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 17209

Description: Lemma for 2503prm 17210. 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 12324	. . . 4 ⊢ 2 ∈ ℕ
2		1nn0 12530	. . . . 5 ⊢ 1 ∈ ℕ₀
3		8nn0 12537	. . . . 5 ⊢ 8 ∈ ℕ₀
4	2, 3	deccl 12736	. . . 4 ⊢ ;18 ∈ ℕ₀
5		nnexpcl 14121	. . . 4 ⊢ ((2 ∈ ℕ ∧ ;18 ∈ ℕ₀) → (2↑;18) ∈ ℕ)
6	1, 4, 5	mp2an 705	. . 3 ⊢ (2↑;18) ∈ ℕ
7		nnm1nn0 12555	. . 3 ⊢ ((2↑;18) ∈ ℕ → ((2↑;18) − 1) ∈ ℕ₀)
8	6, 7	ax-mp 5	. 2 ⊢ ((2↑;18) − 1) ∈ ℕ₀
9		3nn0 12532	. . . 4 ⊢ 3 ∈ ℕ₀
10	4, 9	deccl 12736	. . 3 ⊢ ;;183 ∈ ℕ₀
11	10, 2	deccl 12736	. 2 ⊢ ;;;1831 ∈ ℕ₀
12		2503prm.1	. . 3 ⊢ 𝑁 = ;;;2503
13		2nn0 12531	. . . . . 6 ⊢ 2 ∈ ℕ₀
14		5nn0 12534	. . . . . 6 ⊢ 5 ∈ ℕ₀
15	13, 14	deccl 12736	. . . . 5 ⊢ ;25 ∈ ℕ₀
16		0nn0 12529	. . . . 5 ⊢ 0 ∈ ℕ₀
17	15, 16	deccl 12736	. . . 4 ⊢ ;;250 ∈ ℕ₀
18		3nn 12330	. . . 4 ⊢ 3 ∈ ℕ
19	17, 18	decnncl 12745	. . 3 ⊢ ;;;2503 ∈ ℕ
20	12, 19	eqeltri 2862	. 2 ⊢ 𝑁 ∈ ℕ
21	12	2503lem1 17207	. . 3 ⊢ ((2↑;18) mod 𝑁) = (;;;1832 mod 𝑁)
22		1p1e2 12374	. . . 4 ⊢ (1 + 1) = 2
23		eqid 2766	. . . 4 ⊢ ;;;1831 = ;;;1831
24	10, 2, 22, 23	decsuc 12757	. . 3 ⊢ (;;;1831 + 1) = ;;;1832
25	20, 6, 2, 11, 21, 24	modsubi 17142	. 2 ⊢ (((2↑;18) − 1) mod 𝑁) = (;;;1831 mod 𝑁)
26		6nn0 12535	. . . . 5 ⊢ 6 ∈ ℕ₀
27		7nn0 12536	. . . . 5 ⊢ 7 ∈ ℕ₀
28	26, 27	deccl 12736	. . . 4 ⊢ ;67 ∈ ℕ₀
29	28, 13	deccl 12736	. . 3 ⊢ ;;672 ∈ ℕ₀
30		4nn0 12533	. . . . . 6 ⊢ 4 ∈ ℕ₀
31	30, 3	deccl 12736	. . . . 5 ⊢ ;48 ∈ ℕ₀
32	31, 27	deccl 12736	. . . 4 ⊢ ;;487 ∈ ℕ₀
33	4, 14	deccl 12736	. . . . 5 ⊢ ;;185 ∈ ℕ₀
34	2, 2	deccl 12736	. . . . . . 7 ⊢ ;11 ∈ ℕ₀
35	34, 27	deccl 12736	. . . . . 6 ⊢ ;;117 ∈ ℕ₀
36	26, 3	deccl 12736	. . . . . . 7 ⊢ ;68 ∈ ℕ₀
37		9nn0 12538	. . . . . . . . 9 ⊢ 9 ∈ ℕ₀
38	30, 37	deccl 12736	. . . . . . . 8 ⊢ ;49 ∈ ℕ₀
39	2, 37	deccl 12736	. . . . . . . . 9 ⊢ ;19 ∈ ℕ₀
40	38	nn0zi 12629	. . . . . . . . . . 11 ⊢ ;49 ∈ ℤ
41	39	nn0zi 12629	. . . . . . . . . . 11 ⊢ ;19 ∈ ℤ
42		gcdcom 16581	. . . . . . . . . . 11 ⊢ ((;49 ∈ ℤ ∧ ;19 ∈ ℤ) → (;49 gcd ;19) = (;19 gcd ;49))
43	40, 41, 42	mp2an 705	. . . . . . . . . 10 ⊢ (;49 gcd ;19) = (;19 gcd ;49)
44		9nn 12349	. . . . . . . . . . . . 13 ⊢ 9 ∈ ℕ
45	2, 44	decnncl 12745	. . . . . . . . . . . 12 ⊢ ;19 ∈ ℕ
46		1nn 12254	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℕ
47	2, 46	decnncl 12745	. . . . . . . . . . . 12 ⊢ ;11 ∈ ℕ
48		eqid 2766	. . . . . . . . . . . . 13 ⊢ ;19 = ;19
49		eqid 2766	. . . . . . . . . . . . 13 ⊢ ;11 = ;11
50		2cn 12326	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℂ
51	50	mullidi 11224	. . . . . . . . . . . . . . 15 ⊢ (1 · 2) = 2
52	51, 22	oveq12i 7428	. . . . . . . . . . . . . 14 ⊢ ((1 · 2) + (1 + 1)) = (2 + 2)
53		2p2e4 12385	. . . . . . . . . . . . . 14 ⊢ (2 + 2) = 4
54	52, 53	eqtri 2789	. . . . . . . . . . . . 13 ⊢ ((1 · 2) + (1 + 1)) = 4
55		8p1e9 12400	. . . . . . . . . . . . . 14 ⊢ (8 + 1) = 9
56		9t2e18 12848	. . . . . . . . . . . . . 14 ⊢ (9 · 2) = ;18
57	2, 3, 55, 56	decsuc 12757	. . . . . . . . . . . . 13 ⊢ ((9 · 2) + 1) = ;19
58	2, 37, 2, 2, 48, 49, 13, 37, 2, 54, 57	decmac 12778	. . . . . . . . . . . 12 ⊢ ((;19 · 2) + ;11) = ;49
59		1lt9 12459	. . . . . . . . . . . . 13 ⊢ 1 < 9
60	2, 2, 44, 59	declt 12754	. . . . . . . . . . . 12 ⊢ ;11 < ;19
61	45, 13, 47, 58, 60	ndvdsi 16480	. . . . . . . . . . 11 ⊢ ¬ ;19 ∥ ;49
62		19prm 17188	. . . . . . . . . . . 12 ⊢ ;19 ∈ ℙ
63		coprm 16780	. . . . . . . . . . . 12 ⊢ ((;19 ∈ ℙ ∧ ;49 ∈ ℤ) → (¬ ;19 ∥ ;49 ↔ (;19 gcd ;49) = 1))
64	62, 40, 63	mp2an 705	. . . . . . . . . . 11 ⊢ (¬ ;19 ∥ ;49 ↔ (;19 gcd ;49) = 1)
65	61, 64	mpbi 233	. . . . . . . . . 10 ⊢ (;19 gcd ;49) = 1
66	43, 65	eqtri 2789	. . . . . . . . 9 ⊢ (;49 gcd ;19) = 1
67		eqid 2766	. . . . . . . . . 10 ⊢ ;49 = ;49
68		4cn 12336	. . . . . . . . . . . . 13 ⊢ 4 ∈ ℂ
69	68	mullidi 11224	. . . . . . . . . . . 12 ⊢ (1 · 4) = 4
70	69, 22	oveq12i 7428	. . . . . . . . . . 11 ⊢ ((1 · 4) + (1 + 1)) = (4 + 2)
71		4p2e6 12403	. . . . . . . . . . 11 ⊢ (4 + 2) = 6
72	70, 71	eqtri 2789	. . . . . . . . . 10 ⊢ ((1 · 4) + (1 + 1)) = 6
73		9cn 12351	. . . . . . . . . . . . 13 ⊢ 9 ∈ ℂ
74	73	mullidi 11224	. . . . . . . . . . . 12 ⊢ (1 · 9) = 9
75	74	oveq1i 7426	. . . . . . . . . . 11 ⊢ ((1 · 9) + 9) = (9 + 9)
76		9p9e18 12820	. . . . . . . . . . 11 ⊢ (9 + 9) = ;18
77	75, 76	eqtri 2789	. . . . . . . . . 10 ⊢ ((1 · 9) + 9) = ;18
78	30, 37, 2, 37, 67, 48, 2, 3, 2, 72, 77	decma2c 12779	. . . . . . . . 9 ⊢ ((1 · ;49) + ;19) = ;68
79	2, 39, 38, 66, 78	gcdi 17143	. . . . . . . 8 ⊢ (;68 gcd ;49) = 1
80		eqid 2766	. . . . . . . . 9 ⊢ ;68 = ;68
81		6cn 12342	. . . . . . . . . . . 12 ⊢ 6 ∈ ℂ
82	81	mullidi 11224	. . . . . . . . . . 11 ⊢ (1 · 6) = 6
83		4p1e5 12396	. . . . . . . . . . 11 ⊢ (4 + 1) = 5
84	82, 83	oveq12i 7428	. . . . . . . . . 10 ⊢ ((1 · 6) + (4 + 1)) = (6 + 5)
85		6p5e11 12799	. . . . . . . . . 10 ⊢ (6 + 5) = ;11
86	84, 85	eqtri 2789	. . . . . . . . 9 ⊢ ((1 · 6) + (4 + 1)) = ;11
87		8cn 12348	. . . . . . . . . . . 12 ⊢ 8 ∈ ℂ
88	87	mullidi 11224	. . . . . . . . . . 11 ⊢ (1 · 8) = 8
89	88	oveq1i 7426	. . . . . . . . . 10 ⊢ ((1 · 8) + 9) = (8 + 9)
90		9p8e17 12819	. . . . . . . . . . 11 ⊢ (9 + 8) = ;17
91	73, 87, 90	addcomli 11412	. . . . . . . . . 10 ⊢ (8 + 9) = ;17
92	89, 91	eqtri 2789	. . . . . . . . 9 ⊢ ((1 · 8) + 9) = ;17
93	26, 3, 30, 37, 80, 67, 2, 27, 2, 86, 92	decma2c 12779	. . . . . . . 8 ⊢ ((1 · ;68) + ;49) = ;;117
94	2, 38, 36, 79, 93	gcdi 17143	. . . . . . 7 ⊢ (;;117 gcd ;68) = 1
95		eqid 2766	. . . . . . . 8 ⊢ ;;117 = ;;117
96		6p1e7 12398	. . . . . . . . . 10 ⊢ (6 + 1) = 7
97	27	dec0h 12748	. . . . . . . . . 10 ⊢ 7 = ;07
98	96, 97	eqtri 2789	. . . . . . . . 9 ⊢ (6 + 1) = ;07
99		1t1e1 12412	. . . . . . . . . . 11 ⊢ (1 · 1) = 1
100		00id 11395	. . . . . . . . . . 11 ⊢ (0 + 0) = 0
101	99, 100	oveq12i 7428	. . . . . . . . . 10 ⊢ ((1 · 1) + (0 + 0)) = (1 + 0)
102		ax-1cn 11168	. . . . . . . . . . 11 ⊢ 1 ∈ ℂ
103	102	addridi 11407	. . . . . . . . . 10 ⊢ (1 + 0) = 1
104	101, 103	eqtri 2789	. . . . . . . . 9 ⊢ ((1 · 1) + (0 + 0)) = 1
105	99	oveq1i 7426	. . . . . . . . . 10 ⊢ ((1 · 1) + 7) = (1 + 7)
106		7cn 12345	. . . . . . . . . . 11 ⊢ 7 ∈ ℂ
107		7p1e8 12399	. . . . . . . . . . 11 ⊢ (7 + 1) = 8
108	106, 102, 107	addcomli 11412	. . . . . . . . . 10 ⊢ (1 + 7) = 8
109	3	dec0h 12748	. . . . . . . . . 10 ⊢ 8 = ;08
110	105, 108, 109	3eqtri 2793	. . . . . . . . 9 ⊢ ((1 · 1) + 7) = ;08
111	2, 2, 16, 27, 49, 98, 2, 3, 16, 104, 110	decma2c 12779	. . . . . . . 8 ⊢ ((1 · ;11) + (6 + 1)) = ;18
112	106	mullidi 11224	. . . . . . . . . 10 ⊢ (1 · 7) = 7
113	112	oveq1i 7426	. . . . . . . . 9 ⊢ ((1 · 7) + 8) = (7 + 8)
114		8p7e15 12811	. . . . . . . . . 10 ⊢ (8 + 7) = ;15
115	87, 106, 114	addcomli 11412	. . . . . . . . 9 ⊢ (7 + 8) = ;15
116	113, 115	eqtri 2789	. . . . . . . 8 ⊢ ((1 · 7) + 8) = ;15
117	34, 27, 26, 3, 95, 80, 2, 14, 2, 111, 116	decma2c 12779	. . . . . . 7 ⊢ ((1 · ;;117) + ;68) = ;;185
118	2, 36, 35, 94, 117	gcdi 17143	. . . . . 6 ⊢ (;;185 gcd ;;117) = 1
119		eqid 2766	. . . . . . 7 ⊢ ;;185 = ;;185
120		eqid 2766	. . . . . . . 8 ⊢ ;18 = ;18
121	2, 2, 22, 49	decsuc 12757	. . . . . . . 8 ⊢ (;11 + 1) = ;12
122		2t1e2 12413	. . . . . . . . . 10 ⊢ (2 · 1) = 2
123	122, 22	oveq12i 7428	. . . . . . . . 9 ⊢ ((2 · 1) + (1 + 1)) = (2 + 2)
124	123, 53	eqtri 2789	. . . . . . . 8 ⊢ ((2 · 1) + (1 + 1)) = 4
125		8t2e16 12841	. . . . . . . . . 10 ⊢ (8 · 2) = ;16
126	87, 50, 125	mulcomli 11228	. . . . . . . . 9 ⊢ (2 · 8) = ;16
127		6p2e8 12409	. . . . . . . . 9 ⊢ (6 + 2) = 8
128	2, 26, 13, 126, 127	decaddi 12786	. . . . . . . 8 ⊢ ((2 · 8) + 2) = ;18
129	2, 3, 2, 13, 120, 121, 13, 3, 2, 124, 128	decma2c 12779	. . . . . . 7 ⊢ ((2 · ;18) + (;11 + 1)) = ;48
130		5cn 12339	. . . . . . . . 9 ⊢ 5 ∈ ℂ
131		5t2e10 12826	. . . . . . . . 9 ⊢ (5 · 2) = ;10
132	130, 50, 131	mulcomli 11228	. . . . . . . 8 ⊢ (2 · 5) = ;10
133	106	addlidi 11408	. . . . . . . 8 ⊢ (0 + 7) = 7
134	2, 16, 27, 132, 133	decaddi 12786	. . . . . . 7 ⊢ ((2 · 5) + 7) = ;17
135	4, 14, 34, 27, 119, 95, 13, 27, 2, 129, 134	decma2c 12779	. . . . . 6 ⊢ ((2 · ;;185) + ;;117) = ;;487
136	13, 35, 33, 118, 135	gcdi 17143	. . . . 5 ⊢ (;;487 gcd ;;185) = 1
137		eqid 2766	. . . . . 6 ⊢ ;;487 = ;;487
138		eqid 2766	. . . . . . 7 ⊢ ;48 = ;48
139	2, 3, 55, 120	decsuc 12757	. . . . . . 7 ⊢ (;18 + 1) = ;19
140	30, 3, 2, 37, 138, 139, 2, 27, 2, 72, 92	decma2c 12779	. . . . . 6 ⊢ ((1 · ;48) + (;18 + 1)) = ;67
141	112	oveq1i 7426	. . . . . . 7 ⊢ ((1 · 7) + 5) = (7 + 5)
142		7p5e12 12803	. . . . . . 7 ⊢ (7 + 5) = ;12
143	141, 142	eqtri 2789	. . . . . 6 ⊢ ((1 · 7) + 5) = ;12
144	31, 27, 4, 14, 137, 119, 2, 13, 2, 140, 143	decma2c 12779	. . . . 5 ⊢ ((1 · ;;487) + ;;185) = ;;672
145	2, 33, 32, 136, 144	gcdi 17143	. . . 4 ⊢ (;;672 gcd ;;487) = 1
146		eqid 2766	. . . . 5 ⊢ ;;672 = ;;672
147		eqid 2766	. . . . . 6 ⊢ ;67 = ;67
148	30, 3, 55, 138	decsuc 12757	. . . . . 6 ⊢ (;48 + 1) = ;49
149	71	oveq2i 7427	. . . . . . 7 ⊢ ((2 · 6) + (4 + 2)) = ((2 · 6) + 6)
150		6t2e12 12830	. . . . . . . . 9 ⊢ (6 · 2) = ;12
151	81, 50, 150	mulcomli 11228	. . . . . . . 8 ⊢ (2 · 6) = ;12
152	81, 50, 127	addcomli 11412	. . . . . . . 8 ⊢ (2 + 6) = 8
153	2, 13, 26, 151, 152	decaddi 12786	. . . . . . 7 ⊢ ((2 · 6) + 6) = ;18
154	149, 153	eqtri 2789	. . . . . 6 ⊢ ((2 · 6) + (4 + 2)) = ;18
155		7t2e14 12835	. . . . . . . 8 ⊢ (7 · 2) = ;14
156	106, 50, 155	mulcomli 11228	. . . . . . 7 ⊢ (2 · 7) = ;14
157		9p4e13 12815	. . . . . . . 8 ⊢ (9 + 4) = ;13
158	73, 68, 157	addcomli 11412	. . . . . . 7 ⊢ (4 + 9) = ;13
159	2, 30, 37, 156, 22, 9, 158	decaddci 12787	. . . . . 6 ⊢ ((2 · 7) + 9) = ;23
160	26, 27, 30, 37, 147, 148, 13, 9, 13, 154, 159	decma2c 12779	. . . . 5 ⊢ ((2 · ;67) + (;48 + 1)) = ;;183
161		2t2e4 12414	. . . . . . 7 ⊢ (2 · 2) = 4
162	161	oveq1i 7426	. . . . . 6 ⊢ ((2 · 2) + 7) = (4 + 7)
163		7p4e11 12802	. . . . . . 7 ⊢ (7 + 4) = ;11
164	106, 68, 163	addcomli 11412	. . . . . 6 ⊢ (4 + 7) = ;11
165	162, 164	eqtri 2789	. . . . 5 ⊢ ((2 · 2) + 7) = ;11
166	28, 13, 31, 27, 146, 137, 13, 2, 2, 160, 165	decma2c 12779	. . . 4 ⊢ ((2 · ;;672) + ;;487) = ;;;1831
167	13, 32, 29, 145, 166	gcdi 17143	. . 3 ⊢ (;;;1831 gcd ;;672) = 1
168		eqid 2766	. . . . . 6 ⊢ ;;183 = ;;183
169	28	nn0cni 12526	. . . . . . 7 ⊢ ;67 ∈ ℂ
170	169	addridi 11407	. . . . . 6 ⊢ (;67 + 0) = ;67
171	102	addlidi 11408	. . . . . . . . 9 ⊢ (0 + 1) = 1
172	99, 171	oveq12i 7428	. . . . . . . 8 ⊢ ((1 · 1) + (0 + 1)) = (1 + 1)
173	172, 22	eqtri 2789	. . . . . . 7 ⊢ ((1 · 1) + (0 + 1)) = 2
174	88	oveq1i 7426	. . . . . . . 8 ⊢ ((1 · 8) + 7) = (8 + 7)
175	174, 114	eqtri 2789	. . . . . . 7 ⊢ ((1 · 8) + 7) = ;15
176	2, 3, 16, 27, 120, 98, 2, 14, 2, 173, 175	decma2c 12779	. . . . . 6 ⊢ ((1 · ;18) + (6 + 1)) = ;25
177		3cn 12332	. . . . . . . . 9 ⊢ 3 ∈ ℂ
178	177	mullidi 11224	. . . . . . . 8 ⊢ (1 · 3) = 3
179	178	oveq1i 7426	. . . . . . 7 ⊢ ((1 · 3) + 7) = (3 + 7)
180		7p3e10 12801	. . . . . . . 8 ⊢ (7 + 3) = ;10
181	106, 177, 180	addcomli 11412	. . . . . . 7 ⊢ (3 + 7) = ;10
182	179, 181	eqtri 2789	. . . . . 6 ⊢ ((1 · 3) + 7) = ;10
183	4, 9, 26, 27, 168, 170, 2, 16, 2, 176, 182	decma2c 12779	. . . . 5 ⊢ ((1 · ;;183) + (;67 + 0)) = ;;250
184	99	oveq1i 7426	. . . . . 6 ⊢ ((1 · 1) + 2) = (1 + 2)
185		1p2e3 12393	. . . . . 6 ⊢ (1 + 2) = 3
186	9	dec0h 12748	. . . . . 6 ⊢ 3 = ;03
187	184, 185, 186	3eqtri 2793	. . . . 5 ⊢ ((1 · 1) + 2) = ;03
188	10, 2, 28, 13, 23, 146, 2, 9, 16, 183, 187	decma2c 12779	. . . 4 ⊢ ((1 · ;;;1831) + ;;672) = ;;;2503
189	188, 12	eqtr4i 2792	. . 3 ⊢ ((1 · ;;;1831) + ;;672) = 𝑁
190	2, 29, 11, 167, 189	gcdi 17143	. 2 ⊢ (𝑁 gcd ;;;1831) = 1
191	8, 11, 20, 25, 190	gcdmodi 17144	1 ⊢ (((2↑;18) − 1) gcd 𝑁) = 1

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 ↔ wb 209 = wceq 1570 ∈ wcel 2146 class class class wbr 5112 (class class class)co 7416 0cc0 11110 1c1 11111 + caddc 11113 · cmul 11115 − cmin 11451 ℕcn 12243 2c2 12305 3c3 12306 4c4 12307 5c5 12308 6c6 12309 7c7 12310 8c8 12311 9c9 12312 ℕ₀cn0 12514 ℤcz 12601 cdc 12721 ↑cexp 14108 ∥ cdvds 16320 gcd cgcd 16562 ℙcprime 16739

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-sep 5260 ax-nul 5272 ax-pow 5339 ax-pr 5407 ax-un 7738 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188

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 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4491 df-pw 4567 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4876 df-iun 4961 df-br 5113 df-opab 5177 df-mpt 5196 df-tr 5222 df-id 5559 df-eprel 5564 df-po 5572 df-so 5573 df-fr 5617 df-we 5619 df-xp 5670 df-rel 5671 df-cnv 5672 df-co 5673 df-dm 5674 df-rn 5675 df-res 5676 df-ima 5677 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7865 df-1st 7988 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-2o 8456 df-er 8696 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-sup 9404 df-inf 9405 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11453 df-neg 11454 df-div 11882 df-nn 12244 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12515 df-z 12602 df-dec 12722 df-uz 12873 df-rp 13027 df-fz 13546 df-fl 13836 df-mod 13914 df-seq 14049 df-exp 14109 df-cj 15161 df-re 15162 df-im 15163 df-sqrt 15297 df-abs 15298 df-dvds 16321 df-gcd 16563 df-prm 16740

This theorem is used by: 2503prm 17210

W3C validator