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Theorem metequiv2 15722
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 546 . . . . . . . . . . 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 539 . . . . . . . . . . . 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 533 . . . . . . . . . . . 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 544 . . . . . . . . . . . . 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 10115 . . . . . . . . . . . 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 543 . . . . . . . . . . . . 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 10115 . . . . . . . . . . . 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 545 . . . . . . . . . . . 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 15652 . . . . . . . . . . . 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 1289 . . . . . . . . . . 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 3285 . . . . . . . . . 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 540 . . . . . . . . . . . 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 15652 . . . . . . . . . . . 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 1289 . . . . . . . . . . 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 3284 . . . . . . . . . 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 404 . . . . . . 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 2652 . . . . . 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 2705 . . . . . 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 2617 . . . 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 2677 . . . 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 2617 . 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 15721 . 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
This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   C_ wss 3220   class class class wbr 4131   ` cfv 5380  (class class class)co 6088   RR*cxr 8365   <_ cle 8367   RR+crp 10071   *Metcxmet 14982   ballcbl 14984   MetOpencmopn 14987
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303  ax-arch 8304  ax-caucvg 8305
This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-isom 5389  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-map 6928  df-sup 7330  df-inf 7331  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-n0 9575  df-z 9656  df-uz 9938  df-q 10036  df-rp 10072  df-xneg 10191  df-xadd 10192  df-seqfrec 10906  df-exp 10997  df-cj 11629  df-re 11630  df-im 11631  df-rsqrt 11786  df-abs 11787  df-topgen 13673  df-psmet 14989  df-xmet 14990  df-bl 14992  df-mopn 14993  df-top 15224  df-bases 15269
This theorem is used by:  bdmopn  15730
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