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Theorem metss2lem 18546
Description: Lemma for metss2 18547. (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 708 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  D  e.  ( Met `  X ) )
3 simplrl 738 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  x  e.  X )
4 simpr 449 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  y  e.  X )
5 metcl 18367 . . . . . 6  |-  ( ( D  e.  ( Met `  X )  /\  x  e.  X  /\  y  e.  X )  ->  (
x D y )  e.  RR )
62, 3, 4, 5syl3anc 1185 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  ( x D y )  e.  RR )
7 simplrr 739 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  S  e.  RR+ )
87rpred 10653 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  S  e.  RR )
9 metss2.3 . . . . . 6  |-  ( ph  ->  R  e.  RR+ )
109ad2antrr 708 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  R  e.  RR+ )
116, 8, 10ltmuldiv2d 10697 . . . 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 631 . . . . . 6  |-  ( ( ( ph  /\  x  e.  X )  /\  y  e.  X )  ->  (
x C y )  <_  ( R  x.  ( x D y ) ) )
1413adantlrr 703 . . . . 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 708 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  C  e.  ( Met `  X ) )
17 metcl 18367 . . . . . . 7  |-  ( ( C  e.  ( Met `  X )  /\  x  e.  X  /\  y  e.  X )  ->  (
x C y )  e.  RR )
1816, 3, 4, 17syl3anc 1185 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  ( x C y )  e.  RR )
1910rpred 10653 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  R  e.  RR )
2019, 6remulcld 9121 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  ( R  x.  ( x D y ) )  e.  RR )
21 lelttr 9170 . . . . . 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 1185 . . . . 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 658 . . . 4  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  ( ( R  x.  ( x D y ) )  <  S  ->  (
x C y )  <  S ) )
2411, 23sylbird 228 . . 3  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  ( (
x D y )  <  ( S  /  R )  ->  (
x C y )  <  S ) )
2524ss2rabdv 3426 . 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 18369 . . . . 5  |-  ( D  e.  ( Met `  X
)  ->  D  e.  ( * Met `  X
) )
271, 26syl 16 . . . 4  |-  ( ph  ->  D  e.  ( * Met `  X ) )
2827adantr 453 . . 3  |-  ( (
ph  /\  ( x  e.  X  /\  S  e.  RR+ ) )  ->  D  e.  ( * Met `  X
) )
29 simprl 734 . . 3  |-  ( (
ph  /\  ( x  e.  X  /\  S  e.  RR+ ) )  ->  x  e.  X )
30 simpr 449 . . . . 5  |-  ( ( x  e.  X  /\  S  e.  RR+ )  ->  S  e.  RR+ )
31 rpdivcl 10639 . . . . 5  |-  ( ( S  e.  RR+  /\  R  e.  RR+ )  ->  ( S  /  R )  e.  RR+ )
3230, 9, 31syl2anr 466 . . . 4  |-  ( (
ph  /\  ( x  e.  X  /\  S  e.  RR+ ) )  ->  ( S  /  R )  e.  RR+ )
3332rpxrd 10654 . . 3  |-  ( (
ph  /\  ( x  e.  X  /\  S  e.  RR+ ) )  ->  ( S  /  R )  e. 
RR* )
34 blval 18421 . . 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 1185 . 2  |-  ( (
ph  /\  ( x  e.  X  /\  S  e.  RR+ ) )  ->  (
x ( ball `  D
) ( S  /  R ) )  =  { y  e.  X  |  ( x D y )  <  ( S  /  R ) } )
36 metxmet 18369 . . . . 5  |-  ( C  e.  ( Met `  X
)  ->  C  e.  ( * Met `  X
) )
3715, 36syl 16 . . . 4  |-  ( ph  ->  C  e.  ( * Met `  X ) )
3837adantr 453 . . 3  |-  ( (
ph  /\  ( x  e.  X  /\  S  e.  RR+ ) )  ->  C  e.  ( * Met `  X
) )
39 rpxr 10624 . . . 4  |-  ( S  e.  RR+  ->  S  e. 
RR* )
4039ad2antll 711 . . 3  |-  ( (
ph  /\  ( x  e.  X  /\  S  e.  RR+ ) )  ->  S  e.  RR* )
41 blval 18421 . . 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 1185 . 2  |-  ( (
ph  /\  ( x  e.  X  /\  S  e.  RR+ ) )  ->  (
x ( ball `  C
) S )  =  { y  e.  X  |  ( x C y )  <  S } )
4325, 35, 423sstr4d 3393 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 360    = wceq 1653    e. wcel 1726   {crab 2711    C_ wss 3322   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   RRcr 8994    x. cmul 9000   RR*cxr 9124    < clt 9125    <_ cle 9126    / cdiv 9682   RR+crp 10617   * Metcxmt 16691   Metcme 16692   ballcbl 16693   MetOpencmopn 16696
This theorem is referenced by:  metss2  18547  equivcfil  19257  equivcau  19258  equivtotbnd  26501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-po 4506  df-so 4507  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-er 6908  df-map 7023  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-rp 10618  df-xadd 10716  df-psmet 16699  df-xmet 16700  df-met 16701  df-bl 16702
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