Theorem metustbl 21210

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 996	. . 3 |- ( ( D e. (PsMet ` X ) /\ V e. (metUnif ` D ) /\ P e. X ) -> D e. (PsMet ` X ) )
2		simp3 998	. . 3 |- ( ( D e. (PsMet ` X ) /\ V e. (metUnif ` D ) /\ P e. X ) -> P e. X )
3		simpr 461	. . . . 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 3112	. . . . . . . 8 |- w e. _V
5		eqid 2457	. . . . . . . . 9 |- ( r e. RR+ |-> ( `' D " ( 0 [,) r ) ) ) = ( r e. RR+ |-> ( `' D " ( 0 [,) r ) ) )
6	5	elrnmpt 5259	. . . . . . . 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 194	. . . . . 6 |- ( w e. ran ( r e. RR+ |-> ( `' D " ( 0 [,) r ) ) ) -> E. r e. RR+ w = ( `' D " ( 0 [,) r ) ) )
9	8	ad2antlr 726	. . . . 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 3520	. . . . . . 7 |- ( w = ( `' D " ( 0 [,) r ) ) -> ( w C_ V <-> ( `' D " ( 0 [,) r ) ) C_ V ) )
11	10	biimpcd 224	. . . . . 6 |- ( w C_ V -> ( w = ( `' D " ( 0 [,) r ) ) -> ( `' D " ( 0 [,) r ) ) C_ V ) )
12	11	reximdv 2931	. . . . 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 60	. . . 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 3799	. . . . . 6 |- ( P e. X -> X =/= (/) )
15	2, 14	syl 16	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ V e. (metUnif ` D ) /\ P e. X ) -> X =/= (/) )
16		simp2 997	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ V e. (metUnif ` D ) /\ P e. X ) -> V e. (metUnif ` D ) )
17		metuel 21207	. . . . . 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 624	. . . . 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 1227	. . . 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 2999	. . 3 |- ( ( D e. (PsMet ` X ) /\ V e. (metUnif ` D ) /\ P e. X ) -> E. r e. RR+ ( `' D " ( 0 [,) r ) ) C_ V )
21		imass1 5381	. . . . . 6 |- ( ( `' D " ( 0 [,) r ) ) C_ V -> ( ( `' D " ( 0 [,) r ) ) " { P } ) C_ ( V " { P } ) )
22	21	reximi 2925	. . . . 5 |- ( E. r e. RR+ ( `' D " ( 0 [,) r ) ) C_ V -> E. r e. RR+ ( ( `' D " ( 0 [,) r ) ) " { P } ) C_ ( V " { P } ) )
23		blval2 21204	. . . . . . . 8 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ r e. RR+ ) -> ( P ( ball ` D ) r ) = ( ( `' D " ( 0 [,) r ) ) " { P } ) )
24	23	sseq1d 3526	. . . . . . 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 1196	. . . . . 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 2965	. . . . 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 221	. . . 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 429	. . 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 1227	. 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 21055	. . 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 1015	. 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 232	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 184   /\ wa 369   /\ w3a 973   = wceq 1395   e. wcel 1819   =/= wne 2652   E.wrex 2808   _Vcvv 3109   C_ wss 3471   (/)c0 3793   {csn 4032   |-> cmpt 4515   X. cxp 5006   `'ccnv 5007   ran crn 5009   "cima 5011   ` cfv 5594  (class class class)co 6296   0cc0 9509   RR+crp 11245   [,)cico 11556  PsMetcpsmet 18529   ballcbl 18532  metUnifcmetu 18537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ico 11560  df-psmet 18538  df-bl 18541  df-fbas 18543  df-fg 18544  df-metu 18546

This theorem is referenced by:  psmetutop  21212