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Theorem metequiv2 15219
Description: If there is a sequence of radii approaching zero for which the balls of both metrics coincide, then the generated topologies are equivalent. (Contributed by Mario Carneiro, 26-Aug-2015.)
Hypotheses
Ref Expression
metequiv.3 |- J = ( MetOpen ` C )
metequiv.4 |- K = ( MetOpen ` D )
Assertion
Ref Expression
metequiv2 |- ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` X ) ) -> ( A. x e. X A. r e. RR+ E. s e. RR+ ( s <_ r /\ ( x ( ball ` C ) s ) = ( x ( ball ` D ) s ) ) -> J = K ) )
Distinct variable groups:   s, r, x, C   J, r, s, x   K, r, s, x   D, r, s, x   X, r, s, x
Proof of Theorem metequiv2
StepHypRef Expression
1 simprrr 542 . . . . . . . . . . 11 |- ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` X ) ) /\ x e. X ) /\ ( ( r e. RR+ /\ s e. RR+ ) /\ ( s <_ r /\ ( x ( ball ` C ) s ) = ( x ( ball ` D ) s ) ) ) ) -> ( x ( ball ` C ) s ) = ( x ( ball ` D ) s ) )
2 simplll 535 . . . . . . . . . . . 12 |- ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` X ) ) /\ x e. X ) /\ ( ( r e. RR+ /\ s e. RR+ ) /\ ( s <_ r /\ ( x ( ball ` C ) s ) = ( x ( ball ` D ) s ) ) ) ) -> C e. ( *Met ` X ) )
3 simplr 529 . . . . . . . . . . . 12 |- ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` X ) ) /\ x e. X ) /\ ( ( r e. RR+ /\ s e. RR+ ) /\ ( s <_ r /\ ( x ( ball ` C ) s ) = ( x ( ball ` D ) s ) ) ) ) -> x e. X )
4 simprlr 540 . . . . . . . . . . . . 13 |- ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` X ) ) /\ x e. X ) /\ ( ( r e. RR+ /\ s e. RR+ ) /\ ( s <_ r /\ ( x ( ball ` C ) s ) = ( x ( ball ` D ) s ) ) ) ) -> s e. RR+ )
54rpxrd 9931 . . . . . . . . . . . 12 |- ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` X ) ) /\ x e. X ) /\ ( ( r e. RR+ /\ s e. RR+ ) /\ ( s <_ r /\ ( x ( ball ` C ) s ) = ( x ( ball ` D ) s ) ) ) ) -> s e. RR* )
6 simprll 539 . . . . . . . . . . . . 13 |- ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` X ) ) /\ x e. X ) /\ ( ( r e. RR+ /\ s e. RR+ ) /\ ( s <_ r /\ ( x ( ball ` C ) s ) = ( x ( ball ` D ) s ) ) ) ) -> r e. RR+ )
76rpxrd 9931 . . . . . . . . . . . 12 |- ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` X ) ) /\ x e. X ) /\ ( ( r e. RR+ /\ s e. RR+ ) /\ ( s <_ r /\ ( x ( ball ` C ) s ) = ( x ( ball ` D ) s ) ) ) ) -> r e. RR* )
8 simprrl 541 . . . . . . . . . . . 12 |- ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` X ) ) /\ x e. X ) /\ ( ( r e. RR+ /\ s e. RR+ ) /\ ( s <_ r /\ ( x ( ball ` C ) s ) = ( x ( ball ` D ) s ) ) ) ) -> s <_ r )
9 ssbl 15149 . . . . . . . . . . . 12 |- ( ( ( C e. ( *Met ` X ) /\ x e. X ) /\ ( s e. RR* /\ r e. RR* ) /\ s <_ r ) -> ( x ( ball ` C ) s ) C_ ( x ( ball ` C ) r ) )
102, 3, 5, 7, 8, 9syl221anc 1284 . . . . . . . . . . 11 |- ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` X ) ) /\ x e. X ) /\ ( ( r e. RR+ /\ s e. RR+ ) /\ ( s <_ r /\ ( x ( ball ` C ) s ) = ( x ( ball ` D ) s ) ) ) ) -> ( x ( ball ` C ) s ) C_ ( x ( ball ` C ) r ) )
111, 10eqsstrrd 3264 . . . . . . . . . 10 |- ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` X ) ) /\ x e. X ) /\ ( ( r e. RR+ /\ s e. RR+ ) /\ ( s <_ r /\ ( x ( ball ` C ) s ) = ( x ( ball ` D ) s ) ) ) ) -> ( x ( ball ` D ) s ) C_ ( x ( ball ` C ) r ) )
12 simpllr 536 . . . . . . . . . . . 12 |- ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` X ) ) /\ x e. X ) /\ ( ( r e. RR+ /\ s e. RR+ ) /\ ( s <_ r /\ ( x ( ball ` C ) s ) = ( x ( ball ` D ) s ) ) ) ) -> D e. ( *Met ` X ) )
13 ssbl 15149 . . . . . . . . . . . 12 |- ( ( ( D e. ( *Met ` X ) /\ x e. X ) /\ ( s e. RR* /\ r e. RR* ) /\ s <_ r ) -> ( x ( ball ` D ) s ) C_ ( x ( ball ` D ) r ) )
1412, 3, 5, 7, 8, 13syl221anc 1284 . . . . . . . . . . 11 |- ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` X ) ) /\ x e. X ) /\ ( ( r e. RR+ /\ s e. RR+ ) /\ ( s <_ r /\ ( x ( ball ` C ) s ) = ( x ( ball ` D ) s ) ) ) ) -> ( x ( ball ` D ) s ) C_ ( x ( ball ` D ) r ) )
151, 14eqsstrd 3263 . . . . . . . . . 10 |- ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` X ) ) /\ x e. X ) /\ ( ( r e. RR+ /\ s e. RR+ ) /\ ( s <_ r /\ ( x ( ball ` C ) s ) = ( x ( ball ` D ) s ) ) ) ) -> ( x ( ball ` C ) s ) C_ ( x ( ball ` D ) r ) )
1611, 15jca 306 . . . . . . . . 9 |- ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` X ) ) /\ x e. X ) /\ ( ( r e. RR+ /\ s e. RR+ ) /\ ( s <_ r /\ ( x ( ball ` C ) s ) = ( x ( ball ` D ) s ) ) ) ) -> ( ( x ( ball ` D ) s ) C_ ( x ( ball ` C ) r ) /\ ( x ( ball ` C ) s ) C_ ( x ( ball ` D ) r ) ) )
1716expr 375 . . . . . . . 8 |- ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` X ) ) /\ x e. X ) /\ ( r e. RR+ /\ s e. RR+ ) ) -> ( ( s <_ r /\ ( x ( ball ` C ) s ) = ( x ( ball ` D ) s ) ) -> ( ( x ( ball ` D ) s ) C_ ( x ( ball ` C ) r ) /\ ( x ( ball ` C ) s ) C_ ( x ( ball ` D ) r ) ) ) )
1817anassrs 400 . . . . . . 7 |- ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` X ) ) /\ x e. X ) /\ r e. RR+ ) /\ s e. RR+ ) -> ( ( s <_ r /\ ( x ( ball ` C ) s ) = ( x ( ball ` D ) s ) ) -> ( ( x ( ball ` D ) s ) C_ ( x ( ball ` C ) r ) /\ ( x ( ball ` C ) s ) C_ ( x ( ball ` D ) r ) ) ) )
1918reximdva 2634 . . . . . 6 |- ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` X ) ) /\ x e. X ) /\ r e. RR+ ) -> ( E. s e. RR+ ( s <_ r /\ ( x ( ball ` C ) s ) = ( x ( ball ` D ) s ) ) -> E. s e. RR+ ( ( x ( ball ` D ) s ) C_ ( x ( ball ` C ) r ) /\ ( x ( ball ` C ) s ) C_ ( x ( ball ` D ) r ) ) ) )
20 r19.40 2687 . . . . . 6 |- ( E. s e. RR+ ( ( x ( ball ` D ) s ) C_ ( x ( ball ` C ) r ) /\ ( x ( ball ` C ) s ) C_ ( x ( ball ` D ) r ) ) -> ( E. s e. RR+ ( x ( ball ` D ) s ) C_ ( x ( ball ` C ) r ) /\ E. s e. RR+ ( x ( ball ` C ) s ) C_ ( x ( ball ` D ) r ) ) )
2119, 20syl6 33 . . . . 5 |- ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` X ) ) /\ x e. X ) /\ r e. RR+ ) -> ( E. s e. RR+ ( s <_ r /\ ( x ( ball ` C ) s ) = ( x ( ball ` D ) s ) ) -> ( E. s e. RR+ ( x ( ball ` D ) s ) C_ ( x ( ball ` C ) r ) /\ E. s e. RR+ ( x ( ball ` C ) s ) C_ ( x ( ball ` D ) r ) ) ) )
2221ralimdva 2599 . . . 4 |- ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` X ) ) /\ x e. X ) -> ( A. r e. RR+ E. s e. RR+ ( s <_ r /\ ( x ( ball ` C ) s ) = ( x ( ball ` D ) s ) ) -> A. r e. RR+ ( E. s e. RR+ ( x ( ball ` D ) s ) C_ ( x ( ball ` C ) r ) /\ E. s e. RR+ ( x ( ball ` C ) s ) C_ ( x ( ball ` D ) r ) ) ) )
23 r19.26 2659 . . . 4 |- ( A. r e. RR+ ( E. s e. RR+ ( x ( ball ` D ) s ) C_ ( x ( ball ` C ) r ) /\ E. s e. RR+ ( x ( ball ` C ) s ) C_ ( x ( ball ` D ) r ) ) <-> ( A. r e. RR+ E. s e. RR+ ( x ( ball ` D ) s ) C_ ( x ( ball ` C ) r ) /\ A. r e. RR+ E. s e. RR+ ( x ( ball ` C ) s ) C_ ( x ( ball ` D ) r ) ) )
2422, 23imbitrdi 161 . . 3 |- ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` X ) ) /\ x e. X ) -> ( A. r e. RR+ E. s e. RR+ ( s <_ r /\ ( x ( ball ` C ) s ) = ( x ( ball ` D ) s ) ) -> ( A. r e. RR+ E. s e. RR+ ( x ( ball ` D ) s ) C_ ( x ( ball ` C ) r ) /\ A. r e. RR+ E. s e. RR+ ( x ( ball ` C ) s ) C_ ( x ( ball ` D ) r ) ) ) )
2524ralimdva 2599 . 2 |- ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` X ) ) -> ( A. x e. X A. r e. RR+ E. s e. RR+ ( s <_ r /\ ( x ( ball ` C ) s ) = ( x ( ball ` D ) s ) ) -> A. x e. X ( A. r e. RR+ E. s e. RR+ ( x ( ball ` D ) s ) C_ ( x ( ball ` C ) r ) /\ A. r e. RR+ E. s e. RR+ ( x ( ball ` C ) s ) C_ ( x ( ball ` D ) r ) ) ) )
26 metequiv.3 . . 3 |- J = ( MetOpen ` C )
27 metequiv.4 . . 3 |- K = ( MetOpen ` D )
2826, 27metequiv 15218 . 2 |- ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` X ) ) -> ( J = K <-> A. x e. X ( A. r e. RR+ E. s e. RR+ ( x ( ball ` D ) s ) C_ ( x ( ball ` C ) r ) /\ A. r e. RR+ E. s e. RR+ ( x ( ball ` C ) s ) C_ ( x ( ball ` D ) r ) ) ) )
2925, 28sylibrd 169 1 |- ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` X ) ) -> ( A. x e. X A. r e. RR+ E. s e. RR+ ( s <_ r /\ ( x ( ball ` C ) s ) = ( x ( ball ` D ) s ) ) -> J = K ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   A.wral 2510   E.wrex 2511   C_ wss 3200   class class class wbr 4088   ` cfv 5326  (class class class)co 6017   RR*cxr 8212   <_ cle 8214   RR+crp 9887   *Metcxmet 14549   ballcbl 14551   MetOpencmopn 14554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149  ax-arch 8150  ax-caucvg 8151
This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-isom 5335  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-map 6818  df-sup 7182  df-inf 7183  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-n0 9402  df-z 9479  df-uz 9755  df-q 9853  df-rp 9888  df-xneg 10006  df-xadd 10007  df-seqfrec 10709  df-exp 10800  df-cj 11402  df-re 11403  df-im 11404  df-rsqrt 11558  df-abs 11559  df-topgen 13342  df-psmet 14556  df-xmet 14557  df-bl 14559  df-mopn 14560  df-top 14721  df-bases 14766
This theorem is referenced by:  bdmopn  15227
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