Theorem efif1olem2 23479

Description: Lemma for efif1o 23482. (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 458	. . . . . 6 |- ( ( A e. RR /\ z e. RR ) -> A e. RR )
2		2re 10680	. . . . . . 7 |- 2 e. RR
3		pire 23400	. . . . . . 7 |- pi e. RR
4	2, 3	remulcli 9658	. . . . . 6 |- ( 2 x. pi ) e. RR
5		readdcl 9623	. . . . . 6 |- ( ( A e. RR /\ ( 2 x. pi ) e. RR ) -> ( A + ( 2 x. pi ) ) e. RR )
6	1, 4, 5	sylancl 666	. . . . 5 |- ( ( A e. RR /\ z e. RR ) -> ( A + ( 2 x. pi ) ) e. RR )
7		resubcl 9939	. . . . . 6 |- ( ( A e. RR /\ z e. RR ) -> ( A - z ) e. RR )
8		2pos 10702	. . . . . . . 8 |- 0 < 2
9		pipos 23402	. . . . . . . 8 |- 0 < pi
10	2, 3, 8, 9	mulgt0ii 9769	. . . . . . 7 |- 0 < ( 2 x. pi )
11	4, 10	elrpii 11306	. . . . . 6 |- ( 2 x. pi ) e. RR+
12		modcl 12100	. . . . . 6 |- ( ( ( A - z ) e. RR /\ ( 2 x. pi ) e. RR+ ) -> ( ( A - z ) mod ( 2 x. pi ) ) e. RR )
13	7, 11, 12	sylancl 666	. . . . 5 |- ( ( A e. RR /\ z e. RR ) -> ( ( A - z ) mod ( 2 x. pi ) ) e. RR )
14	6, 13	resubcld 10048	. . . 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 12107	. . . . . . 7 |- ( ( ( A - z ) e. RR /\ ( 2 x. pi ) e. RR+ ) -> ( ( A - z ) mod ( 2 x. pi ) ) < ( 2 x. pi ) )
17	7, 11, 16	sylancl 666	. . . . . 6 |- ( ( A e. RR /\ z e. RR ) -> ( ( A - z ) mod ( 2 x. pi ) ) < ( 2 x. pi ) )
18	13, 15, 1, 17	ltadd2dd 9795	. . . . 5 |- ( ( A e. RR /\ z e. RR ) -> ( A + ( ( A - z ) mod ( 2 x. pi ) ) ) < ( A + ( 2 x. pi ) ) )
19	1, 13, 6	ltaddsubd 10214	. . . . 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 213	. . . 4 |- ( ( A e. RR /\ z e. RR ) -> A < ( ( A + ( 2 x. pi ) ) - ( ( A - z ) mod ( 2 x. pi ) ) ) )
21		modge0 12106	. . . . . 6 |- ( ( ( A - z ) e. RR /\ ( 2 x. pi ) e. RR+ ) -> 0 <_ ( ( A - z ) mod ( 2 x. pi ) ) )
22	7, 11, 21	sylancl 666	. . . . 5 |- ( ( A e. RR /\ z e. RR ) -> 0 <_ ( ( A - z ) mod ( 2 x. pi ) ) )
23	6, 13	subge02d 10206	. . . . 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 213	. . . 4 |- ( ( A e. RR /\ z e. RR ) -> ( ( A + ( 2 x. pi ) ) - ( ( A - z ) mod ( 2 x. pi ) ) ) <_ ( A + ( 2 x. pi ) ) )
25		rexr 9687	. . . . . 6 |- ( A e. RR -> A e. RR* )
26	25	adantr 466	. . . . 5 |- ( ( A e. RR /\ z e. RR ) -> A e. RR* )
27		elioc2 11698	. . . . 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 665	. . . 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 1188	. . 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 2521	. 2 |- ( ( A e. RR /\ z e. RR ) -> ( ( A + ( 2 x. pi ) ) - ( ( A - z ) mod ( 2 x. pi ) ) ) e. D )
32		modval 12098	. . . . . . . . . 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 666	. . . . . . . . 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 6318	. . . . . . . 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 9670	. . . . . . . . 9 |- ( ( A e. RR /\ z e. RR ) -> ( A + ( 2 x. pi ) ) e. CC )
36	7	recnd 9670	. . . . . . . . 9 |- ( ( A e. RR /\ z e. RR ) -> ( A - z ) e. CC )
37	4, 10	gt0ne0ii 10151	. . . . . . . . . . . . . . 15 |- ( 2 x. pi ) =/= 0
38		redivcl 10327	. . . . . . . . . . . . . . 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 1352	. . . . . . . . . . . . . 14 |- ( ( A - z ) e. RR -> ( ( A - z ) / ( 2 x. pi ) ) e. RR )
40	7, 39	syl 17	. . . . . . . . . . . . 13 |- ( ( A e. RR /\ z e. RR ) -> ( ( A - z ) / ( 2 x. pi ) ) e. RR )
41	40	flcld 12034	. . . . . . . . . . . 12 |- ( ( A e. RR /\ z e. RR ) -> ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) e. ZZ )
42	41	zred 11041	. . . . . . . . . . 11 |- ( ( A e. RR /\ z e. RR ) -> ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) e. RR )
43		remulcl 9625	. . . . . . . . . . 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 667	. . . . . . . . . 10 |- ( ( A e. RR /\ z e. RR ) -> ( ( 2 x. pi ) x. ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) ) e. RR )
45	44	recnd 9670	. . . . . . . . 9 |- ( ( A e. RR /\ z e. RR ) -> ( ( 2 x. pi ) x. ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) ) e. CC )
46	35, 36, 45	subsubd 10015	. . . . . . . 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 9670	. . . . . . . . . 10 |- ( ( A e. RR /\ z e. RR ) -> A e. CC )
48	4	recni 9656	. . . . . . . . . . 11 |- ( 2 x. pi ) e. CC
49	48	a1i 11	. . . . . . . . . 10 |- ( ( A e. RR /\ z e. RR ) -> ( 2 x. pi ) e. CC )
50		simpr 462	. . . . . . . . . . 11 |- ( ( A e. RR /\ z e. RR ) -> z e. RR )
51	50	recnd 9670	. . . . . . . . . 10 |- ( ( A e. RR /\ z e. RR ) -> z e. CC )
52	47, 49, 51	pnncand 10026	. . . . . . . . 9 |- ( ( A e. RR /\ z e. RR ) -> ( ( A + ( 2 x. pi ) ) - ( A - z ) ) = ( ( 2 x. pi ) + z ) )
53	52	oveq1d 6317	. . . . . . . 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 2467	. . . . . . 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 6318	. . . . . 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 9622	. . . . . . . 8 |- ( ( ( 2 x. pi ) e. CC /\ z e. CC ) -> ( ( 2 x. pi ) + z ) e. CC )
57	48, 51, 56	sylancr 667	. . . . . . 7 |- ( ( A e. RR /\ z e. RR ) -> ( ( 2 x. pi ) + z ) e. CC )
58	51, 57, 45	subsub4d 10018	. . . . . 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 10003	. . . . . . . . . 10 |- ( ( A e. RR /\ z e. RR ) -> -u ( ( ( 2 x. pi ) + z ) - z ) = ( z - ( ( 2 x. pi ) + z ) ) )
60	49, 51	pncand 9988	. . . . . . . . . . 11 |- ( ( A e. RR /\ z e. RR ) -> ( ( ( 2 x. pi ) + z ) - z ) = ( 2 x. pi ) )
61	60	negeqd 9870	. . . . . . . . . 10 |- ( ( A e. RR /\ z e. RR ) -> -u ( ( ( 2 x. pi ) + z ) - z ) = -u ( 2 x. pi ) )
62	59, 61	eqtr3d 2465	. . . . . . . . 9 |- ( ( A e. RR /\ z e. RR ) -> ( z - ( ( 2 x. pi ) + z ) ) = -u ( 2 x. pi ) )
63		neg1cn 10714	. . . . . . . . . 10 |- -u 1 e. CC
64	48	mulm1i 10064	. . . . . . . . . 10 |- ( -u 1 x. ( 2 x. pi ) ) = -u ( 2 x. pi )
65	63, 48, 64	mulcomli 9651	. . . . . . . . 9 |- ( ( 2 x. pi ) x. -u 1 ) = -u ( 2 x. pi )
66	62, 65	syl6eqr 2481	. . . . . . . 8 |- ( ( A e. RR /\ z e. RR ) -> ( z - ( ( 2 x. pi ) + z ) ) = ( ( 2 x. pi ) x. -u 1 ) )
67	66	oveq1d 6317	. . . . . . 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 11042	. . . . . . . 8 |- ( ( A e. RR /\ z e. RR ) -> ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) e. CC )
70	49, 68, 69	subdid 10075	. . . . . . 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 2466	. . . . . 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 2469	. . . . 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 6317	. . . 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 10974	. . . . . . 7 |- -u 1 e. ZZ
75		zsubcl 10980	. . . . . . 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 667	. . . . . 6 |- ( ( A e. RR /\ z e. RR ) -> ( -u 1 - ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) ) e. ZZ )
77	76	zcnd 11042	. . . . 5 |- ( ( A e. RR /\ z e. RR ) -> ( -u 1 - ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) ) e. CC )
78		divcan3 10295	. . . . . 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 1352	. . . . 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 17	. . . 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 2463	. . 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 2510	. 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 6310	. . . . 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 6317	. . . 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 2491	. . 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 3182	. 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 665	1 |- ( ( A e. RR /\ z e. RR ) -> E. y e. D ( ( z - y ) / ( 2 x. pi ) ) e. ZZ )

Syntax hints:   -> wi 4   <-> wb 187   /\ wa 370   /\ w3a 982   = wceq 1437   e. wcel 1868   =/= wne 2618   E.wrex 2776   class class class wbr 4420   ` cfv 5598  (class class class)co 6302   CCcc 9538   RRcr 9539   0cc0 9540   1c1 9541   + caddc 9543   x. cmul 9545   RR*cxr 9675   < clt 9676   <_ cle 9677   - cmin 9861   -ucneg 9862   / cdiv 10270   2c2 10660   ZZcz 10938   RR+crp 11303   (,]cioc 11637   |_cfl 12026   mod cmo 12096   picpi 14107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-inf2 8149  ax-cnex 9596  ax-resscn 9597  ax-1cn 9598  ax-icn 9599  ax-addcl 9600  ax-addrcl 9601  ax-mulcl 9602  ax-mulrcl 9603  ax-mulcom 9604  ax-addass 9605  ax-mulass 9606  ax-distr 9607  ax-i2m1 9608  ax-1ne0 9609  ax-1rid 9610  ax-rnegex 9611  ax-rrecex 9612  ax-cnre 9613  ax-pre-lttri 9614  ax-pre-lttrn 9615  ax-pre-ltadd 9616  ax-pre-mulgt0 9617  ax-pre-sup 9618  ax-addf 9619  ax-mulf 9620

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-iin 4299  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-se 4810  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-isom 5607  df-riota 6264  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-of 6542  df-om 6704  df-1st 6804  df-2nd 6805  df-supp 6923  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-2o 7188  df-oadd 7191  df-er 7368  df-map 7479  df-pm 7480  df-ixp 7528  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-fsupp 7887  df-fi 7928  df-sup 7959  df-inf 7960  df-oi 8028  df-card 8375  df-cda 8599  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-sub 9863  df-neg 9864  df-div 10271  df-nn 10611  df-2 10669  df-3 10670  df-4 10671  df-5 10672  df-6 10673  df-7 10674  df-8 10675  df-9 10676  df-10 10677  df-n0 10871  df-z 10939  df-dec 11053  df-uz 11161  df-q 11266  df-rp 11304  df-xneg 11410  df-xadd 11411  df-xmul 11412  df-ioo 11640  df-ioc 11641  df-ico 11642  df-icc 11643  df-fz 11786  df-fzo 11917  df-fl 12028  df-mod 12097  df-seq 12214  df-exp 12273  df-fac 12460  df-bc 12488  df-hash 12516  df-shft 13119  df-cj 13151  df-re 13152  df-im 13153  df-sqrt 13287  df-abs 13288  df-limsup 13514  df-clim 13540  df-rlim 13541  df-sum 13741  df-ef 14109  df-sin 14111  df-cos 14112  df-pi 14114  df-struct 15111  df-ndx 15112  df-slot 15113  df-base 15114  df-sets 15115  df-ress 15116  df-plusg 15191  df-mulr 15192  df-starv 15193  df-sca 15194  df-vsca 15195  df-ip 15196  df-tset 15197  df-ple 15198  df-ds 15200  df-unif 15201  df-hom 15202  df-cco 15203  df-rest 15309  df-topn 15310  df-0g 15328  df-gsum 15329  df-topgen 15330  df-pt 15331  df-prds 15334  df-xrs 15388  df-qtop 15394  df-imas 15395  df-xps 15398  df-mre 15480  df-mrc 15481  df-acs 15483  df-mgm 16476  df-sgrp 16515  df-mnd 16525  df-submnd 16571  df-mulg 16664  df-cntz 16959  df-cmn 17420  df-psmet 18950  df-xmet 18951  df-met 18952  df-bl 18953  df-mopn 18954  df-fbas 18955  df-fg 18956  df-cnfld 18959  df-top 19908  df-bases 19909  df-topon 19910  df-topsp 19911  df-cld 20021  df-ntr 20022  df-cls 20023  df-nei 20101  df-lp 20139  df-perf 20140  df-cn 20230  df-cnp 20231  df-haus 20318  df-tx 20564  df-hmeo 20757  df-fil 20848  df-fm 20940  df-flim 20941  df-flf 20942  df-xms 21322  df-ms 21323  df-tms 21324  df-cncf 21897  df-limc 22808  df-dv 22809

This theorem is referenced by:  efif1o  23482  eff1o  23485