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Theorem metss2lem 12680
Description: Lemma for metss2 12681. (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 479 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  D  e.  ( Met `  X ) )
3 simplrl 524 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  x  e.  X )
4 simpr 109 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  y  e.  X )
5 metcl 12536 . . . . . 6  |-  ( ( D  e.  ( Met `  X )  /\  x  e.  X  /\  y  e.  X )  ->  (
x D y )  e.  RR )
62, 3, 4, 5syl3anc 1216 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  ( x D y )  e.  RR )
7 simplrr 525 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  S  e.  RR+ )
87rpred 9495 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  S  e.  RR )
9 metss2.3 . . . . . 6  |-  ( ph  ->  R  e.  RR+ )
109ad2antrr 479 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  R  e.  RR+ )
116, 8, 10ltmuldiv2d 9544 . . . 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 397 . . . . . 6  |-  ( ( ( ph  /\  x  e.  X )  /\  y  e.  X )  ->  (
x C y )  <_  ( R  x.  ( x D y ) ) )
1413adantlrr 474 . . . . 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 479 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  C  e.  ( Met `  X ) )
17 metcl 12536 . . . . . . 7  |-  ( ( C  e.  ( Met `  X )  /\  x  e.  X  /\  y  e.  X )  ->  (
x C y )  e.  RR )
1816, 3, 4, 17syl3anc 1216 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  ( x C y )  e.  RR )
1910rpred 9495 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  R  e.  RR )
2019, 6remulcld 7808 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  ( R  x.  ( x D y ) )  e.  RR )
21 lelttr 7864 . . . . . 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 1216 . . . . 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 425 . . . 4  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  ( ( R  x.  ( x D y ) )  <  S  ->  (
x C y )  <  S ) )
2411, 23sylbird 169 . . 3  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  ( (
x D y )  <  ( S  /  R )  ->  (
x C y )  <  S ) )
2524ss2rabdv 3178 . 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 12538 . . . . 5  |-  ( D  e.  ( Met `  X
)  ->  D  e.  ( *Met `  X
) )
271, 26syl 14 . . . 4  |-  ( ph  ->  D  e.  ( *Met `  X ) )
2827adantr 274 . . 3  |-  ( (
ph  /\  ( x  e.  X  /\  S  e.  RR+ ) )  ->  D  e.  ( *Met `  X ) )
29 simprl 520 . . 3  |-  ( (
ph  /\  ( x  e.  X  /\  S  e.  RR+ ) )  ->  x  e.  X )
30 simpr 109 . . . . 5  |-  ( ( x  e.  X  /\  S  e.  RR+ )  ->  S  e.  RR+ )
31 rpdivcl 9479 . . . . 5  |-  ( ( S  e.  RR+  /\  R  e.  RR+ )  ->  ( S  /  R )  e.  RR+ )
3230, 9, 31syl2anr 288 . . . 4  |-  ( (
ph  /\  ( x  e.  X  /\  S  e.  RR+ ) )  ->  ( S  /  R )  e.  RR+ )
3332rpxrd 9496 . . 3  |-  ( (
ph  /\  ( x  e.  X  /\  S  e.  RR+ ) )  ->  ( S  /  R )  e. 
RR* )
34 blval 12572 . . 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 1216 . 2  |-  ( (
ph  /\  ( x  e.  X  /\  S  e.  RR+ ) )  ->  (
x ( ball `  D
) ( S  /  R ) )  =  { y  e.  X  |  ( x D y )  <  ( S  /  R ) } )
36 metxmet 12538 . . . . 5  |-  ( C  e.  ( Met `  X
)  ->  C  e.  ( *Met `  X
) )
3715, 36syl 14 . . . 4  |-  ( ph  ->  C  e.  ( *Met `  X ) )
3837adantr 274 . . 3  |-  ( (
ph  /\  ( x  e.  X  /\  S  e.  RR+ ) )  ->  C  e.  ( *Met `  X ) )
39 rpxr 9461 . . . 4  |-  ( S  e.  RR+  ->  S  e. 
RR* )
4039ad2antll 482 . . 3  |-  ( (
ph  /\  ( x  e.  X  /\  S  e.  RR+ ) )  ->  S  e.  RR* )
41 blval 12572 . . 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 1216 . 2  |-  ( (
ph  /\  ( x  e.  X  /\  S  e.  RR+ ) )  ->  (
x ( ball `  C
) S )  =  { y  e.  X  |  ( x C y )  <  S } )
4325, 35, 423sstr4d 3142 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 103    = wceq 1331    e. wcel 1480   {crab 2420    C_ wss 3071   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   RRcr 7631    x. cmul 7637   RR*cxr 7811    < clt 7812    <_ cle 7813    / cdiv 8444   RR+crp 9453   *Metcxmet 12163   Metcmet 12164   ballcbl 12165   MetOpencmopn 12168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-mulrcl 7731  ax-addcom 7732  ax-mulcom 7733  ax-addass 7734  ax-mulass 7735  ax-distr 7736  ax-i2m1 7737  ax-0lt1 7738  ax-1rid 7739  ax-0id 7740  ax-rnegex 7741  ax-precex 7742  ax-cnre 7743  ax-pre-ltirr 7744  ax-pre-ltwlin 7745  ax-pre-lttrn 7746  ax-pre-apti 7747  ax-pre-ltadd 7748  ax-pre-mulgt0 7749  ax-pre-mulext 7750
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-po 4218  df-iso 4219  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-map 6544  df-pnf 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-le 7818  df-sub 7947  df-neg 7948  df-reap 8349  df-ap 8356  df-div 8445  df-rp 9454  df-xadd 9572  df-psmet 12170  df-xmet 12171  df-met 12172  df-bl 12173
This theorem is referenced by:  metss2  12681
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