2503lem3 - Metamath Proof Explorer

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Theorem 2503lem3 16655

Description: Lemma for 2503prm 16656. 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 11868	. . . 4 ⊢ 2 ∈ ℕ
2		1nn0 12071	. . . . 5 ⊢ 1 ∈ ℕ₀
3		8nn0 12078	. . . . 5 ⊢ 8 ∈ ℕ₀
4	2, 3	deccl 12273	. . . 4 ⊢ ;18 ∈ ℕ₀
5		nnexpcl 13613	. . . 4 ⊢ ((2 ∈ ℕ ∧ ;18 ∈ ℕ₀) → (2↑;18) ∈ ℕ)
6	1, 4, 5	mp2an 692	. . 3 ⊢ (2↑;18) ∈ ℕ
7		nnm1nn0 12096	. . 3 ⊢ ((2↑;18) ∈ ℕ → ((2↑;18) − 1) ∈ ℕ₀)
8	6, 7	ax-mp 5	. 2 ⊢ ((2↑;18) − 1) ∈ ℕ₀
9		3nn0 12073	. . . 4 ⊢ 3 ∈ ℕ₀
10	4, 9	deccl 12273	. . 3 ⊢ ;;183 ∈ ℕ₀
11	10, 2	deccl 12273	. 2 ⊢ ;;;1831 ∈ ℕ₀
12		2503prm.1	. . 3 ⊢ 𝑁 = ;;;2503
13		2nn0 12072	. . . . . 6 ⊢ 2 ∈ ℕ₀
14		5nn0 12075	. . . . . 6 ⊢ 5 ∈ ℕ₀
15	13, 14	deccl 12273	. . . . 5 ⊢ ;25 ∈ ℕ₀
16		0nn0 12070	. . . . 5 ⊢ 0 ∈ ℕ₀
17	15, 16	deccl 12273	. . . 4 ⊢ ;;250 ∈ ℕ₀
18		3nn 11874	. . . 4 ⊢ 3 ∈ ℕ
19	17, 18	decnncl 12278	. . 3 ⊢ ;;;2503 ∈ ℕ
20	12, 19	eqeltri 2827	. 2 ⊢ 𝑁 ∈ ℕ
21	12	2503lem1 16653	. . 3 ⊢ ((2↑;18) mod 𝑁) = (;;;1832 mod 𝑁)
22		1p1e2 11920	. . . 4 ⊢ (1 + 1) = 2
23		eqid 2736	. . . 4 ⊢ ;;;1831 = ;;;1831
24	10, 2, 22, 23	decsuc 12289	. . 3 ⊢ (;;;1831 + 1) = ;;;1832
25	20, 6, 2, 11, 21, 24	modsubi 16588	. 2 ⊢ (((2↑;18) − 1) mod 𝑁) = (;;;1831 mod 𝑁)
26		6nn0 12076	. . . . 5 ⊢ 6 ∈ ℕ₀
27		7nn0 12077	. . . . 5 ⊢ 7 ∈ ℕ₀
28	26, 27	deccl 12273	. . . 4 ⊢ ;67 ∈ ℕ₀
29	28, 13	deccl 12273	. . 3 ⊢ ;;672 ∈ ℕ₀
30		4nn0 12074	. . . . . 6 ⊢ 4 ∈ ℕ₀
31	30, 3	deccl 12273	. . . . 5 ⊢ ;48 ∈ ℕ₀
32	31, 27	deccl 12273	. . . 4 ⊢ ;;487 ∈ ℕ₀
33	4, 14	deccl 12273	. . . . 5 ⊢ ;;185 ∈ ℕ₀
34	2, 2	deccl 12273	. . . . . . 7 ⊢ ;11 ∈ ℕ₀
35	34, 27	deccl 12273	. . . . . 6 ⊢ ;;117 ∈ ℕ₀
36	26, 3	deccl 12273	. . . . . . 7 ⊢ ;68 ∈ ℕ₀
37		9nn0 12079	. . . . . . . . 9 ⊢ 9 ∈ ℕ₀
38	30, 37	deccl 12273	. . . . . . . 8 ⊢ ;49 ∈ ℕ₀
39	2, 37	deccl 12273	. . . . . . . . 9 ⊢ ;19 ∈ ℕ₀
40	38	nn0zi 12167	. . . . . . . . . . 11 ⊢ ;49 ∈ ℤ
41	39	nn0zi 12167	. . . . . . . . . . 11 ⊢ ;19 ∈ ℤ
42		gcdcom 16035	. . . . . . . . . . 11 ⊢ ((;49 ∈ ℤ ∧ ;19 ∈ ℤ) → (;49 gcd ;19) = (;19 gcd ;49))
43	40, 41, 42	mp2an 692	. . . . . . . . . 10 ⊢ (;49 gcd ;19) = (;19 gcd ;49)
44		9nn 11893	. . . . . . . . . . . . 13 ⊢ 9 ∈ ℕ
45	2, 44	decnncl 12278	. . . . . . . . . . . 12 ⊢ ;19 ∈ ℕ
46		1nn 11806	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℕ
47	2, 46	decnncl 12278	. . . . . . . . . . . 12 ⊢ ;11 ∈ ℕ
48		eqid 2736	. . . . . . . . . . . . 13 ⊢ ;19 = ;19
49		eqid 2736	. . . . . . . . . . . . 13 ⊢ ;11 = ;11
50		2cn 11870	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℂ
51	50	mulid2i 10803	. . . . . . . . . . . . . . 15 ⊢ (1 · 2) = 2
52	51, 22	oveq12i 7203	. . . . . . . . . . . . . 14 ⊢ ((1 · 2) + (1 + 1)) = (2 + 2)
53		2p2e4 11930	. . . . . . . . . . . . . 14 ⊢ (2 + 2) = 4
54	52, 53	eqtri 2759	. . . . . . . . . . . . 13 ⊢ ((1 · 2) + (1 + 1)) = 4
55		8p1e9 11945	. . . . . . . . . . . . . 14 ⊢ (8 + 1) = 9
56		9t2e18 12380	. . . . . . . . . . . . . 14 ⊢ (9 · 2) = ;18
57	2, 3, 55, 56	decsuc 12289	. . . . . . . . . . . . 13 ⊢ ((9 · 2) + 1) = ;19
58	2, 37, 2, 2, 48, 49, 13, 37, 2, 54, 57	decmac 12310	. . . . . . . . . . . 12 ⊢ ((;19 · 2) + ;11) = ;49
59		1lt9 12001	. . . . . . . . . . . . 13 ⊢ 1 < 9
60	2, 2, 44, 59	declt 12286	. . . . . . . . . . . 12 ⊢ ;11 < ;19
61	45, 13, 47, 58, 60	ndvdsi 15936	. . . . . . . . . . 11 ⊢ ¬ ;19 ∥ ;49
62		19prm 16634	. . . . . . . . . . . 12 ⊢ ;19 ∈ ℙ
63		coprm 16231	. . . . . . . . . . . 12 ⊢ ((;19 ∈ ℙ ∧ ;49 ∈ ℤ) → (¬ ;19 ∥ ;49 ↔ (;19 gcd ;49) = 1))
64	62, 40, 63	mp2an 692	. . . . . . . . . . 11 ⊢ (¬ ;19 ∥ ;49 ↔ (;19 gcd ;49) = 1)
65	61, 64	mpbi 233	. . . . . . . . . 10 ⊢ (;19 gcd ;49) = 1
66	43, 65	eqtri 2759	. . . . . . . . 9 ⊢ (;49 gcd ;19) = 1
67		eqid 2736	. . . . . . . . . 10 ⊢ ;49 = ;49
68		4cn 11880	. . . . . . . . . . . . 13 ⊢ 4 ∈ ℂ
69	68	mulid2i 10803	. . . . . . . . . . . 12 ⊢ (1 · 4) = 4
70	69, 22	oveq12i 7203	. . . . . . . . . . 11 ⊢ ((1 · 4) + (1 + 1)) = (4 + 2)
71		4p2e6 11948	. . . . . . . . . . 11 ⊢ (4 + 2) = 6
72	70, 71	eqtri 2759	. . . . . . . . . 10 ⊢ ((1 · 4) + (1 + 1)) = 6
73		9cn 11895	. . . . . . . . . . . . 13 ⊢ 9 ∈ ℂ
74	73	mulid2i 10803	. . . . . . . . . . . 12 ⊢ (1 · 9) = 9
75	74	oveq1i 7201	. . . . . . . . . . 11 ⊢ ((1 · 9) + 9) = (9 + 9)
76		9p9e18 12352	. . . . . . . . . . 11 ⊢ (9 + 9) = ;18
77	75, 76	eqtri 2759	. . . . . . . . . 10 ⊢ ((1 · 9) + 9) = ;18
78	30, 37, 2, 37, 67, 48, 2, 3, 2, 72, 77	decma2c 12311	. . . . . . . . 9 ⊢ ((1 · ;49) + ;19) = ;68
79	2, 39, 38, 66, 78	gcdi 16589	. . . . . . . 8 ⊢ (;68 gcd ;49) = 1
80		eqid 2736	. . . . . . . . 9 ⊢ ;68 = ;68
81		6cn 11886	. . . . . . . . . . . 12 ⊢ 6 ∈ ℂ
82	81	mulid2i 10803	. . . . . . . . . . 11 ⊢ (1 · 6) = 6
83		4p1e5 11941	. . . . . . . . . . 11 ⊢ (4 + 1) = 5
84	82, 83	oveq12i 7203	. . . . . . . . . 10 ⊢ ((1 · 6) + (4 + 1)) = (6 + 5)
85		6p5e11 12331	. . . . . . . . . 10 ⊢ (6 + 5) = ;11
86	84, 85	eqtri 2759	. . . . . . . . 9 ⊢ ((1 · 6) + (4 + 1)) = ;11
87		8cn 11892	. . . . . . . . . . . 12 ⊢ 8 ∈ ℂ
88	87	mulid2i 10803	. . . . . . . . . . 11 ⊢ (1 · 8) = 8
89	88	oveq1i 7201	. . . . . . . . . 10 ⊢ ((1 · 8) + 9) = (8 + 9)
90		9p8e17 12351	. . . . . . . . . . 11 ⊢ (9 + 8) = ;17
91	73, 87, 90	addcomli 10989	. . . . . . . . . 10 ⊢ (8 + 9) = ;17
92	89, 91	eqtri 2759	. . . . . . . . 9 ⊢ ((1 · 8) + 9) = ;17
93	26, 3, 30, 37, 80, 67, 2, 27, 2, 86, 92	decma2c 12311	. . . . . . . 8 ⊢ ((1 · ;68) + ;49) = ;;117
94	2, 38, 36, 79, 93	gcdi 16589	. . . . . . 7 ⊢ (;;117 gcd ;68) = 1
95		eqid 2736	. . . . . . . 8 ⊢ ;;117 = ;;117
96		6p1e7 11943	. . . . . . . . . 10 ⊢ (6 + 1) = 7
97	27	dec0h 12280	. . . . . . . . . 10 ⊢ 7 = ;07
98	96, 97	eqtri 2759	. . . . . . . . 9 ⊢ (6 + 1) = ;07
99		1t1e1 11957	. . . . . . . . . . 11 ⊢ (1 · 1) = 1
100		00id 10972	. . . . . . . . . . 11 ⊢ (0 + 0) = 0
101	99, 100	oveq12i 7203	. . . . . . . . . 10 ⊢ ((1 · 1) + (0 + 0)) = (1 + 0)
102		ax-1cn 10752	. . . . . . . . . . 11 ⊢ 1 ∈ ℂ
103	102	addid1i 10984	. . . . . . . . . 10 ⊢ (1 + 0) = 1
104	101, 103	eqtri 2759	. . . . . . . . 9 ⊢ ((1 · 1) + (0 + 0)) = 1
105	99	oveq1i 7201	. . . . . . . . . 10 ⊢ ((1 · 1) + 7) = (1 + 7)
106		7cn 11889	. . . . . . . . . . 11 ⊢ 7 ∈ ℂ
107		7p1e8 11944	. . . . . . . . . . 11 ⊢ (7 + 1) = 8
108	106, 102, 107	addcomli 10989	. . . . . . . . . 10 ⊢ (1 + 7) = 8
109	3	dec0h 12280	. . . . . . . . . 10 ⊢ 8 = ;08
110	105, 108, 109	3eqtri 2763	. . . . . . . . 9 ⊢ ((1 · 1) + 7) = ;08
111	2, 2, 16, 27, 49, 98, 2, 3, 16, 104, 110	decma2c 12311	. . . . . . . 8 ⊢ ((1 · ;11) + (6 + 1)) = ;18
112	106	mulid2i 10803	. . . . . . . . . 10 ⊢ (1 · 7) = 7
113	112	oveq1i 7201	. . . . . . . . 9 ⊢ ((1 · 7) + 8) = (7 + 8)
114		8p7e15 12343	. . . . . . . . . 10 ⊢ (8 + 7) = ;15
115	87, 106, 114	addcomli 10989	. . . . . . . . 9 ⊢ (7 + 8) = ;15
116	113, 115	eqtri 2759	. . . . . . . 8 ⊢ ((1 · 7) + 8) = ;15
117	34, 27, 26, 3, 95, 80, 2, 14, 2, 111, 116	decma2c 12311	. . . . . . 7 ⊢ ((1 · ;;117) + ;68) = ;;185
118	2, 36, 35, 94, 117	gcdi 16589	. . . . . 6 ⊢ (;;185 gcd ;;117) = 1
119		eqid 2736	. . . . . . 7 ⊢ ;;185 = ;;185
120		eqid 2736	. . . . . . . 8 ⊢ ;18 = ;18
121	2, 2, 22, 49	decsuc 12289	. . . . . . . 8 ⊢ (;11 + 1) = ;12
122		2t1e2 11958	. . . . . . . . . 10 ⊢ (2 · 1) = 2
123	122, 22	oveq12i 7203	. . . . . . . . 9 ⊢ ((2 · 1) + (1 + 1)) = (2 + 2)
124	123, 53	eqtri 2759	. . . . . . . 8 ⊢ ((2 · 1) + (1 + 1)) = 4
125		8t2e16 12373	. . . . . . . . . 10 ⊢ (8 · 2) = ;16
126	87, 50, 125	mulcomli 10807	. . . . . . . . 9 ⊢ (2 · 8) = ;16
127		6p2e8 11954	. . . . . . . . 9 ⊢ (6 + 2) = 8
128	2, 26, 13, 126, 127	decaddi 12318	. . . . . . . 8 ⊢ ((2 · 8) + 2) = ;18
129	2, 3, 2, 13, 120, 121, 13, 3, 2, 124, 128	decma2c 12311	. . . . . . 7 ⊢ ((2 · ;18) + (;11 + 1)) = ;48
130		5cn 11883	. . . . . . . . 9 ⊢ 5 ∈ ℂ
131		5t2e10 12358	. . . . . . . . 9 ⊢ (5 · 2) = ;10
132	130, 50, 131	mulcomli 10807	. . . . . . . 8 ⊢ (2 · 5) = ;10
133	106	addid2i 10985	. . . . . . . 8 ⊢ (0 + 7) = 7
134	2, 16, 27, 132, 133	decaddi 12318	. . . . . . 7 ⊢ ((2 · 5) + 7) = ;17
135	4, 14, 34, 27, 119, 95, 13, 27, 2, 129, 134	decma2c 12311	. . . . . 6 ⊢ ((2 · ;;185) + ;;117) = ;;487
136	13, 35, 33, 118, 135	gcdi 16589	. . . . 5 ⊢ (;;487 gcd ;;185) = 1
137		eqid 2736	. . . . . 6 ⊢ ;;487 = ;;487
138		eqid 2736	. . . . . . 7 ⊢ ;48 = ;48
139	2, 3, 55, 120	decsuc 12289	. . . . . . 7 ⊢ (;18 + 1) = ;19
140	30, 3, 2, 37, 138, 139, 2, 27, 2, 72, 92	decma2c 12311	. . . . . 6 ⊢ ((1 · ;48) + (;18 + 1)) = ;67
141	112	oveq1i 7201	. . . . . . 7 ⊢ ((1 · 7) + 5) = (7 + 5)
142		7p5e12 12335	. . . . . . 7 ⊢ (7 + 5) = ;12
143	141, 142	eqtri 2759	. . . . . 6 ⊢ ((1 · 7) + 5) = ;12
144	31, 27, 4, 14, 137, 119, 2, 13, 2, 140, 143	decma2c 12311	. . . . 5 ⊢ ((1 · ;;487) + ;;185) = ;;672
145	2, 33, 32, 136, 144	gcdi 16589	. . . 4 ⊢ (;;672 gcd ;;487) = 1
146		eqid 2736	. . . . 5 ⊢ ;;672 = ;;672
147		eqid 2736	. . . . . 6 ⊢ ;67 = ;67
148	30, 3, 55, 138	decsuc 12289	. . . . . 6 ⊢ (;48 + 1) = ;49
149	71	oveq2i 7202	. . . . . . 7 ⊢ ((2 · 6) + (4 + 2)) = ((2 · 6) + 6)
150		6t2e12 12362	. . . . . . . . 9 ⊢ (6 · 2) = ;12
151	81, 50, 150	mulcomli 10807	. . . . . . . 8 ⊢ (2 · 6) = ;12
152	81, 50, 127	addcomli 10989	. . . . . . . 8 ⊢ (2 + 6) = 8
153	2, 13, 26, 151, 152	decaddi 12318	. . . . . . 7 ⊢ ((2 · 6) + 6) = ;18
154	149, 153	eqtri 2759	. . . . . 6 ⊢ ((2 · 6) + (4 + 2)) = ;18
155		7t2e14 12367	. . . . . . . 8 ⊢ (7 · 2) = ;14
156	106, 50, 155	mulcomli 10807	. . . . . . 7 ⊢ (2 · 7) = ;14
157		9p4e13 12347	. . . . . . . 8 ⊢ (9 + 4) = ;13
158	73, 68, 157	addcomli 10989	. . . . . . 7 ⊢ (4 + 9) = ;13
159	2, 30, 37, 156, 22, 9, 158	decaddci 12319	. . . . . 6 ⊢ ((2 · 7) + 9) = ;23
160	26, 27, 30, 37, 147, 148, 13, 9, 13, 154, 159	decma2c 12311	. . . . 5 ⊢ ((2 · ;67) + (;48 + 1)) = ;;183
161		2t2e4 11959	. . . . . . 7 ⊢ (2 · 2) = 4
162	161	oveq1i 7201	. . . . . 6 ⊢ ((2 · 2) + 7) = (4 + 7)
163		7p4e11 12334	. . . . . . 7 ⊢ (7 + 4) = ;11
164	106, 68, 163	addcomli 10989	. . . . . 6 ⊢ (4 + 7) = ;11
165	162, 164	eqtri 2759	. . . . 5 ⊢ ((2 · 2) + 7) = ;11
166	28, 13, 31, 27, 146, 137, 13, 2, 2, 160, 165	decma2c 12311	. . . 4 ⊢ ((2 · ;;672) + ;;487) = ;;;1831
167	13, 32, 29, 145, 166	gcdi 16589	. . 3 ⊢ (;;;1831 gcd ;;672) = 1
168		eqid 2736	. . . . . 6 ⊢ ;;183 = ;;183
169	28	nn0cni 12067	. . . . . . 7 ⊢ ;67 ∈ ℂ
170	169	addid1i 10984	. . . . . 6 ⊢ (;67 + 0) = ;67
171	102	addid2i 10985	. . . . . . . . 9 ⊢ (0 + 1) = 1
172	99, 171	oveq12i 7203	. . . . . . . 8 ⊢ ((1 · 1) + (0 + 1)) = (1 + 1)
173	172, 22	eqtri 2759	. . . . . . 7 ⊢ ((1 · 1) + (0 + 1)) = 2
174	88	oveq1i 7201	. . . . . . . 8 ⊢ ((1 · 8) + 7) = (8 + 7)
175	174, 114	eqtri 2759	. . . . . . 7 ⊢ ((1 · 8) + 7) = ;15
176	2, 3, 16, 27, 120, 98, 2, 14, 2, 173, 175	decma2c 12311	. . . . . 6 ⊢ ((1 · ;18) + (6 + 1)) = ;25
177		3cn 11876	. . . . . . . . 9 ⊢ 3 ∈ ℂ
178	177	mulid2i 10803	. . . . . . . 8 ⊢ (1 · 3) = 3
179	178	oveq1i 7201	. . . . . . 7 ⊢ ((1 · 3) + 7) = (3 + 7)
180		7p3e10 12333	. . . . . . . 8 ⊢ (7 + 3) = ;10
181	106, 177, 180	addcomli 10989	. . . . . . 7 ⊢ (3 + 7) = ;10
182	179, 181	eqtri 2759	. . . . . 6 ⊢ ((1 · 3) + 7) = ;10
183	4, 9, 26, 27, 168, 170, 2, 16, 2, 176, 182	decma2c 12311	. . . . 5 ⊢ ((1 · ;;183) + (;67 + 0)) = ;;250
184	99	oveq1i 7201	. . . . . 6 ⊢ ((1 · 1) + 2) = (1 + 2)
185		1p2e3 11938	. . . . . 6 ⊢ (1 + 2) = 3
186	9	dec0h 12280	. . . . . 6 ⊢ 3 = ;03
187	184, 185, 186	3eqtri 2763	. . . . 5 ⊢ ((1 · 1) + 2) = ;03
188	10, 2, 28, 13, 23, 146, 2, 9, 16, 183, 187	decma2c 12311	. . . 4 ⊢ ((1 · ;;;1831) + ;;672) = ;;;2503
189	188, 12	eqtr4i 2762	. . 3 ⊢ ((1 · ;;;1831) + ;;672) = 𝑁
190	2, 29, 11, 167, 189	gcdi 16589	. 2 ⊢ (𝑁 gcd ;;;1831) = 1
191	8, 11, 20, 25, 190	gcdmodi 16590	1 ⊢ (((2↑;18) − 1) gcd 𝑁) = 1

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 209 = wceq 1543 ∈ wcel 2112 class class class wbr 5039 (class class class)co 7191 0cc0 10694 1c1 10695 + caddc 10697 · cmul 10699 − cmin 11027 ℕcn 11795 2c2 11850 3c3 11851 4c4 11852 5c5 11853 6c6 11854 7c7 11855 8c8 11856 9c9 11857 ℕ₀cn0 12055 ℤcz 12141 cdc 12258 ↑cexp 13600 ∥ cdvds 15778 gcd cgcd 16016 ℙcprime 16191

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-2o 8181 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-sup 9036 df-inf 9037 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-dec 12259 df-uz 12404 df-rp 12552 df-fz 13061 df-fl 13332 df-mod 13408 df-seq 13540 df-exp 13601 df-cj 14627 df-re 14628 df-im 14629 df-sqrt 14763 df-abs 14764 df-dvds 15779 df-gcd 16017 df-prm 16192

This theorem is referenced by: 2503prm 16656

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