;;2500
120		eqid 2761	. . . . . . 7 ⊢ ;50 = ;50
121	92	mul02i 11499	. . . . . . . . . 10 ⊢ (0 · 5) = 0
122	11, 11, 39, 120, 70, 121	decmul1 12883	. . . . . . . . 9 ⊢ (;50 · 5) = ;;250
123	122	oveq1i 7430	. . . . . . . 8 ⊢ ((;50 · 5) + 0) = (;;250 + 0)
124	51, 39	deccl 12829	. . . . . . . . . 10 ⊢ ;;250 ∈ ℕ0
125	124	nn0cni 12618	. . . . . . . . 9 ⊢ ;;250 ∈ ℂ
126	125	addridi 11497	. . . . . . . 8 ⊢ (;;250 + 0) = ;;250
127	123, 126	eqtri 2784	. . . . . . 7 ⊢ ((;50 · 5) + 0) = ;;250
128	76	mul01i 11500	. . . . . . . 8 ⊢ (;50 · 0) = 0
129	39	dec0h 12841	. . . . . . . 8 ⊢ 0 = ;00
130	128, 129	eqtri 2784	. . . . . . 7 ⊢ (;50 · 0) = ;00
131	49, 11, 39, 120, 39, 39, 127, 130	decmul2c 12885	. . . . . 6 ⊢ (;50 · ;50) = ;;;2500
132	119, 131	eqtr4i 2787	. . . . 5 ⊢ ((1 · 𝑁) + ;;;1241) = (;50 · ;50)
133	26, 1, 47, 48, 49, 45, 86, 102, 132	mod2xi 17247	. . . 4 ⊢ ((2↑;;306) mod 𝑁) = (;;;1241 mod 𝑁)
134		eqid 2761	. . . . 5 ⊢ ;;306 = ;;306
135		eqid 2761	. . . . . 6 ⊢ ;30 = ;30
136	9	dec0h 12841	. . . . . 6 ⊢ 1 = ;01
137		00id 11485	. . . . . . . 8 ⊢ (0 + 0) = 0
138	99, 137	oveq12i 7432	. . . . . . 7 ⊢ ((2 · 3) + (0 + 0)) = (6 + 0)
139	61	addridi 11497	. . . . . . 7 ⊢ (6 + 0) = 6
140	138, 139	eqtri 2784	. . . . . 6 ⊢ ((2 · 3) + (0 + 0)) = 6
141	57	mul01i 11500	. . . . . . . 8 ⊢ (2 · 0) = 0
142	141	oveq1i 7430	. . . . . . 7 ⊢ ((2 · 0) + 1) = (0 + 1)
143		0p1e1 12463	. . . . . . 7 ⊢ (0 + 1) = 1
144	142, 143, 136	3eqtri 2788	. . . . . 6 ⊢ ((2 · 0) + 1) = ;01
145	29, 39, 39, 9, 135, 136, 3, 9, 39, 140, 144	decma2c 12872	. . . . 5 ⊢ ((2 · ;30) + 1) = ;61
146	3, 40, 2, 134, 3, 9, 145, 63	decmul2c 12885	. . . 4 ⊢ (2 · ;;306) = ;;612
147		eqid 2761	. . . . . 6 ⊢ ;18 = ;18
148	10, 30, 55, 110	decsuc 12850	. . . . . 6 ⊢ (;;124 + 1) = ;;125
149		8cn 12440	. . . . . . 7 ⊢ 8 ∈ ℂ
150	149, 16, 18	addcomli 11502	. . . . . 6 ⊢ (1 + 8) = 9
151	44, 9, 9, 13, 105, 147, 148, 150	decadd 12873	. . . . 5 ⊢ (;;;1241 + ;18) = ;;;1259
152	151, 17	eqtr4i 2787	. . . 4 ⊢ (;;;1241 + ;18) = 𝑁
153	34	nn0cni 12618	. . . . . 6 ⊢ ;;324 ∈ ℂ
154	153	addlidi 11498	. . . . 5 ⊢ (0 + ;;324) = ;;324
155	68	oveq1i 7430	. . . . 5 ⊢ ((0 · 𝑁) + ;;324) = (0 + ;;324)
156	9, 13	deccl 12829	. . . . . 6 ⊢ ;18 ∈ ℕ0
157	9, 30	deccl 12829	. . . . . 6 ⊢ ;14 ∈ ℕ0
158		eqid 2761	. . . . . . 7 ⊢ ;14 = ;14
159	16	mulridi 11313	. . . . . . . . 9 ⊢ (1 · 1) = 1
160	159, 112	oveq12i 7432	. . . . . . . 8 ⊢ ((1 · 1) + (1 + 1)) = (1 + 2)
161		1p2e3 12485	. . . . . . . 8 ⊢ (1 + 2) = 3
162	160, 161	eqtri 2784	. . . . . . 7 ⊢ ((1 · 1) + (1 + 1)) = 3
163	149	mulridi 11313	. . . . . . . . 9 ⊢ (8 · 1) = 8
164	163	oveq1i 7430	. . . . . . . 8 ⊢ ((8 · 1) + 4) = (8 + 4)
165		8p4e12 12901	. . . . . . . 8 ⊢ (8 + 4) = ;12
166	164, 165	eqtri 2784	. . . . . . 7 ⊢ ((8 · 1) + 4) = ;12
167	9, 13, 9, 30, 147, 158, 9, 3, 9, 162, 166	decmac 12871	. . . . . 6 ⊢ ((;18 · 1) + ;14) = ;32
168	149	mullidi 11314	. . . . . . . . 9 ⊢ (1 · 8) = 8
169	168	oveq1i 7430	. . . . . . . 8 ⊢ ((1 · 8) + 6) = (8 + 6)
170		8p6e14 12903	. . . . . . . 8 ⊢ (8 + 6) = ;14
171	169, 170	eqtri 2784	. . . . . . 7 ⊢ ((1 · 8) + 6) = ;14
172		8t8e64 12940	. . . . . . 7 ⊢ (8 · 8) = ;64
173	13, 9, 13, 147, 30, 2, 171, 172	decmul1c 12884	. . . . . 6 ⊢ (;18 · 8) = ;;144
174	156, 9, 13, 147, 30, 157, 167, 173	decmul2c 12885	. . . . 5 ⊢ (;18 · ;18) = ;;324
175	154, 155, 174	3eqtr4i 2794	. . . 4 ⊢ ((0 · 𝑁) + ;;324) = (;18 · ;18)
176	1, 41, 7, 43, 34, 45, 133, 146, 152, 175	mod2xnegi 17249	. . 3 ⊢ ((2↑;;612) mod 𝑁) = (;;324 mod 𝑁)
177	17	1259lem1 17309	. . 3 ⊢ ((2↑;17) mod 𝑁) = (;;136 mod 𝑁)
178		eqid 2761	. . . 4 ⊢ ;;612 = ;;612
179		eqid 2761	. . . 4 ⊢ ;17 = ;17
180		eqid 2761	. . . . 5 ⊢ ;61 = ;61
181	2, 9, 112, 180	decsuc 12850	. . . 4 ⊢ (;61 + 1) = ;62
182		7p2e9 12503	. . . . 5 ⊢ (7 + 2) = 9
183	56, 57, 182	addcomli 11502	. . . 4 ⊢ (2 + 7) = 9
184	27, 3, 9, 35, 178, 179, 181, 183	decadd 12873	. . 3 ⊢ (;;612 + ;17) = ;;629
185	29, 9	deccl 12829	. . . . 5 ⊢ ;31 ∈ ℕ0
186		eqid 2761	. . . . . . 7 ⊢ ;31 = ;31
187		3cn 12424	. . . . . . . . 9 ⊢ 3 ∈ ℂ
188		3p2e5 12493	. . . . . . . . 9 ⊢ (3 + 2) = 5
189	187, 57, 188	addcomli 11502	. . . . . . . 8 ⊢ (2 + 3) = 5
190	9, 3, 29, 111, 189	decaddi 12879	. . . . . . 7 ⊢ (;12 + 3) = ;15
191		5p1e6 12489	. . . . . . 7 ⊢ (5 + 1) = 6
192	10, 11, 29, 9, 109, 186, 190, 191	decadd 12873	. . . . . 6 ⊢ (;;125 + ;31) = ;;156
193	112	oveq1i 7430	. . . . . . . . 9 ⊢ ((1 + 1) + 1) = (2 + 1)
194	193, 73	eqtri 2784	. . . . . . . 8 ⊢ ((1 + 1) + 1) = 3
195		7p5e12 12896	. . . . . . . . 9 ⊢ (7 + 5) = ;12
196	56, 92, 195	addcomli 11502	. . . . . . . 8 ⊢ (5 + 7) = ;12
197	9, 11, 9, 35, 88, 179, 194, 3, 196	decaddc 12874	. . . . . . 7 ⊢ (;15 + ;17) = ;32
198		eqid 2761	. . . . . . . 8 ⊢ ;34 = ;34
199		7p3e10 12894	. . . . . . . . 9 ⊢ (7 + 3) = ;10
200	56, 187, 199	addcomli 11502	. . . . . . . 8 ⊢ (3 + 7) = ;10
201	187	mulridi 11313	. . . . . . . . . 10 ⊢ (3 · 1) = 3
202	16	addridi 11497	. . . . . . . . . 10 ⊢ (1 + 0) = 1
203	201, 202	oveq12i 7432	. . . . . . . . 9 ⊢ ((3 · 1) + (1 + 0)) = (3 + 1)
204		3p1e4 12487	. . . . . . . . 9 ⊢ (3 + 1) = 4
205	203, 204	eqtri 2784	. . . . . . . 8 ⊢ ((3 · 1) + (1 + 0)) = 4
206		4cn 12428	. . . . . . . . . . 11 ⊢ 4 ∈ ℂ
207	206	mulridi 11313	. . . . . . . . . 10 ⊢ (4 · 1) = 4
208	207	oveq1i 7430	. . . . . . . . 9 ⊢ ((4 · 1) + 0) = (4 + 0)
209	206	addridi 11497	. . . . . . . . 9 ⊢ (4 + 0) = 4
210	30	dec0h 12841	. . . . . . . . 9 ⊢ 4 = ;04
211	208, 209, 210	3eqtri 2788	. . . . . . . 8 ⊢ ((4 · 1) + 0) = ;04
212	29, 30, 9, 39, 198, 200, 9, 30, 39, 205, 211	decmac 12871	. . . . . . 7 ⊢ ((;34 · 1) + (3 + 7)) = ;44
213	3	dec0h 12841	. . . . . . . 8 ⊢ 2 = ;02
214		3t2e6 12508	. . . . . . . . . 10 ⊢ (3 · 2) = 6
215	214, 143	oveq12i 7432	. . . . . . . . 9 ⊢ ((3 · 2) + (0 + 1)) = (6 + 1)
216		6p1e7 12490	. . . . . . . . 9 ⊢ (6 + 1) = 7
217	215, 216	eqtri 2784	. . . . . . . 8 ⊢ ((3 · 2) + (0 + 1)) = 7
218		4t2e8 12511	. . . . . . . . . 10 ⊢ (4 · 2) = 8
219	218	oveq1i 7430	. . . . . . . . 9 ⊢ ((4 · 2) + 2) = (8 + 2)
220		8p2e10 12899	. . . . . . . . 9 ⊢ (8 + 2) = ;10
221	219, 220	eqtri 2784	. . . . . . . 8 ⊢ ((4 · 2) + 2) = ;10
222	29, 30, 39, 3, 198, 213, 3, 39, 9, 217, 221	decmac 12871	. . . . . . 7 ⊢ ((;34 · 2) + 2) = ;70
223	9, 3, 29, 3, 111, 197, 31, 39, 35, 212, 222	decma2c 12872	. . . . . 6 ⊢ ((;34 · ;12) + (;15 + ;17)) = ;;440
224		5t3e15 12920	. . . . . . . . 9 ⊢ (5 · 3) = ;15
225	92, 187, 224	mulcomli 11318	. . . . . . . 8 ⊢ (3 · 5) = ;15
226		5p2e7 12498	. . . . . . . 8 ⊢ (5 + 2) = 7
227	9, 11, 3, 225, 226	decaddi 12879	. . . . . . 7 ⊢ ((3 · 5) + 2) = ;17
228		5t4e20 12921	. . . . . . . . 9 ⊢ (5 · 4) = ;20
229	92, 206, 228	mulcomli 11318	. . . . . . . 8 ⊢ (4 · 5) = ;20
230	61	addlidi 11498	. . . . . . . 8 ⊢ (0 + 6) = 6
231	3, 39, 2, 229, 230	decaddi 12879	. . . . . . 7 ⊢ ((4 · 5) + 6) = ;26
232	29, 30, 2, 198, 11, 2, 3, 227, 231	decrmac 12877	. . . . . 6 ⊢ ((;34 · 5) + 6) = ;;176
233	10, 11, 46, 2, 109, 192, 31, 2, 36, 223, 232	decma2c 12872	. . . . 5 ⊢ ((;34 · ;;125) + (;;125 + ;31)) = ;;;4406
234		9cn 12443	. . . . . . . 8 ⊢ 9 ∈ ℂ
235		9t3e27 12942	. . . . . . . 8 ⊢ (9 · 3) = ;27
236	234, 187, 235	mulcomli 11318	. . . . . . 7 ⊢ (3 · 9) = ;27
237		7p4e11 12895	. . . . . . 7 ⊢ (7 + 4) = ;11
238	3, 35, 30, 236, 73, 9, 237	decaddci 12880	. . . . . 6 ⊢ ((3 · 9) + 4) = ;31
239		9t4e36 12943	. . . . . . . 8 ⊢ (9 · 4) = ;36
240	234, 206, 239	mulcomli 11318	. . . . . . 7 ⊢ (4 · 9) = ;36
241	149, 61, 170	addcomli 11502	. . . . . . 7 ⊢ (6 + 8) = ;14
242	29, 2, 13, 240, 204, 30, 241	decaddci 12880	. . . . . 6 ⊢ ((4 · 9) + 8) = ;44
243	29, 30, 13, 198, 5, 30, 30, 238, 242	decrmac 12877	. . . . 5 ⊢ ((;34 · 9) + 8) = ;;314
244	12, 5, 12, 13, 17, 22, 31, 30, 185, 233, 243	decma2c 12872	. . . 4 ⊢ ((;34 · 𝑁) + (𝑁 − 1)) = ;;;;44064
245		eqid 2761	. . . . 5 ⊢ ;;136 = ;;136
246	9, 5	deccl 12829	. . . . . 6 ⊢ ;19 ∈ ℕ0
247	246, 30	deccl 12829	. . . . 5 ⊢ ;;194 ∈ ℕ0
248		eqid 2761	. . . . . 6 ⊢ ;13 = ;13
249		eqid 2761	. . . . . 6 ⊢ ;;194 = ;;194
250	5, 35	deccl 12829	. . . . . 6 ⊢ ;97 ∈ ℕ0
251	9, 9	deccl 12829	. . . . . . 7 ⊢ ;11 ∈ ℕ0
252		eqid 2761	. . . . . . 7 ⊢ ;;324 = ;;324
253		eqid 2761	. . . . . . . 8 ⊢ ;19 = ;19
254		eqid 2761	. . . . . . . 8 ⊢ ;97 = ;97
255	234, 16, 118	addcomli 11502	. . . . . . . . 9 ⊢ (1 + 9) = ;10
256	9, 39, 143, 255	decsuc 12850	. . . . . . . 8 ⊢ ((1 + 9) + 1) = ;11
257		9p7e16 12911	. . . . . . . 8 ⊢ (9 + 7) = ;16
258	9, 5, 5, 35, 253, 254, 256, 2, 257	decaddc 12874	. . . . . . 7 ⊢ (;19 + ;97) = ;;116
259		eqid 2761	. . . . . . . 8 ⊢ ;32 = ;32
260		eqid 2761	. . . . . . . . 9 ⊢ ;11 = ;11
261	9, 9, 112, 260	decsuc 12850	. . . . . . . 8 ⊢ (;11 + 1) = ;12
262	89	oveq1i 7430	. . . . . . . . 9 ⊢ ((2 · 1) + 2) = (2 + 2)
263	262, 113, 210	3eqtri 2788	. . . . . . . 8 ⊢ ((2 · 1) + 2) = ;04
264	29, 3, 9, 3, 259, 261, 9, 30, 39, 205, 263	decmac 12871	. . . . . . 7 ⊢ ((;32 · 1) + (;11 + 1)) = ;44
265	207	oveq1i 7430	. . . . . . . 8 ⊢ ((4 · 1) + 6) = (4 + 6)
266		6p4e10 12891	. . . . . . . . 9 ⊢ (6 + 4) = ;10
267	61, 206, 266	addcomli 11502	. . . . . . . 8 ⊢ (4 + 6) = ;10
268	265, 267	eqtri 2784	. . . . . . 7 ⊢ ((4 · 1) + 6) = ;10
269	33, 30, 251, 2, 252, 258, 9, 39, 9, 264, 268	decmac 12871	. . . . . 6 ⊢ ((;;324 · 1) + (;19 + ;97)) = ;;440
270	143, 136	eqtri 2784	. . . . . . . 8 ⊢ (0 + 1) = ;01
271		3t3e9 12510	. . . . . . . . . 10 ⊢ (3 · 3) = 9
272	271, 137	oveq12i 7432	. . . . . . . . 9 ⊢ ((3 · 3) + (0 + 0)) = (9 + 0)
273	234	addridi 11497	. . . . . . . . 9 ⊢ (9 + 0) = 9
274	272, 273	eqtri 2784	. . . . . . . 8 ⊢ ((3 · 3) + (0 + 0)) = 9
275	99	oveq1i 7430	. . . . . . . . 9 ⊢ ((2 · 3) + 1) = (6 + 1)
276	35	dec0h 12841	. . . . . . . . 9 ⊢ 7 = ;07
277	275, 216, 276	3eqtri 2788	. . . . . . . 8 ⊢ ((2 · 3) + 1) = ;07
278	29, 3, 39, 9, 259, 270, 29, 35, 39, 274, 277	decmac 12871	. . . . . . 7 ⊢ ((;32 · 3) + (0 + 1)) = ;97
279		4t3e12 12917	. . . . . . . 8 ⊢ (4 · 3) = ;12
280		4p2e6 12495	. . . . . . . . 9 ⊢ (4 + 2) = 6
281	206, 57, 280	addcomli 11502	. . . . . . . 8 ⊢ (2 + 4) = 6
282	9, 3, 30, 279, 281	decaddi 12879	. . . . . . 7 ⊢ ((4 · 3) + 4) = ;16
283	33, 30, 39, 30, 252, 210, 29, 2, 9, 278, 282	decmac 12871	. . . . . 6 ⊢ ((;;324 · 3) + 4) = ;;976
284	9, 29, 246, 30, 248, 249, 34, 2, 250, 269, 283	decma2c 12872	. . . . 5 ⊢ ((;;324 · ;13) + ;;194) = ;;;4406
285		6t3e18 12924	. . . . . . . . 9 ⊢ (6 · 3) = ;18
286	61, 187, 285	mulcomli 11318	. . . . . . . 8 ⊢ (3 · 6) = ;18
287	9, 13, 18, 286	decsuc 12850	. . . . . . 7 ⊢ ((3 · 6) + 1) = ;19
288	9, 3, 3, 63, 113	decaddi 12879	. . . . . . 7 ⊢ ((2 · 6) + 2) = ;14
289	29, 3, 3, 259, 2, 30, 9, 287, 288	decrmac 12877	. . . . . 6 ⊢ ((;32 · 6) + 2) = ;;194
290		6t4e24 12925	. . . . . . 7 ⊢ (6 · 4) = ;24
291	61, 206, 290	mulcomli 11318	. . . . . 6 ⊢ (4 · 6) = ;24
292	2, 33, 30, 252, 30, 3, 289, 291	decmul1c 12884	. . . . 5 ⊢ (;;324 · 6) = ;;;1944
293	34, 37, 2, 245, 30, 247, 284, 292	decmul2c 12885	. . . 4 ⊢ (;;324 · ;;136) = ;;;;44064
294	244, 293	eqtr4i 2787	. . 3 ⊢ ((;34 · 𝑁) + (𝑁 − 1)) = (;;324 · ;;136)
295	26, 1, 28, 32, 34, 23, 36, 38, 176, 177, 184, 294	modxai 17246	. 2 ⊢ ((2↑;;629) mod 𝑁) = ((𝑁 − 1) mod 𝑁)
296		eqid 2761	. . . 4 ⊢ ;;629 = ;;629
297		eqid 2761	. . . . 5 ⊢ ;62 = ;62
298	137	oveq2i 7431	. . . . . 6 ⊢ ((2 · 6) + (0 + 0)) = ((2 · 6) + 0)
299	63	oveq1i 7430	. . . . . 6 ⊢ ((2 · 6) + 0) = (;12 + 0)
300	10	nn0cni 12618	. . . . . . 7 ⊢ ;12 ∈ ℂ
301	300	addridi 11497	. . . . . 6 ⊢ (;12 + 0) = ;12
302	298, 299, 301	3eqtri 2788	. . . . 5 ⊢ ((2 · 6) + (0 + 0)) = ;12
303	11	dec0h 12841	. . . . . 6 ⊢ 5 = ;05
304	81, 55, 303	3eqtri 2788	. . . . 5 ⊢ ((2 · 2) + 1) = ;05
305	2, 3, 39, 9, 297, 136, 3, 11, 39, 302, 304	decma2c 12872	. . . 4 ⊢ ((2 · ;62) + 1) = ;;125
306		9t2e18 12941	. . . . 5 ⊢ (9 · 2) = ;18
307	234, 57, 306	mulcomli 11318	. . . 4 ⊢ (2 · 9) = ;18
308	3, 4, 5, 296, 13, 9, 305, 307	decmul2c 12885	. . 3 ⊢ (2 · ;;629) = ;;;1258
309	308, 22	eqtr4i 2787	. 2 ⊢ (2 · ;;629) = (𝑁 − 1)
310		npcan 11566	. . 3 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
311	67, 16, 310	mp2an 705	. 2 ⊢ ((𝑁 − 1) + 1) = 𝑁
312	68	oveq1i 7430	. . 3 ⊢ ((0 · 𝑁) + 1) = (0 + 1)
313	143, 312, 159	3eqtr4i 2794	. 2 ⊢ ((0 · 𝑁) + 1) = (1 · 1)
314	1, 6, 7, 8, 9, 23, 295, 309, 311, 313	mod2xnegi 17249	1 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2145  (class class class)co 7420  ℂcc 11198  0cc0 11200  1c1 11201   + caddc 11203   · cmul 11205   − cmin 11541  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:  1259prm  17314