Theorem dya2iocnrect 30343

Description: For any point of an open rectangle in ( RR X. RR ), there is a closed-below open-above dyadic rational square which contains that point and is included in the rectangle. (Contributed by Thierry Arnoux, 12-Oct-2017.)

Hypotheses

Ref	Expression
sxbrsiga.0	|- J = ( topGen ` ran (,) )
dya2ioc.1	|- I = ( x e. ZZ , n e. ZZ |-> ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) )
dya2ioc.2	|- R = ( u e. ran I , v e. ran I |-> ( u X. v ) )
dya2iocnrect.1	|- B = ran ( e e. ran (,) , f e. ran (,) |-> ( e X. f ) )

Assertion

Ref	Expression
dya2iocnrect	|- ( ( X e. ( RR X. RR ) /\ A e. B /\ X e. A ) -> E. b e. ran R ( X e. b /\ b C_ A ) )

Distinct variable groups:   x, n   x, I   v, u, I, x   e, b, f, A   R, b, e, f   x, b, X, e, f

Allowed substitution hints:   A( x, v, u, n)   B( x, v, u, e, f, n, b)   R( x, v, u, n)   I( e, f, n, b)   J( x, v, u, e, f, n, b)   X( v, u, n)

Proof of Theorem dya2iocnrect

Dummy variables s t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dya2iocnrect.1	. . . . . 6 |- B = ran ( e e. ran (,) , f e. ran (,) |-> ( e X. f ) )
2	1	eleq2i 2693	. . . . 5 |- ( A e. B <-> A e. ran ( e e. ran (,) , f e. ran (,) |-> ( e X. f ) ) )
3		eqid 2622	. . . . . 6 |- ( e e. ran (,) , f e. ran (,) |-> ( e X. f ) ) = ( e e. ran (,) , f e. ran (,) |-> ( e X. f ) )
4		vex 3203	. . . . . . 7 |- e e. _V
5		vex 3203	. . . . . . 7 |- f e. _V
6	4, 5	xpex 6962	. . . . . 6 |- ( e X. f ) e. _V
7	3, 6	elrnmpt2 6773	. . . . 5 |- ( A e. ran ( e e. ran (,) , f e. ran (,) |-> ( e X. f ) ) <-> E. e e. ran (,) E. f e. ran (,) A = ( e X. f ) )
8	2, 7	sylbb 209	. . . 4 |- ( A e. B -> E. e e. ran (,) E. f e. ran (,) A = ( e X. f ) )
9	8	3ad2ant2 1083	. . 3 |- ( ( X e. ( RR X. RR ) /\ A e. B /\ X e. A ) -> E. e e. ran (,) E. f e. ran (,) A = ( e X. f ) )
10		simp1 1061	. . 3 |- ( ( X e. ( RR X. RR ) /\ A e. B /\ X e. A ) -> X e. ( RR X. RR ) )
11		simp3 1063	. . 3 |- ( ( X e. ( RR X. RR ) /\ A e. B /\ X e. A ) -> X e. A )
12	9, 10, 11	jca32 558	. 2 |- ( ( X e. ( RR X. RR ) /\ A e. B /\ X e. A ) -> ( E. e e. ran (,) E. f e. ran (,) A = ( e X. f ) /\ ( X e. ( RR X. RR ) /\ X e. A ) ) )
13		r19.41vv 3091	. . 3 |- ( E. e e. ran (,) E. f e. ran (,) ( A = ( e X. f ) /\ ( X e. ( RR X. RR ) /\ X e. A ) ) <-> ( E. e e. ran (,) E. f e. ran (,) A = ( e X. f ) /\ ( X e. ( RR X. RR ) /\ X e. A ) ) )
14	13	biimpri 218	. 2 |- ( ( E. e e. ran (,) E. f e. ran (,) A = ( e X. f ) /\ ( X e. ( RR X. RR ) /\ X e. A ) ) -> E. e e. ran (,) E. f e. ran (,) ( A = ( e X. f ) /\ ( X e. ( RR X. RR ) /\ X e. A ) ) )
15		simprl 794	. . . . . 6 |- ( ( A = ( e X. f ) /\ ( X e. ( RR X. RR ) /\ X e. A ) ) -> X e. ( RR X. RR ) )
16		simpl 473	. . . . . 6 |- ( ( A = ( e X. f ) /\ ( X e. ( RR X. RR ) /\ X e. A ) ) -> A = ( e X. f ) )
17		simprr 796	. . . . . . 7 |- ( ( A = ( e X. f ) /\ ( X e. ( RR X. RR ) /\ X e. A ) ) -> X e. A )
18	17, 16	eleqtrd 2703	. . . . . 6 |- ( ( A = ( e X. f ) /\ ( X e. ( RR X. RR ) /\ X e. A ) ) -> X e. ( e X. f ) )
19	15, 16, 18	3jca 1242	. . . . 5 |- ( ( A = ( e X. f ) /\ ( X e. ( RR X. RR ) /\ X e. A ) ) -> ( X e. ( RR X. RR ) /\ A = ( e X. f ) /\ X e. ( e X. f ) ) )
20		simpr 477	. . . . . 6 |- ( ( ( e e. ran (,) /\ f e. ran (,) ) /\ ( X e. ( RR X. RR ) /\ A = ( e X. f ) /\ X e. ( e X. f ) ) ) -> ( X e. ( RR X. RR ) /\ A = ( e X. f ) /\ X e. ( e X. f ) ) )
21		xp1st 7198	. . . . . . . . . 10 |- ( X e. ( RR X. RR ) -> ( 1st ` X ) e. RR )
22	21	3ad2ant1 1082	. . . . . . . . 9 |- ( ( X e. ( RR X. RR ) /\ A = ( e X. f ) /\ X e. ( e X. f ) ) -> ( 1st ` X ) e. RR )
23	22	adantl 482	. . . . . . . 8 |- ( ( ( e e. ran (,) /\ f e. ran (,) ) /\ ( X e. ( RR X. RR ) /\ A = ( e X. f ) /\ X e. ( e X. f ) ) ) -> ( 1st ` X ) e. RR )
24		simpll 790	. . . . . . . 8 |- ( ( ( e e. ran (,) /\ f e. ran (,) ) /\ ( X e. ( RR X. RR ) /\ A = ( e X. f ) /\ X e. ( e X. f ) ) ) -> e e. ran (,) )
25		xp1st 7198	. . . . . . . . . 10 |- ( X e. ( e X. f ) -> ( 1st ` X ) e. e )
26	25	3ad2ant3 1084	. . . . . . . . 9 |- ( ( X e. ( RR X. RR ) /\ A = ( e X. f ) /\ X e. ( e X. f ) ) -> ( 1st ` X ) e. e )
27	26	adantl 482	. . . . . . . 8 |- ( ( ( e e. ran (,) /\ f e. ran (,) ) /\ ( X e. ( RR X. RR ) /\ A = ( e X. f ) /\ X e. ( e X. f ) ) ) -> ( 1st ` X ) e. e )
28		sxbrsiga.0	. . . . . . . . 9 |- J = ( topGen ` ran (,) )
29		dya2ioc.1	. . . . . . . . 9 |- I = ( x e. ZZ , n e. ZZ |-> ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) )
30	28, 29	dya2icoseg2 30340	. . . . . . . 8 |- ( ( ( 1st ` X ) e. RR /\ e e. ran (,) /\ ( 1st ` X ) e. e ) -> E. s e. ran I ( ( 1st ` X ) e. s /\ s C_ e ) )
31	23, 24, 27, 30	syl3anc 1326	. . . . . . 7 |- ( ( ( e e. ran (,) /\ f e. ran (,) ) /\ ( X e. ( RR X. RR ) /\ A = ( e X. f ) /\ X e. ( e X. f ) ) ) -> E. s e. ran I ( ( 1st ` X ) e. s /\ s C_ e ) )
32		xp2nd 7199	. . . . . . . . . 10 |- ( X e. ( RR X. RR ) -> ( 2nd ` X ) e. RR )
33	32	3ad2ant1 1082	. . . . . . . . 9 |- ( ( X e. ( RR X. RR ) /\ A = ( e X. f ) /\ X e. ( e X. f ) ) -> ( 2nd ` X ) e. RR )
34	33	adantl 482	. . . . . . . 8 |- ( ( ( e e. ran (,) /\ f e. ran (,) ) /\ ( X e. ( RR X. RR ) /\ A = ( e X. f ) /\ X e. ( e X. f ) ) ) -> ( 2nd ` X ) e. RR )
35		simplr 792	. . . . . . . 8 |- ( ( ( e e. ran (,) /\ f e. ran (,) ) /\ ( X e. ( RR X. RR ) /\ A = ( e X. f ) /\ X e. ( e X. f ) ) ) -> f e. ran (,) )
36		xp2nd 7199	. . . . . . . . . 10 |- ( X e. ( e X. f ) -> ( 2nd ` X ) e. f )
37	36	3ad2ant3 1084	. . . . . . . . 9 |- ( ( X e. ( RR X. RR ) /\ A = ( e X. f ) /\ X e. ( e X. f ) ) -> ( 2nd ` X ) e. f )
38	37	adantl 482	. . . . . . . 8 |- ( ( ( e e. ran (,) /\ f e. ran (,) ) /\ ( X e. ( RR X. RR ) /\ A = ( e X. f ) /\ X e. ( e X. f ) ) ) -> ( 2nd ` X ) e. f )
39	28, 29	dya2icoseg2 30340	. . . . . . . 8 |- ( ( ( 2nd ` X ) e. RR /\ f e. ran (,) /\ ( 2nd ` X ) e. f ) -> E. t e. ran I ( ( 2nd ` X ) e. t /\ t C_ f ) )
40	34, 35, 38, 39	syl3anc 1326	. . . . . . 7 |- ( ( ( e e. ran (,) /\ f e. ran (,) ) /\ ( X e. ( RR X. RR ) /\ A = ( e X. f ) /\ X e. ( e X. f ) ) ) -> E. t e. ran I ( ( 2nd ` X ) e. t /\ t C_ f ) )
41		reeanv 3107	. . . . . . 7 |- ( E. s e. ran I E. t e. ran I ( ( ( 1st ` X ) e. s /\ s C_ e ) /\ ( ( 2nd ` X ) e. t /\ t C_ f ) ) <-> ( E. s e. ran I ( ( 1st ` X ) e. s /\ s C_ e ) /\ E. t e. ran I ( ( 2nd ` X ) e. t /\ t C_ f ) ) )
42	31, 40, 41	sylanbrc 698	. . . . . 6 |- ( ( ( e e. ran (,) /\ f e. ran (,) ) /\ ( X e. ( RR X. RR ) /\ A = ( e X. f ) /\ X e. ( e X. f ) ) ) -> E. s e. ran I E. t e. ran I ( ( ( 1st ` X ) e. s /\ s C_ e ) /\ ( ( 2nd ` X ) e. t /\ t C_ f ) ) )
43		eqid 2622	. . . . . . . . . . . 12 |- ( s X. t ) = ( s X. t )
44		xpeq1 5128	. . . . . . . . . . . . . 14 |- ( u = s -> ( u X. v ) = ( s X. v ) )
45	44	eqeq2d 2632	. . . . . . . . . . . . 13 |- ( u = s -> ( ( s X. t ) = ( u X. v ) <-> ( s X. t ) = ( s X. v ) ) )
46		xpeq2 5129	. . . . . . . . . . . . . 14 |- ( v = t -> ( s X. v ) = ( s X. t ) )
47	46	eqeq2d 2632	. . . . . . . . . . . . 13 |- ( v = t -> ( ( s X. t ) = ( s X. v ) <-> ( s X. t ) = ( s X. t ) ) )
48	45, 47	rspc2ev 3324	. . . . . . . . . . . 12 |- ( ( s e. ran I /\ t e. ran I /\ ( s X. t ) = ( s X. t ) ) -> E. u e. ran I E. v e. ran I ( s X. t ) = ( u X. v ) )
49	43, 48	mp3an3 1413	. . . . . . . . . . 11 |- ( ( s e. ran I /\ t e. ran I ) -> E. u e. ran I E. v e. ran I ( s X. t ) = ( u X. v ) )
50		dya2ioc.2	. . . . . . . . . . . 12 |- R = ( u e. ran I , v e. ran I |-> ( u X. v ) )
51		vex 3203	. . . . . . . . . . . . 13 |- u e. _V
52		vex 3203	. . . . . . . . . . . . 13 |- v e. _V
53	51, 52	xpex 6962	. . . . . . . . . . . 12 |- ( u X. v ) e. _V
54	50, 53	elrnmpt2 6773	. . . . . . . . . . 11 |- ( ( s X. t ) e. ran R <-> E. u e. ran I E. v e. ran I ( s X. t ) = ( u X. v ) )
55	49, 54	sylibr 224	. . . . . . . . . 10 |- ( ( s e. ran I /\ t e. ran I ) -> ( s X. t ) e. ran R )
56	55	ad2antrl 764	. . . . . . . . 9 |- ( ( ( X e. ( RR X. RR ) /\ A = ( e X. f ) /\ X e. ( e X. f ) ) /\ ( ( s e. ran I /\ t e. ran I ) /\ ( ( ( 1st ` X ) e. s /\ s C_ e ) /\ ( ( 2nd ` X ) e. t /\ t C_ f ) ) ) ) -> ( s X. t ) e. ran R )
57		xpss 5226	. . . . . . . . . . 11 |- ( RR X. RR ) C_ ( _V X. _V )
58		simpl1 1064	. . . . . . . . . . 11 |- ( ( ( X e. ( RR X. RR ) /\ A = ( e X. f ) /\ X e. ( e X. f ) ) /\ ( ( s e. ran I /\ t e. ran I ) /\ ( ( ( 1st ` X ) e. s /\ s C_ e ) /\ ( ( 2nd ` X ) e. t /\ t C_ f ) ) ) ) -> X e. ( RR X. RR ) )
59	57, 58	sseldi 3601	. . . . . . . . . 10 |- ( ( ( X e. ( RR X. RR ) /\ A = ( e X. f ) /\ X e. ( e X. f ) ) /\ ( ( s e. ran I /\ t e. ran I ) /\ ( ( ( 1st ` X ) e. s /\ s C_ e ) /\ ( ( 2nd ` X ) e. t /\ t C_ f ) ) ) ) -> X e. ( _V X. _V ) )
60		simprrl 804	. . . . . . . . . . 11 |- ( ( ( X e. ( RR X. RR ) /\ A = ( e X. f ) /\ X e. ( e X. f ) ) /\ ( ( s e. ran I /\ t e. ran I ) /\ ( ( ( 1st ` X ) e. s /\ s C_ e ) /\ ( ( 2nd ` X ) e. t /\ t C_ f ) ) ) ) -> ( ( 1st ` X ) e. s /\ s C_ e ) )
61	60	simpld 475	. . . . . . . . . 10 |- ( ( ( X e. ( RR X. RR ) /\ A = ( e X. f ) /\ X e. ( e X. f ) ) /\ ( ( s e. ran I /\ t e. ran I ) /\ ( ( ( 1st ` X ) e. s /\ s C_ e ) /\ ( ( 2nd ` X ) e. t /\ t C_ f ) ) ) ) -> ( 1st ` X ) e. s )
62		simprrr 805	. . . . . . . . . . 11 |- ( ( ( X e. ( RR X. RR ) /\ A = ( e X. f ) /\ X e. ( e X. f ) ) /\ ( ( s e. ran I /\ t e. ran I ) /\ ( ( ( 1st ` X ) e. s /\ s C_ e ) /\ ( ( 2nd ` X ) e. t /\ t C_ f ) ) ) ) -> ( ( 2nd ` X ) e. t /\ t C_ f ) )
63	62	simpld 475	. . . . . . . . . 10 |- ( ( ( X e. ( RR X. RR ) /\ A = ( e X. f ) /\ X e. ( e X. f ) ) /\ ( ( s e. ran I /\ t e. ran I ) /\ ( ( ( 1st ` X ) e. s /\ s C_ e ) /\ ( ( 2nd ` X ) e. t /\ t C_ f ) ) ) ) -> ( 2nd ` X ) e. t )
64		elxp7 7201	. . . . . . . . . . 11 |- ( X e. ( s X. t ) <-> ( X e. ( _V X. _V ) /\ ( ( 1st ` X ) e. s /\ ( 2nd ` X ) e. t ) ) )
65	64	biimpri 218	. . . . . . . . . 10 |- ( ( X e. ( _V X. _V ) /\ ( ( 1st ` X ) e. s /\ ( 2nd ` X ) e. t ) ) -> X e. ( s X. t ) )
66	59, 61, 63, 65	syl12anc 1324	. . . . . . . . 9 |- ( ( ( X e. ( RR X. RR ) /\ A = ( e X. f ) /\ X e. ( e X. f ) ) /\ ( ( s e. ran I /\ t e. ran I ) /\ ( ( ( 1st ` X ) e. s /\ s C_ e ) /\ ( ( 2nd ` X ) e. t /\ t C_ f ) ) ) ) -> X e. ( s X. t ) )
67	60	simprd 479	. . . . . . . . . . 11 |- ( ( ( X e. ( RR X. RR ) /\ A = ( e X. f ) /\ X e. ( e X. f ) ) /\ ( ( s e. ran I /\ t e. ran I ) /\ ( ( ( 1st ` X ) e. s /\ s C_ e ) /\ ( ( 2nd ` X ) e. t /\ t C_ f ) ) ) ) -> s C_ e )
68	62	simprd 479	. . . . . . . . . . 11 |- ( ( ( X e. ( RR X. RR ) /\ A = ( e X. f ) /\ X e. ( e X. f ) ) /\ ( ( s e. ran I /\ t e. ran I ) /\ ( ( ( 1st ` X ) e. s /\ s C_ e ) /\ ( ( 2nd ` X ) e. t /\ t C_ f ) ) ) ) -> t C_ f )
69		xpss12 5225	. . . . . . . . . . 11 |- ( ( s C_ e /\ t C_ f ) -> ( s X. t ) C_ ( e X. f ) )
70	67, 68, 69	syl2anc 693	. . . . . . . . . 10 |- ( ( ( X e. ( RR X. RR ) /\ A = ( e X. f ) /\ X e. ( e X. f ) ) /\ ( ( s e. ran I /\ t e. ran I ) /\ ( ( ( 1st ` X ) e. s /\ s C_ e ) /\ ( ( 2nd ` X ) e. t /\ t C_ f ) ) ) ) -> ( s X. t ) C_ ( e X. f ) )
71		simpl2 1065	. . . . . . . . . 10 |- ( ( ( X e. ( RR X. RR ) /\ A = ( e X. f ) /\ X e. ( e X. f ) ) /\ ( ( s e. ran I /\ t e. ran I ) /\ ( ( ( 1st ` X ) e. s /\ s C_ e ) /\ ( ( 2nd ` X ) e. t /\ t C_ f ) ) ) ) -> A = ( e X. f ) )
72	70, 71	sseqtr4d 3642	. . . . . . . . 9 |- ( ( ( X e. ( RR X. RR ) /\ A = ( e X. f ) /\ X e. ( e X. f ) ) /\ ( ( s e. ran I /\ t e. ran I ) /\ ( ( ( 1st ` X ) e. s /\ s C_ e ) /\ ( ( 2nd ` X ) e. t /\ t C_ f ) ) ) ) -> ( s X. t ) C_ A )
73		eleq2 2690	. . . . . . . . . . 11 |- ( b = ( s X. t ) -> ( X e. b <-> X e. ( s X. t ) ) )
74		sseq1 3626	. . . . . . . . . . 11 |- ( b = ( s X. t ) -> ( b C_ A <-> ( s X. t ) C_ A ) )
75	73, 74	anbi12d 747	. . . . . . . . . 10 |- ( b = ( s X. t ) -> ( ( X e. b /\ b C_ A ) <-> ( X e. ( s X. t ) /\ ( s X. t ) C_ A ) ) )
76	75	rspcev 3309	. . . . . . . . 9 |- ( ( ( s X. t ) e. ran R /\ ( X e. ( s X. t ) /\ ( s X. t ) C_ A ) ) -> E. b e. ran R ( X e. b /\ b C_ A ) )
77	56, 66, 72, 76	syl12anc 1324	. . . . . . . 8 |- ( ( ( X e. ( RR X. RR ) /\ A = ( e X. f ) /\ X e. ( e X. f ) ) /\ ( ( s e. ran I /\ t e. ran I ) /\ ( ( ( 1st ` X ) e. s /\ s C_ e ) /\ ( ( 2nd ` X ) e. t /\ t C_ f ) ) ) ) -> E. b e. ran R ( X e. b /\ b C_ A ) )
78	77	exp32 631	. . . . . . 7 |- ( ( X e. ( RR X. RR ) /\ A = ( e X. f ) /\ X e. ( e X. f ) ) -> ( ( s e. ran I /\ t e. ran I ) -> ( ( ( ( 1st ` X ) e. s /\ s C_ e ) /\ ( ( 2nd ` X ) e. t /\ t C_ f ) ) -> E. b e. ran R ( X e. b /\ b C_ A ) ) ) )
79	78	rexlimdvv 3037	. . . . . 6 |- ( ( X e. ( RR X. RR ) /\ A = ( e X. f ) /\ X e. ( e X. f ) ) -> ( E. s e. ran I E. t e. ran I ( ( ( 1st ` X ) e. s /\ s C_ e ) /\ ( ( 2nd ` X ) e. t /\ t C_ f ) ) -> E. b e. ran R ( X e. b /\ b C_ A ) ) )
80	20, 42, 79	sylc 65	. . . . 5 |- ( ( ( e e. ran (,) /\ f e. ran (,) ) /\ ( X e. ( RR X. RR ) /\ A = ( e X. f ) /\ X e. ( e X. f ) ) ) -> E. b e. ran R ( X e. b /\ b C_ A ) )
81	19, 80	sylan2 491	. . . 4 |- ( ( ( e e. ran (,) /\ f e. ran (,) ) /\ ( A = ( e X. f ) /\ ( X e. ( RR X. RR ) /\ X e. A ) ) ) -> E. b e. ran R ( X e. b /\ b C_ A ) )
82	81	ex 450	. . 3 |- ( ( e e. ran (,) /\ f e. ran (,) ) -> ( ( A = ( e X. f ) /\ ( X e. ( RR X. RR ) /\ X e. A ) ) -> E. b e. ran R ( X e. b /\ b C_ A ) ) )
83	82	rexlimivv 3036	. 2 |- ( E. e e. ran (,) E. f e. ran (,) ( A = ( e X. f ) /\ ( X e. ( RR X. RR ) /\ X e. A ) ) -> E. b e. ran R ( X e. b /\ b C_ A ) )
84	12, 14, 83	3syl 18	1 |- ( ( X e. ( RR X. RR ) /\ A e. B /\ X e. A ) -> E. b e. ran R ( X e. b /\ b C_ A ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   _Vcvv 3200   C_ wss 3574   X. cxp 5112   ran crn 5115   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   RRcr 9935   1c1 9937   + caddc 9939   / cdiv 10684   2c2 11070   ZZcz 11377   (,)cioo 12175   [,)cico 12177   ^cexp 12860   topGenctg 16098

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-pi 14803  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-refld 19951  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-cmp 21190  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-fcls 21745  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-cfil 23053  df-cmet 23055  df-cms 23132  df-limc 23630  df-dv 23631  df-log 24303  df-cxp 24304  df-logb 24503

This theorem is referenced by:  dya2iocnei  30344