Theorem metuel2 22370

Description: Elementhood in the uniform structure generated by a metric D (Contributed by Thierry Arnoux, 24-Jan-2018.) (Revised by Thierry Arnoux, 11-Feb-2018.)

Hypothesis

Ref	Expression
metuel2.u	|- U = (metUnif ` D )

Assertion

Ref	Expression
metuel2	|- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> ( V e. U <-> ( V C_ ( X X. X ) /\ E. d e. RR+ A. x e. X A. y e. X ( ( x D y ) < d -> x V y ) ) ) )

Distinct variable groups:   x, d, y, D   V, d, x, y   X, d, x, y

Allowed substitution hints:   U( x, y, d)

Proof of Theorem metuel2

Dummy variables a w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		metuel2.u	. . . 4 |- U = (metUnif ` D )
2	1	eleq2i 2693	. . 3 |- ( V e. U <-> V e. (metUnif ` D ) )
3	2	a1i 11	. 2 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> ( V e. U <-> V e. (metUnif ` D ) ) )
4		metuel 22369	. 2 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> ( V e. (metUnif ` D ) <-> ( V C_ ( X X. X ) /\ E. w e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) w C_ V ) ) )
5		vex 3203	. . . . . . . . . . 11 |- w e. _V
6		oveq2 6658	. . . . . . . . . . . . . 14 |- ( a = d -> ( 0 [,) a ) = ( 0 [,) d ) )
7	6	imaeq2d 5466	. . . . . . . . . . . . 13 |- ( a = d -> ( `' D " ( 0 [,) a ) ) = ( `' D " ( 0 [,) d ) ) )
8	7	cbvmptv 4750	. . . . . . . . . . . 12 |- ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) = ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) )
9	8	elrnmpt 5372	. . . . . . . . . . 11 |- ( w e. _V -> ( w e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) <-> E. d e. RR+ w = ( `' D " ( 0 [,) d ) ) ) )
10	5, 9	ax-mp 5	. . . . . . . . . 10 |- ( w e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) <-> E. d e. RR+ w = ( `' D " ( 0 [,) d ) ) )
11	10	anbi1i 731	. . . . . . . . 9 |- ( ( w e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) /\ w C_ V ) <-> ( E. d e. RR+ w = ( `' D " ( 0 [,) d ) ) /\ w C_ V ) )
12		r19.41v 3089	. . . . . . . . 9 |- ( E. d e. RR+ ( w = ( `' D " ( 0 [,) d ) ) /\ w C_ V ) <-> ( E. d e. RR+ w = ( `' D " ( 0 [,) d ) ) /\ w C_ V ) )
13	11, 12	bitr4i 267	. . . . . . . 8 |- ( ( w e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) /\ w C_ V ) <-> E. d e. RR+ ( w = ( `' D " ( 0 [,) d ) ) /\ w C_ V ) )
14	13	exbii 1774	. . . . . . 7 |- ( E. w ( w e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) /\ w C_ V ) <-> E. w E. d e. RR+ ( w = ( `' D " ( 0 [,) d ) ) /\ w C_ V ) )
15		df-rex 2918	. . . . . . 7 |- ( E. w e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) w C_ V <-> E. w ( w e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) /\ w C_ V ) )
16		rexcom4 3225	. . . . . . 7 |- ( E. d e. RR+ E. w ( w = ( `' D " ( 0 [,) d ) ) /\ w C_ V ) <-> E. w E. d e. RR+ ( w = ( `' D " ( 0 [,) d ) ) /\ w C_ V ) )
17	14, 15, 16	3bitr4i 292	. . . . . 6 |- ( E. w e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) w C_ V <-> E. d e. RR+ E. w ( w = ( `' D " ( 0 [,) d ) ) /\ w C_ V ) )
18		cnvexg 7112	. . . . . . . . 9 |- ( D e. (PsMet ` X ) -> `' D e. _V )
19		imaexg 7103	. . . . . . . . 9 |- ( `' D e. _V -> ( `' D " ( 0 [,) d ) ) e. _V )
20		sseq1 3626	. . . . . . . . . 10 |- ( w = ( `' D " ( 0 [,) d ) ) -> ( w C_ V <-> ( `' D " ( 0 [,) d ) ) C_ V ) )
21	20	ceqsexgv 3335	. . . . . . . . 9 |- ( ( `' D " ( 0 [,) d ) ) e. _V -> ( E. w ( w = ( `' D " ( 0 [,) d ) ) /\ w C_ V ) <-> ( `' D " ( 0 [,) d ) ) C_ V ) )
22	18, 19, 21	3syl 18	. . . . . . . 8 |- ( D e. (PsMet ` X ) -> ( E. w ( w = ( `' D " ( 0 [,) d ) ) /\ w C_ V ) <-> ( `' D " ( 0 [,) d ) ) C_ V ) )
23	22	rexbidv 3052	. . . . . . 7 |- ( D e. (PsMet ` X ) -> ( E. d e. RR+ E. w ( w = ( `' D " ( 0 [,) d ) ) /\ w C_ V ) <-> E. d e. RR+ ( `' D " ( 0 [,) d ) ) C_ V ) )
24	23	adantr 481	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ V C_ ( X X. X ) ) -> ( E. d e. RR+ E. w ( w = ( `' D " ( 0 [,) d ) ) /\ w C_ V ) <-> E. d e. RR+ ( `' D " ( 0 [,) d ) ) C_ V ) )
25	17, 24	syl5bb 272	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ V C_ ( X X. X ) ) -> ( E. w e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) w C_ V <-> E. d e. RR+ ( `' D " ( 0 [,) d ) ) C_ V ) )
26		cnvimass 5485	. . . . . . . . 9 |- ( `' D " ( 0 [,) d ) ) C_ dom D
27		simpll 790	. . . . . . . . . 10 |- ( ( ( D e. (PsMet ` X ) /\ V C_ ( X X. X ) ) /\ d e. RR+ ) -> D e. (PsMet ` X ) )
28		psmetf 22111	. . . . . . . . . 10 |- ( D e. (PsMet ` X ) -> D : ( X X. X ) --> RR* )
29		fdm 6051	. . . . . . . . . 10 |- ( D : ( X X. X ) --> RR* -> dom D = ( X X. X ) )
30	27, 28, 29	3syl 18	. . . . . . . . 9 |- ( ( ( D e. (PsMet ` X ) /\ V C_ ( X X. X ) ) /\ d e. RR+ ) -> dom D = ( X X. X ) )
31	26, 30	syl5sseq 3653	. . . . . . . 8 |- ( ( ( D e. (PsMet ` X ) /\ V C_ ( X X. X ) ) /\ d e. RR+ ) -> ( `' D " ( 0 [,) d ) ) C_ ( X X. X ) )
32		ssrel2 5210	. . . . . . . 8 |- ( ( `' D " ( 0 [,) d ) ) C_ ( X X. X ) -> ( ( `' D " ( 0 [,) d ) ) C_ V <-> A. x e. X A. y e. X ( <. x , y >. e. ( `' D " ( 0 [,) d ) ) -> <. x , y >. e. V ) ) )
33	31, 32	syl 17	. . . . . . 7 |- ( ( ( D e. (PsMet ` X ) /\ V C_ ( X X. X ) ) /\ d e. RR+ ) -> ( ( `' D " ( 0 [,) d ) ) C_ V <-> A. x e. X A. y e. X ( <. x , y >. e. ( `' D " ( 0 [,) d ) ) -> <. x , y >. e. V ) ) )
34		simplr 792	. . . . . . . . . . . . 13 |- ( ( ( ( ( D e. (PsMet ` X ) /\ V C_ ( X X. X ) ) /\ d e. RR+ ) /\ x e. X ) /\ y e. X ) -> x e. X )
35		simpr 477	. . . . . . . . . . . . 13 |- ( ( ( ( ( D e. (PsMet ` X ) /\ V C_ ( X X. X ) ) /\ d e. RR+ ) /\ x e. X ) /\ y e. X ) -> y e. X )
36		opelxp 5146	. . . . . . . . . . . . 13 |- ( <. x , y >. e. ( X X. X ) <-> ( x e. X /\ y e. X ) )
37	34, 35, 36	sylanbrc 698	. . . . . . . . . . . 12 |- ( ( ( ( ( D e. (PsMet ` X ) /\ V C_ ( X X. X ) ) /\ d e. RR+ ) /\ x e. X ) /\ y e. X ) -> <. x , y >. e. ( X X. X ) )
38	37	biantrurd 529	. . . . . . . . . . 11 |- ( ( ( ( ( D e. (PsMet ` X ) /\ V C_ ( X X. X ) ) /\ d e. RR+ ) /\ x e. X ) /\ y e. X ) -> ( ( D ` <. x , y >. ) e. ( 0 [,) d ) <-> ( <. x , y >. e. ( X X. X ) /\ ( D ` <. x , y >. ) e. ( 0 [,) d ) ) ) )
39		simp-4l 806	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( D e. (PsMet ` X ) /\ V C_ ( X X. X ) ) /\ d e. RR+ ) /\ x e. X ) /\ y e. X ) -> D e. (PsMet ` X ) )
40		psmetcl 22112	. . . . . . . . . . . . . . 15 |- ( ( D e. (PsMet ` X ) /\ x e. X /\ y e. X ) -> ( x D y ) e. RR* )
41	39, 34, 35, 40	syl3anc 1326	. . . . . . . . . . . . . 14 |- ( ( ( ( ( D e. (PsMet ` X ) /\ V C_ ( X X. X ) ) /\ d e. RR+ ) /\ x e. X ) /\ y e. X ) -> ( x D y ) e. RR* )
42	41	3biant1d 1441	. . . . . . . . . . . . 13 |- ( ( ( ( ( D e. (PsMet ` X ) /\ V C_ ( X X. X ) ) /\ d e. RR+ ) /\ x e. X ) /\ y e. X ) -> ( ( 0 <_ ( x D y ) /\ ( x D y ) < d ) <-> ( ( x D y ) e. RR* /\ 0 <_ ( x D y ) /\ ( x D y ) < d ) ) )
43		psmetge0 22117	. . . . . . . . . . . . . . 15 |- ( ( D e. (PsMet ` X ) /\ x e. X /\ y e. X ) -> 0 <_ ( x D y ) )
44	43	biantrurd 529	. . . . . . . . . . . . . 14 |- ( ( D e. (PsMet ` X ) /\ x e. X /\ y e. X ) -> ( ( x D y ) < d <-> ( 0 <_ ( x D y ) /\ ( x D y ) < d ) ) )
45	39, 34, 35, 44	syl3anc 1326	. . . . . . . . . . . . 13 |- ( ( ( ( ( D e. (PsMet ` X ) /\ V C_ ( X X. X ) ) /\ d e. RR+ ) /\ x e. X ) /\ y e. X ) -> ( ( x D y ) < d <-> ( 0 <_ ( x D y ) /\ ( x D y ) < d ) ) )
46		0xr 10086	. . . . . . . . . . . . . 14 |- 0 e. RR*
47		simpllr 799	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( D e. (PsMet ` X ) /\ V C_ ( X X. X ) ) /\ d e. RR+ ) /\ x e. X ) /\ y e. X ) -> d e. RR+ )
48	47	rpxrd 11873	. . . . . . . . . . . . . 14 |- ( ( ( ( ( D e. (PsMet ` X ) /\ V C_ ( X X. X ) ) /\ d e. RR+ ) /\ x e. X ) /\ y e. X ) -> d e. RR* )
49		elico1 12218	. . . . . . . . . . . . . 14 |- ( ( 0 e. RR* /\ d e. RR* ) -> ( ( x D y ) e. ( 0 [,) d ) <-> ( ( x D y ) e. RR* /\ 0 <_ ( x D y ) /\ ( x D y ) < d ) ) )
50	46, 48, 49	sylancr 695	. . . . . . . . . . . . 13 |- ( ( ( ( ( D e. (PsMet ` X ) /\ V C_ ( X X. X ) ) /\ d e. RR+ ) /\ x e. X ) /\ y e. X ) -> ( ( x D y ) e. ( 0 [,) d ) <-> ( ( x D y ) e. RR* /\ 0 <_ ( x D y ) /\ ( x D y ) < d ) ) )
51	42, 45, 50	3bitr4d 300	. . . . . . . . . . . 12 |- ( ( ( ( ( D e. (PsMet ` X ) /\ V C_ ( X X. X ) ) /\ d e. RR+ ) /\ x e. X ) /\ y e. X ) -> ( ( x D y ) < d <-> ( x D y ) e. ( 0 [,) d ) ) )
52		df-ov 6653	. . . . . . . . . . . . 13 |- ( x D y ) = ( D ` <. x , y >. )
53	52	eleq1i 2692	. . . . . . . . . . . 12 |- ( ( x D y ) e. ( 0 [,) d ) <-> ( D ` <. x , y >. ) e. ( 0 [,) d ) )
54	51, 53	syl6bb 276	. . . . . . . . . . 11 |- ( ( ( ( ( D e. (PsMet ` X ) /\ V C_ ( X X. X ) ) /\ d e. RR+ ) /\ x e. X ) /\ y e. X ) -> ( ( x D y ) < d <-> ( D ` <. x , y >. ) e. ( 0 [,) d ) ) )
55		ffn 6045	. . . . . . . . . . . 12 |- ( D : ( X X. X ) --> RR* -> D Fn ( X X. X ) )
56		elpreima 6337	. . . . . . . . . . . 12 |- ( D Fn ( X X. X ) -> ( <. x , y >. e. ( `' D " ( 0 [,) d ) ) <-> ( <. x , y >. e. ( X X. X ) /\ ( D ` <. x , y >. ) e. ( 0 [,) d ) ) ) )
57	39, 28, 55, 56	4syl 19	. . . . . . . . . . 11 |- ( ( ( ( ( D e. (PsMet ` X ) /\ V C_ ( X X. X ) ) /\ d e. RR+ ) /\ x e. X ) /\ y e. X ) -> ( <. x , y >. e. ( `' D " ( 0 [,) d ) ) <-> ( <. x , y >. e. ( X X. X ) /\ ( D ` <. x , y >. ) e. ( 0 [,) d ) ) ) )
58	38, 54, 57	3bitr4d 300	. . . . . . . . . 10 |- ( ( ( ( ( D e. (PsMet ` X ) /\ V C_ ( X X. X ) ) /\ d e. RR+ ) /\ x e. X ) /\ y e. X ) -> ( ( x D y ) < d <-> <. x , y >. e. ( `' D " ( 0 [,) d ) ) ) )
59	58	anasss 679	. . . . . . . . 9 |- ( ( ( ( D e. (PsMet ` X ) /\ V C_ ( X X. X ) ) /\ d e. RR+ ) /\ ( x e. X /\ y e. X ) ) -> ( ( x D y ) < d <-> <. x , y >. e. ( `' D " ( 0 [,) d ) ) ) )
60		df-br 4654	. . . . . . . . . 10 |- ( x V y <-> <. x , y >. e. V )
61	60	a1i 11	. . . . . . . . 9 |- ( ( ( ( D e. (PsMet ` X ) /\ V C_ ( X X. X ) ) /\ d e. RR+ ) /\ ( x e. X /\ y e. X ) ) -> ( x V y <-> <. x , y >. e. V ) )
62	59, 61	imbi12d 334	. . . . . . . 8 |- ( ( ( ( D e. (PsMet ` X ) /\ V C_ ( X X. X ) ) /\ d e. RR+ ) /\ ( x e. X /\ y e. X ) ) -> ( ( ( x D y ) < d -> x V y ) <-> ( <. x , y >. e. ( `' D " ( 0 [,) d ) ) -> <. x , y >. e. V ) ) )
63	62	2ralbidva 2988	. . . . . . 7 |- ( ( ( D e. (PsMet ` X ) /\ V C_ ( X X. X ) ) /\ d e. RR+ ) -> ( A. x e. X A. y e. X ( ( x D y ) < d -> x V y ) <-> A. x e. X A. y e. X ( <. x , y >. e. ( `' D " ( 0 [,) d ) ) -> <. x , y >. e. V ) ) )
64	33, 63	bitr4d 271	. . . . . 6 |- ( ( ( D e. (PsMet ` X ) /\ V C_ ( X X. X ) ) /\ d e. RR+ ) -> ( ( `' D " ( 0 [,) d ) ) C_ V <-> A. x e. X A. y e. X ( ( x D y ) < d -> x V y ) ) )
65	64	rexbidva 3049	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ V C_ ( X X. X ) ) -> ( E. d e. RR+ ( `' D " ( 0 [,) d ) ) C_ V <-> E. d e. RR+ A. x e. X A. y e. X ( ( x D y ) < d -> x V y ) ) )
66	25, 65	bitrd 268	. . . 4 |- ( ( D e. (PsMet ` X ) /\ V C_ ( X X. X ) ) -> ( E. w e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) w C_ V <-> E. d e. RR+ A. x e. X A. y e. X ( ( x D y ) < d -> x V y ) ) )
67	66	pm5.32da 673	. . 3 |- ( D e. (PsMet ` X ) -> ( ( V C_ ( X X. X ) /\ E. w e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) w C_ V ) <-> ( V C_ ( X X. X ) /\ E. d e. RR+ A. x e. X A. y e. X ( ( x D y ) < d -> x V y ) ) ) )
68	67	adantl 482	. 2 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> ( ( V C_ ( X X. X ) /\ E. w e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) w C_ V ) <-> ( V C_ ( X X. X ) /\ E. d e. RR+ A. x e. X A. y e. X ( ( x D y ) < d -> x V y ) ) ) )
69	3, 4, 68	3bitrd 294	1 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> ( V e. U <-> ( V C_ ( X X. X ) /\ E. d e. RR+ A. x e. X A. y e. X ( ( x D y ) < d -> x V y ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   (/)c0 3915   <.cop 4183   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   0cc0 9936   RR*cxr 10073   < clt 10074   <_ cle 10075   RR+crp 11832   [,)cico 12177  PsMetcpsmet 19730  metUnifcmetu 19737

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  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-2 11079  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ico 12181  df-psmet 19738  df-fbas 19743  df-fg 19744  df-metu 19745

This theorem is referenced by: (None)