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Theorem metequiv 20202
Description: Two ways of saying that two metrics generate the same topology. Two metrics satisfying the right-hand side are said to be (topologically) equivalent. (Contributed by Jeff Hankins, 21-Jun-2009.) (Revised by Mario Carneiro, 12-Nov-2013.)
Hypotheses
Ref Expression
metequiv.3  |-  J  =  ( MetOpen `  C )
metequiv.4  |-  K  =  ( MetOpen `  D )
Assertion
Ref Expression
metequiv  |-  ( ( 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. a  e.  RR+  E. b  e.  RR+  ( x (
ball `  C )
b )  C_  (
x ( ball `  D
) a ) ) ) )
Distinct variable groups:    s, r, x, C    J, r, s, x    K, r, s, x    D, r, s, x    X, r, s, x    a, b, x, C    D, a,
b    J, a, b    K, a, b    X, a, b

Proof of Theorem metequiv
StepHypRef Expression
1 metequiv.3 . . . 4  |-  J  =  ( MetOpen `  C )
2 metequiv.4 . . . 4  |-  K  =  ( MetOpen `  D )
31, 2metss 20201 . . 3  |-  ( ( C  e.  ( *Met `  X )  /\  D  e.  ( *Met `  X
) )  ->  ( J  C_  K  <->  A. x  e.  X  A. r  e.  RR+  E. s  e.  RR+  ( x ( ball `  D ) s ) 
C_  ( x (
ball `  C )
r ) ) )
42, 1metss 20201 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  C  e.  ( *Met `  X
) )  ->  ( K  C_  J  <->  A. x  e.  X  A. a  e.  RR+  E. b  e.  RR+  ( x ( ball `  C ) b ) 
C_  ( x (
ball `  D )
a ) ) )
54ancoms 453 . . 3  |-  ( ( C  e.  ( *Met `  X )  /\  D  e.  ( *Met `  X
) )  ->  ( K  C_  J  <->  A. x  e.  X  A. a  e.  RR+  E. b  e.  RR+  ( x ( ball `  C ) b ) 
C_  ( x (
ball `  D )
a ) ) )
63, 5anbi12d 710 . 2  |-  ( ( C  e.  ( *Met `  X )  /\  D  e.  ( *Met `  X
) )  ->  (
( J  C_  K  /\  K  C_  J )  <-> 
( A. x  e.  X  A. r  e.  RR+  E. s  e.  RR+  ( x ( ball `  D ) s ) 
C_  ( x (
ball `  C )
r )  /\  A. x  e.  X  A. a  e.  RR+  E. b  e.  RR+  ( x (
ball `  C )
b )  C_  (
x ( ball `  D
) a ) ) ) )
7 eqss 3471 . 2  |-  ( J  =  K  <->  ( J  C_  K  /\  K  C_  J ) )
8 r19.26 2947 . 2  |-  ( A. x  e.  X  ( A. r  e.  RR+  E. s  e.  RR+  ( x (
ball `  D )
s )  C_  (
x ( ball `  C
) r )  /\  A. a  e.  RR+  E. b  e.  RR+  ( x (
ball `  C )
b )  C_  (
x ( ball `  D
) a ) )  <-> 
( A. x  e.  X  A. r  e.  RR+  E. s  e.  RR+  ( x ( ball `  D ) s ) 
C_  ( x (
ball `  C )
r )  /\  A. x  e.  X  A. a  e.  RR+  E. b  e.  RR+  ( x (
ball `  C )
b )  C_  (
x ( ball `  D
) a ) ) )
96, 7, 83bitr4g 288 1  |-  ( ( 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. a  e.  RR+  E. b  e.  RR+  ( x (
ball `  C )
b )  C_  (
x ( ball `  D
) a ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2795   E.wrex 2796    C_ wss 3428   ` cfv 5518  (class class class)co 6192   RR+crp 11094   *Metcxmt 17912   ballcbl 17914   MetOpencmopn 17917
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462  ax-pre-sup 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-recs 6934  df-rdg 6968  df-er 7203  df-map 7318  df-en 7413  df-dom 7414  df-sdom 7415  df-sup 7794  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-div 10097  df-nn 10426  df-2 10483  df-n0 10683  df-z 10750  df-uz 10965  df-q 11057  df-rp 11095  df-xneg 11192  df-xadd 11193  df-xmul 11194  df-topgen 14486  df-psmet 17920  df-xmet 17921  df-bl 17923  df-mopn 17924  df-bases 18623
This theorem is referenced by:  metequiv2  20203
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