1259lem4 - Intuitionistic Logic Explorer

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Theorem 1259lem4 13265

Description: Lemma for 1259prm 13267. 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 9470	. 2 ⊢ 2 ∈ ℕ
2		6nn0 9588	. . . 4 ⊢ 6 ∈ ℕ₀
3		2nn0 9584	. . . 4 ⊢ 2 ∈ ℕ₀
4	2, 3	deccl 9795	. . 3 ⊢ ;62 ∈ ℕ₀
5		9nn0 9591	. . 3 ⊢ 9 ∈ ℕ₀
6	4, 5	deccl 9795	. 2 ⊢ ;;629 ∈ ℕ₀
7		0z 9659	. 2 ⊢ 0 ∈ ℤ
8		1nn 9317	. 2 ⊢ 1 ∈ ℕ
9		1nn0 9583	. 2 ⊢ 1 ∈ ℕ₀
10		12nn0 9797	. . . . . . 7 ⊢ ;12 ∈ ℕ₀
11		5nn0 9587	. . . . . . 7 ⊢ 5 ∈ ℕ₀
12	10, 11	deccl 9795	. . . . . 6 ⊢ ;;125 ∈ ℕ₀
13		8nn0 9590	. . . . . 6 ⊢ 8 ∈ ℕ₀
14	12, 13	deccl 9795	. . . . 5 ⊢ ;;;1258 ∈ ℕ₀
15	14	nn0cni 9579	. . . 4 ⊢ ;;;1258 ∈ ℂ
16		ax-1cn 8272	. . . 4 ⊢ 1 ∈ ℂ
17		1259prm.1	. . . . 5 ⊢ 𝑁 = ;;;1259
18		8p1e9 9447	. . . . . 6 ⊢ (8 + 1) = 9
19		eqid 2238	. . . . . 6 ⊢ ;;;1258 = ;;;1258
20	12, 13, 18, 19	decsuc 9816	. . . . 5 ⊢ (;;;1258 + 1) = ;;;1259
21	17, 20	eqtr4i 2262	. . . 4 ⊢ 𝑁 = (;;;1258 + 1)
22	15, 16, 21	mvrraddi 8544	. . 3 ⊢ (𝑁 − 1) = ;;;1258
23	22, 14	eqeltri 2311	. 2 ⊢ (𝑁 − 1) ∈ ℕ₀
24		9nn 9477	. . . . 5 ⊢ 9 ∈ ℕ
25	12, 24	decnncl 9804	. . . 4 ⊢ ;;;1259 ∈ ℕ
26	17, 25	eqeltri 2311	. . 3 ⊢ 𝑁 ∈ ℕ
27	2, 9	deccl 9795	. . . 4 ⊢ ;61 ∈ ℕ₀
28	27, 3	deccl 9795	. . 3 ⊢ ;;612 ∈ ℕ₀
29		3nn0 9585	. . . . 5 ⊢ 3 ∈ ℕ₀
30		4nn0 9586	. . . . 5 ⊢ 4 ∈ ℕ₀
31	29, 30	deccl 9795	. . . 4 ⊢ ;34 ∈ ℕ₀
32	31	nn0zi 9670	. . 3 ⊢ ;34 ∈ ℤ
33	29, 3	deccl 9795	. . . 4 ⊢ ;32 ∈ ℕ₀
34	33, 30	deccl 9795	. . 3 ⊢ ;;324 ∈ ℕ₀
35		7nn0 9589	. . . 4 ⊢ 7 ∈ ℕ₀
36	9, 35	deccl 9795	. . 3 ⊢ ;17 ∈ ℕ₀
37	9, 29	deccl 9795	. . . 4 ⊢ ;13 ∈ ℕ₀
38	37, 2	deccl 9795	. . 3 ⊢ ;;136 ∈ ℕ₀
39		0nn0 9582	. . . . . 6 ⊢ 0 ∈ ℕ₀
40	29, 39	deccl 9795	. . . . 5 ⊢ ;30 ∈ ℕ₀
41	40, 2	deccl 9795	. . . 4 ⊢ ;;306 ∈ ℕ₀
42		8nn 9476	. . . . 5 ⊢ 8 ∈ ℕ
43	9, 42	decnncl 9804	. . . 4 ⊢ ;18 ∈ ℕ
44	10, 30	deccl 9795	. . . . 5 ⊢ ;;124 ∈ ℕ₀
45	44, 9	deccl 9795	. . . 4 ⊢ ;;;1241 ∈ ℕ₀
46	9, 11	deccl 9795	. . . . . 6 ⊢ ;15 ∈ ℕ₀
47	46, 29	deccl 9795	. . . . 5 ⊢ ;;153 ∈ ℕ₀
48		1z 9674	. . . . 5 ⊢ 1 ∈ ℤ
49	11, 39	deccl 9795	. . . . 5 ⊢ ;50 ∈ ℕ₀
50	46, 3	deccl 9795	. . . . . 6 ⊢ ;;152 ∈ ℕ₀
51		25nn0 9799	. . . . . 6 ⊢ ;25 ∈ ℕ₀
52	35, 2	deccl 9795	. . . . . . 7 ⊢ ;76 ∈ ℕ₀
53	17	1259lem3 13264	. . . . . . 7 ⊢ ((2↑;76) mod 𝑁) = (5 mod 𝑁)
54		eqid 2238	. . . . . . . 8 ⊢ ;76 = ;76
55		4p1e5 9443	. . . . . . . . 9 ⊢ (4 + 1) = 5
56		7cn 9390	. . . . . . . . . 10 ⊢ 7 ∈ ℂ
57		2cn 9377	. . . . . . . . . 10 ⊢ 2 ∈ ℂ
58		7t2e14 9894	. . . . . . . . . 10 ⊢ (7 · 2) = ;14
59	56, 57, 58	mulcomli 8333	. . . . . . . . 9 ⊢ (2 · 7) = ;14
60	9, 30, 55, 59	decsuc 9816	. . . . . . . 8 ⊢ ((2 · 7) + 1) = ;15
61		6cn 9388	. . . . . . . . 9 ⊢ 6 ∈ ℂ
62		6t2e12 9889	. . . . . . . . 9 ⊢ (6 · 2) = ;12
63	61, 57, 62	mulcomli 8333	. . . . . . . 8 ⊢ (2 · 6) = ;12
64	3, 35, 2, 54, 3, 9, 60, 63	decmul2c 9851	. . . . . . 7 ⊢ (2 · ;76) = ;;152
65	51	nn0cni 9579	. . . . . . . . 9 ⊢ ;25 ∈ ℂ
66	65	addlidi 8470	. . . . . . . 8 ⊢ (0 + ;25) = ;25
67	26	nncni 9316	. . . . . . . . . 10 ⊢ 𝑁 ∈ ℂ
68	67	mul02i 8718	. . . . . . . . 9 ⊢ (0 · 𝑁) = 0
69	68	oveq1i 6095	. . . . . . . 8 ⊢ ((0 · 𝑁) + ;25) = (0 + ;25)
70		5t5e25 9888	. . . . . . . 8 ⊢ (5 · 5) = ;25
71	66, 69, 70	3eqtr4i 2269	. . . . . . 7 ⊢ ((0 · 𝑁) + ;25) = (5 · 5)
72	26, 1, 52, 7, 11, 51, 53, 64, 71	mod2xi 13216	. . . . . 6 ⊢ ((2↑;;152) mod 𝑁) = (;25 mod 𝑁)
73		2p1e3 9440	. . . . . . 7 ⊢ (2 + 1) = 3
74		eqid 2238	. . . . . . 7 ⊢ ;;152 = ;;152
75	46, 3, 73, 74	decsuc 9816	. . . . . 6 ⊢ (;;152 + 1) = ;;153
76	49	nn0cni 9579	. . . . . . . 8 ⊢ ;50 ∈ ℂ
77	76	addlidi 8470	. . . . . . 7 ⊢ (0 + ;50) = ;50
78	68	oveq1i 6095	. . . . . . 7 ⊢ ((0 · 𝑁) + ;50) = (0 + ;50)
79		eqid 2238	. . . . . . . 8 ⊢ ;25 = ;25
80		2t2e4 9461	. . . . . . . . . 10 ⊢ (2 · 2) = 4
81	80	oveq1i 6095	. . . . . . . . 9 ⊢ ((2 · 2) + 1) = (4 + 1)
82	81, 55	eqtri 2259	. . . . . . . 8 ⊢ ((2 · 2) + 1) = 5
83		5t2e10 9885	. . . . . . . 8 ⊢ (5 · 2) = ;10
84	3, 3, 11, 79, 39, 9, 82, 83	decmul1c 9850	. . . . . . 7 ⊢ (;25 · 2) = ;50
85	77, 78, 84	3eqtr4i 2269	. . . . . 6 ⊢ ((0 · 𝑁) + ;50) = (;25 · 2)
86	26, 1, 50, 7, 51, 49, 72, 75, 85	modxp1i 13217	. . . . 5 ⊢ ((2↑;;153) mod 𝑁) = (;50 mod 𝑁)
87		eqid 2238	. . . . . 6 ⊢ ;;153 = ;;153
88		eqid 2238	. . . . . . . . 9 ⊢ ;15 = ;15
89	57	mulridi 8328	. . . . . . . . . . 11 ⊢ (2 · 1) = 2
90	89	oveq1i 6095	. . . . . . . . . 10 ⊢ ((2 · 1) + 1) = (2 + 1)
91	90, 73	eqtri 2259	. . . . . . . . 9 ⊢ ((2 · 1) + 1) = 3
92		5cn 9386	. . . . . . . . . 10 ⊢ 5 ∈ ℂ
93	92, 57, 83	mulcomli 8333	. . . . . . . . 9 ⊢ (2 · 5) = ;10
94	3, 9, 11, 88, 39, 9, 91, 93	decmul2c 9851	. . . . . . . 8 ⊢ (2 · ;15) = ;30
95	94	oveq1i 6095	. . . . . . 7 ⊢ ((2 · ;15) + 0) = (;30 + 0)
96	40	nn0cni 9579	. . . . . . . 8 ⊢ ;30 ∈ ℂ
97	96	addridi 8469	. . . . . . 7 ⊢ (;30 + 0) = ;30
98	95, 97	eqtri 2259	. . . . . 6 ⊢ ((2 · ;15) + 0) = ;30
99		2t3e6 9464	. . . . . . 7 ⊢ (2 · 3) = 6
100	2	dec0h 9807	. . . . . . 7 ⊢ 6 = ;06
101	99, 100	eqtri 2259	. . . . . 6 ⊢ (2 · 3) = ;06
102	3, 46, 29, 87, 2, 39, 98, 101	decmul2c 9851	. . . . 5 ⊢ (2 · ;;153) = ;;306
103	67	mullidi 8329	. . . . . . . 8 ⊢ (1 · 𝑁) = 𝑁
104	103, 17	eqtri 2259	. . . . . . 7 ⊢ (1 · 𝑁) = ;;;1259
105		eqid 2238	. . . . . . 7 ⊢ ;;;1241 = ;;;1241
106	3, 30	deccl 9795	. . . . . . . 8 ⊢ ;24 ∈ ℕ₀
107		eqid 2238	. . . . . . . . 9 ⊢ ;24 = ;24
108	3, 30, 55, 107	decsuc 9816	. . . . . . . 8 ⊢ (;24 + 1) = ;25
109		eqid 2238	. . . . . . . . 9 ⊢ ;;125 = ;;125
110		eqid 2238	. . . . . . . . 9 ⊢ ;;124 = ;;124
111		eqid 2238	. . . . . . . . . 10 ⊢ ;12 = ;12
112		1p1e2 9423	. . . . . . . . . 10 ⊢ (1 + 1) = 2
113		2p2e4 9433	. . . . . . . . . 10 ⊢ (2 + 2) = 4
114	9, 3, 9, 3, 111, 111, 112, 113	decadd 9839	. . . . . . . . 9 ⊢ (;12 + ;12) = ;24
115		5p4e9 9455	. . . . . . . . 9 ⊢ (5 + 4) = 9
116	10, 11, 10, 30, 109, 110, 114, 115	decadd 9839	. . . . . . . 8 ⊢ (;;125 + ;;124) = ;;249
117	106, 108, 116	decsucc 9826	. . . . . . 7 ⊢ ((;;125 + ;;124) + 1) = ;;250
118		9p1e10 9783	. . . . . . 7 ⊢ (9 + 1) = ;10
119	12, 5, 44, 9, 104, 105, 117, 118	decaddc2 9841	. . . . . 6 ⊢ ((1 · 𝑁) + ;;;1241) = ;;;2500
120		eqid 2238	. . . . . . 7 ⊢ ;50 = ;50
121	92	mul02i 8718	. . . . . . . . . 10 ⊢ (0 · 5) = 0
122	11, 11, 39, 120, 39, 70, 121	decmul1 9849	. . . . . . . . 9 ⊢ (;50 · 5) = ;;250
123	122	oveq1i 6095	. . . . . . . 8 ⊢ ((;50 · 5) + 0) = (;;250 + 0)
124	51, 39	deccl 9795	. . . . . . . . . 10 ⊢ ;;250 ∈ ℕ₀
125	124	nn0cni 9579	. . . . . . . . 9 ⊢ ;;250 ∈ ℂ
126	125	addridi 8469	. . . . . . . 8 ⊢ (;;250 + 0) = ;;250
127	123, 126	eqtri 2259	. . . . . . 7 ⊢ ((;50 · 5) + 0) = ;;250
128	76	mul01i 8719	. . . . . . . 8 ⊢ (;50 · 0) = 0
129	39	dec0h 9807	. . . . . . . 8 ⊢ 0 = ;00
130	128, 129	eqtri 2259	. . . . . . 7 ⊢ (;50 · 0) = ;00
131	49, 11, 39, 120, 39, 39, 127, 130	decmul2c 9851	. . . . . 6 ⊢ (;50 · ;50) = ;;;2500
132	119, 131	eqtr4i 2262	. . . . 5 ⊢ ((1 · 𝑁) + ;;;1241) = (;50 · ;50)
133	26, 1, 47, 48, 49, 45, 86, 102, 132	mod2xi 13216	. . . 4 ⊢ ((2↑;;306) mod 𝑁) = (;;;1241 mod 𝑁)
134		eqid 2238	. . . . 5 ⊢ ;;306 = ;;306
135		eqid 2238	. . . . . 6 ⊢ ;30 = ;30
136	9	dec0h 9807	. . . . . 6 ⊢ 1 = ;01
137		00id 8468	. . . . . . . 8 ⊢ (0 + 0) = 0
138	99, 137	oveq12i 6097	. . . . . . 7 ⊢ ((2 · 3) + (0 + 0)) = (6 + 0)
139	61	addridi 8469	. . . . . . 7 ⊢ (6 + 0) = 6
140	138, 139	eqtri 2259	. . . . . 6 ⊢ ((2 · 3) + (0 + 0)) = 6
141	57	mul01i 8719	. . . . . . . 8 ⊢ (2 · 0) = 0
142	141	oveq1i 6095	. . . . . . 7 ⊢ ((2 · 0) + 1) = (0 + 1)
143		0p1e1 9420	. . . . . . 7 ⊢ (0 + 1) = 1
144	142, 143, 136	3eqtri 2263	. . . . . 6 ⊢ ((2 · 0) + 1) = ;01
145	29, 39, 39, 9, 135, 136, 3, 9, 39, 140, 144	decma2c 9838	. . . . 5 ⊢ ((2 · ;30) + 1) = ;61
146	3, 40, 2, 134, 3, 9, 145, 63	decmul2c 9851	. . . 4 ⊢ (2 · ;;306) = ;;612
147		eqid 2238	. . . . . 6 ⊢ ;18 = ;18
148	10, 30, 55, 110	decsuc 9816	. . . . . 6 ⊢ (;;124 + 1) = ;;125
149		8cn 9392	. . . . . . 7 ⊢ 8 ∈ ℂ
150	149, 16, 18	addcomli 8472	. . . . . 6 ⊢ (1 + 8) = 9
151	44, 9, 9, 13, 105, 147, 148, 150	decadd 9839	. . . . 5 ⊢ (;;;1241 + ;18) = ;;;1259
152	151, 17	eqtr4i 2262	. . . 4 ⊢ (;;;1241 + ;18) = 𝑁
153	34	nn0cni 9579	. . . . . 6 ⊢ ;;324 ∈ ℂ
154	153	addlidi 8470	. . . . 5 ⊢ (0 + ;;324) = ;;324
155	68	oveq1i 6095	. . . . 5 ⊢ ((0 · 𝑁) + ;;324) = (0 + ;;324)
156	9, 13	deccl 9795	. . . . . 6 ⊢ ;18 ∈ ℕ₀
157	9, 30	deccl 9795	. . . . . 6 ⊢ ;14 ∈ ℕ₀
158		eqid 2238	. . . . . . 7 ⊢ ;14 = ;14
159	16	mulridi 8328	. . . . . . . . 9 ⊢ (1 · 1) = 1
160	159, 112	oveq12i 6097	. . . . . . . 8 ⊢ ((1 · 1) + (1 + 1)) = (1 + 2)
161		1p2e3 9441	. . . . . . . 8 ⊢ (1 + 2) = 3
162	160, 161	eqtri 2259	. . . . . . 7 ⊢ ((1 · 1) + (1 + 1)) = 3
163	149	mulridi 8328	. . . . . . . . 9 ⊢ (8 · 1) = 8
164	163	oveq1i 6095	. . . . . . . 8 ⊢ ((8 · 1) + 4) = (8 + 4)
165		8p4e12 9867	. . . . . . . 8 ⊢ (8 + 4) = ;12
166	164, 165	eqtri 2259	. . . . . . 7 ⊢ ((8 · 1) + 4) = ;12
167	9, 13, 9, 30, 147, 158, 9, 3, 9, 162, 166	decmac 9837	. . . . . 6 ⊢ ((;18 · 1) + ;14) = ;32
168	149	mullidi 8329	. . . . . . . . 9 ⊢ (1 · 8) = 8
169	168	oveq1i 6095	. . . . . . . 8 ⊢ ((1 · 8) + 6) = (8 + 6)
170		8p6e14 9869	. . . . . . . 8 ⊢ (8 + 6) = ;14
171	169, 170	eqtri 2259	. . . . . . 7 ⊢ ((1 · 8) + 6) = ;14
172		8t8e64 9906	. . . . . . 7 ⊢ (8 · 8) = ;64
173	13, 9, 13, 147, 30, 2, 171, 172	decmul1c 9850	. . . . . 6 ⊢ (;18 · 8) = ;;144
174	156, 9, 13, 147, 30, 157, 167, 173	decmul2c 9851	. . . . 5 ⊢ (;18 · ;18) = ;;324
175	154, 155, 174	3eqtr4i 2269	. . . 4 ⊢ ((0 · 𝑁) + ;;324) = (;18 · ;18)
176	1, 41, 7, 43, 34, 45, 133, 146, 152, 175	mod2xnegi 13218	. . 3 ⊢ ((2↑;;612) mod 𝑁) = (;;324 mod 𝑁)
177	17	1259lem1 13262	. . 3 ⊢ ((2↑;17) mod 𝑁) = (;;136 mod 𝑁)
178		eqid 2238	. . . 4 ⊢ ;;612 = ;;612
179		eqid 2238	. . . 4 ⊢ ;17 = ;17
180		eqid 2238	. . . . 5 ⊢ ;61 = ;61
181	2, 9, 112, 180	decsuc 9816	. . . 4 ⊢ (;61 + 1) = ;62
182		7p2e9 9458	. . . . 5 ⊢ (7 + 2) = 9
183	56, 57, 182	addcomli 8472	. . . 4 ⊢ (2 + 7) = 9
184	27, 3, 9, 35, 178, 179, 181, 183	decadd 9839	. . 3 ⊢ (;;612 + ;17) = ;;629
185	29, 9	deccl 9795	. . . . 5 ⊢ ;31 ∈ ℕ₀
186		eqid 2238	. . . . . . 7 ⊢ ;31 = ;31
187		3cn 9381	. . . . . . . . 9 ⊢ 3 ∈ ℂ
188		3p2e5 9448	. . . . . . . . 9 ⊢ (3 + 2) = 5
189	187, 57, 188	addcomli 8472	. . . . . . . 8 ⊢ (2 + 3) = 5
190	9, 3, 29, 111, 189	decaddi 9845	. . . . . . 7 ⊢ (;12 + 3) = ;15
191		5p1e6 9444	. . . . . . 7 ⊢ (5 + 1) = 6
192	10, 11, 29, 9, 109, 186, 190, 191	decadd 9839	. . . . . 6 ⊢ (;;125 + ;31) = ;;156
193	112	oveq1i 6095	. . . . . . . . 9 ⊢ ((1 + 1) + 1) = (2 + 1)
194	193, 73	eqtri 2259	. . . . . . . 8 ⊢ ((1 + 1) + 1) = 3
195		7p5e12 9862	. . . . . . . . 9 ⊢ (7 + 5) = ;12
196	56, 92, 195	addcomli 8472	. . . . . . . 8 ⊢ (5 + 7) = ;12
197	9, 11, 9, 35, 88, 179, 194, 3, 196	decaddc 9840	. . . . . . 7 ⊢ (;15 + ;17) = ;32
198		eqid 2238	. . . . . . . 8 ⊢ ;34 = ;34
199		7p3e10 9860	. . . . . . . . 9 ⊢ (7 + 3) = ;10
200	56, 187, 199	addcomli 8472	. . . . . . . 8 ⊢ (3 + 7) = ;10
201	187	mulridi 8328	. . . . . . . . . 10 ⊢ (3 · 1) = 3
202	16	addridi 8469	. . . . . . . . . 10 ⊢ (1 + 0) = 1
203	201, 202	oveq12i 6097	. . . . . . . . 9 ⊢ ((3 · 1) + (1 + 0)) = (3 + 1)
204		3p1e4 9442	. . . . . . . . 9 ⊢ (3 + 1) = 4
205	203, 204	eqtri 2259	. . . . . . . 8 ⊢ ((3 · 1) + (1 + 0)) = 4
206		4cn 9384	. . . . . . . . . . 11 ⊢ 4 ∈ ℂ
207	206	mulridi 8328	. . . . . . . . . 10 ⊢ (4 · 1) = 4
208	207	oveq1i 6095	. . . . . . . . 9 ⊢ ((4 · 1) + 0) = (4 + 0)
209	206	addridi 8469	. . . . . . . . 9 ⊢ (4 + 0) = 4
210	30	dec0h 9807	. . . . . . . . 9 ⊢ 4 = ;04
211	208, 209, 210	3eqtri 2263	. . . . . . . 8 ⊢ ((4 · 1) + 0) = ;04
212	29, 30, 9, 39, 198, 200, 9, 30, 39, 205, 211	decmac 9837	. . . . . . 7 ⊢ ((;34 · 1) + (3 + 7)) = ;44
213	3	dec0h 9807	. . . . . . . 8 ⊢ 2 = ;02
214		3t2e6 9463	. . . . . . . . . 10 ⊢ (3 · 2) = 6
215	214, 143	oveq12i 6097	. . . . . . . . 9 ⊢ ((3 · 2) + (0 + 1)) = (6 + 1)
216		6p1e7 9445	. . . . . . . . 9 ⊢ (6 + 1) = 7
217	215, 216	eqtri 2259	. . . . . . . 8 ⊢ ((3 · 2) + (0 + 1)) = 7
218		4t2e8 9466	. . . . . . . . . 10 ⊢ (4 · 2) = 8
219	218	oveq1i 6095	. . . . . . . . 9 ⊢ ((4 · 2) + 2) = (8 + 2)
220		8p2e10 9865	. . . . . . . . 9 ⊢ (8 + 2) = ;10
221	219, 220	eqtri 2259	. . . . . . . 8 ⊢ ((4 · 2) + 2) = ;10
222	29, 30, 39, 3, 198, 213, 3, 39, 9, 217, 221	decmac 9837	. . . . . . 7 ⊢ ((;34 · 2) + 2) = ;70
223	9, 3, 29, 3, 111, 197, 31, 39, 35, 212, 222	decma2c 9838	. . . . . 6 ⊢ ((;34 · ;12) + (;15 + ;17)) = ;;440
224		5t3e15 9886	. . . . . . . . 9 ⊢ (5 · 3) = ;15
225	92, 187, 224	mulcomli 8333	. . . . . . . 8 ⊢ (3 · 5) = ;15
226		5p2e7 9453	. . . . . . . 8 ⊢ (5 + 2) = 7
227	9, 11, 3, 225, 226	decaddi 9845	. . . . . . 7 ⊢ ((3 · 5) + 2) = ;17
228		5t4e20 9887	. . . . . . . . 9 ⊢ (5 · 4) = ;20
229	92, 206, 228	mulcomli 8333	. . . . . . . 8 ⊢ (4 · 5) = ;20
230	61	addlidi 8470	. . . . . . . 8 ⊢ (0 + 6) = 6
231	3, 39, 2, 229, 230	decaddi 9845	. . . . . . 7 ⊢ ((4 · 5) + 6) = ;26
232	29, 30, 2, 198, 11, 2, 3, 227, 231	decrmac 9843	. . . . . 6 ⊢ ((;34 · 5) + 6) = ;;176
233	10, 11, 46, 2, 109, 192, 31, 2, 36, 223, 232	decma2c 9838	. . . . 5 ⊢ ((;34 · ;;125) + (;;125 + ;31)) = ;;;4406
234		9cn 9394	. . . . . . . 8 ⊢ 9 ∈ ℂ
235		9t3e27 9908	. . . . . . . 8 ⊢ (9 · 3) = ;27
236	234, 187, 235	mulcomli 8333	. . . . . . 7 ⊢ (3 · 9) = ;27
237		7p4e11 9861	. . . . . . 7 ⊢ (7 + 4) = ;11
238	3, 35, 30, 236, 73, 9, 237	decaddci 9846	. . . . . 6 ⊢ ((3 · 9) + 4) = ;31
239		9t4e36 9909	. . . . . . . 8 ⊢ (9 · 4) = ;36
240	234, 206, 239	mulcomli 8333	. . . . . . 7 ⊢ (4 · 9) = ;36
241	149, 61, 170	addcomli 8472	. . . . . . 7 ⊢ (6 + 8) = ;14
242	29, 2, 13, 240, 204, 30, 241	decaddci 9846	. . . . . 6 ⊢ ((4 · 9) + 8) = ;44
243	29, 30, 13, 198, 5, 30, 30, 238, 242	decrmac 9843	. . . . 5 ⊢ ((;34 · 9) + 8) = ;;314
244	12, 5, 12, 13, 17, 22, 31, 30, 185, 233, 243	decma2c 9838	. . . 4 ⊢ ((;34 · 𝑁) + (𝑁 − 1)) = ;;;;44064
245		eqid 2238	. . . . 5 ⊢ ;;136 = ;;136
246	9, 5	deccl 9795	. . . . . 6 ⊢ ;19 ∈ ℕ₀
247	246, 30	deccl 9795	. . . . 5 ⊢ ;;194 ∈ ℕ₀
248		eqid 2238	. . . . . 6 ⊢ ;13 = ;13
249		eqid 2238	. . . . . 6 ⊢ ;;194 = ;;194
250	5, 35	deccl 9795	. . . . . 6 ⊢ ;97 ∈ ℕ₀
251	9, 9	deccl 9795	. . . . . . 7 ⊢ ;11 ∈ ℕ₀
252		eqid 2238	. . . . . . 7 ⊢ ;;324 = ;;324
253		eqid 2238	. . . . . . . 8 ⊢ ;19 = ;19
254		eqid 2238	. . . . . . . 8 ⊢ ;97 = ;97
255	234, 16, 118	addcomli 8472	. . . . . . . . 9 ⊢ (1 + 9) = ;10
256	9, 39, 143, 255	decsuc 9816	. . . . . . . 8 ⊢ ((1 + 9) + 1) = ;11
257		9p7e16 9877	. . . . . . . 8 ⊢ (9 + 7) = ;16
258	9, 5, 5, 35, 253, 254, 256, 2, 257	decaddc 9840	. . . . . . 7 ⊢ (;19 + ;97) = ;;116
259		eqid 2238	. . . . . . . 8 ⊢ ;32 = ;32
260		eqid 2238	. . . . . . . . 9 ⊢ ;11 = ;11
261	9, 9, 112, 260	decsuc 9816	. . . . . . . 8 ⊢ (;11 + 1) = ;12
262	89	oveq1i 6095	. . . . . . . . 9 ⊢ ((2 · 1) + 2) = (2 + 2)
263	262, 113, 210	3eqtri 2263	. . . . . . . 8 ⊢ ((2 · 1) + 2) = ;04
264	29, 3, 9, 3, 259, 261, 9, 30, 39, 205, 263	decmac 9837	. . . . . . 7 ⊢ ((;32 · 1) + (;11 + 1)) = ;44
265	207	oveq1i 6095	. . . . . . . 8 ⊢ ((4 · 1) + 6) = (4 + 6)
266		6p4e10 9857	. . . . . . . . 9 ⊢ (6 + 4) = ;10
267	61, 206, 266	addcomli 8472	. . . . . . . 8 ⊢ (4 + 6) = ;10
268	265, 267	eqtri 2259	. . . . . . 7 ⊢ ((4 · 1) + 6) = ;10
269	33, 30, 251, 2, 252, 258, 9, 39, 9, 264, 268	decmac 9837	. . . . . 6 ⊢ ((;;324 · 1) + (;19 + ;97)) = ;;440
270	143, 136	eqtri 2259	. . . . . . . 8 ⊢ (0 + 1) = ;01
271		3t3e9 9465	. . . . . . . . . 10 ⊢ (3 · 3) = 9
272	271, 137	oveq12i 6097	. . . . . . . . 9 ⊢ ((3 · 3) + (0 + 0)) = (9 + 0)
273	234	addridi 8469	. . . . . . . . 9 ⊢ (9 + 0) = 9
274	272, 273	eqtri 2259	. . . . . . . 8 ⊢ ((3 · 3) + (0 + 0)) = 9
275	99	oveq1i 6095	. . . . . . . . 9 ⊢ ((2 · 3) + 1) = (6 + 1)
276	35	dec0h 9807	. . . . . . . . 9 ⊢ 7 = ;07
277	275, 216, 276	3eqtri 2263	. . . . . . . 8 ⊢ ((2 · 3) + 1) = ;07
278	29, 3, 39, 9, 259, 270, 29, 35, 39, 274, 277	decmac 9837	. . . . . . 7 ⊢ ((;32 · 3) + (0 + 1)) = ;97
279		4t3e12 9883	. . . . . . . 8 ⊢ (4 · 3) = ;12
280		4p2e6 9450	. . . . . . . . 9 ⊢ (4 + 2) = 6
281	206, 57, 280	addcomli 8472	. . . . . . . 8 ⊢ (2 + 4) = 6
282	9, 3, 30, 279, 281	decaddi 9845	. . . . . . 7 ⊢ ((4 · 3) + 4) = ;16
283	33, 30, 39, 30, 252, 210, 29, 2, 9, 278, 282	decmac 9837	. . . . . 6 ⊢ ((;;324 · 3) + 4) = ;;976
284	9, 29, 246, 30, 248, 249, 34, 2, 250, 269, 283	decma2c 9838	. . . . 5 ⊢ ((;;324 · ;13) + ;;194) = ;;;4406
285		6t3e18 9890	. . . . . . . . 9 ⊢ (6 · 3) = ;18
286	61, 187, 285	mulcomli 8333	. . . . . . . 8 ⊢ (3 · 6) = ;18
287	9, 13, 18, 286	decsuc 9816	. . . . . . 7 ⊢ ((3 · 6) + 1) = ;19
288	9, 3, 3, 63, 113	decaddi 9845	. . . . . . 7 ⊢ ((2 · 6) + 2) = ;14
289	29, 3, 3, 259, 2, 30, 9, 287, 288	decrmac 9843	. . . . . 6 ⊢ ((;32 · 6) + 2) = ;;194
290		6t4e24 9891	. . . . . . 7 ⊢ (6 · 4) = ;24
291	61, 206, 290	mulcomli 8333	. . . . . 6 ⊢ (4 · 6) = ;24
292	2, 33, 30, 252, 30, 3, 289, 291	decmul1c 9850	. . . . 5 ⊢ (;;324 · 6) = ;;;1944
293	34, 37, 2, 245, 30, 247, 284, 292	decmul2c 9851	. . . 4 ⊢ (;;324 · ;;136) = ;;;;44064
294	244, 293	eqtr4i 2262	. . 3 ⊢ ((;34 · 𝑁) + (𝑁 − 1)) = (;;324 · ;;136)
295	26, 1, 28, 32, 34, 23, 36, 38, 176, 177, 184, 294	modxai 13215	. 2 ⊢ ((2↑;;629) mod 𝑁) = ((𝑁 − 1) mod 𝑁)
296		eqid 2238	. . . 4 ⊢ ;;629 = ;;629
297		eqid 2238	. . . . 5 ⊢ ;62 = ;62
298	137	oveq2i 6096	. . . . . 6 ⊢ ((2 · 6) + (0 + 0)) = ((2 · 6) + 0)
299	63	oveq1i 6095	. . . . . 6 ⊢ ((2 · 6) + 0) = (;12 + 0)
300	10	nn0cni 9579	. . . . . . 7 ⊢ ;12 ∈ ℂ
301	300	addridi 8469	. . . . . 6 ⊢ (;12 + 0) = ;12
302	298, 299, 301	3eqtri 2263	. . . . 5 ⊢ ((2 · 6) + (0 + 0)) = ;12
303	11	dec0h 9807	. . . . . 6 ⊢ 5 = ;05
304	81, 55, 303	3eqtri 2263	. . . . 5 ⊢ ((2 · 2) + 1) = ;05
305	2, 3, 39, 9, 297, 136, 3, 11, 39, 302, 304	decma2c 9838	. . . 4 ⊢ ((2 · ;62) + 1) = ;;125
306		9t2e18 9907	. . . . 5 ⊢ (9 · 2) = ;18
307	234, 57, 306	mulcomli 8333	. . . 4 ⊢ (2 · 9) = ;18
308	3, 4, 5, 296, 13, 9, 305, 307	decmul2c 9851	. . 3 ⊢ (2 · ;;629) = ;;;1258
309	308, 22	eqtr4i 2262	. 2 ⊢ (2 · ;;629) = (𝑁 − 1)
310		npcan 8536	. . 3 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
311	67, 16, 310	mp2an 430	. 2 ⊢ ((𝑁 − 1) + 1) = 𝑁
312	68	oveq1i 6095	. . 3 ⊢ ((0 · 𝑁) + 1) = (0 + 1)
313	143, 312, 159	3eqtr4i 2269	. 2 ⊢ ((0 · 𝑁) + 1) = (1 · 1)
314	1, 6, 7, 8, 9, 23, 295, 309, 311, 313	mod2xnegi 13218	1 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)

Colors of variables: wff set class

This proof depends on syntax axioms: = wceq 1402 ∈ wcel 2209 (class class class)co 6085 ℂcc 8177 0cc0 8179 1c1 8180 + caddc 8182 · cmul 8184 − cmin 8498 ℕcn 9306 2c2 9357 3c3 9358 4c4 9359 5c5 9360 6c6 9361 7c7 9362 8c8 9363 9c9 9364 ℕ₀cn0 9567 cdc 9781 mod cmo 10772 ↑cexp 10988

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735 ax-cnex 8270 ax-resscn 8271 ax-1cn 8272 ax-1re 8273 ax-icn 8274 ax-addcl 8275 ax-addrcl 8276 ax-mulcl 8277 ax-mulrcl 8278 ax-addcom 8279 ax-mulcom 8280 ax-addass 8281 ax-mulass 8282 ax-distr 8283 ax-i2m1 8284 ax-0lt1 8285 ax-1rid 8286 ax-0id 8287 ax-rnegex 8288 ax-precex 8289 ax-cnre 8290 ax-pre-ltirr 8291 ax-pre-ltwlin 8292 ax-pre-lttrn 8293 ax-pre-apti 8294 ax-pre-ltadd 8295 ax-pre-mulgt0 8296 ax-pre-mulext 8297 ax-arch 8298

This proof depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-id 4438 df-po 4441 df-iso 4442 df-iord 4511 df-on 4513 df-ilim 4514 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-1st 6374 df-2nd 6375 df-recs 6576 df-frec 6662 df-pnf 8362 df-mnf 8363 df-xr 8364 df-ltxr 8365 df-le 8366 df-sub 8500 df-neg 8501 df-reap 8905 df-ap 8912 df-div 9005 df-inn 9307 df-2 9365 df-3 9366 df-4 9367 df-5 9368 df-6 9369 df-7 9370 df-8 9371 df-9 9372 df-n0 9568 df-z 9649 df-dec 9782 df-uz 9931 df-q 10029 df-rp 10065 df-fl 10715 df-mod 10773 df-seqfrec 10898 df-exp 10989

This theorem is used by: 1259prm 13267

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