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Theorem metdscnlem 18916
Description: Lemma for metdscn 18917. (Contributed by Mario Carneiro, 4-Sep-2015.)
Hypotheses
Ref Expression
metdscn.f  |-  F  =  ( x  e.  X  |->  sup ( ran  (
y  e.  S  |->  ( x D y ) ) ,  RR* ,  `'  <  ) )
metdscn.j  |-  J  =  ( MetOpen `  D )
metdscn.c  |-  C  =  ( dist `  RR* s
)
metdscn.k  |-  K  =  ( MetOpen `  C )
metdscnlem.1  |-  ( ph  ->  D  e.  ( * Met `  X ) )
metdscnlem.2  |-  ( ph  ->  S  C_  X )
metdscnlem.3  |-  ( ph  ->  A  e.  X )
metdscnlem.4  |-  ( ph  ->  B  e.  X )
metdscnlem.5  |-  ( ph  ->  R  e.  RR+ )
metdscnlem.6  |-  ( ph  ->  ( A D B )  <  R )
Assertion
Ref Expression
metdscnlem  |-  ( ph  ->  ( ( F `  A ) + e  - e ( F `  B ) )  < 
R )
Distinct variable groups:    x, y, A    x, D, y    y, J    x, B, y    x, S, y    x, X, y
Allowed substitution hints:    ph( x, y)    C( x, y)    R( x, y)    F( x, y)    J( x)    K( x, y)

Proof of Theorem metdscnlem
StepHypRef Expression
1 metdscnlem.1 . . . . . 6  |-  ( ph  ->  D  e.  ( * Met `  X ) )
2 metdscnlem.2 . . . . . 6  |-  ( ph  ->  S  C_  X )
3 metdscn.f . . . . . . 7  |-  F  =  ( x  e.  X  |->  sup ( ran  (
y  e.  S  |->  ( x D y ) ) ,  RR* ,  `'  <  ) )
43metdsf 18909 . . . . . 6  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X
)  ->  F : X
--> ( 0 [,]  +oo ) )
51, 2, 4syl2anc 644 . . . . 5  |-  ( ph  ->  F : X --> ( 0 [,]  +oo ) )
6 metdscnlem.3 . . . . 5  |-  ( ph  ->  A  e.  X )
75, 6ffvelrnd 5900 . . . 4  |-  ( ph  ->  ( F `  A
)  e.  ( 0 [,]  +oo ) )
8 elxrge0 11039 . . . . 5  |-  ( ( F `  A )  e.  ( 0 [,] 
+oo )  <->  ( ( F `  A )  e.  RR*  /\  0  <_ 
( F `  A
) ) )
98simplbi 448 . . . 4  |-  ( ( F `  A )  e.  ( 0 [,] 
+oo )  ->  ( F `  A )  e.  RR* )
107, 9syl 16 . . 3  |-  ( ph  ->  ( F `  A
)  e.  RR* )
11 metdscnlem.4 . . . . . 6  |-  ( ph  ->  B  e.  X )
125, 11ffvelrnd 5900 . . . . 5  |-  ( ph  ->  ( F `  B
)  e.  ( 0 [,]  +oo ) )
13 elxrge0 11039 . . . . . 6  |-  ( ( F `  B )  e.  ( 0 [,] 
+oo )  <->  ( ( F `  B )  e.  RR*  /\  0  <_ 
( F `  B
) ) )
1413simplbi 448 . . . . 5  |-  ( ( F `  B )  e.  ( 0 [,] 
+oo )  ->  ( F `  B )  e.  RR* )
1512, 14syl 16 . . . 4  |-  ( ph  ->  ( F `  B
)  e.  RR* )
1615xnegcld 10910 . . 3  |-  ( ph  -> 
- e ( F `
 B )  e. 
RR* )
1710, 16xaddcld 10911 . 2  |-  ( ph  ->  ( ( F `  A ) + e  - e ( F `  B ) )  e. 
RR* )
18 xmetcl 18392 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( A D B )  e.  RR* )
191, 6, 11, 18syl3anc 1185 . 2  |-  ( ph  ->  ( A D B )  e.  RR* )
20 metdscnlem.5 . . 3  |-  ( ph  ->  R  e.  RR+ )
2120rpxrd 10680 . 2  |-  ( ph  ->  R  e.  RR* )
223metdstri 18912 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X
)  /\  ( A  e.  X  /\  B  e.  X ) )  -> 
( F `  A
)  <_  ( ( A D B ) + e ( F `  B ) ) )
231, 2, 6, 11, 22syl22anc 1186 . . 3  |-  ( ph  ->  ( F `  A
)  <_  ( ( A D B ) + e ( F `  B ) ) )
248simprbi 452 . . . . 5  |-  ( ( F `  A )  e.  ( 0 [,] 
+oo )  ->  0  <_  ( F `  A
) )
257, 24syl 16 . . . 4  |-  ( ph  ->  0  <_  ( F `  A ) )
2613simprbi 452 . . . . . 6  |-  ( ( F `  B )  e.  ( 0 [,] 
+oo )  ->  0  <_  ( F `  B
) )
2712, 26syl 16 . . . . 5  |-  ( ph  ->  0  <_  ( F `  B ) )
28 ge0nemnf 10792 . . . . 5  |-  ( ( ( F `  B
)  e.  RR*  /\  0  <_  ( F `  B
) )  ->  ( F `  B )  =/=  -oo )
2915, 27, 28syl2anc 644 . . . 4  |-  ( ph  ->  ( F `  B
)  =/=  -oo )
30 xmetge0 18405 . . . . 5  |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  0  <_  ( A D B ) )
311, 6, 11, 30syl3anc 1185 . . . 4  |-  ( ph  ->  0  <_  ( A D B ) )
32 xlesubadd 10873 . . . 4  |-  ( ( ( ( F `  A )  e.  RR*  /\  ( F `  B
)  e.  RR*  /\  ( A D B )  e. 
RR* )  /\  (
0  <_  ( F `  A )  /\  ( F `  B )  =/=  -oo  /\  0  <_ 
( A D B ) ) )  -> 
( ( ( F `
 A ) + e  - e ( F `  B ) )  <_  ( A D B )  <->  ( F `  A )  <_  (
( A D B ) + e ( F `  B ) ) ) )
3310, 15, 19, 25, 29, 31, 32syl33anc 1200 . . 3  |-  ( ph  ->  ( ( ( F `
 A ) + e  - e ( F `  B ) )  <_  ( A D B )  <->  ( F `  A )  <_  (
( A D B ) + e ( F `  B ) ) ) )
3423, 33mpbird 225 . 2  |-  ( ph  ->  ( ( F `  A ) + e  - e ( F `  B ) )  <_ 
( A D B ) )
35 metdscnlem.6 . 2  |-  ( ph  ->  ( A D B )  <  R )
3617, 19, 21, 34, 35xrlelttrd 10781 1  |-  ( ph  ->  ( ( F `  A ) + e  - e ( F `  B ) )  < 
R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    = wceq 1653    e. wcel 1727    =/= wne 2605    C_ wss 3306   class class class wbr 4237    e. cmpt 4291   `'ccnv 4906   ran crn 4908   -->wf 5479   ` cfv 5483  (class class class)co 6110   supcsup 7474   0cc0 9021    +oocpnf 9148    -oocmnf 9149   RR*cxr 9150    < clt 9151    <_ cle 9152   RR+crp 10643    - ecxne 10738   + ecxad 10739   [,]cicc 10950   distcds 13569   RR* scxrs 13753   * Metcxmt 16717   MetOpencmopn 16722
This theorem is referenced by:  metdscn  18917
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 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-cnex 9077  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-pre-mulgt0 9098  ax-pre-sup 9099
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 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-po 4532  df-so 4533  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-riota 6578  df-er 6934  df-ec 6936  df-map 7049  df-en 7139  df-dom 7140  df-sdom 7141  df-sup 7475  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-sub 9324  df-neg 9325  df-div 9709  df-2 10089  df-rp 10644  df-xneg 10741  df-xadd 10742  df-xmul 10743  df-icc 10954  df-psmet 16725  df-xmet 16726  df-bl 16728
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