Theorem efif1olem2 22131

Description: Lemma for efif1o 22134. (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 10501	. . . . . . 7 |- 2 e. RR
3		pire 22053	. . . . . . 7 |- pi e. RR
4	2, 3	remulcli 9510	. . . . . 6 |- ( 2 x. pi ) e. RR
5		readdcl 9475	. . . . . 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 9783	. . . . . 6 |- ( ( A e. RR /\ z e. RR ) -> ( A - z ) e. RR )
8		2pos 10523	. . . . . . . 8 |- 0 < 2
9		pipos 22055	. . . . . . . 8 |- 0 < pi
10	2, 3, 8, 9	mulgt0ii 9617	. . . . . . 7 |- 0 < ( 2 x. pi )
11	4, 10	elrpii 11104	. . . . . 6 |- ( 2 x. pi ) e. RR+
12		modcl 11828	. . . . . 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 9886	. . . 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 11834	. . . . . . 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 9640	. . . . 5 |- ( ( A e. RR /\ z e. RR ) -> ( A + ( ( A - z ) mod ( 2 x. pi ) ) ) < ( A + ( 2 x. pi ) ) )
19	1, 13, 6	ltaddsubd 10049	. . . . 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 11833	. . . . . 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 10041	. . . . 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 9539	. . . . . 6 |- ( A e. RR -> A e. RR* )
26	25	adantr 465	. . . . 5 |- ( ( A e. RR /\ z e. RR ) -> A e. RR* )
27		elioc2 11468	. . . . 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 2553	. 2 |- ( ( A e. RR /\ z e. RR ) -> ( ( A + ( 2 x. pi ) ) - ( ( A - z ) mod ( 2 x. pi ) ) ) e. D )
32		modval 11826	. . . . . . . . . 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 6215	. . . . . . . 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 9522	. . . . . . . . 9 |- ( ( A e. RR /\ z e. RR ) -> ( A + ( 2 x. pi ) ) e. CC )
36	7	recnd 9522	. . . . . . . . 9 |- ( ( A e. RR /\ z e. RR ) -> ( A - z ) e. CC )
37	4, 10	gt0ne0ii 9986	. . . . . . . . . . . . . . 15 |- ( 2 x. pi ) =/= 0
38		redivcl 10160	. . . . . . . . . . . . . . 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 1307	. . . . . . . . . . . . . 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 11764	. . . . . . . . . . . 12 |- ( ( A e. RR /\ z e. RR ) -> ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) e. ZZ )
42	41	zred 10857	. . . . . . . . . . 11 |- ( ( A e. RR /\ z e. RR ) -> ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) e. RR )
43		remulcl 9477	. . . . . . . . . . 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 9522	. . . . . . . . 9 |- ( ( A e. RR /\ z e. RR ) -> ( ( 2 x. pi ) x. ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) ) e. CC )
46	35, 36, 45	subsubd 9857	. . . . . . . 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 9522	. . . . . . . . . 10 |- ( ( A e. RR /\ z e. RR ) -> A e. CC )
48	4	recni 9508	. . . . . . . . . . 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 9522	. . . . . . . . . 10 |- ( ( A e. RR /\ z e. RR ) -> z e. CC )
52	47, 49, 51	pnncand 9868	. . . . . . . . 9 |- ( ( A e. RR /\ z e. RR ) -> ( ( A + ( 2 x. pi ) ) - ( A - z ) ) = ( ( 2 x. pi ) + z ) )
53	52	oveq1d 6214	. . . . . . . 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 6215	. . . . . 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 9474	. . . . . . . 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 9860	. . . . . 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 9845	. . . . . . . . . 10 |- ( ( A e. RR /\ z e. RR ) -> -u ( ( ( 2 x. pi ) + z ) - z ) = ( z - ( ( 2 x. pi ) + z ) ) )
60	49, 51	pncand 9830	. . . . . . . . . . 11 |- ( ( A e. RR /\ z e. RR ) -> ( ( ( 2 x. pi ) + z ) - z ) = ( 2 x. pi ) )
61	60	negeqd 9714	. . . . . . . . . 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 10535	. . . . . . . . . 10 |- -u 1 e. CC
64	48	mulm1i 9899	. . . . . . . . . 10 |- ( -u 1 x. ( 2 x. pi ) ) = -u ( 2 x. pi )
65	63, 48, 64	mulcomli 9503	. . . . . . . . 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 6214	. . . . . . 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 10858	. . . . . . . 8 |- ( ( A e. RR /\ z e. RR ) -> ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) e. CC )
70	49, 68, 69	subdid 9910	. . . . . . 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 6214	. . . 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 10791	. . . . . . 7 |- -u 1 e. ZZ
75		zsubcl 10797	. . . . . . 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 10858	. . . . 5 |- ( ( A e. RR /\ z e. RR ) -> ( -u 1 - ( |_ ` ( ( A - z ) / ( 2 x. pi ) ) ) ) e. CC )
78		divcan3 10128	. . . . . 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 1307	. . . . 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 6207	. . . . 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 6214	. . . 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 3177	. 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 1370   e. wcel 1758   =/= wne 2647   E.wrex 2799   class class class wbr 4399   ` cfv 5525  (class class class)co 6199   CCcc 9390   RRcr 9391   0cc0 9392   1c1 9393   + caddc 9395   x. cmul 9397   RR*cxr 9527   < clt 9528   <_ cle 9529   - cmin 9705   -ucneg 9706   / cdiv 10103   2c2 10481   ZZcz 10756   RR+crp 11101   (,]cioc 11411   |_cfl 11756   mod cmo 11824   picpi 13469

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-inf2 7957  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469  ax-pre-sup 9470  ax-addf 9471  ax-mulf 9472

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-iin 4281  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-se 4787  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-isom 5534  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-of 6429  df-om 6586  df-1st 6686  df-2nd 6687  df-supp 6800  df-recs 6941  df-rdg 6975  df-1o 7029  df-2o 7030  df-oadd 7033  df-er 7210  df-map 7325  df-pm 7326  df-ixp 7373  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-fsupp 7731  df-fi 7771  df-sup 7801  df-oi 7834  df-card 8219  df-cda 8447  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-div 10104  df-nn 10433  df-2 10490  df-3 10491  df-4 10492  df-5 10493  df-6 10494  df-7 10495  df-8 10496  df-9 10497  df-10 10498  df-n0 10690  df-z 10757  df-dec 10866  df-uz 10972  df-q 11064  df-rp 11102  df-xneg 11199  df-xadd 11200  df-xmul 11201  df-ioo 11414  df-ioc 11415  df-ico 11416  df-icc 11417  df-fz 11554  df-fzo 11665  df-fl 11758  df-mod 11825  df-seq 11923  df-exp 11982  df-fac 12168  df-bc 12195  df-hash 12220  df-shft 12673  df-cj 12705  df-re 12706  df-im 12707  df-sqr 12841  df-abs 12842  df-limsup 13066  df-clim 13083  df-rlim 13084  df-sum 13281  df-ef 13470  df-sin 13472  df-cos 13473  df-pi 13475  df-struct 14293  df-ndx 14294  df-slot 14295  df-base 14296  df-sets 14297  df-ress 14298  df-plusg 14369  df-mulr 14370  df-starv 14371  df-sca 14372  df-vsca 14373  df-ip 14374  df-tset 14375  df-ple 14376  df-ds 14378  df-unif 14379  df-hom 14380  df-cco 14381  df-rest 14479  df-topn 14480  df-0g 14498  df-gsum 14499  df-topgen 14500  df-pt 14501  df-prds 14504  df-xrs 14558  df-qtop 14563  df-imas 14564  df-xps 14566  df-mre 14642  df-mrc 14643  df-acs 14645  df-mnd 15533  df-submnd 15583  df-mulg 15666  df-cntz 15953  df-cmn 16399  df-psmet 17933  df-xmet 17934  df-met 17935  df-bl 17936  df-mopn 17937  df-fbas 17938  df-fg 17939  df-cnfld 17943  df-top 18634  df-bases 18636  df-topon 18637  df-topsp 18638  df-cld 18754  df-ntr 18755  df-cls 18756  df-nei 18833  df-lp 18871  df-perf 18872  df-cn 18962  df-cnp 18963  df-haus 19050  df-tx 19266  df-hmeo 19459  df-fil 19550  df-fm 19642  df-flim 19643  df-flf 19644  df-xms 20026  df-ms 20027  df-tms 20028  df-cncf 20585  df-limc 21473  df-dv 21474

This theorem is referenced by:  efif1o  22134  eff1o  22137