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Theorem metdscnlem 21091
Description: Lemma for metdscn 21092. (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 21084 . . . . . 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 6020 . . . 4  |-  ( ph  ->  ( F `  A
)  e.  ( 0 [,] +oo ) )
8 elxrge0 11625 . . . . 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 6020 . . . . 5  |-  ( ph  ->  ( F `  B
)  e.  ( 0 [,] +oo ) )
13 elxrge0 11625 . . . . . 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 11488 . . 3  |-  ( ph  -> 
-e ( F `
 B )  e. 
RR* )
1710, 16xaddcld 11489 . 2  |-  ( ph  ->  ( ( F `  A ) +e  -e ( F `  B ) )  e. 
RR* )
18 xmetcl 20566 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( A D B )  e.  RR* )
191, 6, 11, 18syl3anc 1228 . 2  |-  ( ph  ->  ( A D B )  e.  RR* )
20 metdscnlem.5 . . 3  |-  ( ph  ->  R  e.  RR+ )
2120rpxrd 11253 . 2  |-  ( ph  ->  R  e.  RR* )
223metdstri 21087 . . . 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 1229 . . 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 11370 . . . . 5  |-  ( ( ( F `  B
)  e.  RR*  /\  0  <_  ( F `  B
) )  ->  ( F `  B )  =/= -oo )
2915, 27, 28syl2anc 661 . . . 4  |-  ( ph  ->  ( F `  B
)  =/= -oo )
30 xmetge0 20579 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  0  <_  ( A D B ) )
311, 6, 11, 30syl3anc 1228 . . . 4  |-  ( ph  ->  0  <_  ( A D B ) )
32 xlesubadd 11451 . . . 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 1243 . . 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 11359 1  |-  ( ph  ->  ( ( F `  A ) +e  -e ( F `  B ) )  < 
R )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1379    e. wcel 1767    =/= wne 2662    C_ wss 3476   class class class wbr 4447    |-> cmpt 4505   `'ccnv 4998   ran crn 5000   -->wf 5582   ` cfv 5586  (class class class)co 6282   supcsup 7896   0cc0 9488   +oocpnf 9621   -oocmnf 9622   RR*cxr 9623    < clt 9624    <_ cle 9625   RR+crp 11216    -ecxne 11311   +ecxad 11312   [,]cicc 11528   distcds 14557   RR*scxrs 14748   *Metcxmt 18171   MetOpencmopn 18176
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-er 7308  df-ec 7310  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-sup 7897  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-2 10590  df-rp 11217  df-xneg 11314  df-xadd 11315  df-xmul 11316  df-icc 11532  df-psmet 18179  df-xmet 18180  df-bl 18182
This theorem is referenced by:  metdscn  21092
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