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Theorem metss2 18074
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 447 . . . . 5  |-  ( ( x  e.  X  /\  r  e.  RR+ )  -> 
r  e.  RR+ )
2 metss2.3 . . . . 5  |-  ( ph  ->  R  e.  RR+ )
3 rpdivcl 10392 . . . . 5  |-  ( ( r  e.  RR+  /\  R  e.  RR+ )  ->  (
r  /  R )  e.  RR+ )
41, 2, 3syl2anr 464 . . . 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 18073 . . . 4  |-  ( (
ph  /\  ( x  e.  X  /\  r  e.  RR+ ) )  -> 
( x ( ball `  D ) ( r  /  R ) ) 
C_  ( x (
ball `  C )
r ) )
11 oveq2 5882 . . . . . 6  |-  ( s  =  ( r  /  R )  ->  (
x ( ball `  D
) s )  =  ( x ( ball `  D ) ( r  /  R ) ) )
1211sseq1d 3218 . . . . 5  |-  ( s  =  ( r  /  R )  ->  (
( x ( ball `  D ) s ) 
C_  ( x (
ball `  C )
r )  <->  ( x
( ball `  D )
( r  /  R
) )  C_  (
x ( ball `  C
) r ) ) )
1312rspcev 2897 . . . 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 642 . . 3  |-  ( (
ph  /\  ( x  e.  X  /\  r  e.  RR+ ) )  ->  E. s  e.  RR+  (
x ( ball `  D
) s )  C_  ( x ( ball `  C ) r ) )
1514ralrimivva 2648 . 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 17915 . . . 4  |-  ( C  e.  ( Met `  X
)  ->  C  e.  ( * Met `  X
) )
177, 16syl 15 . . 3  |-  ( ph  ->  C  e.  ( * Met `  X ) )
18 metxmet 17915 . . . 4  |-  ( D  e.  ( Met `  X
)  ->  D  e.  ( * Met `  X
) )
198, 18syl 15 . . 3  |-  ( ph  ->  D  e.  ( * Met `  X ) )
205, 6metss 18070 . . 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 642 . 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 223 1  |-  ( ph  ->  J  C_  K )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557    C_ wss 3165   class class class wbr 4039   ` cfv 5271  (class class class)co 5874    x. cmul 8758    <_ cle 8884    / cdiv 9439   RR+crp 10370   * Metcxmt 16385   Metcme 16386   ballcbl 16387   MetOpencmopn 16388
This theorem is referenced by:  equivcmet  18757
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-topgen 13360  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-bases 16654
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