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Theorem metdscnlem 18359
Description: Lemma for metdscn 18360. (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 18352 . . . . . 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 5663 . . . . 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 10747 . . . . 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 5663 . . . . . 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 10747 . . . . . 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 10620 . . 3  |-  ( ph  -> 
- e ( F `
 B )  e. 
RR* )
1911, 18xaddcld 10621 . 2  |-  ( ph  ->  ( ( F `  A ) + e  - e ( F `  B ) )  e. 
RR* )
20 xmetcl 17896 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( A D B )  e.  RR* )
211, 6, 12, 20syl3anc 1182 . 2  |-  ( ph  ->  ( A D B )  e.  RR* )
22 metdscnlem.5 . . 3  |-  ( ph  ->  R  e.  RR+ )
2322rpxrd 10391 . 2  |-  ( ph  ->  R  e.  RR* )
243metdstri 18355 . . . 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 1183 . . 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 10502 . . . . 5  |-  ( ( ( F `  B
)  e.  RR*  /\  0  <_  ( F `  B
) )  ->  ( F `  B )  =/=  -oo )
3117, 29, 30syl2anc 642 . . . 4  |-  ( ph  ->  ( F `  B
)  =/=  -oo )
32 xmetge0 17909 . . . . 5  |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  0  <_  ( A D B ) )
331, 6, 12, 32syl3anc 1182 . . . 4  |-  ( ph  ->  0  <_  ( A D B ) )
34 xlesubadd 10583 . . . 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 1197 . . 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 10491 1  |-  ( ph  ->  ( ( F `  A ) + e  - e ( F `  B ) )  < 
R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684    =/= wne 2446    C_ wss 3152   class class class wbr 4023    e. cmpt 4077   `'ccnv 4688   ran crn 4690   -->wf 5251   ` cfv 5255  (class class class)co 5858   supcsup 7193   0cc0 8737    +oocpnf 8864    -oocmnf 8865   RR*cxr 8866    < clt 8867    <_ cle 8868   RR+crp 10354    - ecxne 10449   + ecxad 10450   [,]cicc 10659   distcds 13217   RR* scxrs 13399   * Metcxmt 16369   MetOpencmopn 16372
This theorem is referenced by:  metdscn  18360
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-er 6660  df-ec 6662  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-2 9804  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-icc 10663  df-xmet 16373  df-bl 16375
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