Theorem efif1olem2 21974

Description: Lemma for efif1o 21977. (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 10383	. . . . . . 7 |- 2 e. RR
3		pire 21896	. . . . . . 7 |- pi e. RR
4	2, 3	remulcli 9392	. . . . . 6 |- ( 2 x. pi ) e. RR
5		readdcl 9357	. . . . . 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 9665	. . . . . 6 |- ( ( A e. RR /\ z e. RR ) -> ( A - z ) e. RR )
8		2pos 10405	. . . . . . . 8 |- 0 < 2
9		pipos 21898	. . . . . . . 8 |- 0 < pi
10	2, 3, 8, 9	mulgt0ii 9499	. . . . . . 7 |- 0 < ( 2 x. pi )
11	4, 10	elrpii 10986	. . . . . 6 |- ( 2 x. pi ) e. RR+
12		modcl 11704	. . . . . 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 9768	. . . 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 11710	. . . . . . 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 9522	. . . . 5 |- ( ( A e. RR /\ z e. RR ) -> ( A + ( ( A - z ) mod ( 2 x. pi ) ) ) < ( A + ( 2 x. pi ) ) )
19	1, 13, 6	ltaddsubd 9931	. . . . 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 11709	. . . . . 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 9923	. . . . 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 9421	. . . . . 6 |- ( A e. RR -> A e. RR* )
26	25	adantr 465	. . . . 5 |- ( ( A e. RR /\ z e. RR ) -> A e. RR* )
27		elioc2 11350	. . . . 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 1171	. . 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 2529	. 2 |- ( ( A e. RR /\ z e. RR ) -> ( ( A + ( 2 x. pi ) ) - ( ( A - z ) mod ( 2 x. pi ) ) ) e. D )
32		modval 11702	. . . . . . . . . 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 6102	. . . . . . . 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 9404	. . . . . . . . 9 |- ( ( A e. RR /\ z e. RR ) -> ( A + ( 2 x. pi ) ) e. CC )
36	7	recnd 9404	. . . . . . . . 9 |- ( ( A e. RR /\ z e. RR ) -> ( A - z ) e. CC )
37	4, 10	gt0ne0ii 9868	. . . . . . . . . . . . . . 15 |- ( 2 x. pi ) =/= 0
38		redivcl 10042	. . . . . . . . . . . . . . 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 1306	. . . . . . . . . . . . . 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 11640	. . . . . . . . . . . 12 |- ( ( A e. RR /\ z e. RR ) -> ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) e. ZZ )
42	41	zred 10739	. . . . . . . . . . 11 |- ( ( A e. RR /\ z e. RR ) -> ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) e. RR )
43		remulcl 9359	. . . . . . . . . . 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 9404	. . . . . . . . 9 |- ( ( A e. RR /\ z e. RR ) -> ( ( 2 x. pi ) x. ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) ) e. CC )
46	35, 36, 45	subsubd 9739	. . . . . . . 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 9404	. . . . . . . . . 10 |- ( ( A e. RR /\ z e. RR ) -> A e. CC )
48	4	recni 9390	. . . . . . . . . . 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 9404	. . . . . . . . . 10 |- ( ( A e. RR /\ z e. RR ) -> z e. CC )
52	47, 49, 51	pnncand 9750	. . . . . . . . 9 |- ( ( A e. RR /\ z e. RR ) -> ( ( A + ( 2 x. pi ) ) - ( A - z ) ) = ( ( 2 x. pi ) + z ) )
53	52	oveq1d 6101	. . . . . . . 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 2474	. . . . . . 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 6102	. . . . . 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 9356	. . . . . . . 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 9742	. . . . . 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 9727	. . . . . . . . . 10 |- ( ( A e. RR /\ z e. RR ) -> -u ( ( ( 2 x. pi ) + z ) - z ) = ( z - ( ( 2 x. pi ) + z ) ) )
60	49, 51	pncand 9712	. . . . . . . . . . 11 |- ( ( A e. RR /\ z e. RR ) -> ( ( ( 2 x. pi ) + z ) - z ) = ( 2 x. pi ) )
61	60	negeqd 9596	. . . . . . . . . 10 |- ( ( A e. RR /\ z e. RR ) -> -u ( ( ( 2 x. pi ) + z ) - z ) = -u ( 2 x. pi ) )
62	59, 61	eqtr3d 2472	. . . . . . . . 9 |- ( ( A e. RR /\ z e. RR ) -> ( z - ( ( 2 x. pi ) + z ) ) = -u ( 2 x. pi ) )
63		neg1cn 10417	. . . . . . . . . 10 |- -u 1 e. CC
64	48	mulm1i 9781	. . . . . . . . . 10 |- ( -u 1 x. ( 2 x. pi ) ) = -u ( 2 x. pi )
65	63, 48, 64	mulcomli 9385	. . . . . . . . 9 |- ( ( 2 x. pi ) x. -u 1 ) = -u ( 2 x. pi )
66	62, 65	syl6eqr 2488	. . . . . . . 8 |- ( ( A e. RR /\ z e. RR ) -> ( z - ( ( 2 x. pi ) + z ) ) = ( ( 2 x. pi ) x. -u 1 ) )
67	66	oveq1d 6101	. . . . . . 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 10740	. . . . . . . 8 |- ( ( A e. RR /\ z e. RR ) -> ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) e. CC )
70	49, 68, 69	subdid 9792	. . . . . . 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 2473	. . . . . 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 2476	. . . . 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 6101	. . . 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 10673	. . . . . . 7 |- -u 1 e. ZZ
75		zsubcl 10679	. . . . . . 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 10740	. . . . 5 |- ( ( A e. RR /\ z e. RR ) -> ( -u 1 - ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) ) e. CC )
78		divcan3 10010	. . . . . 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 1306	. . . . 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 2470	. . 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 2512	. 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 6094	. . . . 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 6101	. . . 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 2504	. . 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 3068	. 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 965   = wceq 1369   e. wcel 1756   =/= wne 2601   E.wrex 2711   class class class wbr 4287   ` cfv 5413  (class class class)co 6086   CCcc 9272   RRcr 9273   0cc0 9274   1c1 9275   + caddc 9277   x. cmul 9279   RR*cxr 9409   < clt 9410   <_ cle 9411   - cmin 9587   -ucneg 9588   / cdiv 9985   2c2 10363   ZZcz 10638   RR+crp 10983   (,]cioc 11293   |_cfl 11632   mod cmo 11700   picpi 13344

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353  ax-mulf 9354

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fsupp 7613  df-fi 7653  df-sup 7683  df-oi 7716  df-card 8101  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ioo 11296  df-ioc 11297  df-ico 11298  df-icc 11299  df-fz 11430  df-fzo 11541  df-fl 11634  df-mod 11701  df-seq 11799  df-exp 11858  df-fac 12044  df-bc 12071  df-hash 12096  df-shft 12548  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-limsup 12941  df-clim 12958  df-rlim 12959  df-sum 13156  df-ef 13345  df-sin 13347  df-cos 13348  df-pi 13350  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-starv 14245  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-hom 14254  df-cco 14255  df-rest 14353  df-topn 14354  df-0g 14372  df-gsum 14373  df-topgen 14374  df-pt 14375  df-prds 14378  df-xrs 14432  df-qtop 14437  df-imas 14438  df-xps 14440  df-mre 14516  df-mrc 14517  df-acs 14519  df-mnd 15407  df-submnd 15457  df-mulg 15539  df-cntz 15826  df-cmn 16270  df-psmet 17784  df-xmet 17785  df-met 17786  df-bl 17787  df-mopn 17788  df-fbas 17789  df-fg 17790  df-cnfld 17794  df-top 18478  df-bases 18480  df-topon 18481  df-topsp 18482  df-cld 18598  df-ntr 18599  df-cls 18600  df-nei 18677  df-lp 18715  df-perf 18716  df-cn 18806  df-cnp 18807  df-haus 18894  df-tx 19110  df-hmeo 19303  df-fil 19394  df-fm 19486  df-flim 19487  df-flf 19488  df-xms 19870  df-ms 19871  df-tms 19872  df-cncf 20429  df-limc 21316  df-dv 21317

This theorem is referenced by:  efif1o  21977  eff1o  21980