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Theorem metdscnlem 21858
Description: Lemma for metdscn 21859. (Contributed by Mario Carneiro, 4-Sep-2015.)
Hypotheses
Ref Expression
metdscn.f  |-  F  =  ( x  e.  X  |-> inf ( 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  |-> inf ( ran  ( y  e.  S  |->  ( x D y ) ) ,  RR* ,  <  )
)
43metdsf 21851 . . . . . 6  |-  ( ( D  e.  ( *Met `  X )  /\  S  C_  X
)  ->  F : X
--> ( 0 [,] +oo ) )
51, 2, 4syl2anc 665 . . . . 5  |-  ( ph  ->  F : X --> ( 0 [,] +oo ) )
6 metdscnlem.3 . . . . 5  |-  ( ph  ->  A  e.  X )
75, 6ffvelrnd 6034 . . . 4  |-  ( ph  ->  ( F `  A
)  e.  ( 0 [,] +oo ) )
8 elxrge0 11741 . . . . 5  |-  ( ( F `  A )  e.  ( 0 [,] +oo )  <->  ( ( F `
 A )  e. 
RR*  /\  0  <_  ( F `  A ) ) )
98simplbi 461 . . . 4  |-  ( ( F `  A )  e.  ( 0 [,] +oo )  ->  ( F `
 A )  e. 
RR* )
107, 9syl 17 . . 3  |-  ( ph  ->  ( F `  A
)  e.  RR* )
11 metdscnlem.4 . . . . . 6  |-  ( ph  ->  B  e.  X )
125, 11ffvelrnd 6034 . . . . 5  |-  ( ph  ->  ( F `  B
)  e.  ( 0 [,] +oo ) )
13 elxrge0 11741 . . . . . 6  |-  ( ( F `  B )  e.  ( 0 [,] +oo )  <->  ( ( F `
 B )  e. 
RR*  /\  0  <_  ( F `  B ) ) )
1413simplbi 461 . . . . 5  |-  ( ( F `  B )  e.  ( 0 [,] +oo )  ->  ( F `
 B )  e. 
RR* )
1512, 14syl 17 . . . 4  |-  ( ph  ->  ( F `  B
)  e.  RR* )
1615xnegcld 11586 . . 3  |-  ( ph  -> 
-e ( F `
 B )  e. 
RR* )
1710, 16xaddcld 11587 . 2  |-  ( ph  ->  ( ( F `  A ) +e  -e ( F `  B ) )  e. 
RR* )
18 xmetcl 21332 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( A D B )  e.  RR* )
191, 6, 11, 18syl3anc 1264 . 2  |-  ( ph  ->  ( A D B )  e.  RR* )
20 metdscnlem.5 . . 3  |-  ( ph  ->  R  e.  RR+ )
2120rpxrd 11342 . 2  |-  ( ph  ->  R  e.  RR* )
223metdstri 21854 . . . 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 1265 . . 3  |-  ( ph  ->  ( F `  A
)  <_  ( ( A D B ) +e ( F `  B ) ) )
248simprbi 465 . . . . 5  |-  ( ( F `  A )  e.  ( 0 [,] +oo )  ->  0  <_ 
( F `  A
) )
257, 24syl 17 . . . 4  |-  ( ph  ->  0  <_  ( F `  A ) )
2613simprbi 465 . . . . . 6  |-  ( ( F `  B )  e.  ( 0 [,] +oo )  ->  0  <_ 
( F `  B
) )
2712, 26syl 17 . . . . 5  |-  ( ph  ->  0  <_  ( F `  B ) )
28 ge0nemnf 11468 . . . . 5  |-  ( ( ( F `  B
)  e.  RR*  /\  0  <_  ( F `  B
) )  ->  ( F `  B )  =/= -oo )
2915, 27, 28syl2anc 665 . . . 4  |-  ( ph  ->  ( F `  B
)  =/= -oo )
30 xmetge0 21345 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  0  <_  ( A D B ) )
311, 6, 11, 30syl3anc 1264 . . . 4  |-  ( ph  ->  0  <_  ( A D B ) )
32 xlesubadd 11549 . . . 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 1279 . . 3  |-  ( ph  ->  ( ( ( F `
 A ) +e  -e ( F `  B ) )  <_  ( A D B )  <->  ( F `  A )  <_  (
( A D B ) +e ( F `  B ) ) ) )
3423, 33mpbird 235 . 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 11457 1  |-  ( ph  ->  ( ( F `  A ) +e  -e ( F `  B ) )  < 
R )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    = wceq 1437    e. wcel 1868    =/= wne 2618    C_ wss 3436   class class class wbr 4420    |-> cmpt 4479   ran crn 4850   -->wf 5593   ` cfv 5597  (class class class)co 6301  infcinf 7957   0cc0 9539   +oocpnf 9672   -oocmnf 9673   RR*cxr 9674    < clt 9675    <_ cle 9676   RR+crp 11302    -ecxne 11406   +ecxad 11407   [,]cicc 11638   distcds 15186   RR*scxrs 15385   *Metcxmt 18942   MetOpencmopn 18947
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4764  df-po 4770  df-so 4771  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-1st 6803  df-2nd 6804  df-er 7367  df-ec 7369  df-map 7478  df-en 7574  df-dom 7575  df-sdom 7576  df-sup 7958  df-inf 7959  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-2 10668  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-icc 11642  df-psmet 18949  df-xmet 18950  df-bl 18952
This theorem is referenced by:  metdscn  21859
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