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Theorem metdscnlem 20429
Description: Lemma for metdscn 20430. (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 20422 . . . . . 6  |-  ( ( D  e.  ( *Met `  X )  /\  S  C_  X
)  ->  F : X
--> ( 0 [,] +oo ) )
51, 2, 4syl2anc 661 . . . . 5  |-  ( ph  ->  F : X --> ( 0 [,] +oo ) )
6 metdscnlem.3 . . . . 5  |-  ( ph  ->  A  e.  X )
75, 6ffvelrnd 5842 . . . 4  |-  ( ph  ->  ( F `  A
)  e.  ( 0 [,] +oo ) )
8 elxrge0 11392 . . . . 5  |-  ( ( F `  A )  e.  ( 0 [,] +oo )  <->  ( ( F `
 A )  e. 
RR*  /\  0  <_  ( F `  A ) ) )
98simplbi 460 . . . 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 5842 . . . . 5  |-  ( ph  ->  ( F `  B
)  e.  ( 0 [,] +oo ) )
13 elxrge0 11392 . . . . . 6  |-  ( ( F `  B )  e.  ( 0 [,] +oo )  <->  ( ( F `
 B )  e. 
RR*  /\  0  <_  ( F `  B ) ) )
1413simplbi 460 . . . . 5  |-  ( ( F `  B )  e.  ( 0 [,] +oo )  ->  ( F `
 B )  e. 
RR* )
1512, 14syl 16 . . . 4  |-  ( ph  ->  ( F `  B
)  e.  RR* )
1615xnegcld 11261 . . 3  |-  ( ph  -> 
-e ( F `
 B )  e. 
RR* )
1710, 16xaddcld 11262 . 2  |-  ( ph  ->  ( ( F `  A ) +e  -e ( F `  B ) )  e. 
RR* )
18 xmetcl 19904 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( A D B )  e.  RR* )
191, 6, 11, 18syl3anc 1218 . 2  |-  ( ph  ->  ( A D B )  e.  RR* )
20 metdscnlem.5 . . 3  |-  ( ph  ->  R  e.  RR+ )
2120rpxrd 11026 . 2  |-  ( ph  ->  R  e.  RR* )
223metdstri 20425 . . . 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 1219 . . 3  |-  ( ph  ->  ( F `  A
)  <_  ( ( A D B ) +e ( F `  B ) ) )
248simprbi 464 . . . . 5  |-  ( ( F `  A )  e.  ( 0 [,] +oo )  ->  0  <_ 
( F `  A
) )
257, 24syl 16 . . . 4  |-  ( ph  ->  0  <_  ( F `  A ) )
2613simprbi 464 . . . . . 6  |-  ( ( F `  B )  e.  ( 0 [,] +oo )  ->  0  <_ 
( F `  B
) )
2712, 26syl 16 . . . . 5  |-  ( ph  ->  0  <_  ( F `  B ) )
28 ge0nemnf 11143 . . . . 5  |-  ( ( ( F `  B
)  e.  RR*  /\  0  <_  ( F `  B
) )  ->  ( F `  B )  =/= -oo )
2915, 27, 28syl2anc 661 . . . 4  |-  ( ph  ->  ( F `  B
)  =/= -oo )
30 xmetge0 19917 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  0  <_  ( A D B ) )
311, 6, 11, 30syl3anc 1218 . . . 4  |-  ( ph  ->  0  <_  ( A D B ) )
32 xlesubadd 11224 . . . 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 1233 . . 3  |-  ( ph  ->  ( ( ( F `
 A ) +e  -e ( F `  B ) )  <_  ( A D B )  <->  ( F `  A )  <_  (
( A D B ) +e ( F `  B ) ) ) )
3423, 33mpbird 232 . 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 11132 1  |-  ( ph  ->  ( ( F `  A ) +e  -e ( F `  B ) )  < 
R )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1369    e. wcel 1756    =/= wne 2604    C_ wss 3326   class class class wbr 4290    e. cmpt 4348   `'ccnv 4837   ran crn 4839   -->wf 5412   ` cfv 5416  (class class class)co 6089   supcsup 7688   0cc0 9280   +oocpnf 9413   -oocmnf 9414   RR*cxr 9415    < clt 9416    <_ cle 9417   RR+crp 10989    -ecxne 11084   +ecxad 11085   [,]cicc 11301   distcds 14245   RR*scxrs 14436   *Metcxmt 17799   MetOpencmopn 17804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357  ax-pre-sup 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-po 4639  df-so 4640  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-1st 6575  df-2nd 6576  df-er 7099  df-ec 7101  df-map 7214  df-en 7309  df-dom 7310  df-sdom 7311  df-sup 7689  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-div 9992  df-2 10378  df-rp 10990  df-xneg 11087  df-xadd 11088  df-xmul 11089  df-icc 11305  df-psmet 17807  df-xmet 17808  df-bl 17810
This theorem is referenced by:  metdscn  20430
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