Theorem metustbl 21593

Description: The "section" image of an entourage at a point P always contains a ball (centered on this point). (Contributed by Thierry Arnoux, 8-Dec-2017.)

Assertion

Ref	Expression
metustbl	|- ( ( D e. (PsMet ` X ) /\ V e. (metUnif ` D ) /\ P e. X ) -> E. a e. ran ( ball ` D ) ( P e. a /\ a C_ ( V " { P } ) ) )

Distinct variable groups:   D, a   P, a   V, a   X, a

Proof of Theorem metustbl

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

Step	Hyp	Ref	Expression
1		simp1 1009	. . 3 |- ( ( D e. (PsMet ` X ) /\ V e. (metUnif ` D ) /\ P e. X ) -> D e. (PsMet ` X ) )
2		simp3 1011	. . 3 |- ( ( D e. (PsMet ` X ) /\ V e. (metUnif ` D ) /\ P e. X ) -> P e. X )
3		simpr 463	. . . . 5 |- ( ( ( ( D e. (PsMet ` X ) /\ V e. (metUnif ` D ) /\ P e. X ) /\ w e. ran ( r e. RR+ |-> ( `' D " ( 0 [,) r ) ) ) ) /\ w C_ V ) -> w C_ V )
4		vex 3050	. . . . . . . 8 |- w e. _V
5		eqid 2453	. . . . . . . . 9 |- ( r e. RR+ |-> ( `' D " ( 0 [,) r ) ) ) = ( r e. RR+ |-> ( `' D " ( 0 [,) r ) ) )
6	5	elrnmpt 5084	. . . . . . . 8 |- ( w e. _V -> ( w e. ran ( r e. RR+ |-> ( `' D " ( 0 [,) r ) ) ) <-> E. r e. RR+ w = ( `' D " ( 0 [,) r ) ) ) )
7	4, 6	ax-mp 5	. . . . . . 7 |- ( w e. ran ( r e. RR+ |-> ( `' D " ( 0 [,) r ) ) ) <-> E. r e. RR+ w = ( `' D " ( 0 [,) r ) ) )
8	7	biimpi 198	. . . . . 6 |- ( w e. ran ( r e. RR+ |-> ( `' D " ( 0 [,) r ) ) ) -> E. r e. RR+ w = ( `' D " ( 0 [,) r ) ) )
9	8	ad2antlr 734	. . . . 5 |- ( ( ( ( D e. (PsMet ` X ) /\ V e. (metUnif ` D ) /\ P e. X ) /\ w e. ran ( r e. RR+ |-> ( `' D " ( 0 [,) r ) ) ) ) /\ w C_ V ) -> E. r e. RR+ w = ( `' D " ( 0 [,) r ) ) )
10		sseq1 3455	. . . . . . 7 |- ( w = ( `' D " ( 0 [,) r ) ) -> ( w C_ V <-> ( `' D " ( 0 [,) r ) ) C_ V ) )
11	10	biimpcd 228	. . . . . 6 |- ( w C_ V -> ( w = ( `' D " ( 0 [,) r ) ) -> ( `' D " ( 0 [,) r ) ) C_ V ) )
12	11	reximdv 2863	. . . . 5 |- ( w C_ V -> ( E. r e. RR+ w = ( `' D " ( 0 [,) r ) ) -> E. r e. RR+ ( `' D " ( 0 [,) r ) ) C_ V ) )
13	3, 9, 12	sylc 62	. . . 4 |- ( ( ( ( D e. (PsMet ` X ) /\ V e. (metUnif ` D ) /\ P e. X ) /\ w e. ran ( r e. RR+ |-> ( `' D " ( 0 [,) r ) ) ) ) /\ w C_ V ) -> E. r e. RR+ ( `' D " ( 0 [,) r ) ) C_ V )
14		ne0i 3739	. . . . . 6 |- ( P e. X -> X =/= (/) )
15	2, 14	syl 17	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ V e. (metUnif ` D ) /\ P e. X ) -> X =/= (/) )
16		simp2 1010	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ V e. (metUnif ` D ) /\ P e. X ) -> V e. (metUnif ` D ) )
17		metuel 21591	. . . . . 6 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> ( V e. (metUnif ` D ) <-> ( V C_ ( X X. X ) /\ E. w e. ran ( r e. RR+ |-> ( `' D " ( 0 [,) r ) ) ) w C_ V ) ) )
18	17	simplbda 630	. . . . 5 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ V e. (metUnif ` D ) ) -> E. w e. ran ( r e. RR+ |-> ( `' D " ( 0 [,) r ) ) ) w C_ V )
19	15, 1, 16, 18	syl21anc 1268	. . . 4 |- ( ( D e. (PsMet ` X ) /\ V e. (metUnif ` D ) /\ P e. X ) -> E. w e. ran ( r e. RR+ |-> ( `' D " ( 0 [,) r ) ) ) w C_ V )
20	13, 19	r19.29a 2934	. . 3 |- ( ( D e. (PsMet ` X ) /\ V e. (metUnif ` D ) /\ P e. X ) -> E. r e. RR+ ( `' D " ( 0 [,) r ) ) C_ V )
21		imass1 5206	. . . . . 6 |- ( ( `' D " ( 0 [,) r ) ) C_ V -> ( ( `' D " ( 0 [,) r ) ) " { P } ) C_ ( V " { P } ) )
22	21	reximi 2857	. . . . 5 |- ( E. r e. RR+ ( `' D " ( 0 [,) r ) ) C_ V -> E. r e. RR+ ( ( `' D " ( 0 [,) r ) ) " { P } ) C_ ( V " { P } ) )
23		blval2 21589	. . . . . . . 8 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ r e. RR+ ) -> ( P ( ball ` D ) r ) = ( ( `' D " ( 0 [,) r ) ) " { P } ) )
24	23	sseq1d 3461	. . . . . . 7 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ r e. RR+ ) -> ( ( P ( ball ` D ) r ) C_ ( V " { P } ) <-> ( ( `' D " ( 0 [,) r ) ) " { P } ) C_ ( V " { P } ) ) )
25	24	3expa 1209	. . . . . 6 |- ( ( ( D e. (PsMet ` X ) /\ P e. X ) /\ r e. RR+ ) -> ( ( P ( ball ` D ) r ) C_ ( V " { P } ) <-> ( ( `' D " ( 0 [,) r ) ) " { P } ) C_ ( V " { P } ) ) )
26	25	rexbidva 2900	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ P e. X ) -> ( E. r e. RR+ ( P ( ball ` D ) r ) C_ ( V " { P } ) <-> E. r e. RR+ ( ( `' D " ( 0 [,) r ) ) " { P } ) C_ ( V " { P } ) ) )
27	22, 26	syl5ibr 225	. . . 4 |- ( ( D e. (PsMet ` X ) /\ P e. X ) -> ( E. r e. RR+ ( `' D " ( 0 [,) r ) ) C_ V -> E. r e. RR+ ( P ( ball ` D ) r ) C_ ( V " { P } ) ) )
28	27	imp 431	. . 3 |- ( ( ( D e. (PsMet ` X ) /\ P e. X ) /\ E. r e. RR+ ( `' D " ( 0 [,) r ) ) C_ V ) -> E. r e. RR+ ( P ( ball ` D ) r ) C_ ( V " { P } ) )
29	1, 2, 20, 28	syl21anc 1268	. 2 |- ( ( D e. (PsMet ` X ) /\ V e. (metUnif ` D ) /\ P e. X ) -> E. r e. RR+ ( P ( ball ` D ) r ) C_ ( V " { P } ) )
30		blssexps 21453	. . 3 |- ( ( D e. (PsMet ` X ) /\ P e. X ) -> ( E. a e. ran ( ball ` D ) ( P e. a /\ a C_ ( V " { P } ) ) <-> E. r e. RR+ ( P ( ball ` D ) r ) C_ ( V " { P } ) ) )
31	30	3adant2 1028	. 2 |- ( ( D e. (PsMet ` X ) /\ V e. (metUnif ` D ) /\ P e. X ) -> ( E. a e. ran ( ball ` D ) ( P e. a /\ a C_ ( V " { P } ) ) <-> E. r e. RR+ ( P ( ball ` D ) r ) C_ ( V " { P } ) ) )
32	29, 31	mpbird 236	1 |- ( ( D e. (PsMet ` X ) /\ V e. (metUnif ` D ) /\ P e. X ) -> E. a e. ran ( ball ` D ) ( P e. a /\ a C_ ( V " { P } ) ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   /\ w3a 986   = wceq 1446   e. wcel 1889   =/= wne 2624   E.wrex 2740   _Vcvv 3047   C_ wss 3406   (/)c0 3733   {csn 3970   |-> cmpt 4464   X. cxp 4835   `'ccnv 4836   ran crn 4838   "cima 4840   ` cfv 5585  (class class class)co 6295   0cc0 9544   RR+crp 11309   [,)cico 11644  PsMetcpsmet 18966   ballcbl 18969  metUnifcmetu 18973

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621  ax-pre-sup 9622

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-er 7368  df-map 7479  df-en 7575  df-dom 7576  df-sdom 7577  df-sup 7961  df-inf 7962  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-div 10277  df-nn 10617  df-2 10675  df-n0 10877  df-z 10945  df-uz 11167  df-q 11272  df-rp 11310  df-xneg 11416  df-xadd 11417  df-xmul 11418  df-ico 11648  df-psmet 18974  df-bl 18977  df-fbas 18979  df-fg 18980  df-metu 18981

This theorem is referenced by:  psmetutop  21594