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Theorem metequiv 21097
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 21096 . . 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 21096 . . . 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 451 . . 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 708 . 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 3432 . 2  |-  ( J  =  K  <->  ( J  C_  K  /\  K  C_  J ) )
8 r19.26 2909 . 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 367    = wceq 1399    e. wcel 1826   A.wral 2732   E.wrex 2733    C_ wss 3389   ` cfv 5496  (class class class)co 6196   RR+crp 11139   *Metcxmt 18516   ballcbl 18518   MetOpencmopn 18521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-pre-sup 9481
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-er 7229  df-map 7340  df-en 7436  df-dom 7437  df-sdom 7438  df-sup 7816  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-nn 10453  df-2 10511  df-n0 10713  df-z 10782  df-uz 11002  df-q 11102  df-rp 11140  df-xneg 11239  df-xadd 11240  df-xmul 11241  df-topgen 14851  df-psmet 18524  df-xmet 18525  df-bl 18527  df-mopn 18528  df-bases 19486
This theorem is referenced by:  metequiv2  21098
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