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Theorem metss2lem 18159
Description: Lemma for metss2 18160. (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 17999 . . . . . 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 10482 . . . . 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 10526 . . . 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 17999 . . . . . . 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 10482 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  R  e.  RR )
2019, 6remulcld 8953 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  ( R  x.  ( x D y ) )  e.  RR )
21 lelttr 9002 . . . . . 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 3330 . 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 18001 . . . . 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 10468 . . . . 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 10483 . . 3  |-  ( (
ph  /\  ( x  e.  X  /\  S  e.  RR+ ) )  ->  ( S  /  R )  e. 
RR* )
34 blval 18050 . . 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 18001 . . . . 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 10453 . . . 4  |-  ( S  e.  RR+  ->  S  e. 
RR* )
4039ad2antll 709 . . 3  |-  ( (
ph  /\  ( x  e.  X  /\  S  e.  RR+ ) )  ->  S  e.  RR* )
41 blval 18050 . . 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 3297 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 1642    e. wcel 1710   {crab 2623    C_ wss 3228   class class class wbr 4104   ` cfv 5337  (class class class)co 5945   RRcr 8826    x. cmul 8832   RR*cxr 8956    < clt 8957    <_ cle 8958    / cdiv 9513   RR+crp 10446   * Metcxmt 16468   Metcme 16469   ballcbl 16470   MetOpencmopn 16473
This theorem is referenced by:  metss2  18160  equivcfil  18829  equivcau  18830  equivtotbnd  25825
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-po 4396  df-so 4397  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-riota 6391  df-er 6747  df-map 6862  df-en 6952  df-dom 6953  df-sdom 6954  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514  df-rp 10447  df-xadd 10545  df-xmet 16475  df-met 16476  df-bl 16477
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