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Theorem metdscnlem 18573
Description: Lemma for metdscn 18574. (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 18566 . . . . . 6  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X
)  ->  F : X
--> ( 0 [,]  +oo ) )
51, 2, 4syl2anc 642 . . . . 5  |-  ( ph  ->  F : X --> ( 0 [,]  +oo ) )
6 metdscnlem.3 . . . . 5  |-  ( ph  ->  A  e.  X )
7 ffvelrn 5770 . . . . 5  |-  ( ( F : X --> ( 0 [,]  +oo )  /\  A  e.  X )  ->  ( F `  A )  e.  ( 0 [,]  +oo ) )
85, 6, 7syl2anc 642 . . . 4  |-  ( ph  ->  ( F `  A
)  e.  ( 0 [,]  +oo ) )
9 elxrge0 10900 . . . . 5  |-  ( ( F `  A )  e.  ( 0 [,] 
+oo )  <->  ( ( F `  A )  e.  RR*  /\  0  <_ 
( F `  A
) ) )
109simplbi 446 . . . 4  |-  ( ( F `  A )  e.  ( 0 [,] 
+oo )  ->  ( F `  A )  e.  RR* )
118, 10syl 15 . . 3  |-  ( ph  ->  ( F `  A
)  e.  RR* )
12 metdscnlem.4 . . . . . 6  |-  ( ph  ->  B  e.  X )
13 ffvelrn 5770 . . . . . 6  |-  ( ( F : X --> ( 0 [,]  +oo )  /\  B  e.  X )  ->  ( F `  B )  e.  ( 0 [,]  +oo ) )
145, 12, 13syl2anc 642 . . . . 5  |-  ( ph  ->  ( F `  B
)  e.  ( 0 [,]  +oo ) )
15 elxrge0 10900 . . . . . 6  |-  ( ( F `  B )  e.  ( 0 [,] 
+oo )  <->  ( ( F `  B )  e.  RR*  /\  0  <_ 
( F `  B
) ) )
1615simplbi 446 . . . . 5  |-  ( ( F `  B )  e.  ( 0 [,] 
+oo )  ->  ( F `  B )  e.  RR* )
1714, 16syl 15 . . . 4  |-  ( ph  ->  ( F `  B
)  e.  RR* )
1817xnegcld 10772 . . 3  |-  ( ph  -> 
- e ( F `
 B )  e. 
RR* )
1911, 18xaddcld 10773 . 2  |-  ( ph  ->  ( ( F `  A ) + e  - e ( F `  B ) )  e. 
RR* )
20 xmetcl 18109 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( A D B )  e.  RR* )
211, 6, 12, 20syl3anc 1183 . 2  |-  ( ph  ->  ( A D B )  e.  RR* )
22 metdscnlem.5 . . 3  |-  ( ph  ->  R  e.  RR+ )
2322rpxrd 10542 . 2  |-  ( ph  ->  R  e.  RR* )
243metdstri 18569 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X
)  /\  ( A  e.  X  /\  B  e.  X ) )  -> 
( F `  A
)  <_  ( ( A D B ) + e ( F `  B ) ) )
251, 2, 6, 12, 24syl22anc 1184 . . 3  |-  ( ph  ->  ( F `  A
)  <_  ( ( A D B ) + e ( F `  B ) ) )
269simprbi 450 . . . . 5  |-  ( ( F `  A )  e.  ( 0 [,] 
+oo )  ->  0  <_  ( F `  A
) )
278, 26syl 15 . . . 4  |-  ( ph  ->  0  <_  ( F `  A ) )
2815simprbi 450 . . . . . 6  |-  ( ( F `  B )  e.  ( 0 [,] 
+oo )  ->  0  <_  ( F `  B
) )
2914, 28syl 15 . . . . 5  |-  ( ph  ->  0  <_  ( F `  B ) )
30 ge0nemnf 10654 . . . . 5  |-  ( ( ( F `  B
)  e.  RR*  /\  0  <_  ( F `  B
) )  ->  ( F `  B )  =/=  -oo )
3117, 29, 30syl2anc 642 . . . 4  |-  ( ph  ->  ( F `  B
)  =/=  -oo )
32 xmetge0 18122 . . . . 5  |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  0  <_  ( A D B ) )
331, 6, 12, 32syl3anc 1183 . . . 4  |-  ( ph  ->  0  <_  ( A D B ) )
34 xlesubadd 10735 . . . 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 ) ) ) )
3511, 17, 21, 27, 31, 33, 34syl33anc 1198 . . 3  |-  ( ph  ->  ( ( ( F `
 A ) + e  - e ( F `  B ) )  <_  ( A D B )  <->  ( F `  A )  <_  (
( A D B ) + e ( F `  B ) ) ) )
3625, 35mpbird 223 . 2  |-  ( ph  ->  ( ( F `  A ) + e  - e ( F `  B ) )  <_ 
( A D B ) )
37 metdscnlem.6 . 2  |-  ( ph  ->  ( A D B )  <  R )
3819, 21, 23, 36, 37xrlelttrd 10643 1  |-  ( ph  ->  ( ( F `  A ) + e  - e ( F `  B ) )  < 
R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1647    e. wcel 1715    =/= wne 2529    C_ wss 3238   class class class wbr 4125    e. cmpt 4179   `'ccnv 4791   ran crn 4793   -->wf 5354   ` cfv 5358  (class class class)co 5981   supcsup 7340   0cc0 8884    +oocpnf 9011    -oocmnf 9012   RR*cxr 9013    < clt 9014    <_ cle 9015   RR+crp 10505    - ecxne 10600   + ecxad 10601   [,]cicc 10812   distcds 13425   RR* scxrs 13609   * Metcxmt 16579   MetOpencmopn 16584
This theorem is referenced by:  metdscn  18574
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961  ax-pre-sup 8962
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-po 4417  df-so 4418  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-riota 6446  df-er 6802  df-ec 6804  df-map 6917  df-en 7007  df-dom 7008  df-sdom 7009  df-sup 7341  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-div 9571  df-2 9951  df-rp 10506  df-xneg 10603  df-xadd 10604  df-xmul 10605  df-icc 10816  df-xmet 16586  df-bl 16588
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