Theorem leibpilem2 24668

Description: The Leibniz formula for pi. (Contributed by Mario Carneiro, 7-Apr-2015.)

Hypotheses

Ref	Expression
leibpi.1	|- F = ( n e. NN0 |-> ( ( -u 1 ^ n ) / ( ( 2 x. n ) + 1 ) ) )
leibpilem2.2	|- G = ( k e. NN0 |-> if ( ( k = 0 \/ 2 || k ) , 0 , ( ( -u 1 ^ ( ( k - 1 ) / 2 ) ) / k ) ) )
leibpilem2.3	|- A e. _V

Assertion

Ref	Expression
leibpilem2	|- ( seq 0 ( + , F ) ~~> A <-> seq 0 ( + , G ) ~~> A )

Distinct variable groups:   k, n   n, G

Allowed substitution hints:   A( k, n)   F( k, n)   G( k)

Proof of Theorem leibpilem2

Step	Hyp	Ref	Expression
1		leibpi.1	. . . . 5 |- F = ( n e. NN0 |-> ( ( -u 1 ^ n ) / ( ( 2 x. n ) + 1 ) ) )
2		2cn 11091	. . . . . . . . . . . 12 |- 2 e. CC
3		nn0cn 11302	. . . . . . . . . . . 12 |- ( n e. NN0 -> n e. CC )
4		mulcl 10020	. . . . . . . . . . . 12 |- ( ( 2 e. CC /\ n e. CC ) -> ( 2 x. n ) e. CC )
5	2, 3, 4	sylancr 695	. . . . . . . . . . 11 |- ( n e. NN0 -> ( 2 x. n ) e. CC )
6		ax-1cn 9994	. . . . . . . . . . 11 |- 1 e. CC
7		pncan 10287	. . . . . . . . . . 11 |- ( ( ( 2 x. n ) e. CC /\ 1 e. CC ) -> ( ( ( 2 x. n ) + 1 ) - 1 ) = ( 2 x. n ) )
8	5, 6, 7	sylancl 694	. . . . . . . . . 10 |- ( n e. NN0 -> ( ( ( 2 x. n ) + 1 ) - 1 ) = ( 2 x. n ) )
9	8	oveq1d 6665	. . . . . . . . 9 |- ( n e. NN0 -> ( ( ( ( 2 x. n ) + 1 ) - 1 ) / 2 ) = ( ( 2 x. n ) / 2 ) )
10		2ne0 11113	. . . . . . . . . . 11 |- 2 =/= 0
11		divcan3 10711	. . . . . . . . . . 11 |- ( ( n e. CC /\ 2 e. CC /\ 2 =/= 0 ) -> ( ( 2 x. n ) / 2 ) = n )
12	2, 10, 11	mp3an23 1416	. . . . . . . . . 10 |- ( n e. CC -> ( ( 2 x. n ) / 2 ) = n )
13	3, 12	syl 17	. . . . . . . . 9 |- ( n e. NN0 -> ( ( 2 x. n ) / 2 ) = n )
14	9, 13	eqtrd 2656	. . . . . . . 8 |- ( n e. NN0 -> ( ( ( ( 2 x. n ) + 1 ) - 1 ) / 2 ) = n )
15	14	oveq2d 6666	. . . . . . 7 |- ( n e. NN0 -> ( -u 1 ^ ( ( ( ( 2 x. n ) + 1 ) - 1 ) / 2 ) ) = ( -u 1 ^ n ) )
16	15	oveq1d 6665	. . . . . 6 |- ( n e. NN0 -> ( ( -u 1 ^ ( ( ( ( 2 x. n ) + 1 ) - 1 ) / 2 ) ) / ( ( 2 x. n ) + 1 ) ) = ( ( -u 1 ^ n ) / ( ( 2 x. n ) + 1 ) ) )
17	16	mpteq2ia 4740	. . . . 5 |- ( n e. NN0 |-> ( ( -u 1 ^ ( ( ( ( 2 x. n ) + 1 ) - 1 ) / 2 ) ) / ( ( 2 x. n ) + 1 ) ) ) = ( n e. NN0 |-> ( ( -u 1 ^ n ) / ( ( 2 x. n ) + 1 ) ) )
18	1, 17	eqtr4i 2647	. . . 4 |- F = ( n e. NN0 |-> ( ( -u 1 ^ ( ( ( ( 2 x. n ) + 1 ) - 1 ) / 2 ) ) / ( ( 2 x. n ) + 1 ) ) )
19		seqeq3 12806	. . . 4 |- ( F = ( n e. NN0 |-> ( ( -u 1 ^ ( ( ( ( 2 x. n ) + 1 ) - 1 ) / 2 ) ) / ( ( 2 x. n ) + 1 ) ) ) -> seq 0 ( + , F ) = seq 0 ( + , ( n e. NN0 |-> ( ( -u 1 ^ ( ( ( ( 2 x. n ) + 1 ) - 1 ) / 2 ) ) / ( ( 2 x. n ) + 1 ) ) ) ) )
20	18, 19	ax-mp 5	. . 3 |- seq 0 ( + , F ) = seq 0 ( + , ( n e. NN0 |-> ( ( -u 1 ^ ( ( ( ( 2 x. n ) + 1 ) - 1 ) / 2 ) ) / ( ( 2 x. n ) + 1 ) ) ) )
21	20	breq1i 4660	. 2 |- ( seq 0 ( + , F ) ~~> A <-> seq 0 ( + , ( n e. NN0 |-> ( ( -u 1 ^ ( ( ( ( 2 x. n ) + 1 ) - 1 ) / 2 ) ) / ( ( 2 x. n ) + 1 ) ) ) ) ~~> A )
22		neg1rr 11125	. . . . . . . . 9 |- -u 1 e. RR
23		reexpcl 12877	. . . . . . . . 9 |- ( ( -u 1 e. RR /\ n e. NN0 ) -> ( -u 1 ^ n ) e. RR )
24	22, 23	mpan 706	. . . . . . . 8 |- ( n e. NN0 -> ( -u 1 ^ n ) e. RR )
25		2nn0 11309	. . . . . . . . . 10 |- 2 e. NN0
26		nn0mulcl 11329	. . . . . . . . . 10 |- ( ( 2 e. NN0 /\ n e. NN0 ) -> ( 2 x. n ) e. NN0 )
27	25, 26	mpan 706	. . . . . . . . 9 |- ( n e. NN0 -> ( 2 x. n ) e. NN0 )
28		nn0p1nn 11332	. . . . . . . . 9 |- ( ( 2 x. n ) e. NN0 -> ( ( 2 x. n ) + 1 ) e. NN )
29	27, 28	syl 17	. . . . . . . 8 |- ( n e. NN0 -> ( ( 2 x. n ) + 1 ) e. NN )
30	24, 29	nndivred 11069	. . . . . . 7 |- ( n e. NN0 -> ( ( -u 1 ^ n ) / ( ( 2 x. n ) + 1 ) ) e. RR )
31	30	recnd 10068	. . . . . 6 |- ( n e. NN0 -> ( ( -u 1 ^ n ) / ( ( 2 x. n ) + 1 ) ) e. CC )
32	16, 31	eqeltrd 2701	. . . . 5 |- ( n e. NN0 -> ( ( -u 1 ^ ( ( ( ( 2 x. n ) + 1 ) - 1 ) / 2 ) ) / ( ( 2 x. n ) + 1 ) ) e. CC )
33	32	adantl 482	. . . 4 |- ( ( T. /\ n e. NN0 ) -> ( ( -u 1 ^ ( ( ( ( 2 x. n ) + 1 ) - 1 ) / 2 ) ) / ( ( 2 x. n ) + 1 ) ) e. CC )
34		oveq1 6657	. . . . . . 7 |- ( k = ( ( 2 x. n ) + 1 ) -> ( k - 1 ) = ( ( ( 2 x. n ) + 1 ) - 1 ) )
35	34	oveq1d 6665	. . . . . 6 |- ( k = ( ( 2 x. n ) + 1 ) -> ( ( k - 1 ) / 2 ) = ( ( ( ( 2 x. n ) + 1 ) - 1 ) / 2 ) )
36	35	oveq2d 6666	. . . . 5 |- ( k = ( ( 2 x. n ) + 1 ) -> ( -u 1 ^ ( ( k - 1 ) / 2 ) ) = ( -u 1 ^ ( ( ( ( 2 x. n ) + 1 ) - 1 ) / 2 ) ) )
37		id 22	. . . . 5 |- ( k = ( ( 2 x. n ) + 1 ) -> k = ( ( 2 x. n ) + 1 ) )
38	36, 37	oveq12d 6668	. . . 4 |- ( k = ( ( 2 x. n ) + 1 ) -> ( ( -u 1 ^ ( ( k - 1 ) / 2 ) ) / k ) = ( ( -u 1 ^ ( ( ( ( 2 x. n ) + 1 ) - 1 ) / 2 ) ) / ( ( 2 x. n ) + 1 ) ) )
39	33, 38	iserodd 15540	. . 3 |- ( T. -> ( seq 0 ( + , ( n e. NN0 |-> ( ( -u 1 ^ ( ( ( ( 2 x. n ) + 1 ) - 1 ) / 2 ) ) / ( ( 2 x. n ) + 1 ) ) ) ) ~~> A <-> seq 1 ( + , ( k e. NN |-> if ( 2 || k , 0 , ( ( -u 1 ^ ( ( k - 1 ) / 2 ) ) / k ) ) ) ) ~~> A ) )
40	39	trud 1493	. 2 |- ( seq 0 ( + , ( n e. NN0 |-> ( ( -u 1 ^ ( ( ( ( 2 x. n ) + 1 ) - 1 ) / 2 ) ) / ( ( 2 x. n ) + 1 ) ) ) ) ~~> A <-> seq 1 ( + , ( k e. NN |-> if ( 2 || k , 0 , ( ( -u 1 ^ ( ( k - 1 ) / 2 ) ) / k ) ) ) ) ~~> A )
41		addid2 10219	. . . . . . . 8 |- ( n e. CC -> ( 0 + n ) = n )
42	41	adantl 482	. . . . . . 7 |- ( ( T. /\ n e. CC ) -> ( 0 + n ) = n )
43		0cnd 10033	. . . . . . 7 |- ( T. -> 0 e. CC )
44		1eluzge0 11732	. . . . . . . 8 |- 1 e. ( ZZ>= ` 0 )
45	44	a1i 11	. . . . . . 7 |- ( T. -> 1 e. ( ZZ>= ` 0 ) )
46		1nn0 11308	. . . . . . . 8 |- 1 e. NN0
47		leibpilem2.2	. . . . . . . . . 10 |- G = ( k e. NN0 |-> if ( ( k = 0 \/ 2 || k ) , 0 , ( ( -u 1 ^ ( ( k - 1 ) / 2 ) ) / k ) ) )
48		0cnd 10033	. . . . . . . . . . 11 |- ( ( k e. NN0 /\ ( k = 0 \/ 2 || k ) ) -> 0 e. CC )
49		ioran 511	. . . . . . . . . . . 12 |- ( -. ( k = 0 \/ 2 || k ) <-> ( -. k = 0 /\ -. 2 || k ) )
50		leibpilem1 24667	. . . . . . . . . . . . . . . 16 |- ( ( k e. NN0 /\ ( -. k = 0 /\ -. 2 || k ) ) -> ( k e. NN /\ ( ( k - 1 ) / 2 ) e. NN0 ) )
51	50	simprd 479	. . . . . . . . . . . . . . 15 |- ( ( k e. NN0 /\ ( -. k = 0 /\ -. 2 || k ) ) -> ( ( k - 1 ) / 2 ) e. NN0 )
52		reexpcl 12877	. . . . . . . . . . . . . . 15 |- ( ( -u 1 e. RR /\ ( ( k - 1 ) / 2 ) e. NN0 ) -> ( -u 1 ^ ( ( k - 1 ) / 2 ) ) e. RR )
53	22, 51, 52	sylancr 695	. . . . . . . . . . . . . 14 |- ( ( k e. NN0 /\ ( -. k = 0 /\ -. 2 || k ) ) -> ( -u 1 ^ ( ( k - 1 ) / 2 ) ) e. RR )
54	50	simpld 475	. . . . . . . . . . . . . 14 |- ( ( k e. NN0 /\ ( -. k = 0 /\ -. 2 || k ) ) -> k e. NN )
55	53, 54	nndivred 11069	. . . . . . . . . . . . 13 |- ( ( k e. NN0 /\ ( -. k = 0 /\ -. 2 || k ) ) -> ( ( -u 1 ^ ( ( k - 1 ) / 2 ) ) / k ) e. RR )
56	55	recnd 10068	. . . . . . . . . . . 12 |- ( ( k e. NN0 /\ ( -. k = 0 /\ -. 2 || k ) ) -> ( ( -u 1 ^ ( ( k - 1 ) / 2 ) ) / k ) e. CC )
57	49, 56	sylan2b 492	. . . . . . . . . . 11 |- ( ( k e. NN0 /\ -. ( k = 0 \/ 2 || k ) ) -> ( ( -u 1 ^ ( ( k - 1 ) / 2 ) ) / k ) e. CC )
58	48, 57	ifclda 4120	. . . . . . . . . 10 |- ( k e. NN0 -> if ( ( k = 0 \/ 2 || k ) , 0 , ( ( -u 1 ^ ( ( k - 1 ) / 2 ) ) / k ) ) e. CC )
59	47, 58	fmpti 6383	. . . . . . . . 9 |- G : NN0 --> CC
60	59	ffvelrni 6358	. . . . . . . 8 |- ( 1 e. NN0 -> ( G ` 1 ) e. CC )
61	46, 60	mp1i 13	. . . . . . 7 |- ( T. -> ( G ` 1 ) e. CC )
62		simpr 477	. . . . . . . . . . 11 |- ( ( T. /\ n e. ( 0 ... ( 1 - 1 ) ) ) -> n e. ( 0 ... ( 1 - 1 ) ) )
63		1m1e0 11089	. . . . . . . . . . . 12 |- ( 1 - 1 ) = 0
64	63	oveq2i 6661	. . . . . . . . . . 11 |- ( 0 ... ( 1 - 1 ) ) = ( 0 ... 0 )
65	62, 64	syl6eleq 2711	. . . . . . . . . 10 |- ( ( T. /\ n e. ( 0 ... ( 1 - 1 ) ) ) -> n e. ( 0 ... 0 ) )
66		elfz1eq 12352	. . . . . . . . . 10 |- ( n e. ( 0 ... 0 ) -> n = 0 )
67	65, 66	syl 17	. . . . . . . . 9 |- ( ( T. /\ n e. ( 0 ... ( 1 - 1 ) ) ) -> n = 0 )
68	67	fveq2d 6195	. . . . . . . 8 |- ( ( T. /\ n e. ( 0 ... ( 1 - 1 ) ) ) -> ( G ` n ) = ( G ` 0 ) )
69		0nn0 11307	. . . . . . . . 9 |- 0 e. NN0
70		iftrue 4092	. . . . . . . . . . 11 |- ( ( k = 0 \/ 2 || k ) -> if ( ( k = 0 \/ 2 || k ) , 0 , ( ( -u 1 ^ ( ( k - 1 ) / 2 ) ) / k ) ) = 0 )
71	70	orcs 409	. . . . . . . . . 10 |- ( k = 0 -> if ( ( k = 0 \/ 2 || k ) , 0 , ( ( -u 1 ^ ( ( k - 1 ) / 2 ) ) / k ) ) = 0 )
72		c0ex 10034	. . . . . . . . . 10 |- 0 e. _V
73	71, 47, 72	fvmpt 6282	. . . . . . . . 9 |- ( 0 e. NN0 -> ( G ` 0 ) = 0 )
74	69, 73	ax-mp 5	. . . . . . . 8 |- ( G ` 0 ) = 0
75	68, 74	syl6eq 2672	. . . . . . 7 |- ( ( T. /\ n e. ( 0 ... ( 1 - 1 ) ) ) -> ( G ` n ) = 0 )
76	42, 43, 45, 61, 75	seqid 12846	. . . . . 6 |- ( T. -> ( seq 0 ( + , G ) |` ( ZZ>= ` 1 ) ) = seq 1 ( + , G ) )
77		1zzd 11408	. . . . . . 7 |- ( T. -> 1 e. ZZ )
78		simpr 477	. . . . . . . . 9 |- ( ( T. /\ n e. ( ZZ>= ` 1 ) ) -> n e. ( ZZ>= ` 1 ) )
79		nnuz 11723	. . . . . . . . 9 |- NN = ( ZZ>= ` 1 )
80	78, 79	syl6eleqr 2712	. . . . . . . 8 |- ( ( T. /\ n e. ( ZZ>= ` 1 ) ) -> n e. NN )
81		nnne0 11053	. . . . . . . . . . . 12 |- ( n e. NN -> n =/= 0 )
82	81	neneqd 2799	. . . . . . . . . . 11 |- ( n e. NN -> -. n = 0 )
83		biorf 420	. . . . . . . . . . 11 |- ( -. n = 0 -> ( 2 || n <-> ( n = 0 \/ 2 || n ) ) )
84	82, 83	syl 17	. . . . . . . . . 10 |- ( n e. NN -> ( 2 || n <-> ( n = 0 \/ 2 || n ) ) )
85	84	ifbid 4108	. . . . . . . . 9 |- ( n e. NN -> if ( 2 || n , 0 , ( ( -u 1 ^ ( ( n - 1 ) / 2 ) ) / n ) ) = if ( ( n = 0 \/ 2 || n ) , 0 , ( ( -u 1 ^ ( ( n - 1 ) / 2 ) ) / n ) ) )
86		breq2 4657	. . . . . . . . . . 11 |- ( k = n -> ( 2 || k <-> 2 || n ) )
87		oveq1 6657	. . . . . . . . . . . . . 14 |- ( k = n -> ( k - 1 ) = ( n - 1 ) )
88	87	oveq1d 6665	. . . . . . . . . . . . 13 |- ( k = n -> ( ( k - 1 ) / 2 ) = ( ( n - 1 ) / 2 ) )
89	88	oveq2d 6666	. . . . . . . . . . . 12 |- ( k = n -> ( -u 1 ^ ( ( k - 1 ) / 2 ) ) = ( -u 1 ^ ( ( n - 1 ) / 2 ) ) )
90		id 22	. . . . . . . . . . . 12 |- ( k = n -> k = n )
91	89, 90	oveq12d 6668	. . . . . . . . . . 11 |- ( k = n -> ( ( -u 1 ^ ( ( k - 1 ) / 2 ) ) / k ) = ( ( -u 1 ^ ( ( n - 1 ) / 2 ) ) / n ) )
92	86, 91	ifbieq2d 4111	. . . . . . . . . 10 |- ( k = n -> if ( 2 || k , 0 , ( ( -u 1 ^ ( ( k - 1 ) / 2 ) ) / k ) ) = if ( 2 || n , 0 , ( ( -u 1 ^ ( ( n - 1 ) / 2 ) ) / n ) ) )
93		eqid 2622	. . . . . . . . . 10 |- ( k e. NN |-> if ( 2 || k , 0 , ( ( -u 1 ^ ( ( k - 1 ) / 2 ) ) / k ) ) ) = ( k e. NN |-> if ( 2 || k , 0 , ( ( -u 1 ^ ( ( k - 1 ) / 2 ) ) / k ) ) )
94		ovex 6678	. . . . . . . . . . 11 |- ( ( -u 1 ^ ( ( n - 1 ) / 2 ) ) / n ) e. _V
95	72, 94	ifex 4156	. . . . . . . . . 10 |- if ( 2 || n , 0 , ( ( -u 1 ^ ( ( n - 1 ) / 2 ) ) / n ) ) e. _V
96	92, 93, 95	fvmpt 6282	. . . . . . . . 9 |- ( n e. NN -> ( ( k e. NN |-> if ( 2 || k , 0 , ( ( -u 1 ^ ( ( k - 1 ) / 2 ) ) / k ) ) ) ` n ) = if ( 2 || n , 0 , ( ( -u 1 ^ ( ( n - 1 ) / 2 ) ) / n ) ) )
97		nnnn0 11299	. . . . . . . . . 10 |- ( n e. NN -> n e. NN0 )
98		eqeq1 2626	. . . . . . . . . . . . 13 |- ( k = n -> ( k = 0 <-> n = 0 ) )
99	98, 86	orbi12d 746	. . . . . . . . . . . 12 |- ( k = n -> ( ( k = 0 \/ 2 || k ) <-> ( n = 0 \/ 2 || n ) ) )
100	99, 91	ifbieq2d 4111	. . . . . . . . . . 11 |- ( k = n -> if ( ( k = 0 \/ 2 || k ) , 0 , ( ( -u 1 ^ ( ( k - 1 ) / 2 ) ) / k ) ) = if ( ( n = 0 \/ 2 || n ) , 0 , ( ( -u 1 ^ ( ( n - 1 ) / 2 ) ) / n ) ) )
101	72, 94	ifex 4156	. . . . . . . . . . 11 |- if ( ( n = 0 \/ 2 || n ) , 0 , ( ( -u 1 ^ ( ( n - 1 ) / 2 ) ) / n ) ) e. _V
102	100, 47, 101	fvmpt 6282	. . . . . . . . . 10 |- ( n e. NN0 -> ( G ` n ) = if ( ( n = 0 \/ 2 || n ) , 0 , ( ( -u 1 ^ ( ( n - 1 ) / 2 ) ) / n ) ) )
103	97, 102	syl 17	. . . . . . . . 9 |- ( n e. NN -> ( G ` n ) = if ( ( n = 0 \/ 2 || n ) , 0 , ( ( -u 1 ^ ( ( n - 1 ) / 2 ) ) / n ) ) )
104	85, 96, 103	3eqtr4d 2666	. . . . . . . 8 |- ( n e. NN -> ( ( k e. NN |-> if ( 2 || k , 0 , ( ( -u 1 ^ ( ( k - 1 ) / 2 ) ) / k ) ) ) ` n ) = ( G ` n ) )
105	80, 104	syl 17	. . . . . . 7 |- ( ( T. /\ n e. ( ZZ>= ` 1 ) ) -> ( ( k e. NN |-> if ( 2 || k , 0 , ( ( -u 1 ^ ( ( k - 1 ) / 2 ) ) / k ) ) ) ` n ) = ( G ` n ) )
106	77, 105	seqfeq 12826	. . . . . 6 |- ( T. -> seq 1 ( + , ( k e. NN |-> if ( 2 || k , 0 , ( ( -u 1 ^ ( ( k - 1 ) / 2 ) ) / k ) ) ) ) = seq 1 ( + , G ) )
107	76, 106	eqtr4d 2659	. . . . 5 |- ( T. -> ( seq 0 ( + , G ) |` ( ZZ>= ` 1 ) ) = seq 1 ( + , ( k e. NN |-> if ( 2 || k , 0 , ( ( -u 1 ^ ( ( k - 1 ) / 2 ) ) / k ) ) ) ) )
108	107	trud 1493	. . . 4 |- ( seq 0 ( + , G ) |` ( ZZ>= ` 1 ) ) = seq 1 ( + , ( k e. NN |-> if ( 2 || k , 0 , ( ( -u 1 ^ ( ( k - 1 ) / 2 ) ) / k ) ) ) )
109	108	breq1i 4660	. . 3 |- ( ( seq 0 ( + , G ) |` ( ZZ>= ` 1 ) ) ~~> A <-> seq 1 ( + , ( k e. NN |-> if ( 2 || k , 0 , ( ( -u 1 ^ ( ( k - 1 ) / 2 ) ) / k ) ) ) ) ~~> A )
110		1z 11407	. . . 4 |- 1 e. ZZ
111		seqex 12803	. . . 4 |- seq 0 ( + , G ) e. _V
112		climres 14306	. . . 4 |- ( ( 1 e. ZZ /\ seq 0 ( + , G ) e. _V ) -> ( ( seq 0 ( + , G ) |` ( ZZ>= ` 1 ) ) ~~> A <-> seq 0 ( + , G ) ~~> A ) )
113	110, 111, 112	mp2an 708	. . 3 |- ( ( seq 0 ( + , G ) |` ( ZZ>= ` 1 ) ) ~~> A <-> seq 0 ( + , G ) ~~> A )
114	109, 113	bitr3i 266	. 2 |- ( seq 1 ( + , ( k e. NN |-> if ( 2 || k , 0 , ( ( -u 1 ^ ( ( k - 1 ) / 2 ) ) / k ) ) ) ) ~~> A <-> seq 0 ( + , G ) ~~> A )
115	21, 40, 114	3bitri 286	1 |- ( seq 0 ( + , F ) ~~> A <-> seq 0 ( + , G ) ~~> A )

Syntax hints:   -. wn 3   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   T. wtru 1484   e. wcel 1990   =/= wne 2794   _Vcvv 3200   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   |` cres 5116   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   -ucneg 10267   / cdiv 10684   NNcn 11020   2c2 11070   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   seqcseq 12801   ^cexp 12860   ~~> cli 14215   || cdvds 14983

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  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-int 4476  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-isom 5897  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-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-card 8765  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-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-seq 12802  df-exp 12861  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-dvds 14984

This theorem is referenced by:  leibpi  24669