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Theorem metss2lem 18057
Description: Lemma for metss2 18058. (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
metss2lem  |-  ( (
ph  /\  ( x  e.  X  /\  S  e.  RR+ ) )  ->  (
x ( ball `  D
) ( S  /  R ) )  C_  ( x ( ball `  C ) S ) )
Distinct variable groups:    x, y, C    x, J, y    x, K, y    y, R    y, S    x, D, y    ph, x, y    x, X, y
Allowed substitution hints:    R( x)    S( x)

Proof of Theorem metss2lem
StepHypRef Expression
1 metss2.2 . . . . . . 7  |-  ( ph  ->  D  e.  ( Met `  X ) )
21ad2antrr 706 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  D  e.  ( Met `  X ) )
3 simplrl 736 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  x  e.  X )
4 simpr 447 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  y  e.  X )
5 metcl 17897 . . . . . 6  |-  ( ( D  e.  ( Met `  X )  /\  x  e.  X  /\  y  e.  X )  ->  (
x D y )  e.  RR )
62, 3, 4, 5syl3anc 1182 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  ( x D y )  e.  RR )
7 simplrr 737 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  S  e.  RR+ )
87rpred 10390 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  S  e.  RR )
9 metss2.3 . . . . . 6  |-  ( ph  ->  R  e.  RR+ )
109ad2antrr 706 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  R  e.  RR+ )
116, 8, 10ltmuldiv2d 10434 . . . 4  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  ( ( R  x.  ( x D y ) )  <  S  <->  ( x D y )  < 
( S  /  R
) ) )
12 metss2.4 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( x C y )  <_  ( R  x.  ( x D y ) ) )
1312anassrs 629 . . . . . 6  |-  ( ( ( ph  /\  x  e.  X )  /\  y  e.  X )  ->  (
x C y )  <_  ( R  x.  ( x D y ) ) )
1413adantlrr 701 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  ( x C y )  <_ 
( R  x.  (
x D y ) ) )
15 metss2.1 . . . . . . . 8  |-  ( ph  ->  C  e.  ( Met `  X ) )
1615ad2antrr 706 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  C  e.  ( Met `  X ) )
17 metcl 17897 . . . . . . 7  |-  ( ( C  e.  ( Met `  X )  /\  x  e.  X  /\  y  e.  X )  ->  (
x C y )  e.  RR )
1816, 3, 4, 17syl3anc 1182 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  ( x C y )  e.  RR )
1910rpred 10390 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  R  e.  RR )
2019, 6remulcld 8863 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  ( R  x.  ( x D y ) )  e.  RR )
21 lelttr 8912 . . . . . 6  |-  ( ( ( x C y )  e.  RR  /\  ( R  x.  (
x D y ) )  e.  RR  /\  S  e.  RR )  ->  ( ( ( x C y )  <_ 
( R  x.  (
x D y ) )  /\  ( R  x.  ( x D y ) )  < 
S )  ->  (
x C y )  <  S ) )
2218, 20, 8, 21syl3anc 1182 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  ( (
( x C y )  <_  ( R  x.  ( x D y ) )  /\  ( R  x.  ( x D y ) )  <  S )  -> 
( x C y )  <  S ) )
2314, 22mpand 656 . . . 4  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  ( ( R  x.  ( x D y ) )  <  S  ->  (
x C y )  <  S ) )
2411, 23sylbird 226 . . 3  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  ( (
x D y )  <  ( S  /  R )  ->  (
x C y )  <  S ) )
2524ss2rabdv 3254 . 2  |-  ( (
ph  /\  ( x  e.  X  /\  S  e.  RR+ ) )  ->  { y  e.  X  |  ( x D y )  <  ( S  /  R ) }  C_  { y  e.  X  | 
( x C y )  <  S }
)
26 metxmet 17899 . . . . 5  |-  ( D  e.  ( Met `  X
)  ->  D  e.  ( * Met `  X
) )
271, 26syl 15 . . . 4  |-  ( ph  ->  D  e.  ( * Met `  X ) )
2827adantr 451 . . 3  |-  ( (
ph  /\  ( x  e.  X  /\  S  e.  RR+ ) )  ->  D  e.  ( * Met `  X
) )
29 simprl 732 . . 3  |-  ( (
ph  /\  ( x  e.  X  /\  S  e.  RR+ ) )  ->  x  e.  X )
30 simpr 447 . . . . 5  |-  ( ( x  e.  X  /\  S  e.  RR+ )  ->  S  e.  RR+ )
31 rpdivcl 10376 . . . . 5  |-  ( ( S  e.  RR+  /\  R  e.  RR+ )  ->  ( S  /  R )  e.  RR+ )
3230, 9, 31syl2anr 464 . . . 4  |-  ( (
ph  /\  ( x  e.  X  /\  S  e.  RR+ ) )  ->  ( S  /  R )  e.  RR+ )
3332rpxrd 10391 . . 3  |-  ( (
ph  /\  ( x  e.  X  /\  S  e.  RR+ ) )  ->  ( S  /  R )  e. 
RR* )
34 blval 17948 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  x  e.  X  /\  ( S  /  R
)  e.  RR* )  ->  ( x ( ball `  D ) ( S  /  R ) )  =  { y  e.  X  |  ( x D y )  < 
( S  /  R
) } )
3528, 29, 33, 34syl3anc 1182 . 2  |-  ( (
ph  /\  ( x  e.  X  /\  S  e.  RR+ ) )  ->  (
x ( ball `  D
) ( S  /  R ) )  =  { y  e.  X  |  ( x D y )  <  ( S  /  R ) } )
36 metxmet 17899 . . . . 5  |-  ( C  e.  ( Met `  X
)  ->  C  e.  ( * Met `  X
) )
3715, 36syl 15 . . . 4  |-  ( ph  ->  C  e.  ( * Met `  X ) )
3837adantr 451 . . 3  |-  ( (
ph  /\  ( x  e.  X  /\  S  e.  RR+ ) )  ->  C  e.  ( * Met `  X
) )
39 rpxr 10361 . . . 4  |-  ( S  e.  RR+  ->  S  e. 
RR* )
4039ad2antll 709 . . 3  |-  ( (
ph  /\  ( x  e.  X  /\  S  e.  RR+ ) )  ->  S  e.  RR* )
41 blval 17948 . . 3  |-  ( ( C  e.  ( * Met `  X )  /\  x  e.  X  /\  S  e.  RR* )  ->  ( x ( ball `  C ) S )  =  { y  e.  X  |  ( x C y )  < 
S } )
4238, 29, 40, 41syl3anc 1182 . 2  |-  ( (
ph  /\  ( x  e.  X  /\  S  e.  RR+ ) )  ->  (
x ( ball `  C
) S )  =  { y  e.  X  |  ( x C y )  <  S } )
4325, 35, 423sstr4d 3221 1  |-  ( (
ph  /\  ( x  e.  X  /\  S  e.  RR+ ) )  ->  (
x ( ball `  D
) ( S  /  R ) )  C_  ( x ( ball `  C ) S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547    C_ wss 3152   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   RRcr 8736    x. cmul 8742   RR*cxr 8866    < clt 8867    <_ cle 8868    / cdiv 9423   RR+crp 10354   * Metcxmt 16369   Metcme 16370   ballcbl 16371   MetOpencmopn 16372
This theorem is referenced by:  metss2  18058  equivcfil  18725  equivcau  18726  equivtotbnd  26502
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-rp 10355  df-xadd 10453  df-xmet 16373  df-met 16374  df-bl 16375
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