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Theorem metss2 21100
Description: If the metric  D is "strongly finer" than  C (meaning that there is a positive real constant 
R such that  C ( x ,  y )  <_  R  x.  D (
x ,  y )), then  D generates a finer topology. (Using this theorem twice in each direction states that if two metrics are strongly equivalent, then they generate the same topology.) (Contributed by Mario Carneiro, 14-Sep-2015.)
Hypotheses
Ref Expression
metequiv.3  |-  J  =  ( MetOpen `  C )
metequiv.4  |-  K  =  ( MetOpen `  D )
metss2.1  |-  ( ph  ->  C  e.  ( Met `  X ) )
metss2.2  |-  ( ph  ->  D  e.  ( Met `  X ) )
metss2.3  |-  ( ph  ->  R  e.  RR+ )
metss2.4  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( x C y )  <_  ( R  x.  ( x D y ) ) )
Assertion
Ref Expression
metss2  |-  ( ph  ->  J  C_  K )
Distinct variable groups:    x, y, C    x, J, y    x, K, y    y, R    x, D, y    ph, x, y   
x, X, y
Allowed substitution hint:    R( x)

Proof of Theorem metss2
Dummy variables  s 
r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 459 . . . . 5  |-  ( ( x  e.  X  /\  r  e.  RR+ )  -> 
r  e.  RR+ )
2 metss2.3 . . . . 5  |-  ( ph  ->  R  e.  RR+ )
3 rpdivcl 11162 . . . . 5  |-  ( ( r  e.  RR+  /\  R  e.  RR+ )  ->  (
r  /  R )  e.  RR+ )
41, 2, 3syl2anr 476 . . . 4  |-  ( (
ph  /\  ( x  e.  X  /\  r  e.  RR+ ) )  -> 
( r  /  R
)  e.  RR+ )
5 metequiv.3 . . . . 5  |-  J  =  ( MetOpen `  C )
6 metequiv.4 . . . . 5  |-  K  =  ( MetOpen `  D )
7 metss2.1 . . . . 5  |-  ( ph  ->  C  e.  ( Met `  X ) )
8 metss2.2 . . . . 5  |-  ( ph  ->  D  e.  ( Met `  X ) )
9 metss2.4 . . . . 5  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( x C y )  <_  ( R  x.  ( x D y ) ) )
105, 6, 7, 8, 2, 9metss2lem 21099 . . . 4  |-  ( (
ph  /\  ( x  e.  X  /\  r  e.  RR+ ) )  -> 
( x ( ball `  D ) ( r  /  R ) ) 
C_  ( x (
ball `  C )
r ) )
11 oveq2 6204 . . . . . 6  |-  ( s  =  ( r  /  R )  ->  (
x ( ball `  D
) s )  =  ( x ( ball `  D ) ( r  /  R ) ) )
1211sseq1d 3444 . . . . 5  |-  ( s  =  ( r  /  R )  ->  (
( x ( ball `  D ) s ) 
C_  ( x (
ball `  C )
r )  <->  ( x
( ball `  D )
( r  /  R
) )  C_  (
x ( ball `  C
) r ) ) )
1312rspcev 3135 . . . 4  |-  ( ( ( r  /  R
)  e.  RR+  /\  (
x ( ball `  D
) ( r  /  R ) )  C_  ( x ( ball `  C ) r ) )  ->  E. s  e.  RR+  ( x (
ball `  D )
s )  C_  (
x ( ball `  C
) r ) )
144, 10, 13syl2anc 659 . . 3  |-  ( (
ph  /\  ( x  e.  X  /\  r  e.  RR+ ) )  ->  E. s  e.  RR+  (
x ( ball `  D
) s )  C_  ( x ( ball `  C ) r ) )
1514ralrimivva 2803 . 2  |-  ( ph  ->  A. x  e.  X  A. r  e.  RR+  E. s  e.  RR+  ( x (
ball `  D )
s )  C_  (
x ( ball `  C
) r ) )
16 metxmet 20922 . . . 4  |-  ( C  e.  ( Met `  X
)  ->  C  e.  ( *Met `  X
) )
177, 16syl 16 . . 3  |-  ( ph  ->  C  e.  ( *Met `  X ) )
18 metxmet 20922 . . . 4  |-  ( D  e.  ( Met `  X
)  ->  D  e.  ( *Met `  X
) )
198, 18syl 16 . . 3  |-  ( ph  ->  D  e.  ( *Met `  X ) )
205, 6metss 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 ) ) )
2117, 19, 20syl2anc 659 . 2  |-  ( ph  ->  ( J  C_  K  <->  A. x  e.  X  A. r  e.  RR+  E. s  e.  RR+  ( x (
ball `  D )
s )  C_  (
x ( ball `  C
) r ) ) )
2215, 21mpbird 232 1  |-  ( ph  ->  J  C_  K )
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   class class class wbr 4367   ` cfv 5496  (class class class)co 6196    x. cmul 9408    <_ cle 9540    / cdiv 10123   RR+crp 11139   *Metcxmt 18516   Metcme 18517   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-met 18526  df-bl 18527  df-mopn 18528  df-bases 19486
This theorem is referenced by:  equivcmet  21839
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