Theorem efif1olem2 23096

Description: Lemma for efif1o 23099. (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 455	. . . . . 6 |- ( ( A e. RR /\ z e. RR ) -> A e. RR )
2		2re 10601	. . . . . . 7 |- 2 e. RR
3		pire 23017	. . . . . . 7 |- pi e. RR
4	2, 3	remulcli 9599	. . . . . 6 |- ( 2 x. pi ) e. RR
5		readdcl 9564	. . . . . 6 |- ( ( A e. RR /\ ( 2 x. pi ) e. RR ) -> ( A + ( 2 x. pi ) ) e. RR )
6	1, 4, 5	sylancl 660	. . . . 5 |- ( ( A e. RR /\ z e. RR ) -> ( A + ( 2 x. pi ) ) e. RR )
7		resubcl 9874	. . . . . 6 |- ( ( A e. RR /\ z e. RR ) -> ( A - z ) e. RR )
8		2pos 10623	. . . . . . . 8 |- 0 < 2
9		pipos 23019	. . . . . . . 8 |- 0 < pi
10	2, 3, 8, 9	mulgt0ii 9707	. . . . . . 7 |- 0 < ( 2 x. pi )
11	4, 10	elrpii 11224	. . . . . 6 |- ( 2 x. pi ) e. RR+
12		modcl 11982	. . . . . 6 |- ( ( ( A - z ) e. RR /\ ( 2 x. pi ) e. RR+ ) -> ( ( A - z ) mod ( 2 x. pi ) ) e. RR )
13	7, 11, 12	sylancl 660	. . . . 5 |- ( ( A e. RR /\ z e. RR ) -> ( ( A - z ) mod ( 2 x. pi ) ) e. RR )
14	6, 13	resubcld 9983	. . . 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 11988	. . . . . . 7 |- ( ( ( A - z ) e. RR /\ ( 2 x. pi ) e. RR+ ) -> ( ( A - z ) mod ( 2 x. pi ) ) < ( 2 x. pi ) )
17	7, 11, 16	sylancl 660	. . . . . 6 |- ( ( A e. RR /\ z e. RR ) -> ( ( A - z ) mod ( 2 x. pi ) ) < ( 2 x. pi ) )
18	13, 15, 1, 17	ltadd2dd 9730	. . . . 5 |- ( ( A e. RR /\ z e. RR ) -> ( A + ( ( A - z ) mod ( 2 x. pi ) ) ) < ( A + ( 2 x. pi ) ) )
19	1, 13, 6	ltaddsubd 10148	. . . . 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 11987	. . . . . 6 |- ( ( ( A - z ) e. RR /\ ( 2 x. pi ) e. RR+ ) -> 0 <_ ( ( A - z ) mod ( 2 x. pi ) ) )
22	7, 11, 21	sylancl 660	. . . . 5 |- ( ( A e. RR /\ z e. RR ) -> 0 <_ ( ( A - z ) mod ( 2 x. pi ) ) )
23	6, 13	subge02d 10140	. . . . 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 9628	. . . . . 6 |- ( A e. RR -> A e. RR* )
26	25	adantr 463	. . . . 5 |- ( ( A e. RR /\ z e. RR ) -> A e. RR* )
27		elioc2 11590	. . . . 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 659	. . . 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 1177	. . 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 2553	. 2 |- ( ( A e. RR /\ z e. RR ) -> ( ( A + ( 2 x. pi ) ) - ( ( A - z ) mod ( 2 x. pi ) ) ) e. D )
32		modval 11980	. . . . . . . . . 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 660	. . . . . . . . 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 6286	. . . . . . . 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 9611	. . . . . . . . 9 |- ( ( A e. RR /\ z e. RR ) -> ( A + ( 2 x. pi ) ) e. CC )
36	7	recnd 9611	. . . . . . . . 9 |- ( ( A e. RR /\ z e. RR ) -> ( A - z ) e. CC )
37	4, 10	gt0ne0ii 10085	. . . . . . . . . . . . . . 15 |- ( 2 x. pi ) =/= 0
38		redivcl 10259	. . . . . . . . . . . . . . 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 1314	. . . . . . . . . . . . . 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 11916	. . . . . . . . . . . 12 |- ( ( A e. RR /\ z e. RR ) -> ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) e. ZZ )
42	41	zred 10965	. . . . . . . . . . 11 |- ( ( A e. RR /\ z e. RR ) -> ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) e. RR )
43		remulcl 9566	. . . . . . . . . . 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 661	. . . . . . . . . 10 |- ( ( A e. RR /\ z e. RR ) -> ( ( 2 x. pi ) x. ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) ) e. RR )
45	44	recnd 9611	. . . . . . . . 9 |- ( ( A e. RR /\ z e. RR ) -> ( ( 2 x. pi ) x. ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) ) e. CC )
46	35, 36, 45	subsubd 9950	. . . . . . . 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 9611	. . . . . . . . . 10 |- ( ( A e. RR /\ z e. RR ) -> A e. CC )
48	4	recni 9597	. . . . . . . . . . 11 |- ( 2 x. pi ) e. CC
49	48	a1i 11	. . . . . . . . . 10 |- ( ( A e. RR /\ z e. RR ) -> ( 2 x. pi ) e. CC )
50		simpr 459	. . . . . . . . . . 11 |- ( ( A e. RR /\ z e. RR ) -> z e. RR )
51	50	recnd 9611	. . . . . . . . . 10 |- ( ( A e. RR /\ z e. RR ) -> z e. CC )
52	47, 49, 51	pnncand 9961	. . . . . . . . 9 |- ( ( A e. RR /\ z e. RR ) -> ( ( A + ( 2 x. pi ) ) - ( A - z ) ) = ( ( 2 x. pi ) + z ) )
53	52	oveq1d 6285	. . . . . . . 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 2499	. . . . . . 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 6286	. . . . . 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 9563	. . . . . . . 8 |- ( ( ( 2 x. pi ) e. CC /\ z e. CC ) -> ( ( 2 x. pi ) + z ) e. CC )
57	48, 51, 56	sylancr 661	. . . . . . 7 |- ( ( A e. RR /\ z e. RR ) -> ( ( 2 x. pi ) + z ) e. CC )
58	51, 57, 45	subsub4d 9953	. . . . . 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 9938	. . . . . . . . . 10 |- ( ( A e. RR /\ z e. RR ) -> -u ( ( ( 2 x. pi ) + z ) - z ) = ( z - ( ( 2 x. pi ) + z ) ) )
60	49, 51	pncand 9923	. . . . . . . . . . 11 |- ( ( A e. RR /\ z e. RR ) -> ( ( ( 2 x. pi ) + z ) - z ) = ( 2 x. pi ) )
61	60	negeqd 9805	. . . . . . . . . 10 |- ( ( A e. RR /\ z e. RR ) -> -u ( ( ( 2 x. pi ) + z ) - z ) = -u ( 2 x. pi ) )
62	59, 61	eqtr3d 2497	. . . . . . . . 9 |- ( ( A e. RR /\ z e. RR ) -> ( z - ( ( 2 x. pi ) + z ) ) = -u ( 2 x. pi ) )
63		neg1cn 10635	. . . . . . . . . 10 |- -u 1 e. CC
64	48	mulm1i 9997	. . . . . . . . . 10 |- ( -u 1 x. ( 2 x. pi ) ) = -u ( 2 x. pi )
65	63, 48, 64	mulcomli 9592	. . . . . . . . 9 |- ( ( 2 x. pi ) x. -u 1 ) = -u ( 2 x. pi )
66	62, 65	syl6eqr 2513	. . . . . . . 8 |- ( ( A e. RR /\ z e. RR ) -> ( z - ( ( 2 x. pi ) + z ) ) = ( ( 2 x. pi ) x. -u 1 ) )
67	66	oveq1d 6285	. . . . . . 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 10966	. . . . . . . 8 |- ( ( A e. RR /\ z e. RR ) -> ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) e. CC )
70	49, 68, 69	subdid 10008	. . . . . . 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 2498	. . . . . 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 2501	. . . . 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 6285	. . . 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 10896	. . . . . . 7 |- -u 1 e. ZZ
75		zsubcl 10902	. . . . . . 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 661	. . . . . 6 |- ( ( A e. RR /\ z e. RR ) -> ( -u 1 - ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) ) e. ZZ )
77	76	zcnd 10966	. . . . 5 |- ( ( A e. RR /\ z e. RR ) -> ( -u 1 - ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) ) e. CC )
78		divcan3 10227	. . . . . 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 1314	. . . . 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 2495	. . 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 2542	. 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 6278	. . . . 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 6285	. . . 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 2523	. . 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 3207	. 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 659	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 367   /\ w3a 971   = wceq 1398   e. wcel 1823   =/= wne 2649   E.wrex 2805   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   CCcc 9479   RRcr 9480   0cc0 9481   1c1 9482   + caddc 9484   x. cmul 9486   RR*cxr 9616   < clt 9617   <_ cle 9618   - cmin 9796   -ucneg 9797   / cdiv 10202   2c2 10581   ZZcz 10860   RR+crp 11221   (,]cioc 11533   |_cfl 11908   mod cmo 11978   picpi 13884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-fi 7863  df-sup 7893  df-oi 7927  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-ioo 11536  df-ioc 11537  df-ico 11538  df-icc 11539  df-fz 11676  df-fzo 11800  df-fl 11910  df-mod 11979  df-seq 12090  df-exp 12149  df-fac 12336  df-bc 12363  df-hash 12388  df-shft 12982  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-limsup 13376  df-clim 13393  df-rlim 13394  df-sum 13591  df-ef 13885  df-sin 13887  df-cos 13888  df-pi 13890  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-mulr 14798  df-starv 14799  df-sca 14800  df-vsca 14801  df-ip 14802  df-tset 14803  df-ple 14804  df-ds 14806  df-unif 14807  df-hom 14808  df-cco 14809  df-rest 14912  df-topn 14913  df-0g 14931  df-gsum 14932  df-topgen 14933  df-pt 14934  df-prds 14937  df-xrs 14991  df-qtop 14996  df-imas 14997  df-xps 14999  df-mre 15075  df-mrc 15076  df-acs 15078  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-submnd 16166  df-mulg 16259  df-cntz 16554  df-cmn 16999  df-psmet 18606  df-xmet 18607  df-met 18608  df-bl 18609  df-mopn 18610  df-fbas 18611  df-fg 18612  df-cnfld 18616  df-top 19566  df-bases 19568  df-topon 19569  df-topsp 19570  df-cld 19687  df-ntr 19688  df-cls 19689  df-nei 19766  df-lp 19804  df-perf 19805  df-cn 19895  df-cnp 19896  df-haus 19983  df-tx 20229  df-hmeo 20422  df-fil 20513  df-fm 20605  df-flim 20606  df-flf 20607  df-xms 20989  df-ms 20990  df-tms 20991  df-cncf 21548  df-limc 22436  df-dv 22437

This theorem is referenced by:  efif1o  23099  eff1o  23102