2503lem3 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > 2503lem3		Structured version Visualization version GIF version

Theorem 2503lem3 17101

Description: Lemma for 2503prm 17102. Calculate the GCD of 2↑18 − 1≡1831 with 𝑁 = 2503. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 15-Sep-2021.)

**Hypothesis**
Ref	Expression
2503prm.1	⊢ 𝑁 = ;;;2503

**Assertion**
Ref	Expression
2503lem3	⊢ (((2↑;18) − 1) gcd 𝑁) = 1

**Proof of Theorem 2503lem3**
Step	Hyp	Ref	Expression
1		2nn 12246	. . . 4 ⊢ 2 ∈ ℕ
2		1nn0 12445	. . . . 5 ⊢ 1 ∈ ℕ₀
3		8nn0 12452	. . . . 5 ⊢ 8 ∈ ℕ₀
4	2, 3	deccl 12651	. . . 4 ⊢ ;18 ∈ ℕ₀
5		nnexpcl 14028	. . . 4 ⊢ ((2 ∈ ℕ ∧ ;18 ∈ ℕ₀) → (2↑;18) ∈ ℕ)
6	1, 4, 5	mp2an 698	. . 3 ⊢ (2↑;18) ∈ ℕ
7		nnm1nn0 12470	. . 3 ⊢ ((2↑;18) ∈ ℕ → ((2↑;18) − 1) ∈ ℕ₀)
8	6, 7	ax-mp 5	. 2 ⊢ ((2↑;18) − 1) ∈ ℕ₀
9		3nn0 12447	. . . 4 ⊢ 3 ∈ ℕ₀
10	4, 9	deccl 12651	. . 3 ⊢ ;;183 ∈ ℕ₀
11	10, 2	deccl 12651	. 2 ⊢ ;;;1831 ∈ ℕ₀
12		2503prm.1	. . 3 ⊢ 𝑁 = ;;;2503
13		2nn0 12446	. . . . . 6 ⊢ 2 ∈ ℕ₀
14		5nn0 12449	. . . . . 6 ⊢ 5 ∈ ℕ₀
15	13, 14	deccl 12651	. . . . 5 ⊢ ;25 ∈ ℕ₀
16		0nn0 12444	. . . . 5 ⊢ 0 ∈ ℕ₀
17	15, 16	deccl 12651	. . . 4 ⊢ ;;250 ∈ ℕ₀
18		3nn 12252	. . . 4 ⊢ 3 ∈ ℕ
19	17, 18	decnncl 12656	. . 3 ⊢ ;;;2503 ∈ ℕ
20	12, 19	eqeltri 2835	. 2 ⊢ 𝑁 ∈ ℕ
21	12	2503lem1 17099	. . 3 ⊢ ((2↑;18) mod 𝑁) = (;;;1832 mod 𝑁)
22		1p1e2 12293	. . . 4 ⊢ (1 + 1) = 2
23		eqid 2739	. . . 4 ⊢ ;;;1831 = ;;;1831
24	10, 2, 22, 23	decsuc 12667	. . 3 ⊢ (;;;1831 + 1) = ;;;1832
25	20, 6, 2, 11, 21, 24	modsubi 17035	. 2 ⊢ (((2↑;18) − 1) mod 𝑁) = (;;;1831 mod 𝑁)
26		6nn0 12450	. . . . 5 ⊢ 6 ∈ ℕ₀
27		7nn0 12451	. . . . 5 ⊢ 7 ∈ ℕ₀
28	26, 27	deccl 12651	. . . 4 ⊢ ;67 ∈ ℕ₀
29	28, 13	deccl 12651	. . 3 ⊢ ;;672 ∈ ℕ₀
30		4nn0 12448	. . . . . 6 ⊢ 4 ∈ ℕ₀
31	30, 3	deccl 12651	. . . . 5 ⊢ ;48 ∈ ℕ₀
32	31, 27	deccl 12651	. . . 4 ⊢ ;;487 ∈ ℕ₀
33	4, 14	deccl 12651	. . . . 5 ⊢ ;;185 ∈ ℕ₀
34	2, 2	deccl 12651	. . . . . . 7 ⊢ ;11 ∈ ℕ₀
35	34, 27	deccl 12651	. . . . . 6 ⊢ ;;117 ∈ ℕ₀
36	26, 3	deccl 12651	. . . . . . 7 ⊢ ;68 ∈ ℕ₀
37		9nn0 12453	. . . . . . . . 9 ⊢ 9 ∈ ℕ₀
38	30, 37	deccl 12651	. . . . . . . 8 ⊢ ;49 ∈ ℕ₀
39	2, 37	deccl 12651	. . . . . . . . 9 ⊢ ;19 ∈ ℕ₀
40	38	nn0zi 12544	. . . . . . . . . . 11 ⊢ ;49 ∈ ℤ
41	39	nn0zi 12544	. . . . . . . . . . 11 ⊢ ;19 ∈ ℤ
42		gcdcom 16474	. . . . . . . . . . 11 ⊢ ((;49 ∈ ℤ ∧ ;19 ∈ ℤ) → (;49 gcd ;19) = (;19 gcd ;49))
43	40, 41, 42	mp2an 698	. . . . . . . . . 10 ⊢ (;49 gcd ;19) = (;19 gcd ;49)
44		9nn 12271	. . . . . . . . . . . . 13 ⊢ 9 ∈ ℕ
45	2, 44	decnncl 12656	. . . . . . . . . . . 12 ⊢ ;19 ∈ ℕ
46		1nn 12177	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℕ
47	2, 46	decnncl 12656	. . . . . . . . . . . 12 ⊢ ;11 ∈ ℕ
48		eqid 2739	. . . . . . . . . . . . 13 ⊢ ;19 = ;19
49		eqid 2739	. . . . . . . . . . . . 13 ⊢ ;11 = ;11
50		2cn 12248	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℂ
51	50	mullidi 11142	. . . . . . . . . . . . . . 15 ⊢ (1 · 2) = 2
52	51, 22	oveq12i 7369	. . . . . . . . . . . . . 14 ⊢ ((1 · 2) + (1 + 1)) = (2 + 2)
53		2p2e4 12303	. . . . . . . . . . . . . 14 ⊢ (2 + 2) = 4
54	52, 53	eqtri 2762	. . . . . . . . . . . . 13 ⊢ ((1 · 2) + (1 + 1)) = 4
55		8p1e9 12318	. . . . . . . . . . . . . 14 ⊢ (8 + 1) = 9
56		9t2e18 12758	. . . . . . . . . . . . . 14 ⊢ (9 · 2) = ;18
57	2, 3, 55, 56	decsuc 12667	. . . . . . . . . . . . 13 ⊢ ((9 · 2) + 1) = ;19
58	2, 37, 2, 2, 48, 49, 13, 37, 2, 54, 57	decmac 12688	. . . . . . . . . . . 12 ⊢ ((;19 · 2) + ;11) = ;49
59		1lt9 12374	. . . . . . . . . . . . 13 ⊢ 1 < 9
60	2, 2, 44, 59	declt 12664	. . . . . . . . . . . 12 ⊢ ;11 < ;19
61	45, 13, 47, 58, 60	ndvdsi 16373	. . . . . . . . . . 11 ⊢ ¬ ;19 ∥ ;49
62		19prm 17080	. . . . . . . . . . . 12 ⊢ ;19 ∈ ℙ
63		coprm 16673	. . . . . . . . . . . 12 ⊢ ((;19 ∈ ℙ ∧ ;49 ∈ ℤ) → (¬ ;19 ∥ ;49 ↔ (;19 gcd ;49) = 1))
64	62, 40, 63	mp2an 698	. . . . . . . . . . 11 ⊢ (¬ ;19 ∥ ;49 ↔ (;19 gcd ;49) = 1)
65	61, 64	mpbi 231	. . . . . . . . . 10 ⊢ (;19 gcd ;49) = 1
66	43, 65	eqtri 2762	. . . . . . . . 9 ⊢ (;49 gcd ;19) = 1
67		eqid 2739	. . . . . . . . . 10 ⊢ ;49 = ;49
68		4cn 12258	. . . . . . . . . . . . 13 ⊢ 4 ∈ ℂ
69	68	mullidi 11142	. . . . . . . . . . . 12 ⊢ (1 · 4) = 4
70	69, 22	oveq12i 7369	. . . . . . . . . . 11 ⊢ ((1 · 4) + (1 + 1)) = (4 + 2)
71		4p2e6 12321	. . . . . . . . . . 11 ⊢ (4 + 2) = 6
72	70, 71	eqtri 2762	. . . . . . . . . 10 ⊢ ((1 · 4) + (1 + 1)) = 6
73		9cn 12273	. . . . . . . . . . . . 13 ⊢ 9 ∈ ℂ
74	73	mullidi 11142	. . . . . . . . . . . 12 ⊢ (1 · 9) = 9
75	74	oveq1i 7367	. . . . . . . . . . 11 ⊢ ((1 · 9) + 9) = (9 + 9)
76		9p9e18 12730	. . . . . . . . . . 11 ⊢ (9 + 9) = ;18
77	75, 76	eqtri 2762	. . . . . . . . . 10 ⊢ ((1 · 9) + 9) = ;18
78	30, 37, 2, 37, 67, 48, 2, 3, 2, 72, 77	decma2c 12689	. . . . . . . . 9 ⊢ ((1 · ;49) + ;19) = ;68
79	2, 39, 38, 66, 78	gcdi 17036	. . . . . . . 8 ⊢ (;68 gcd ;49) = 1
80		eqid 2739	. . . . . . . . 9 ⊢ ;68 = ;68
81		6cn 12264	. . . . . . . . . . . 12 ⊢ 6 ∈ ℂ
82	81	mullidi 11142	. . . . . . . . . . 11 ⊢ (1 · 6) = 6
83		4p1e5 12314	. . . . . . . . . . 11 ⊢ (4 + 1) = 5
84	82, 83	oveq12i 7369	. . . . . . . . . 10 ⊢ ((1 · 6) + (4 + 1)) = (6 + 5)
85		6p5e11 12709	. . . . . . . . . 10 ⊢ (6 + 5) = ;11
86	84, 85	eqtri 2762	. . . . . . . . 9 ⊢ ((1 · 6) + (4 + 1)) = ;11
87		8cn 12270	. . . . . . . . . . . 12 ⊢ 8 ∈ ℂ
88	87	mullidi 11142	. . . . . . . . . . 11 ⊢ (1 · 8) = 8
89	88	oveq1i 7367	. . . . . . . . . 10 ⊢ ((1 · 8) + 9) = (8 + 9)
90		9p8e17 12729	. . . . . . . . . . 11 ⊢ (9 + 8) = ;17
91	73, 87, 90	addcomli 11330	. . . . . . . . . 10 ⊢ (8 + 9) = ;17
92	89, 91	eqtri 2762	. . . . . . . . 9 ⊢ ((1 · 8) + 9) = ;17
93	26, 3, 30, 37, 80, 67, 2, 27, 2, 86, 92	decma2c 12689	. . . . . . . 8 ⊢ ((1 · ;68) + ;49) = ;;117
94	2, 38, 36, 79, 93	gcdi 17036	. . . . . . 7 ⊢ (;;117 gcd ;68) = 1
95		eqid 2739	. . . . . . . 8 ⊢ ;;117 = ;;117
96		6p1e7 12316	. . . . . . . . . 10 ⊢ (6 + 1) = 7
97	27	dec0h 12658	. . . . . . . . . 10 ⊢ 7 = ;07
98	96, 97	eqtri 2762	. . . . . . . . 9 ⊢ (6 + 1) = ;07
99		1t1e1 12330	. . . . . . . . . . 11 ⊢ (1 · 1) = 1
100		00id 11313	. . . . . . . . . . 11 ⊢ (0 + 0) = 0
101	99, 100	oveq12i 7369	. . . . . . . . . 10 ⊢ ((1 · 1) + (0 + 0)) = (1 + 0)
102		ax-1cn 11088	. . . . . . . . . . 11 ⊢ 1 ∈ ℂ
103	102	addridi 11325	. . . . . . . . . 10 ⊢ (1 + 0) = 1
104	101, 103	eqtri 2762	. . . . . . . . 9 ⊢ ((1 · 1) + (0 + 0)) = 1
105	99	oveq1i 7367	. . . . . . . . . 10 ⊢ ((1 · 1) + 7) = (1 + 7)
106		7cn 12267	. . . . . . . . . . 11 ⊢ 7 ∈ ℂ
107		7p1e8 12317	. . . . . . . . . . 11 ⊢ (7 + 1) = 8
108	106, 102, 107	addcomli 11330	. . . . . . . . . 10 ⊢ (1 + 7) = 8
109	3	dec0h 12658	. . . . . . . . . 10 ⊢ 8 = ;08
110	105, 108, 109	3eqtri 2766	. . . . . . . . 9 ⊢ ((1 · 1) + 7) = ;08
111	2, 2, 16, 27, 49, 98, 2, 3, 16, 104, 110	decma2c 12689	. . . . . . . 8 ⊢ ((1 · ;11) + (6 + 1)) = ;18
112	106	mullidi 11142	. . . . . . . . . 10 ⊢ (1 · 7) = 7
113	112	oveq1i 7367	. . . . . . . . 9 ⊢ ((1 · 7) + 8) = (7 + 8)
114		8p7e15 12721	. . . . . . . . . 10 ⊢ (8 + 7) = ;15
115	87, 106, 114	addcomli 11330	. . . . . . . . 9 ⊢ (7 + 8) = ;15
116	113, 115	eqtri 2762	. . . . . . . 8 ⊢ ((1 · 7) + 8) = ;15
117	34, 27, 26, 3, 95, 80, 2, 14, 2, 111, 116	decma2c 12689	. . . . . . 7 ⊢ ((1 · ;;117) + ;68) = ;;185
118	2, 36, 35, 94, 117	gcdi 17036	. . . . . 6 ⊢ (;;185 gcd ;;117) = 1
119		eqid 2739	. . . . . . 7 ⊢ ;;185 = ;;185
120		eqid 2739	. . . . . . . 8 ⊢ ;18 = ;18
121	2, 2, 22, 49	decsuc 12667	. . . . . . . 8 ⊢ (;11 + 1) = ;12
122		2t1e2 12331	. . . . . . . . . 10 ⊢ (2 · 1) = 2
123	122, 22	oveq12i 7369	. . . . . . . . 9 ⊢ ((2 · 1) + (1 + 1)) = (2 + 2)
124	123, 53	eqtri 2762	. . . . . . . 8 ⊢ ((2 · 1) + (1 + 1)) = 4
125		8t2e16 12751	. . . . . . . . . 10 ⊢ (8 · 2) = ;16
126	87, 50, 125	mulcomli 11146	. . . . . . . . 9 ⊢ (2 · 8) = ;16
127		6p2e8 12327	. . . . . . . . 9 ⊢ (6 + 2) = 8
128	2, 26, 13, 126, 127	decaddi 12696	. . . . . . . 8 ⊢ ((2 · 8) + 2) = ;18
129	2, 3, 2, 13, 120, 121, 13, 3, 2, 124, 128	decma2c 12689	. . . . . . 7 ⊢ ((2 · ;18) + (;11 + 1)) = ;48
130		5cn 12261	. . . . . . . . 9 ⊢ 5 ∈ ℂ
131		5t2e10 12736	. . . . . . . . 9 ⊢ (5 · 2) = ;10
132	130, 50, 131	mulcomli 11146	. . . . . . . 8 ⊢ (2 · 5) = ;10
133	106	addlidi 11326	. . . . . . . 8 ⊢ (0 + 7) = 7
134	2, 16, 27, 132, 133	decaddi 12696	. . . . . . 7 ⊢ ((2 · 5) + 7) = ;17
135	4, 14, 34, 27, 119, 95, 13, 27, 2, 129, 134	decma2c 12689	. . . . . 6 ⊢ ((2 · ;;185) + ;;117) = ;;487
136	13, 35, 33, 118, 135	gcdi 17036	. . . . 5 ⊢ (;;487 gcd ;;185) = 1
137		eqid 2739	. . . . . 6 ⊢ ;;487 = ;;487
138		eqid 2739	. . . . . . 7 ⊢ ;48 = ;48
139	2, 3, 55, 120	decsuc 12667	. . . . . . 7 ⊢ (;18 + 1) = ;19
140	30, 3, 2, 37, 138, 139, 2, 27, 2, 72, 92	decma2c 12689	. . . . . 6 ⊢ ((1 · ;48) + (;18 + 1)) = ;67
141	112	oveq1i 7367	. . . . . . 7 ⊢ ((1 · 7) + 5) = (7 + 5)
142		7p5e12 12713	. . . . . . 7 ⊢ (7 + 5) = ;12
143	141, 142	eqtri 2762	. . . . . 6 ⊢ ((1 · 7) + 5) = ;12
144	31, 27, 4, 14, 137, 119, 2, 13, 2, 140, 143	decma2c 12689	. . . . 5 ⊢ ((1 · ;;487) + ;;185) = ;;672
145	2, 33, 32, 136, 144	gcdi 17036	. . . 4 ⊢ (;;672 gcd ;;487) = 1
146		eqid 2739	. . . . 5 ⊢ ;;672 = ;;672
147		eqid 2739	. . . . . 6 ⊢ ;67 = ;67
148	30, 3, 55, 138	decsuc 12667	. . . . . 6 ⊢ (;48 + 1) = ;49
149	71	oveq2i 7368	. . . . . . 7 ⊢ ((2 · 6) + (4 + 2)) = ((2 · 6) + 6)
150		6t2e12 12740	. . . . . . . . 9 ⊢ (6 · 2) = ;12
151	81, 50, 150	mulcomli 11146	. . . . . . . 8 ⊢ (2 · 6) = ;12
152	81, 50, 127	addcomli 11330	. . . . . . . 8 ⊢ (2 + 6) = 8
153	2, 13, 26, 151, 152	decaddi 12696	. . . . . . 7 ⊢ ((2 · 6) + 6) = ;18
154	149, 153	eqtri 2762	. . . . . 6 ⊢ ((2 · 6) + (4 + 2)) = ;18
155		7t2e14 12745	. . . . . . . 8 ⊢ (7 · 2) = ;14
156	106, 50, 155	mulcomli 11146	. . . . . . 7 ⊢ (2 · 7) = ;14
157		9p4e13 12725	. . . . . . . 8 ⊢ (9 + 4) = ;13
158	73, 68, 157	addcomli 11330	. . . . . . 7 ⊢ (4 + 9) = ;13
159	2, 30, 37, 156, 22, 9, 158	decaddci 12697	. . . . . 6 ⊢ ((2 · 7) + 9) = ;23
160	26, 27, 30, 37, 147, 148, 13, 9, 13, 154, 159	decma2c 12689	. . . . 5 ⊢ ((2 · ;67) + (;48 + 1)) = ;;183
161		2t2e4 12332	. . . . . . 7 ⊢ (2 · 2) = 4
162	161	oveq1i 7367	. . . . . 6 ⊢ ((2 · 2) + 7) = (4 + 7)
163		7p4e11 12712	. . . . . . 7 ⊢ (7 + 4) = ;11
164	106, 68, 163	addcomli 11330	. . . . . 6 ⊢ (4 + 7) = ;11
165	162, 164	eqtri 2762	. . . . 5 ⊢ ((2 · 2) + 7) = ;11
166	28, 13, 31, 27, 146, 137, 13, 2, 2, 160, 165	decma2c 12689	. . . 4 ⊢ ((2 · ;;672) + ;;487) = ;;;1831
167	13, 32, 29, 145, 166	gcdi 17036	. . 3 ⊢ (;;;1831 gcd ;;672) = 1
168		eqid 2739	. . . . . 6 ⊢ ;;183 = ;;183
169	28	nn0cni 12441	. . . . . . 7 ⊢ ;67 ∈ ℂ
170	169	addridi 11325	. . . . . 6 ⊢ (;67 + 0) = ;67
171	102	addlidi 11326	. . . . . . . . 9 ⊢ (0 + 1) = 1
172	99, 171	oveq12i 7369	. . . . . . . 8 ⊢ ((1 · 1) + (0 + 1)) = (1 + 1)
173	172, 22	eqtri 2762	. . . . . . 7 ⊢ ((1 · 1) + (0 + 1)) = 2
174	88	oveq1i 7367	. . . . . . . 8 ⊢ ((1 · 8) + 7) = (8 + 7)
175	174, 114	eqtri 2762	. . . . . . 7 ⊢ ((1 · 8) + 7) = ;15
176	2, 3, 16, 27, 120, 98, 2, 14, 2, 173, 175	decma2c 12689	. . . . . 6 ⊢ ((1 · ;18) + (6 + 1)) = ;25
177		3cn 12254	. . . . . . . . 9 ⊢ 3 ∈ ℂ
178	177	mullidi 11142	. . . . . . . 8 ⊢ (1 · 3) = 3
179	178	oveq1i 7367	. . . . . . 7 ⊢ ((1 · 3) + 7) = (3 + 7)
180		7p3e10 12711	. . . . . . . 8 ⊢ (7 + 3) = ;10
181	106, 177, 180	addcomli 11330	. . . . . . 7 ⊢ (3 + 7) = ;10
182	179, 181	eqtri 2762	. . . . . 6 ⊢ ((1 · 3) + 7) = ;10
183	4, 9, 26, 27, 168, 170, 2, 16, 2, 176, 182	decma2c 12689	. . . . 5 ⊢ ((1 · ;;183) + (;67 + 0)) = ;;250
184	99	oveq1i 7367	. . . . . 6 ⊢ ((1 · 1) + 2) = (1 + 2)
185		1p2e3 12311	. . . . . 6 ⊢ (1 + 2) = 3
186	9	dec0h 12658	. . . . . 6 ⊢ 3 = ;03
187	184, 185, 186	3eqtri 2766	. . . . 5 ⊢ ((1 · 1) + 2) = ;03
188	10, 2, 28, 13, 23, 146, 2, 9, 16, 183, 187	decma2c 12689	. . . 4 ⊢ ((1 · ;;;1831) + ;;672) = ;;;2503
189	188, 12	eqtr4i 2765	. . 3 ⊢ ((1 · ;;;1831) + ;;672) = 𝑁
190	2, 29, 11, 167, 189	gcdi 17036	. 2 ⊢ (𝑁 gcd ;;;1831) = 1
191	8, 11, 20, 25, 190	gcdmodi 17037	1 ⊢ (((2↑;18) − 1) gcd 𝑁) = 1

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 207 = wceq 1547 ∈ wcel 2119 class class class wbr 5073 (class class class)co 7357 0cc0 11030 1c1 11031 + caddc 11033 · cmul 11035 − cmin 11369 ℕcn 12166 2c2 12228 3c3 12229 4c4 12230 5c5 12231 6c6 12232 7c7 12233 8c8 12234 9c9 12235 ℕ₀cn0 12429 ℤcz 12516 cdc 12636 ↑cexp 14015 ∥ cdvds 16213 gcd cgcd 16455 ℙcprime 16632

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 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-sep 5219 ax-nul 5229 ax-pow 5295 ax-pr 5363 ax-un 7679 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107 ax-pre-sup 11108

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4263 df-if 4456 df-pw 4532 df-sn 4557 df-pr 4559 df-op 4563 df-uni 4840 df-iun 4924 df-br 5074 df-opab 5136 df-mpt 5155 df-tr 5181 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7314 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7808 df-1st 7932 df-2nd 7933 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-2o 8397 df-er 8634 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-sup 9346 df-inf 9347 df-pnf 11173 df-mnf 11174 df-xr 11175 df-ltxr 11176 df-le 11177 df-sub 11371 df-neg 11372 df-div 11800 df-nn 12167 df-2 12236 df-3 12237 df-4 12238 df-5 12239 df-6 12240 df-7 12241 df-8 12242 df-9 12243 df-n0 12430 df-z 12517 df-dec 12637 df-uz 12781 df-rp 12935 df-fz 13454 df-fl 13743 df-mod 13821 df-seq 13956 df-exp 14016 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-dvds 16214 df-gcd 16456 df-prm 16633

This theorem is referenced by: 2503prm 17102

W3C validator