2503lem3 - Metamath Proof Explorer

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Theorem 2503lem3 17234

Description: Lemma for 2503prm 17235. 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 12341	. . . 4 ⊢ 2 ∈ ℕ
2		1nn0 12547	. . . . 5 ⊢ 1 ∈ ℕ₀
3		8nn0 12554	. . . . 5 ⊢ 8 ∈ ℕ₀
4	2, 3	deccl 12754	. . . 4 ⊢ ;18 ∈ ℕ₀
5		nnexpcl 14141	. . . 4 ⊢ ((2 ∈ ℕ ∧ ;18 ∈ ℕ₀) → (2↑;18) ∈ ℕ)
6	1, 4, 5	mp2an 705	. . 3 ⊢ (2↑;18) ∈ ℕ
7		nnm1nn0 12572	. . 3 ⊢ ((2↑;18) ∈ ℕ → ((2↑;18) − 1) ∈ ℕ₀)
8	6, 7	ax-mp 5	. 2 ⊢ ((2↑;18) − 1) ∈ ℕ₀
9		3nn0 12549	. . . 4 ⊢ 3 ∈ ℕ₀
10	4, 9	deccl 12754	. . 3 ⊢ ;;183 ∈ ℕ₀
11	10, 2	deccl 12754	. 2 ⊢ ;;;1831 ∈ ℕ₀
12		2503prm.1	. . 3 ⊢ 𝑁 = ;;;2503
13		2nn0 12548	. . . . . 6 ⊢ 2 ∈ ℕ₀
14		5nn0 12551	. . . . . 6 ⊢ 5 ∈ ℕ₀
15	13, 14	deccl 12754	. . . . 5 ⊢ ;25 ∈ ℕ₀
16		0nn0 12546	. . . . 5 ⊢ 0 ∈ ℕ₀
17	15, 16	deccl 12754	. . . 4 ⊢ ;;250 ∈ ℕ₀
18		3nn 12347	. . . 4 ⊢ 3 ∈ ℕ
19	17, 18	decnncl 12763	. . 3 ⊢ ;;;2503 ∈ ℕ
20	12, 19	eqeltri 2856	. 2 ⊢ 𝑁 ∈ ℕ
21	12	2503lem1 17232	. . 3 ⊢ ((2↑;18) mod 𝑁) = (;;;1832 mod 𝑁)
22		1p1e2 12391	. . . 4 ⊢ (1 + 1) = 2
23		eqid 2760	. . . 4 ⊢ ;;;1831 = ;;;1831
24	10, 2, 22, 23	decsuc 12775	. . 3 ⊢ (;;;1831 + 1) = ;;;1832
25	20, 6, 2, 11, 21, 24	modsubi 17167	. 2 ⊢ (((2↑;18) − 1) mod 𝑁) = (;;;1831 mod 𝑁)
26		6nn0 12552	. . . . 5 ⊢ 6 ∈ ℕ₀
27		7nn0 12553	. . . . 5 ⊢ 7 ∈ ℕ₀
28	26, 27	deccl 12754	. . . 4 ⊢ ;67 ∈ ℕ₀
29	28, 13	deccl 12754	. . 3 ⊢ ;;672 ∈ ℕ₀
30		4nn0 12550	. . . . . 6 ⊢ 4 ∈ ℕ₀
31	30, 3	deccl 12754	. . . . 5 ⊢ ;48 ∈ ℕ₀
32	31, 27	deccl 12754	. . . 4 ⊢ ;;487 ∈ ℕ₀
33	4, 14	deccl 12754	. . . . 5 ⊢ ;;185 ∈ ℕ₀
34	2, 2	deccl 12754	. . . . . . 7 ⊢ ;11 ∈ ℕ₀
35	34, 27	deccl 12754	. . . . . 6 ⊢ ;;117 ∈ ℕ₀
36	26, 3	deccl 12754	. . . . . . 7 ⊢ ;68 ∈ ℕ₀
37		9nn0 12555	. . . . . . . . 9 ⊢ 9 ∈ ℕ₀
38	30, 37	deccl 12754	. . . . . . . 8 ⊢ ;49 ∈ ℕ₀
39	2, 37	deccl 12754	. . . . . . . . 9 ⊢ ;19 ∈ ℕ₀
40	38	nn0zi 12646	. . . . . . . . . . 11 ⊢ ;49 ∈ ℤ
41	39	nn0zi 12646	. . . . . . . . . . 11 ⊢ ;19 ∈ ℤ
42		gcdcom 16606	. . . . . . . . . . 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 12366	. . . . . . . . . . . . 13 ⊢ 9 ∈ ℕ
45	2, 44	decnncl 12763	. . . . . . . . . . . 12 ⊢ ;19 ∈ ℕ
46		1nn 12271	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℕ
47	2, 46	decnncl 12763	. . . . . . . . . . . 12 ⊢ ;11 ∈ ℕ
48		eqid 2760	. . . . . . . . . . . . 13 ⊢ ;19 = ;19
49		eqid 2760	. . . . . . . . . . . . 13 ⊢ ;11 = ;11
50		2cn 12343	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℂ
51	50	mullidi 11241	. . . . . . . . . . . . . . 15 ⊢ (1 · 2) = 2
52	51, 22	oveq12i 7426	. . . . . . . . . . . . . 14 ⊢ ((1 · 2) + (1 + 1)) = (2 + 2)
53		2p2e4 12402	. . . . . . . . . . . . . 14 ⊢ (2 + 2) = 4
54	52, 53	eqtri 2783	. . . . . . . . . . . . 13 ⊢ ((1 · 2) + (1 + 1)) = 4
55		8p1e9 12417	. . . . . . . . . . . . . 14 ⊢ (8 + 1) = 9
56		9t2e18 12866	. . . . . . . . . . . . . 14 ⊢ (9 · 2) = ;18
57	2, 3, 55, 56	decsuc 12775	. . . . . . . . . . . . 13 ⊢ ((9 · 2) + 1) = ;19
58	2, 37, 2, 2, 48, 49, 13, 37, 2, 54, 57	decmac 12796	. . . . . . . . . . . 12 ⊢ ((;19 · 2) + ;11) = ;49
59		1lt9 12476	. . . . . . . . . . . . 13 ⊢ 1 < 9
60	2, 2, 44, 59	declt 12772	. . . . . . . . . . . 12 ⊢ ;11 < ;19
61	45, 13, 47, 58, 60	ndvdsi 16505	. . . . . . . . . . 11 ⊢ ¬ ;19 ∥ ;49
62		19prm 17213	. . . . . . . . . . . 12 ⊢ ;19 ∈ ℙ
63		coprm 16805	. . . . . . . . . . . 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 2783	. . . . . . . . 9 ⊢ (;49 gcd ;19) = 1
67		eqid 2760	. . . . . . . . . 10 ⊢ ;49 = ;49
68		4cn 12353	. . . . . . . . . . . . 13 ⊢ 4 ∈ ℂ
69	68	mullidi 11241	. . . . . . . . . . . 12 ⊢ (1 · 4) = 4
70	69, 22	oveq12i 7426	. . . . . . . . . . 11 ⊢ ((1 · 4) + (1 + 1)) = (4 + 2)
71		4p2e6 12420	. . . . . . . . . . 11 ⊢ (4 + 2) = 6
72	70, 71	eqtri 2783	. . . . . . . . . 10 ⊢ ((1 · 4) + (1 + 1)) = 6
73		9cn 12368	. . . . . . . . . . . . 13 ⊢ 9 ∈ ℂ
74	73	mullidi 11241	. . . . . . . . . . . 12 ⊢ (1 · 9) = 9
75	74	oveq1i 7424	. . . . . . . . . . 11 ⊢ ((1 · 9) + 9) = (9 + 9)
76		9p9e18 12838	. . . . . . . . . . 11 ⊢ (9 + 9) = ;18
77	75, 76	eqtri 2783	. . . . . . . . . 10 ⊢ ((1 · 9) + 9) = ;18
78	30, 37, 2, 37, 67, 48, 2, 3, 2, 72, 77	decma2c 12797	. . . . . . . . 9 ⊢ ((1 · ;49) + ;19) = ;68
79	2, 39, 38, 66, 78	gcdi 17168	. . . . . . . 8 ⊢ (;68 gcd ;49) = 1
80		eqid 2760	. . . . . . . . 9 ⊢ ;68 = ;68
81		6cn 12359	. . . . . . . . . . . 12 ⊢ 6 ∈ ℂ
82	81	mullidi 11241	. . . . . . . . . . 11 ⊢ (1 · 6) = 6
83		4p1e5 12413	. . . . . . . . . . 11 ⊢ (4 + 1) = 5
84	82, 83	oveq12i 7426	. . . . . . . . . 10 ⊢ ((1 · 6) + (4 + 1)) = (6 + 5)
85		6p5e11 12817	. . . . . . . . . 10 ⊢ (6 + 5) = ;11
86	84, 85	eqtri 2783	. . . . . . . . 9 ⊢ ((1 · 6) + (4 + 1)) = ;11
87		8cn 12365	. . . . . . . . . . . 12 ⊢ 8 ∈ ℂ
88	87	mullidi 11241	. . . . . . . . . . 11 ⊢ (1 · 8) = 8
89	88	oveq1i 7424	. . . . . . . . . 10 ⊢ ((1 · 8) + 9) = (8 + 9)
90		9p8e17 12837	. . . . . . . . . . 11 ⊢ (9 + 8) = ;17
91	73, 87, 90	addcomli 11429	. . . . . . . . . 10 ⊢ (8 + 9) = ;17
92	89, 91	eqtri 2783	. . . . . . . . 9 ⊢ ((1 · 8) + 9) = ;17
93	26, 3, 30, 37, 80, 67, 2, 27, 2, 86, 92	decma2c 12797	. . . . . . . 8 ⊢ ((1 · ;68) + ;49) = ;;117
94	2, 38, 36, 79, 93	gcdi 17168	. . . . . . 7 ⊢ (;;117 gcd ;68) = 1
95		eqid 2760	. . . . . . . 8 ⊢ ;;117 = ;;117
96		6p1e7 12415	. . . . . . . . . 10 ⊢ (6 + 1) = 7
97	27	dec0h 12766	. . . . . . . . . 10 ⊢ 7 = ;07
98	96, 97	eqtri 2783	. . . . . . . . 9 ⊢ (6 + 1) = ;07
99		1t1e1 12429	. . . . . . . . . . 11 ⊢ (1 · 1) = 1
100		00id 11412	. . . . . . . . . . 11 ⊢ (0 + 0) = 0
101	99, 100	oveq12i 7426	. . . . . . . . . 10 ⊢ ((1 · 1) + (0 + 0)) = (1 + 0)
102		ax-1cn 11185	. . . . . . . . . . 11 ⊢ 1 ∈ ℂ
103	102	addridi 11424	. . . . . . . . . 10 ⊢ (1 + 0) = 1
104	101, 103	eqtri 2783	. . . . . . . . 9 ⊢ ((1 · 1) + (0 + 0)) = 1
105	99	oveq1i 7424	. . . . . . . . . 10 ⊢ ((1 · 1) + 7) = (1 + 7)
106		7cn 12362	. . . . . . . . . . 11 ⊢ 7 ∈ ℂ
107		7p1e8 12416	. . . . . . . . . . 11 ⊢ (7 + 1) = 8
108	106, 102, 107	addcomli 11429	. . . . . . . . . 10 ⊢ (1 + 7) = 8
109	3	dec0h 12766	. . . . . . . . . 10 ⊢ 8 = ;08
110	105, 108, 109	3eqtri 2787	. . . . . . . . 9 ⊢ ((1 · 1) + 7) = ;08
111	2, 2, 16, 27, 49, 98, 2, 3, 16, 104, 110	decma2c 12797	. . . . . . . 8 ⊢ ((1 · ;11) + (6 + 1)) = ;18
112	106	mullidi 11241	. . . . . . . . . 10 ⊢ (1 · 7) = 7
113	112	oveq1i 7424	. . . . . . . . 9 ⊢ ((1 · 7) + 8) = (7 + 8)
114		8p7e15 12829	. . . . . . . . . 10 ⊢ (8 + 7) = ;15
115	87, 106, 114	addcomli 11429	. . . . . . . . 9 ⊢ (7 + 8) = ;15
116	113, 115	eqtri 2783	. . . . . . . 8 ⊢ ((1 · 7) + 8) = ;15
117	34, 27, 26, 3, 95, 80, 2, 14, 2, 111, 116	decma2c 12797	. . . . . . 7 ⊢ ((1 · ;;117) + ;68) = ;;185
118	2, 36, 35, 94, 117	gcdi 17168	. . . . . 6 ⊢ (;;185 gcd ;;117) = 1
119		eqid 2760	. . . . . . 7 ⊢ ;;185 = ;;185
120		eqid 2760	. . . . . . . 8 ⊢ ;18 = ;18
121	2, 2, 22, 49	decsuc 12775	. . . . . . . 8 ⊢ (;11 + 1) = ;12
122		2t1e2 12430	. . . . . . . . . 10 ⊢ (2 · 1) = 2
123	122, 22	oveq12i 7426	. . . . . . . . 9 ⊢ ((2 · 1) + (1 + 1)) = (2 + 2)
124	123, 53	eqtri 2783	. . . . . . . 8 ⊢ ((2 · 1) + (1 + 1)) = 4
125		8t2e16 12859	. . . . . . . . . 10 ⊢ (8 · 2) = ;16
126	87, 50, 125	mulcomli 11245	. . . . . . . . 9 ⊢ (2 · 8) = ;16
127		6p2e8 12426	. . . . . . . . 9 ⊢ (6 + 2) = 8
128	2, 26, 13, 126, 127	decaddi 12804	. . . . . . . 8 ⊢ ((2 · 8) + 2) = ;18
129	2, 3, 2, 13, 120, 121, 13, 3, 2, 124, 128	decma2c 12797	. . . . . . 7 ⊢ ((2 · ;18) + (;11 + 1)) = ;48
130		5cn 12356	. . . . . . . . 9 ⊢ 5 ∈ ℂ
131		5t2e10 12844	. . . . . . . . 9 ⊢ (5 · 2) = ;10
132	130, 50, 131	mulcomli 11245	. . . . . . . 8 ⊢ (2 · 5) = ;10
133	106	addlidi 11425	. . . . . . . 8 ⊢ (0 + 7) = 7
134	2, 16, 27, 132, 133	decaddi 12804	. . . . . . 7 ⊢ ((2 · 5) + 7) = ;17
135	4, 14, 34, 27, 119, 95, 13, 27, 2, 129, 134	decma2c 12797	. . . . . 6 ⊢ ((2 · ;;185) + ;;117) = ;;487
136	13, 35, 33, 118, 135	gcdi 17168	. . . . 5 ⊢ (;;487 gcd ;;185) = 1
137		eqid 2760	. . . . . 6 ⊢ ;;487 = ;;487
138		eqid 2760	. . . . . . 7 ⊢ ;48 = ;48
139	2, 3, 55, 120	decsuc 12775	. . . . . . 7 ⊢ (;18 + 1) = ;19
140	30, 3, 2, 37, 138, 139, 2, 27, 2, 72, 92	decma2c 12797	. . . . . 6 ⊢ ((1 · ;48) + (;18 + 1)) = ;67
141	112	oveq1i 7424	. . . . . . 7 ⊢ ((1 · 7) + 5) = (7 + 5)
142		7p5e12 12821	. . . . . . 7 ⊢ (7 + 5) = ;12
143	141, 142	eqtri 2783	. . . . . 6 ⊢ ((1 · 7) + 5) = ;12
144	31, 27, 4, 14, 137, 119, 2, 13, 2, 140, 143	decma2c 12797	. . . . 5 ⊢ ((1 · ;;487) + ;;185) = ;;672
145	2, 33, 32, 136, 144	gcdi 17168	. . . 4 ⊢ (;;672 gcd ;;487) = 1
146		eqid 2760	. . . . 5 ⊢ ;;672 = ;;672
147		eqid 2760	. . . . . 6 ⊢ ;67 = ;67
148	30, 3, 55, 138	decsuc 12775	. . . . . 6 ⊢ (;48 + 1) = ;49
149	71	oveq2i 7425	. . . . . . 7 ⊢ ((2 · 6) + (4 + 2)) = ((2 · 6) + 6)
150		6t2e12 12848	. . . . . . . . 9 ⊢ (6 · 2) = ;12
151	81, 50, 150	mulcomli 11245	. . . . . . . 8 ⊢ (2 · 6) = ;12
152	81, 50, 127	addcomli 11429	. . . . . . . 8 ⊢ (2 + 6) = 8
153	2, 13, 26, 151, 152	decaddi 12804	. . . . . . 7 ⊢ ((2 · 6) + 6) = ;18
154	149, 153	eqtri 2783	. . . . . 6 ⊢ ((2 · 6) + (4 + 2)) = ;18
155		7t2e14 12853	. . . . . . . 8 ⊢ (7 · 2) = ;14
156	106, 50, 155	mulcomli 11245	. . . . . . 7 ⊢ (2 · 7) = ;14
157		9p4e13 12833	. . . . . . . 8 ⊢ (9 + 4) = ;13
158	73, 68, 157	addcomli 11429	. . . . . . 7 ⊢ (4 + 9) = ;13
159	2, 30, 37, 156, 22, 9, 158	decaddci 12805	. . . . . 6 ⊢ ((2 · 7) + 9) = ;23
160	26, 27, 30, 37, 147, 148, 13, 9, 13, 154, 159	decma2c 12797	. . . . 5 ⊢ ((2 · ;67) + (;48 + 1)) = ;;183
161		2t2e4 12431	. . . . . . 7 ⊢ (2 · 2) = 4
162	161	oveq1i 7424	. . . . . 6 ⊢ ((2 · 2) + 7) = (4 + 7)
163		7p4e11 12820	. . . . . . 7 ⊢ (7 + 4) = ;11
164	106, 68, 163	addcomli 11429	. . . . . 6 ⊢ (4 + 7) = ;11
165	162, 164	eqtri 2783	. . . . 5 ⊢ ((2 · 2) + 7) = ;11
166	28, 13, 31, 27, 146, 137, 13, 2, 2, 160, 165	decma2c 12797	. . . 4 ⊢ ((2 · ;;672) + ;;487) = ;;;1831
167	13, 32, 29, 145, 166	gcdi 17168	. . 3 ⊢ (;;;1831 gcd ;;672) = 1
168		eqid 2760	. . . . . 6 ⊢ ;;183 = ;;183
169	28	nn0cni 12543	. . . . . . 7 ⊢ ;67 ∈ ℂ
170	169	addridi 11424	. . . . . 6 ⊢ (;67 + 0) = ;67
171	102	addlidi 11425	. . . . . . . . 9 ⊢ (0 + 1) = 1
172	99, 171	oveq12i 7426	. . . . . . . 8 ⊢ ((1 · 1) + (0 + 1)) = (1 + 1)
173	172, 22	eqtri 2783	. . . . . . 7 ⊢ ((1 · 1) + (0 + 1)) = 2
174	88	oveq1i 7424	. . . . . . . 8 ⊢ ((1 · 8) + 7) = (8 + 7)
175	174, 114	eqtri 2783	. . . . . . 7 ⊢ ((1 · 8) + 7) = ;15
176	2, 3, 16, 27, 120, 98, 2, 14, 2, 173, 175	decma2c 12797	. . . . . 6 ⊢ ((1 · ;18) + (6 + 1)) = ;25
177		3cn 12349	. . . . . . . . 9 ⊢ 3 ∈ ℂ
178	177	mullidi 11241	. . . . . . . 8 ⊢ (1 · 3) = 3
179	178	oveq1i 7424	. . . . . . 7 ⊢ ((1 · 3) + 7) = (3 + 7)
180		7p3e10 12819	. . . . . . . 8 ⊢ (7 + 3) = ;10
181	106, 177, 180	addcomli 11429	. . . . . . 7 ⊢ (3 + 7) = ;10
182	179, 181	eqtri 2783	. . . . . 6 ⊢ ((1 · 3) + 7) = ;10
183	4, 9, 26, 27, 168, 170, 2, 16, 2, 176, 182	decma2c 12797	. . . . 5 ⊢ ((1 · ;;183) + (;67 + 0)) = ;;250
184	99	oveq1i 7424	. . . . . 6 ⊢ ((1 · 1) + 2) = (1 + 2)
185		1p2e3 12410	. . . . . 6 ⊢ (1 + 2) = 3
186	9	dec0h 12766	. . . . . 6 ⊢ 3 = ;03
187	184, 185, 186	3eqtri 2787	. . . . 5 ⊢ ((1 · 1) + 2) = ;03
188	10, 2, 28, 13, 23, 146, 2, 9, 16, 183, 187	decma2c 12797	. . . 4 ⊢ ((1 · ;;;1831) + ;;672) = ;;;2503
189	188, 12	eqtr4i 2786	. . 3 ⊢ ((1 · ;;;1831) + ;;672) = 𝑁
190	2, 29, 11, 167, 189	gcdi 17168	. 2 ⊢ (𝑁 gcd ;;;1831) = 1
191	8, 11, 20, 25, 190	gcdmodi 17169	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 2145 class class class wbr 5103 (class class class)co 7414 0cc0 11127 1c1 11128 + caddc 11130 · cmul 11132 − cmin 11468 ℕcn 12260 2c2 12322 3c3 12323 4c4 12324 5c5 12325 6c6 12326 7c7 12327 8c8 12328 9c9 12329 ℕ₀cn0 12531 ℤcz 12618 cdc 12739 ↑cexp 14128 ∥ cdvds 16345 gcd cgcd 16587 ℙcprime 16764

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 2732 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205

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 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 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 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-1o 8458 df-2o 8459 df-er 8699 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-sup 9415 df-inf 9416 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-div 11899 df-nn 12261 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12532 df-z 12619 df-dec 12740 df-uz 12891 df-rp 13046 df-fz 13565 df-fl 13856 df-mod 13934 df-seq 14069 df-exp 14129 df-cj 15189 df-re 15190 df-im 15191 df-sqrt 15325 df-abs 15326 df-dvds 16346 df-gcd 16588 df-prm 16765

This theorem is used by: 2503prm 17235

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