Theorem efif1olem2 22691

Description: Lemma for efif1o 22694. (Contributed by Mario Carneiro, 13-May-2014.)

Hypothesis

Ref	Expression
efif1olem1.1	|- D = ( A (,] ( A + ( 2 x. pi ) ) )

Assertion

Ref	Expression
efif1olem2	|- ( ( A e. RR /\ z e. RR ) -> E. y e. D ( ( z - y ) / ( 2 x. pi ) ) e. ZZ )

Distinct variable groups:   y, z   y, A   y, D

Allowed substitution hints:   A( z)   D( z)

Proof of Theorem efif1olem2

Step	Hyp	Ref	Expression
1		simpl 457	. . . . . 6 |- ( ( A e. RR /\ z e. RR ) -> A e. RR )
2		2re 10605	. . . . . . 7 |- 2 e. RR
3		pire 22613	. . . . . . 7 |- pi e. RR
4	2, 3	remulcli 9610	. . . . . 6 |- ( 2 x. pi ) e. RR
5		readdcl 9575	. . . . . 6 |- ( ( A e. RR /\ ( 2 x. pi ) e. RR ) -> ( A + ( 2 x. pi ) ) e. RR )
6	1, 4, 5	sylancl 662	. . . . 5 |- ( ( A e. RR /\ z e. RR ) -> ( A + ( 2 x. pi ) ) e. RR )
7		resubcl 9883	. . . . . 6 |- ( ( A e. RR /\ z e. RR ) -> ( A - z ) e. RR )
8		2pos 10627	. . . . . . . 8 |- 0 < 2
9		pipos 22615	. . . . . . . 8 |- 0 < pi
10	2, 3, 8, 9	mulgt0ii 9717	. . . . . . 7 |- 0 < ( 2 x. pi )
11	4, 10	elrpii 11223	. . . . . 6 |- ( 2 x. pi ) e. RR+
12		modcl 11968	. . . . . 6 |- ( ( ( A - z ) e. RR /\ ( 2 x. pi ) e. RR+ ) -> ( ( A - z ) mod ( 2 x. pi ) ) e. RR )
13	7, 11, 12	sylancl 662	. . . . 5 |- ( ( A e. RR /\ z e. RR ) -> ( ( A - z ) mod ( 2 x. pi ) ) e. RR )
14	6, 13	resubcld 9987	. . . 4 |- ( ( A e. RR /\ z e. RR ) -> ( ( A + ( 2 x. pi ) ) - ( ( A - z ) mod ( 2 x. pi ) ) ) e. RR )
15	4	a1i 11	. . . . . 6 |- ( ( A e. RR /\ z e. RR ) -> ( 2 x. pi ) e. RR )
16		modlt 11974	. . . . . . 7 |- ( ( ( A - z ) e. RR /\ ( 2 x. pi ) e. RR+ ) -> ( ( A - z ) mod ( 2 x. pi ) ) < ( 2 x. pi ) )
17	7, 11, 16	sylancl 662	. . . . . 6 |- ( ( A e. RR /\ z e. RR ) -> ( ( A - z ) mod ( 2 x. pi ) ) < ( 2 x. pi ) )
18	13, 15, 1, 17	ltadd2dd 9740	. . . . 5 |- ( ( A e. RR /\ z e. RR ) -> ( A + ( ( A - z ) mod ( 2 x. pi ) ) ) < ( A + ( 2 x. pi ) ) )
19	1, 13, 6	ltaddsubd 10152	. . . . 5 |- ( ( A e. RR /\ z e. RR ) -> ( ( A + ( ( A - z ) mod ( 2 x. pi ) ) ) < ( A + ( 2 x. pi ) ) <-> A < ( ( A + ( 2 x. pi ) ) - ( ( A - z ) mod ( 2 x. pi ) ) ) ) )
20	18, 19	mpbid 210	. . . 4 |- ( ( A e. RR /\ z e. RR ) -> A < ( ( A + ( 2 x. pi ) ) - ( ( A - z ) mod ( 2 x. pi ) ) ) )
21		modge0 11973	. . . . . 6 |- ( ( ( A - z ) e. RR /\ ( 2 x. pi ) e. RR+ ) -> 0 <_ ( ( A - z ) mod ( 2 x. pi ) ) )
22	7, 11, 21	sylancl 662	. . . . 5 |- ( ( A e. RR /\ z e. RR ) -> 0 <_ ( ( A - z ) mod ( 2 x. pi ) ) )
23	6, 13	subge02d 10144	. . . . 5 |- ( ( A e. RR /\ z e. RR ) -> ( 0 <_ ( ( A - z ) mod ( 2 x. pi ) ) <-> ( ( A + ( 2 x. pi ) ) - ( ( A - z ) mod ( 2 x. pi ) ) ) <_ ( A + ( 2 x. pi ) ) ) )
24	22, 23	mpbid 210	. . . 4 |- ( ( A e. RR /\ z e. RR ) -> ( ( A + ( 2 x. pi ) ) - ( ( A - z ) mod ( 2 x. pi ) ) ) <_ ( A + ( 2 x. pi ) ) )
25		rexr 9639	. . . . . 6 |- ( A e. RR -> A e. RR* )
26	25	adantr 465	. . . . 5 |- ( ( A e. RR /\ z e. RR ) -> A e. RR* )
27		elioc2 11587	. . . . 5 |- ( ( A e. RR* /\ ( A + ( 2 x. pi ) ) e. RR ) -> ( ( ( A + ( 2 x. pi ) ) - ( ( A - z ) mod ( 2 x. pi ) ) ) e. ( A (,] ( A + ( 2 x. pi ) ) ) <-> ( ( ( A + ( 2 x. pi ) ) - ( ( A - z ) mod ( 2 x. pi ) ) ) e. RR /\ A < ( ( A + ( 2 x. pi ) ) - ( ( A - z ) mod ( 2 x. pi ) ) ) /\ ( ( A + ( 2 x. pi ) ) - ( ( A - z ) mod ( 2 x. pi ) ) ) <_ ( A + ( 2 x. pi ) ) ) ) )
28	26, 6, 27	syl2anc 661	. . . 4 |- ( ( A e. RR /\ z e. RR ) -> ( ( ( A + ( 2 x. pi ) ) - ( ( A - z ) mod ( 2 x. pi ) ) ) e. ( A (,] ( A + ( 2 x. pi ) ) ) <-> ( ( ( A + ( 2 x. pi ) ) - ( ( A - z ) mod ( 2 x. pi ) ) ) e. RR /\ A < ( ( A + ( 2 x. pi ) ) - ( ( A - z ) mod ( 2 x. pi ) ) ) /\ ( ( A + ( 2 x. pi ) ) - ( ( A - z ) mod ( 2 x. pi ) ) ) <_ ( A + ( 2 x. pi ) ) ) ) )
29	14, 20, 24, 28	mpbir3and 1179	. . 3 |- ( ( A e. RR /\ z e. RR ) -> ( ( A + ( 2 x. pi ) ) - ( ( A - z ) mod ( 2 x. pi ) ) ) e. ( A (,] ( A + ( 2 x. pi ) ) ) )
30		efif1olem1.1	. . 3 |- D = ( A (,] ( A + ( 2 x. pi ) ) )
31	29, 30	syl6eleqr 2566	. 2 |- ( ( A e. RR /\ z e. RR ) -> ( ( A + ( 2 x. pi ) ) - ( ( A - z ) mod ( 2 x. pi ) ) ) e. D )
32		modval 11966	. . . . . . . . . 10 |- ( ( ( A - z ) e. RR /\ ( 2 x. pi ) e. RR+ ) -> ( ( A - z ) mod ( 2 x. pi ) ) = ( ( A - z ) - ( ( 2 x. pi ) x. ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) ) ) )
33	7, 11, 32	sylancl 662	. . . . . . . . 9 |- ( ( A e. RR /\ z e. RR ) -> ( ( A - z ) mod ( 2 x. pi ) ) = ( ( A - z ) - ( ( 2 x. pi ) x. ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) ) ) )
34	33	oveq2d 6300	. . . . . . . 8 |- ( ( A e. RR /\ z e. RR ) -> ( ( A + ( 2 x. pi ) ) - ( ( A - z ) mod ( 2 x. pi ) ) ) = ( ( A + ( 2 x. pi ) ) - ( ( A - z ) - ( ( 2 x. pi ) x. ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) ) ) ) )
35	6	recnd 9622	. . . . . . . . 9 |- ( ( A e. RR /\ z e. RR ) -> ( A + ( 2 x. pi ) ) e. CC )
36	7	recnd 9622	. . . . . . . . 9 |- ( ( A e. RR /\ z e. RR ) -> ( A - z ) e. CC )
37	4, 10	gt0ne0ii 10089	. . . . . . . . . . . . . . 15 |- ( 2 x. pi ) =/= 0
38		redivcl 10263	. . . . . . . . . . . . . . 15 |- ( ( ( A - z ) e. RR /\ ( 2 x. pi ) e. RR /\ ( 2 x. pi ) =/= 0 ) -> ( ( A - z ) / ( 2 x. pi ) ) e. RR )
39	4, 37, 38	mp3an23 1316	. . . . . . . . . . . . . 14 |- ( ( A - z ) e. RR -> ( ( A - z ) / ( 2 x. pi ) ) e. RR )
40	7, 39	syl 16	. . . . . . . . . . . . 13 |- ( ( A e. RR /\ z e. RR ) -> ( ( A - z ) / ( 2 x. pi ) ) e. RR )
41	40	flcld 11903	. . . . . . . . . . . 12 |- ( ( A e. RR /\ z e. RR ) -> ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) e. ZZ )
42	41	zred 10966	. . . . . . . . . . 11 |- ( ( A e. RR /\ z e. RR ) -> ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) e. RR )
43		remulcl 9577	. . . . . . . . . . 11 |- ( ( ( 2 x. pi ) e. RR /\ ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) e. RR ) -> ( ( 2 x. pi ) x. ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) ) e. RR )
44	4, 42, 43	sylancr 663	. . . . . . . . . 10 |- ( ( A e. RR /\ z e. RR ) -> ( ( 2 x. pi ) x. ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) ) e. RR )
45	44	recnd 9622	. . . . . . . . 9 |- ( ( A e. RR /\ z e. RR ) -> ( ( 2 x. pi ) x. ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) ) e. CC )
46	35, 36, 45	subsubd 9958	. . . . . . . 8 |- ( ( A e. RR /\ z e. RR ) -> ( ( A + ( 2 x. pi ) ) - ( ( A - z ) - ( ( 2 x. pi ) x. ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) ) ) ) = ( ( ( A + ( 2 x. pi ) ) - ( A - z ) ) + ( ( 2 x. pi ) x. ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) ) ) )
47	1	recnd 9622	. . . . . . . . . 10 |- ( ( A e. RR /\ z e. RR ) -> A e. CC )
48	4	recni 9608	. . . . . . . . . . 11 |- ( 2 x. pi ) e. CC
49	48	a1i 11	. . . . . . . . . 10 |- ( ( A e. RR /\ z e. RR ) -> ( 2 x. pi ) e. CC )
50		simpr 461	. . . . . . . . . . 11 |- ( ( A e. RR /\ z e. RR ) -> z e. RR )
51	50	recnd 9622	. . . . . . . . . 10 |- ( ( A e. RR /\ z e. RR ) -> z e. CC )
52	47, 49, 51	pnncand 9969	. . . . . . . . 9 |- ( ( A e. RR /\ z e. RR ) -> ( ( A + ( 2 x. pi ) ) - ( A - z ) ) = ( ( 2 x. pi ) + z ) )
53	52	oveq1d 6299	. . . . . . . 8 |- ( ( A e. RR /\ z e. RR ) -> ( ( ( A + ( 2 x. pi ) ) - ( A - z ) ) + ( ( 2 x. pi ) x. ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) ) ) = ( ( ( 2 x. pi ) + z ) + ( ( 2 x. pi ) x. ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) ) ) )
54	34, 46, 53	3eqtrd 2512	. . . . . . 7 |- ( ( A e. RR /\ z e. RR ) -> ( ( A + ( 2 x. pi ) ) - ( ( A - z ) mod ( 2 x. pi ) ) ) = ( ( ( 2 x. pi ) + z ) + ( ( 2 x. pi ) x. ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) ) ) )
55	54	oveq2d 6300	. . . . . 6 |- ( ( A e. RR /\ z e. RR ) -> ( z - ( ( A + ( 2 x. pi ) ) - ( ( A - z ) mod ( 2 x. pi ) ) ) ) = ( z - ( ( ( 2 x. pi ) + z ) + ( ( 2 x. pi ) x. ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) ) ) ) )
56		addcl 9574	. . . . . . . 8 |- ( ( ( 2 x. pi ) e. CC /\ z e. CC ) -> ( ( 2 x. pi ) + z ) e. CC )
57	48, 51, 56	sylancr 663	. . . . . . 7 |- ( ( A e. RR /\ z e. RR ) -> ( ( 2 x. pi ) + z ) e. CC )
58	51, 57, 45	subsub4d 9961	. . . . . 6 |- ( ( A e. RR /\ z e. RR ) -> ( ( z - ( ( 2 x. pi ) + z ) ) - ( ( 2 x. pi ) x. ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) ) ) = ( z - ( ( ( 2 x. pi ) + z ) + ( ( 2 x. pi ) x. ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) ) ) ) )
59	57, 51	negsubdi2d 9946	. . . . . . . . . 10 |- ( ( A e. RR /\ z e. RR ) -> -u ( ( ( 2 x. pi ) + z ) - z ) = ( z - ( ( 2 x. pi ) + z ) ) )
60	49, 51	pncand 9931	. . . . . . . . . . 11 |- ( ( A e. RR /\ z e. RR ) -> ( ( ( 2 x. pi ) + z ) - z ) = ( 2 x. pi ) )
61	60	negeqd 9814	. . . . . . . . . 10 |- ( ( A e. RR /\ z e. RR ) -> -u ( ( ( 2 x. pi ) + z ) - z ) = -u ( 2 x. pi ) )
62	59, 61	eqtr3d 2510	. . . . . . . . 9 |- ( ( A e. RR /\ z e. RR ) -> ( z - ( ( 2 x. pi ) + z ) ) = -u ( 2 x. pi ) )
63		neg1cn 10639	. . . . . . . . . 10 |- -u 1 e. CC
64	48	mulm1i 10001	. . . . . . . . . 10 |- ( -u 1 x. ( 2 x. pi ) ) = -u ( 2 x. pi )
65	63, 48, 64	mulcomli 9603	. . . . . . . . 9 |- ( ( 2 x. pi ) x. -u 1 ) = -u ( 2 x. pi )
66	62, 65	syl6eqr 2526	. . . . . . . 8 |- ( ( A e. RR /\ z e. RR ) -> ( z - ( ( 2 x. pi ) + z ) ) = ( ( 2 x. pi ) x. -u 1 ) )
67	66	oveq1d 6299	. . . . . . 7 |- ( ( A e. RR /\ z e. RR ) -> ( ( z - ( ( 2 x. pi ) + z ) ) - ( ( 2 x. pi ) x. ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) ) ) = ( ( ( 2 x. pi ) x. -u 1 ) - ( ( 2 x. pi ) x. ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) ) ) )
68	63	a1i 11	. . . . . . . 8 |- ( ( A e. RR /\ z e. RR ) -> -u 1 e. CC )
69	41	zcnd 10967	. . . . . . . 8 |- ( ( A e. RR /\ z e. RR ) -> ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) e. CC )
70	49, 68, 69	subdid 10012	. . . . . . 7 |- ( ( A e. RR /\ z e. RR ) -> ( ( 2 x. pi ) x. ( -u 1 - ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) ) ) = ( ( ( 2 x. pi ) x. -u 1 ) - ( ( 2 x. pi ) x. ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) ) ) )
71	67, 70	eqtr4d 2511	. . . . . 6 |- ( ( A e. RR /\ z e. RR ) -> ( ( z - ( ( 2 x. pi ) + z ) ) - ( ( 2 x. pi ) x. ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) ) ) = ( ( 2 x. pi ) x. ( -u 1 - ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) ) ) )
72	55, 58, 71	3eqtr2d 2514	. . . . 5 |- ( ( A e. RR /\ z e. RR ) -> ( z - ( ( A + ( 2 x. pi ) ) - ( ( A - z ) mod ( 2 x. pi ) ) ) ) = ( ( 2 x. pi ) x. ( -u 1 - ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) ) ) )
73	72	oveq1d 6299	. . . 4 |- ( ( A e. RR /\ z e. RR ) -> ( ( z - ( ( A + ( 2 x. pi ) ) - ( ( A - z ) mod ( 2 x. pi ) ) ) ) / ( 2 x. pi ) ) = ( ( ( 2 x. pi ) x. ( -u 1 - ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) ) ) / ( 2 x. pi ) ) )
74		neg1z 10899	. . . . . . 7 |- -u 1 e. ZZ
75		zsubcl 10905	. . . . . . 7 |- ( ( -u 1 e. ZZ /\ ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) e. ZZ ) -> ( -u 1 - ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) ) e. ZZ )
76	74, 41, 75	sylancr 663	. . . . . 6 |- ( ( A e. RR /\ z e. RR ) -> ( -u 1 - ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) ) e. ZZ )
77	76	zcnd 10967	. . . . 5 |- ( ( A e. RR /\ z e. RR ) -> ( -u 1 - ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) ) e. CC )
78		divcan3 10231	. . . . . 6 |- ( ( ( -u 1 - ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) ) e. CC /\ ( 2 x. pi ) e. CC /\ ( 2 x. pi ) =/= 0 ) -> ( ( ( 2 x. pi ) x. ( -u 1 - ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) ) ) / ( 2 x. pi ) ) = ( -u 1 - ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) ) )
79	48, 37, 78	mp3an23 1316	. . . . 5 |- ( ( -u 1 - ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) ) e. CC -> ( ( ( 2 x. pi ) x. ( -u 1 - ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) ) ) / ( 2 x. pi ) ) = ( -u 1 - ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) ) )
80	77, 79	syl 16	. . . 4 |- ( ( A e. RR /\ z e. RR ) -> ( ( ( 2 x. pi ) x. ( -u 1 - ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) ) ) / ( 2 x. pi ) ) = ( -u 1 - ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) ) )
81	73, 80	eqtrd 2508	. . 3 |- ( ( A e. RR /\ z e. RR ) -> ( ( z - ( ( A + ( 2 x. pi ) ) - ( ( A - z ) mod ( 2 x. pi ) ) ) ) / ( 2 x. pi ) ) = ( -u 1 - ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) ) )
82	81, 76	eqeltrd 2555	. 2 |- ( ( A e. RR /\ z e. RR ) -> ( ( z - ( ( A + ( 2 x. pi ) ) - ( ( A - z ) mod ( 2 x. pi ) ) ) ) / ( 2 x. pi ) ) e. ZZ )
83		oveq2 6292	. . . . 5 |- ( y = ( ( A + ( 2 x. pi ) ) - ( ( A - z ) mod ( 2 x. pi ) ) ) -> ( z - y ) = ( z - ( ( A + ( 2 x. pi ) ) - ( ( A - z ) mod ( 2 x. pi ) ) ) ) )
84	83	oveq1d 6299	. . . 4 |- ( y = ( ( A + ( 2 x. pi ) ) - ( ( A - z ) mod ( 2 x. pi ) ) ) -> ( ( z - y ) / ( 2 x. pi ) ) = ( ( z - ( ( A + ( 2 x. pi ) ) - ( ( A - z ) mod ( 2 x. pi ) ) ) ) / ( 2 x. pi ) ) )
85	84	eleq1d 2536	. . 3 |- ( y = ( ( A + ( 2 x. pi ) ) - ( ( A - z ) mod ( 2 x. pi ) ) ) -> ( ( ( z - y ) / ( 2 x. pi ) ) e. ZZ <-> ( ( z - ( ( A + ( 2 x. pi ) ) - ( ( A - z ) mod ( 2 x. pi ) ) ) ) / ( 2 x. pi ) ) e. ZZ ) )
86	85	rspcev 3214	. 2 |- ( ( ( ( A + ( 2 x. pi ) ) - ( ( A - z ) mod ( 2 x. pi ) ) ) e. D /\ ( ( z - ( ( A + ( 2 x. pi ) ) - ( ( A - z ) mod ( 2 x. pi ) ) ) ) / ( 2 x. pi ) ) e. ZZ ) -> E. y e. D ( ( z - y ) / ( 2 x. pi ) ) e. ZZ )
87	31, 82, 86	syl2anc 661	1 |- ( ( A e. RR /\ z e. RR ) -> E. y e. D ( ( z - y ) / ( 2 x. pi ) ) e. ZZ )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   = wceq 1379   e. wcel 1767   =/= wne 2662   E.wrex 2815   class class class wbr 4447   ` cfv 5588  (class class class)co 6284   CCcc 9490   RRcr 9491   0cc0 9492   1c1 9493   + caddc 9495   x. cmul 9497   RR*cxr 9627   < clt 9628   <_ cle 9629   - cmin 9805   -ucneg 9806   / cdiv 10206   2c2 10585   ZZcz 10864   RR+crp 11220   (,]cioc 11530   |_cfl 11895   mod cmo 11964   picpi 13664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570  ax-addf 9571  ax-mulf 9572

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-om 6685  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7830  df-fi 7871  df-sup 7901  df-oi 7935  df-card 8320  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-q 11183  df-rp 11221  df-xneg 11318  df-xadd 11319  df-xmul 11320  df-ioo 11533  df-ioc 11534  df-ico 11535  df-icc 11536  df-fz 11673  df-fzo 11793  df-fl 11897  df-mod 11965  df-seq 12076  df-exp 12135  df-fac 12322  df-bc 12349  df-hash 12374  df-shft 12863  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-limsup 13257  df-clim 13274  df-rlim 13275  df-sum 13472  df-ef 13665  df-sin 13667  df-cos 13668  df-pi 13670  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-starv 14570  df-sca 14571  df-vsca 14572  df-ip 14573  df-tset 14574  df-ple 14575  df-ds 14577  df-unif 14578  df-hom 14579  df-cco 14580  df-rest 14678  df-topn 14679  df-0g 14697  df-gsum 14698  df-topgen 14699  df-pt 14700  df-prds 14703  df-xrs 14757  df-qtop 14762  df-imas 14763  df-xps 14765  df-mre 14841  df-mrc 14842  df-acs 14844  df-mnd 15732  df-submnd 15787  df-mulg 15870  df-cntz 16160  df-cmn 16606  df-psmet 18210  df-xmet 18211  df-met 18212  df-bl 18213  df-mopn 18214  df-fbas 18215  df-fg 18216  df-cnfld 18220  df-top 19194  df-bases 19196  df-topon 19197  df-topsp 19198  df-cld 19314  df-ntr 19315  df-cls 19316  df-nei 19393  df-lp 19431  df-perf 19432  df-cn 19522  df-cnp 19523  df-haus 19610  df-tx 19826  df-hmeo 20019  df-fil 20110  df-fm 20202  df-flim 20203  df-flf 20204  df-xms 20586  df-ms 20587  df-tms 20588  df-cncf 21145  df-limc 22033  df-dv 22034

This theorem is referenced by:  efif1o  22694  eff1o  22697