MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  stdbdmopn Structured version   Visualization version   Unicode version
Theorem stdbdmopn 22323
Description: The standard bounded metric corresponding to C generates the same topology as C. (Contributed by Mario Carneiro, 26-Aug-2015.)
Hypotheses
Ref Expression
stdbdmet.1 |- D = ( x e. X , y e. X |-> if ( ( x C y ) <_ R , ( x C y ) , R ) )
stdbdmopn.2 |- J = ( MetOpen ` C )
Assertion
Ref Expression
stdbdmopn |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> J = ( MetOpen ` D ) )
Distinct variable groups:   x, y, C   x, R, y   x, X, y
Allowed substitution hints:   D( x, y)   J( x, y)
Proof of Theorem stdbdmopn
Dummy variables r s z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rpxr 11840 . . . . . . . 8 |- ( r e. RR+ -> r e. RR* )
21ad2antll 765 . . . . . . 7 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> r e. RR* )
3 simpl2 1065 . . . . . . 7 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> R e. RR* )
42, 3ifcld 4131 . . . . . 6 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> if ( r <_ R , r , R ) e. RR* )
5 rpre 11839 . . . . . . 7 |- ( r e. RR+ -> r e. RR )
65ad2antll 765 . . . . . 6 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> r e. RR )
7 rpgt0 11844 . . . . . . . . 9 |- ( r e. RR+ -> 0 < r )
87ad2antll 765 . . . . . . . 8 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> 0 < r )
9 simpl3 1066 . . . . . . . 8 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> 0 < R )
10 breq2 4657 . . . . . . . . 9 |- ( r = if ( r <_ R , r , R ) -> ( 0 < r <-> 0 < if ( r <_ R , r , R ) ) )
11 breq2 4657 . . . . . . . . 9 |- ( R = if ( r <_ R , r , R ) -> ( 0 < R <-> 0 < if ( r <_ R , r , R ) ) )
1210, 11ifboth 4124 . . . . . . . 8 |- ( ( 0 < r /\ 0 < R ) -> 0 < if ( r <_ R , r , R ) )
138, 9, 12syl2anc 693 . . . . . . 7 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> 0 < if ( r <_ R , r , R ) )
14 0xr 10086 . . . . . . . 8 |- 0 e. RR*
15 xrltle 11982 . . . . . . . 8 |- ( ( 0 e. RR* /\ if ( r <_ R , r , R ) e. RR* ) -> ( 0 < if ( r <_ R , r , R ) -> 0 <_ if ( r <_ R , r , R ) ) )
1614, 4, 15sylancr 695 . . . . . . 7 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> ( 0 < if ( r <_ R , r , R ) -> 0 <_ if ( r <_ R , r , R ) ) )
1713, 16mpd 15 . . . . . 6 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> 0 <_ if ( r <_ R , r , R ) )
18 xrmin1 12008 . . . . . . 7 |- ( ( r e. RR* /\ R e. RR* ) -> if ( r <_ R , r , R ) <_ r )
192, 3, 18syl2anc 693 . . . . . 6 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> if ( r <_ R , r , R ) <_ r )
20 xrrege0 12005 . . . . . 6 |- ( ( ( if ( r <_ R , r , R ) e. RR* /\ r e. RR ) /\ ( 0 <_ if ( r <_ R , r , R ) /\ if ( r <_ R , r , R ) <_ r ) ) -> if ( r <_ R , r , R ) e. RR )
214, 6, 17, 19, 20syl22anc 1327 . . . . 5 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> if ( r <_ R , r , R ) e. RR )
2221, 13elrpd 11869 . . . 4 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> if ( r <_ R , r , R ) e. RR+ )
23 simprl 794 . . . . . . 7 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> z e. X )
24 xrmin2 12009 . . . . . . . 8 |- ( ( r e. RR* /\ R e. RR* ) -> if ( r <_ R , r , R ) <_ R )
252, 3, 24syl2anc 693 . . . . . . 7 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> if ( r <_ R , r , R ) <_ R )
2623, 4, 253jca 1242 . . . . . 6 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> ( z e. X /\ if ( r <_ R , r , R ) e. RR* /\ if ( r <_ R , r , R ) <_ R ) )
27 stdbdmet.1 . . . . . . 7 |- D = ( x e. X , y e. X |-> if ( ( x C y ) <_ R , ( x C y ) , R ) )
2827stdbdbl 22322 . . . . . 6 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ if ( r <_ R , r , R ) e. RR* /\ if ( r <_ R , r , R ) <_ R ) ) -> ( z ( ball ` D ) if ( r <_ R , r , R ) ) = ( z ( ball ` C ) if ( r <_ R , r , R ) ) )
2926, 28syldan 487 . . . . 5 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> ( z ( ball ` D ) if ( r <_ R , r , R ) ) = ( z ( ball ` C ) if ( r <_ R , r , R ) ) )
3029eqcomd 2628 . . . 4 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> ( z ( ball ` C ) if ( r <_ R , r , R ) ) = ( z ( ball ` D ) if ( r <_ R , r , R ) ) )
31 breq1 4656 . . . . . 6 |- ( s = if ( r <_ R , r , R ) -> ( s <_ r <-> if ( r <_ R , r , R ) <_ r ) )
32 oveq2 6658 . . . . . . 7 |- ( s = if ( r <_ R , r , R ) -> ( z ( ball ` C ) s ) = ( z ( ball ` C ) if ( r <_ R , r , R ) ) )
33 oveq2 6658 . . . . . . 7 |- ( s = if ( r <_ R , r , R ) -> ( z ( ball ` D ) s ) = ( z ( ball ` D ) if ( r <_ R , r , R ) ) )
3432, 33eqeq12d 2637 . . . . . 6 |- ( s = if ( r <_ R , r , R ) -> ( ( z ( ball ` C ) s ) = ( z ( ball ` D ) s ) <-> ( z ( ball ` C ) if ( r <_ R , r , R ) ) = ( z ( ball ` D ) if ( r <_ R , r , R ) ) ) )
3531, 34anbi12d 747 . . . . 5 |- ( s = if ( r <_ R , r , R ) -> ( ( s <_ r /\ ( z ( ball ` C ) s ) = ( z ( ball ` D ) s ) ) <-> ( if ( r <_ R , r , R ) <_ r /\ ( z ( ball ` C ) if ( r <_ R , r , R ) ) = ( z ( ball ` D ) if ( r <_ R , r , R ) ) ) ) )
3635rspcev 3309 . . . 4 |- ( ( if ( r <_ R , r , R ) e. RR+ /\ ( if ( r <_ R , r , R ) <_ r /\ ( z ( ball ` C ) if ( r <_ R , r , R ) ) = ( z ( ball ` D ) if ( r <_ R , r , R ) ) ) ) -> E. s e. RR+ ( s <_ r /\ ( z ( ball ` C ) s ) = ( z ( ball ` D ) s ) ) )
3722, 19, 30, 36syl12anc 1324 . . 3 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> E. s e. RR+ ( s <_ r /\ ( z ( ball ` C ) s ) = ( z ( ball ` D ) s ) ) )
3837ralrimivva 2971 . 2 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> A. z e. X A. r e. RR+ E. s e. RR+ ( s <_ r /\ ( z ( ball ` C ) s ) = ( z ( ball ` D ) s ) ) )
39 simp1 1061 . . 3 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> C e. ( *Met ` X ) )
4027stdbdxmet 22320 . . 3 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> D e. ( *Met ` X ) )
41 stdbdmopn.2 . . . 4 |- J = ( MetOpen ` C )
42 eqid 2622 . . . 4 |- ( MetOpen ` D ) = ( MetOpen ` D )
4341, 42metequiv2 22315 . . 3 |- ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` X ) ) -> ( A. z e. X A. r e. RR+ E. s e. RR+ ( s <_ r /\ ( z ( ball ` C ) s ) = ( z ( ball ` D ) s ) ) -> J = ( MetOpen ` D ) ) )
4439, 40, 43syl2anc 693 . 2 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> ( A. z e. X A. r e. RR+ E. s e. RR+ ( s <_ r /\ ( z ( ball ` C ) s ) = ( z ( ball ` D ) s ) ) -> J = ( MetOpen ` D ) ) )
4538, 44mpd 15 1 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> J = ( MetOpen ` D ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   ifcif 4086   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   RRcr 9935   0cc0 9936   RR*cxr 10073   < clt 10074   <_ cle 10075   RR+crp 11832   *Metcxmt 19731   ballcbl 19733   MetOpencmopn 19736
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  ax-pre-sup 10014
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-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-iun 4522  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-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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  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-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-icc 12182  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-bl 19741  df-mopn 19742  df-bases 20750
This theorem is referenced by:  mopnex  22324  xlebnum  22764
  Copyright terms: Public domain W3C validator