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Theorem metdscnlem 18846
Description: Lemma for metdscn 18847. (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 18839 . . . . . 6  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X
)  ->  F : X
--> ( 0 [,]  +oo ) )
51, 2, 4syl2anc 643 . . . . 5  |-  ( ph  ->  F : X --> ( 0 [,]  +oo ) )
6 metdscnlem.3 . . . . 5  |-  ( ph  ->  A  e.  X )
75, 6ffvelrnd 5838 . . . 4  |-  ( ph  ->  ( F `  A
)  e.  ( 0 [,]  +oo ) )
8 elxrge0 10972 . . . . 5  |-  ( ( F `  A )  e.  ( 0 [,] 
+oo )  <->  ( ( F `  A )  e.  RR*  /\  0  <_ 
( F `  A
) ) )
98simplbi 447 . . . 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 5838 . . . . 5  |-  ( ph  ->  ( F `  B
)  e.  ( 0 [,]  +oo ) )
13 elxrge0 10972 . . . . . 6  |-  ( ( F `  B )  e.  ( 0 [,] 
+oo )  <->  ( ( F `  B )  e.  RR*  /\  0  <_ 
( F `  B
) ) )
1413simplbi 447 . . . . 5  |-  ( ( F `  B )  e.  ( 0 [,] 
+oo )  ->  ( F `  B )  e.  RR* )
1512, 14syl 16 . . . 4  |-  ( ph  ->  ( F `  B
)  e.  RR* )
1615xnegcld 10843 . . 3  |-  ( ph  -> 
- e ( F `
 B )  e. 
RR* )
1710, 16xaddcld 10844 . 2  |-  ( ph  ->  ( ( F `  A ) + e  - e ( F `  B ) )  e. 
RR* )
18 xmetcl 18322 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( A D B )  e.  RR* )
191, 6, 11, 18syl3anc 1184 . 2  |-  ( ph  ->  ( A D B )  e.  RR* )
20 metdscnlem.5 . . 3  |-  ( ph  ->  R  e.  RR+ )
2120rpxrd 10613 . 2  |-  ( ph  ->  R  e.  RR* )
223metdstri 18842 . . . 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 1185 . . 3  |-  ( ph  ->  ( F `  A
)  <_  ( ( A D B ) + e ( F `  B ) ) )
248simprbi 451 . . . . 5  |-  ( ( F `  A )  e.  ( 0 [,] 
+oo )  ->  0  <_  ( F `  A
) )
257, 24syl 16 . . . 4  |-  ( ph  ->  0  <_  ( F `  A ) )
2613simprbi 451 . . . . . 6  |-  ( ( F `  B )  e.  ( 0 [,] 
+oo )  ->  0  <_  ( F `  B
) )
2712, 26syl 16 . . . . 5  |-  ( ph  ->  0  <_  ( F `  B ) )
28 ge0nemnf 10725 . . . . 5  |-  ( ( ( F `  B
)  e.  RR*  /\  0  <_  ( F `  B
) )  ->  ( F `  B )  =/=  -oo )
2915, 27, 28syl2anc 643 . . . 4  |-  ( ph  ->  ( F `  B
)  =/=  -oo )
30 xmetge0 18335 . . . . 5  |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  0  <_  ( A D B ) )
311, 6, 11, 30syl3anc 1184 . . . 4  |-  ( ph  ->  0  <_  ( A D B ) )
32 xlesubadd 10806 . . . 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 1199 . . 3  |-  ( ph  ->  ( ( ( F `
 A ) + e  - e ( F `  B ) )  <_  ( A D B )  <->  ( F `  A )  <_  (
( A D B ) + e ( F `  B ) ) ) )
3423, 33mpbird 224 . 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 10714 1  |-  ( ph  ->  ( ( F `  A ) + e  - e ( F `  B ) )  < 
R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1721    =/= wne 2575    C_ wss 3288   class class class wbr 4180    e. cmpt 4234   `'ccnv 4844   ran crn 4846   -->wf 5417   ` cfv 5421  (class class class)co 6048   supcsup 7411   0cc0 8954    +oocpnf 9081    -oocmnf 9082   RR*cxr 9083    < clt 9084    <_ cle 9085   RR+crp 10576    - ecxne 10671   + ecxad 10672   [,]cicc 10883   distcds 13501   RR* scxrs 13685   * Metcxmt 16649   MetOpencmopn 16654
This theorem is referenced by:  metdscn  18847
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  ax-pre-sup 9032
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-ec 6874  df-map 6987  df-en 7077  df-dom 7078  df-sdom 7079  df-sup 7412  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-2 10022  df-rp 10577  df-xneg 10674  df-xadd 10675  df-xmul 10676  df-icc 10887  df-psmet 16657  df-xmet 16658  df-bl 16660
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