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Theorem metss2lem 18502
Description: Lemma for metss2 18503. (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 707 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  D  e.  ( Met `  X ) )
3 simplrl 737 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  x  e.  X )
4 simpr 448 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  y  e.  X )
5 metcl 18323 . . . . . 6  |-  ( ( D  e.  ( Met `  X )  /\  x  e.  X  /\  y  e.  X )  ->  (
x D y )  e.  RR )
62, 3, 4, 5syl3anc 1184 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  ( x D y )  e.  RR )
7 simplrr 738 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  S  e.  RR+ )
87rpred 10612 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  S  e.  RR )
9 metss2.3 . . . . . 6  |-  ( ph  ->  R  e.  RR+ )
109ad2antrr 707 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  R  e.  RR+ )
116, 8, 10ltmuldiv2d 10656 . . . 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 630 . . . . . 6  |-  ( ( ( ph  /\  x  e.  X )  /\  y  e.  X )  ->  (
x C y )  <_  ( R  x.  ( x D y ) ) )
1413adantlrr 702 . . . . 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 707 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  C  e.  ( Met `  X ) )
17 metcl 18323 . . . . . . 7  |-  ( ( C  e.  ( Met `  X )  /\  x  e.  X  /\  y  e.  X )  ->  (
x C y )  e.  RR )
1816, 3, 4, 17syl3anc 1184 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  ( x C y )  e.  RR )
1910rpred 10612 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  R  e.  RR )
2019, 6remulcld 9080 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  ( R  x.  ( x D y ) )  e.  RR )
21 lelttr 9129 . . . . . 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 1184 . . . . 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 657 . . . 4  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  ( ( R  x.  ( x D y ) )  <  S  ->  (
x C y )  <  S ) )
2411, 23sylbird 227 . . 3  |-  ( ( ( ph  /\  (
x  e.  X  /\  S  e.  RR+ ) )  /\  y  e.  X
)  ->  ( (
x D y )  <  ( S  /  R )  ->  (
x C y )  <  S ) )
2524ss2rabdv 3392 . 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 18325 . . . . 5  |-  ( D  e.  ( Met `  X
)  ->  D  e.  ( * Met `  X
) )
271, 26syl 16 . . . 4  |-  ( ph  ->  D  e.  ( * Met `  X ) )
2827adantr 452 . . 3  |-  ( (
ph  /\  ( x  e.  X  /\  S  e.  RR+ ) )  ->  D  e.  ( * Met `  X
) )
29 simprl 733 . . 3  |-  ( (
ph  /\  ( x  e.  X  /\  S  e.  RR+ ) )  ->  x  e.  X )
30 simpr 448 . . . . 5  |-  ( ( x  e.  X  /\  S  e.  RR+ )  ->  S  e.  RR+ )
31 rpdivcl 10598 . . . . 5  |-  ( ( S  e.  RR+  /\  R  e.  RR+ )  ->  ( S  /  R )  e.  RR+ )
3230, 9, 31syl2anr 465 . . . 4  |-  ( (
ph  /\  ( x  e.  X  /\  S  e.  RR+ ) )  ->  ( S  /  R )  e.  RR+ )
3332rpxrd 10613 . . 3  |-  ( (
ph  /\  ( x  e.  X  /\  S  e.  RR+ ) )  ->  ( S  /  R )  e. 
RR* )
34 blval 18377 . . 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 1184 . 2  |-  ( (
ph  /\  ( x  e.  X  /\  S  e.  RR+ ) )  ->  (
x ( ball `  D
) ( S  /  R ) )  =  { y  e.  X  |  ( x D y )  <  ( S  /  R ) } )
36 metxmet 18325 . . . . 5  |-  ( C  e.  ( Met `  X
)  ->  C  e.  ( * Met `  X
) )
3715, 36syl 16 . . . 4  |-  ( ph  ->  C  e.  ( * Met `  X ) )
3837adantr 452 . . 3  |-  ( (
ph  /\  ( x  e.  X  /\  S  e.  RR+ ) )  ->  C  e.  ( * Met `  X
) )
39 rpxr 10583 . . . 4  |-  ( S  e.  RR+  ->  S  e. 
RR* )
4039ad2antll 710 . . 3  |-  ( (
ph  /\  ( x  e.  X  /\  S  e.  RR+ ) )  ->  S  e.  RR* )
41 blval 18377 . . 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 1184 . 2  |-  ( (
ph  /\  ( x  e.  X  /\  S  e.  RR+ ) )  ->  (
x ( ball `  C
) S )  =  { y  e.  X  |  ( x C y )  <  S } )
4325, 35, 423sstr4d 3359 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 359    = wceq 1649    e. wcel 1721   {crab 2678    C_ wss 3288   class class class wbr 4180   ` cfv 5421  (class class class)co 6048   RRcr 8953    x. cmul 8959   RR*cxr 9083    < clt 9084    <_ cle 9085    / cdiv 9641   RR+crp 10576   * Metcxmt 16649   Metcme 16650   ballcbl 16651   MetOpencmopn 16654
This theorem is referenced by:  metss2  18503  equivcfil  19213  equivcau  19214  equivtotbnd  26385
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-po 4471  df-so 4472  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-er 6872  df-map 6987  df-en 7077  df-dom 7078  df-sdom 7079  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-rp 10577  df-xadd 10675  df-psmet 16657  df-xmet 16658  df-met 16659  df-bl 16660
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