Theorem 2503lem3 15846

Description: Lemma for 2503prm 15847. Calculate the GCD of 2 ^ 1 8 - 1 == 1 8 3 1 with N = 2 5 0 3. (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 = ;;; 2 5 0 3

Assertion

Ref	Expression
2503lem3	|- ( ( ( 2 ^; 1 8 ) - 1 ) gcd N ) = 1

Proof of Theorem 2503lem3

Step	Hyp	Ref	Expression
1		2nn 11185	. . . 4 |- 2 e. NN
2		1nn0 11308	. . . . 5 |- 1 e. NN0
3		8nn0 11315	. . . . 5 |- 8 e. NN0
4	2, 3	deccl 11512	. . . 4 |- ; 1 8 e. NN0
5		nnexpcl 12873	. . . 4 |- ( ( 2 e. NN /\ ; 1 8 e. NN0 ) -> ( 2 ^; 1 8 ) e. NN )
6	1, 4, 5	mp2an 708	. . 3 |- ( 2 ^; 1 8 ) e. NN
7		nnm1nn0 11334	. . 3 |- ( ( 2 ^; 1 8 ) e. NN -> ( ( 2 ^; 1 8 ) - 1 ) e. NN0 )
8	6, 7	ax-mp 5	. 2 |- ( ( 2 ^; 1 8 ) - 1 ) e. NN0
9		3nn0 11310	. . . 4 |- 3 e. NN0
10	4, 9	deccl 11512	. . 3 |- ;; 1 8 3 e. NN0
11	10, 2	deccl 11512	. 2 |- ;;; 1 8 3 1 e. NN0
12		2503prm.1	. . 3 |- N = ;;; 2 5 0 3
13		2nn0 11309	. . . . . 6 |- 2 e. NN0
14		5nn0 11312	. . . . . 6 |- 5 e. NN0
15	13, 14	deccl 11512	. . . . 5 |- ; 2 5 e. NN0
16		0nn0 11307	. . . . 5 |- 0 e. NN0
17	15, 16	deccl 11512	. . . 4 |- ;; 2 5 0 e. NN0
18		3nn 11186	. . . 4 |- 3 e. NN
19	17, 18	decnncl 11518	. . 3 |- ;;; 2 5 0 3 e. NN
20	12, 19	eqeltri 2697	. 2 |- N e. NN
21	12	2503lem1 15844	. . 3 |- ( ( 2 ^; 1 8 ) mod N ) = (;;; 1 8 3 2 mod N )
22		1p1e2 11134	. . . 4 |- ( 1 + 1 ) = 2
23		eqid 2622	. . . 4 |- ;;; 1 8 3 1 = ;;; 1 8 3 1
24	10, 2, 22, 23	decsuc 11535	. . 3 |- (;;; 1 8 3 1 + 1 ) = ;;; 1 8 3 2
25	20, 6, 2, 11, 21, 24	modsubi 15776	. 2 |- ( ( ( 2 ^; 1 8 ) - 1 ) mod N ) = (;;; 1 8 3 1 mod N )
26		6nn0 11313	. . . . 5 |- 6 e. NN0
27		7nn0 11314	. . . . 5 |- 7 e. NN0
28	26, 27	deccl 11512	. . . 4 |- ; 6 7 e. NN0
29	28, 13	deccl 11512	. . 3 |- ;; 6 7 2 e. NN0
30		4nn0 11311	. . . . . 6 |- 4 e. NN0
31	30, 3	deccl 11512	. . . . 5 |- ; 4 8 e. NN0
32	31, 27	deccl 11512	. . . 4 |- ;; 4 8 7 e. NN0
33	4, 14	deccl 11512	. . . . 5 |- ;; 1 8 5 e. NN0
34	2, 2	deccl 11512	. . . . . . 7 |- ; 1 1 e. NN0
35	34, 27	deccl 11512	. . . . . 6 |- ;; 1 1 7 e. NN0
36	26, 3	deccl 11512	. . . . . . 7 |- ; 6 8 e. NN0
37		9nn0 11316	. . . . . . . . 9 |- 9 e. NN0
38	30, 37	deccl 11512	. . . . . . . 8 |- ; 4 9 e. NN0
39	2, 37	deccl 11512	. . . . . . . . 9 |- ; 1 9 e. NN0
40	38	nn0zi 11402	. . . . . . . . . . 11 |- ; 4 9 e. ZZ
41	39	nn0zi 11402	. . . . . . . . . . 11 |- ; 1 9 e. ZZ
42		gcdcom 15235	. . . . . . . . . . 11 |- ( (; 4 9 e. ZZ /\ ; 1 9 e. ZZ ) -> (; 4 9 gcd ; 1 9 ) = (; 1 9 gcd ; 4 9 ) )
43	40, 41, 42	mp2an 708	. . . . . . . . . 10 |- (; 4 9 gcd ; 1 9 ) = (; 1 9 gcd ; 4 9 )
44		9nn 11192	. . . . . . . . . . . . 13 |- 9 e. NN
45	2, 44	decnncl 11518	. . . . . . . . . . . 12 |- ; 1 9 e. NN
46		1nn 11031	. . . . . . . . . . . . 13 |- 1 e. NN
47	2, 46	decnncl 11518	. . . . . . . . . . . 12 |- ; 1 1 e. NN
48		eqid 2622	. . . . . . . . . . . . 13 |- ; 1 9 = ; 1 9
49		eqid 2622	. . . . . . . . . . . . 13 |- ; 1 1 = ; 1 1
50		2cn 11091	. . . . . . . . . . . . . . . 16 |- 2 e. CC
51	50	mulid2i 10043	. . . . . . . . . . . . . . 15 |- ( 1 x. 2 ) = 2
52	51, 22	oveq12i 6662	. . . . . . . . . . . . . 14 |- ( ( 1 x. 2 ) + ( 1 + 1 ) ) = ( 2 + 2 )
53		2p2e4 11144	. . . . . . . . . . . . . 14 |- ( 2 + 2 ) = 4
54	52, 53	eqtri 2644	. . . . . . . . . . . . 13 |- ( ( 1 x. 2 ) + ( 1 + 1 ) ) = 4
55		8p1e9 11158	. . . . . . . . . . . . . 14 |- ( 8 + 1 ) = 9
56		9t2e18 11663	. . . . . . . . . . . . . 14 |- ( 9 x. 2 ) = ; 1 8
57	2, 3, 55, 56	decsuc 11535	. . . . . . . . . . . . 13 |- ( ( 9 x. 2 ) + 1 ) = ; 1 9
58	2, 37, 2, 2, 48, 49, 13, 37, 2, 54, 57	decmac 11566	. . . . . . . . . . . 12 |- ( (; 1 9 x. 2 ) + ; 1 1 ) = ; 4 9
59		1lt9 11229	. . . . . . . . . . . . 13 |- 1 < 9
60	2, 2, 44, 59	declt 11530	. . . . . . . . . . . 12 |- ; 1 1 < ; 1 9
61	45, 13, 47, 58, 60	ndvdsi 15136	. . . . . . . . . . 11 |- -. ; 1 9 || ; 4 9
62		19prm 15825	. . . . . . . . . . . 12 |- ; 1 9 e. Prime
63		coprm 15423	. . . . . . . . . . . 12 |- ( (; 1 9 e. Prime /\ ; 4 9 e. ZZ ) -> ( -. ; 1 9 || ; 4 9 <-> (; 1 9 gcd ; 4 9 ) = 1 ) )
64	62, 40, 63	mp2an 708	. . . . . . . . . . 11 |- ( -. ; 1 9 || ; 4 9 <-> (; 1 9 gcd ; 4 9 ) = 1 )
65	61, 64	mpbi 220	. . . . . . . . . 10 |- (; 1 9 gcd ; 4 9 ) = 1
66	43, 65	eqtri 2644	. . . . . . . . 9 |- (; 4 9 gcd ; 1 9 ) = 1
67		eqid 2622	. . . . . . . . . 10 |- ; 4 9 = ; 4 9
68		4cn 11098	. . . . . . . . . . . . 13 |- 4 e. CC
69	68	mulid2i 10043	. . . . . . . . . . . 12 |- ( 1 x. 4 ) = 4
70	69, 22	oveq12i 6662	. . . . . . . . . . 11 |- ( ( 1 x. 4 ) + ( 1 + 1 ) ) = ( 4 + 2 )
71		4p2e6 11162	. . . . . . . . . . 11 |- ( 4 + 2 ) = 6
72	70, 71	eqtri 2644	. . . . . . . . . 10 |- ( ( 1 x. 4 ) + ( 1 + 1 ) ) = 6
73		9cn 11108	. . . . . . . . . . . . 13 |- 9 e. CC
74	73	mulid2i 10043	. . . . . . . . . . . 12 |- ( 1 x. 9 ) = 9
75	74	oveq1i 6660	. . . . . . . . . . 11 |- ( ( 1 x. 9 ) + 9 ) = ( 9 + 9 )
76		9p9e18 11627	. . . . . . . . . . 11 |- ( 9 + 9 ) = ; 1 8
77	75, 76	eqtri 2644	. . . . . . . . . 10 |- ( ( 1 x. 9 ) + 9 ) = ; 1 8
78	30, 37, 2, 37, 67, 48, 2, 3, 2, 72, 77	decma2c 11568	. . . . . . . . 9 |- ( ( 1 x. ; 4 9 ) + ; 1 9 ) = ; 6 8
79	2, 39, 38, 66, 78	gcdi 15777	. . . . . . . 8 |- (; 6 8 gcd ; 4 9 ) = 1
80		eqid 2622	. . . . . . . . 9 |- ; 6 8 = ; 6 8
81		6cn 11102	. . . . . . . . . . . 12 |- 6 e. CC
82	81	mulid2i 10043	. . . . . . . . . . 11 |- ( 1 x. 6 ) = 6
83		4p1e5 11154	. . . . . . . . . . 11 |- ( 4 + 1 ) = 5
84	82, 83	oveq12i 6662	. . . . . . . . . 10 |- ( ( 1 x. 6 ) + ( 4 + 1 ) ) = ( 6 + 5 )
85		6p5e11 11600	. . . . . . . . . 10 |- ( 6 + 5 ) = ; 1 1
86	84, 85	eqtri 2644	. . . . . . . . 9 |- ( ( 1 x. 6 ) + ( 4 + 1 ) ) = ; 1 1
87		8cn 11106	. . . . . . . . . . . 12 |- 8 e. CC
88	87	mulid2i 10043	. . . . . . . . . . 11 |- ( 1 x. 8 ) = 8
89	88	oveq1i 6660	. . . . . . . . . 10 |- ( ( 1 x. 8 ) + 9 ) = ( 8 + 9 )
90		9p8e17 11626	. . . . . . . . . . 11 |- ( 9 + 8 ) = ; 1 7
91	73, 87, 90	addcomli 10228	. . . . . . . . . 10 |- ( 8 + 9 ) = ; 1 7
92	89, 91	eqtri 2644	. . . . . . . . 9 |- ( ( 1 x. 8 ) + 9 ) = ; 1 7
93	26, 3, 30, 37, 80, 67, 2, 27, 2, 86, 92	decma2c 11568	. . . . . . . 8 |- ( ( 1 x. ; 6 8 ) + ; 4 9 ) = ;; 1 1 7
94	2, 38, 36, 79, 93	gcdi 15777	. . . . . . 7 |- (;; 1 1 7 gcd ; 6 8 ) = 1
95		eqid 2622	. . . . . . . 8 |- ;; 1 1 7 = ;; 1 1 7
96		6p1e7 11156	. . . . . . . . . 10 |- ( 6 + 1 ) = 7
97	27	dec0h 11522	. . . . . . . . . 10 |- 7 = ; 0 7
98	96, 97	eqtri 2644	. . . . . . . . 9 |- ( 6 + 1 ) = ; 0 7
99		1t1e1 11175	. . . . . . . . . . 11 |- ( 1 x. 1 ) = 1
100		00id 10211	. . . . . . . . . . 11 |- ( 0 + 0 ) = 0
101	99, 100	oveq12i 6662	. . . . . . . . . 10 |- ( ( 1 x. 1 ) + ( 0 + 0 ) ) = ( 1 + 0 )
102		ax-1cn 9994	. . . . . . . . . . 11 |- 1 e. CC
103	102	addid1i 10223	. . . . . . . . . 10 |- ( 1 + 0 ) = 1
104	101, 103	eqtri 2644	. . . . . . . . 9 |- ( ( 1 x. 1 ) + ( 0 + 0 ) ) = 1
105	99	oveq1i 6660	. . . . . . . . . 10 |- ( ( 1 x. 1 ) + 7 ) = ( 1 + 7 )
106		7cn 11104	. . . . . . . . . . 11 |- 7 e. CC
107		7p1e8 11157	. . . . . . . . . . 11 |- ( 7 + 1 ) = 8
108	106, 102, 107	addcomli 10228	. . . . . . . . . 10 |- ( 1 + 7 ) = 8
109	3	dec0h 11522	. . . . . . . . . 10 |- 8 = ; 0 8
110	105, 108, 109	3eqtri 2648	. . . . . . . . 9 |- ( ( 1 x. 1 ) + 7 ) = ; 0 8
111	2, 2, 16, 27, 49, 98, 2, 3, 16, 104, 110	decma2c 11568	. . . . . . . 8 |- ( ( 1 x. ; 1 1 ) + ( 6 + 1 ) ) = ; 1 8
112	106	mulid2i 10043	. . . . . . . . . 10 |- ( 1 x. 7 ) = 7
113	112	oveq1i 6660	. . . . . . . . 9 |- ( ( 1 x. 7 ) + 8 ) = ( 7 + 8 )
114		8p7e15 11617	. . . . . . . . . 10 |- ( 8 + 7 ) = ; 1 5
115	87, 106, 114	addcomli 10228	. . . . . . . . 9 |- ( 7 + 8 ) = ; 1 5
116	113, 115	eqtri 2644	. . . . . . . 8 |- ( ( 1 x. 7 ) + 8 ) = ; 1 5
117	34, 27, 26, 3, 95, 80, 2, 14, 2, 111, 116	decma2c 11568	. . . . . . 7 |- ( ( 1 x. ;; 1 1 7 ) + ; 6 8 ) = ;; 1 8 5
118	2, 36, 35, 94, 117	gcdi 15777	. . . . . 6 |- (;; 1 8 5 gcd ;; 1 1 7 ) = 1
119		eqid 2622	. . . . . . 7 |- ;; 1 8 5 = ;; 1 8 5
120		eqid 2622	. . . . . . . 8 |- ; 1 8 = ; 1 8
121	2, 2, 22, 49	decsuc 11535	. . . . . . . 8 |- (; 1 1 + 1 ) = ; 1 2
122		2t1e2 11176	. . . . . . . . . 10 |- ( 2 x. 1 ) = 2
123	122, 22	oveq12i 6662	. . . . . . . . 9 |- ( ( 2 x. 1 ) + ( 1 + 1 ) ) = ( 2 + 2 )
124	123, 53	eqtri 2644	. . . . . . . 8 |- ( ( 2 x. 1 ) + ( 1 + 1 ) ) = 4
125		8t2e16 11654	. . . . . . . . . 10 |- ( 8 x. 2 ) = ; 1 6
126	87, 50, 125	mulcomli 10047	. . . . . . . . 9 |- ( 2 x. 8 ) = ; 1 6
127		6p2e8 11169	. . . . . . . . 9 |- ( 6 + 2 ) = 8
128	2, 26, 13, 126, 127	decaddi 11579	. . . . . . . 8 |- ( ( 2 x. 8 ) + 2 ) = ; 1 8
129	2, 3, 2, 13, 120, 121, 13, 3, 2, 124, 128	decma2c 11568	. . . . . . 7 |- ( ( 2 x. ; 1 8 ) + (; 1 1 + 1 ) ) = ; 4 8
130		5cn 11100	. . . . . . . . 9 |- 5 e. CC
131		5t2e10 11634	. . . . . . . . 9 |- ( 5 x. 2 ) = ; 1 0
132	130, 50, 131	mulcomli 10047	. . . . . . . 8 |- ( 2 x. 5 ) = ; 1 0
133	106	addid2i 10224	. . . . . . . 8 |- ( 0 + 7 ) = 7
134	2, 16, 27, 132, 133	decaddi 11579	. . . . . . 7 |- ( ( 2 x. 5 ) + 7 ) = ; 1 7
135	4, 14, 34, 27, 119, 95, 13, 27, 2, 129, 134	decma2c 11568	. . . . . 6 |- ( ( 2 x. ;; 1 8 5 ) + ;; 1 1 7 ) = ;; 4 8 7
136	13, 35, 33, 118, 135	gcdi 15777	. . . . 5 |- (;; 4 8 7 gcd ;; 1 8 5 ) = 1
137		eqid 2622	. . . . . 6 |- ;; 4 8 7 = ;; 4 8 7
138		eqid 2622	. . . . . . 7 |- ; 4 8 = ; 4 8
139	2, 3, 55, 120	decsuc 11535	. . . . . . 7 |- (; 1 8 + 1 ) = ; 1 9
140	30, 3, 2, 37, 138, 139, 2, 27, 2, 72, 92	decma2c 11568	. . . . . 6 |- ( ( 1 x. ; 4 8 ) + (; 1 8 + 1 ) ) = ; 6 7
141	112	oveq1i 6660	. . . . . . 7 |- ( ( 1 x. 7 ) + 5 ) = ( 7 + 5 )
142		7p5e12 11607	. . . . . . 7 |- ( 7 + 5 ) = ; 1 2
143	141, 142	eqtri 2644	. . . . . 6 |- ( ( 1 x. 7 ) + 5 ) = ; 1 2
144	31, 27, 4, 14, 137, 119, 2, 13, 2, 140, 143	decma2c 11568	. . . . 5 |- ( ( 1 x. ;; 4 8 7 ) + ;; 1 8 5 ) = ;; 6 7 2
145	2, 33, 32, 136, 144	gcdi 15777	. . . 4 |- (;; 6 7 2 gcd ;; 4 8 7 ) = 1
146		eqid 2622	. . . . 5 |- ;; 6 7 2 = ;; 6 7 2
147		eqid 2622	. . . . . 6 |- ; 6 7 = ; 6 7
148	30, 3, 55, 138	decsuc 11535	. . . . . 6 |- (; 4 8 + 1 ) = ; 4 9
149	71	oveq2i 6661	. . . . . . 7 |- ( ( 2 x. 6 ) + ( 4 + 2 ) ) = ( ( 2 x. 6 ) + 6 )
150		6t2e12 11641	. . . . . . . . 9 |- ( 6 x. 2 ) = ; 1 2
151	81, 50, 150	mulcomli 10047	. . . . . . . 8 |- ( 2 x. 6 ) = ; 1 2
152	81, 50, 127	addcomli 10228	. . . . . . . 8 |- ( 2 + 6 ) = 8
153	2, 13, 26, 151, 152	decaddi 11579	. . . . . . 7 |- ( ( 2 x. 6 ) + 6 ) = ; 1 8
154	149, 153	eqtri 2644	. . . . . 6 |- ( ( 2 x. 6 ) + ( 4 + 2 ) ) = ; 1 8
155		7t2e14 11648	. . . . . . . 8 |- ( 7 x. 2 ) = ; 1 4
156	106, 50, 155	mulcomli 10047	. . . . . . 7 |- ( 2 x. 7 ) = ; 1 4
157		9p4e13 11622	. . . . . . . 8 |- ( 9 + 4 ) = ; 1 3
158	73, 68, 157	addcomli 10228	. . . . . . 7 |- ( 4 + 9 ) = ; 1 3
159	2, 30, 37, 156, 22, 9, 158	decaddci 11580	. . . . . 6 |- ( ( 2 x. 7 ) + 9 ) = ; 2 3
160	26, 27, 30, 37, 147, 148, 13, 9, 13, 154, 159	decma2c 11568	. . . . 5 |- ( ( 2 x. ; 6 7 ) + (; 4 8 + 1 ) ) = ;; 1 8 3
161		2t2e4 11177	. . . . . . 7 |- ( 2 x. 2 ) = 4
162	161	oveq1i 6660	. . . . . 6 |- ( ( 2 x. 2 ) + 7 ) = ( 4 + 7 )
163		7p4e11 11605	. . . . . . 7 |- ( 7 + 4 ) = ; 1 1
164	106, 68, 163	addcomli 10228	. . . . . 6 |- ( 4 + 7 ) = ; 1 1
165	162, 164	eqtri 2644	. . . . 5 |- ( ( 2 x. 2 ) + 7 ) = ; 1 1
166	28, 13, 31, 27, 146, 137, 13, 2, 2, 160, 165	decma2c 11568	. . . 4 |- ( ( 2 x. ;; 6 7 2 ) + ;; 4 8 7 ) = ;;; 1 8 3 1
167	13, 32, 29, 145, 166	gcdi 15777	. . 3 |- (;;; 1 8 3 1 gcd ;; 6 7 2 ) = 1
168		eqid 2622	. . . . . 6 |- ;; 1 8 3 = ;; 1 8 3
169	28	nn0cni 11304	. . . . . . 7 |- ; 6 7 e. CC
170	169	addid1i 10223	. . . . . 6 |- (; 6 7 + 0 ) = ; 6 7
171	102	addid2i 10224	. . . . . . . . 9 |- ( 0 + 1 ) = 1
172	99, 171	oveq12i 6662	. . . . . . . 8 |- ( ( 1 x. 1 ) + ( 0 + 1 ) ) = ( 1 + 1 )
173	172, 22	eqtri 2644	. . . . . . 7 |- ( ( 1 x. 1 ) + ( 0 + 1 ) ) = 2
174	88	oveq1i 6660	. . . . . . . 8 |- ( ( 1 x. 8 ) + 7 ) = ( 8 + 7 )
175	174, 114	eqtri 2644	. . . . . . 7 |- ( ( 1 x. 8 ) + 7 ) = ; 1 5
176	2, 3, 16, 27, 120, 98, 2, 14, 2, 173, 175	decma2c 11568	. . . . . 6 |- ( ( 1 x. ; 1 8 ) + ( 6 + 1 ) ) = ; 2 5
177		3cn 11095	. . . . . . . . 9 |- 3 e. CC
178	177	mulid2i 10043	. . . . . . . 8 |- ( 1 x. 3 ) = 3
179	178	oveq1i 6660	. . . . . . 7 |- ( ( 1 x. 3 ) + 7 ) = ( 3 + 7 )
180		7p3e10 11603	. . . . . . . 8 |- ( 7 + 3 ) = ; 1 0
181	106, 177, 180	addcomli 10228	. . . . . . 7 |- ( 3 + 7 ) = ; 1 0
182	179, 181	eqtri 2644	. . . . . 6 |- ( ( 1 x. 3 ) + 7 ) = ; 1 0
183	4, 9, 26, 27, 168, 170, 2, 16, 2, 176, 182	decma2c 11568	. . . . 5 |- ( ( 1 x. ;; 1 8 3 ) + (; 6 7 + 0 ) ) = ;; 2 5 0
184	99	oveq1i 6660	. . . . . 6 |- ( ( 1 x. 1 ) + 2 ) = ( 1 + 2 )
185		1p2e3 11152	. . . . . 6 |- ( 1 + 2 ) = 3
186	9	dec0h 11522	. . . . . 6 |- 3 = ; 0 3
187	184, 185, 186	3eqtri 2648	. . . . 5 |- ( ( 1 x. 1 ) + 2 ) = ; 0 3
188	10, 2, 28, 13, 23, 146, 2, 9, 16, 183, 187	decma2c 11568	. . . 4 |- ( ( 1 x. ;;; 1 8 3 1 ) + ;; 6 7 2 ) = ;;; 2 5 0 3
189	188, 12	eqtr4i 2647	. . 3 |- ( ( 1 x. ;;; 1 8 3 1 ) + ;; 6 7 2 ) = N
190	2, 29, 11, 167, 189	gcdi 15777	. 2 |- ( N gcd ;;; 1 8 3 1 ) = 1
191	8, 11, 20, 25, 190	gcdmodi 15778	1 |- ( ( ( 2 ^; 1 8 ) - 1 ) gcd N ) = 1

Syntax hints:   -. wn 3   <-> wb 196   = wceq 1483   e. wcel 1990   class class class wbr 4653  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   NNcn 11020   2c2 11070   3c3 11071   4c4 11072   5c5 11073   6c6 11074   7c7 11075   8c8 11076   9c9 11077   NN0cn0 11292   ZZcz 11377  ;cdc 11493   ^cexp 12860   || cdvds 14983   gcd cgcd 15216   Primecprime 15385

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-rp 11833  df-fz 12327  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-gcd 15217  df-prm 15386

This theorem is referenced by:  2503prm  15847