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Theorem metnrmlem1 18762
Description: Lemma for metnrm 18765. (Contributed by Mario Carneiro, 14-Jan-2014.) (Revised 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 )
metnrmlem.1  |-  ( ph  ->  D  e.  ( * Met `  X ) )
metnrmlem.2  |-  ( ph  ->  S  e.  ( Clsd `  J ) )
metnrmlem.3  |-  ( ph  ->  T  e.  ( Clsd `  J ) )
metnrmlem.4  |-  ( ph  ->  ( S  i^i  T
)  =  (/) )
Assertion
Ref Expression
metnrmlem1  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  if ( 1  <_  ( F `  B ) ,  1 ,  ( F `  B ) )  <_  ( A D B ) )
Distinct variable groups:    x, y, A    x, D, y    y, J    x, B, y    x, T, y    x, S, y   
x, X, y
Allowed substitution hints:    ph( x, y)    F( x, y)    J( x)

Proof of Theorem metnrmlem1
StepHypRef Expression
1 1re 9025 . . . 4  |-  1  e.  RR
21rexri 9072 . . 3  |-  1  e.  RR*
3 metnrmlem.1 . . . . . . 7  |-  ( ph  ->  D  e.  ( * Met `  X ) )
43adantr 452 . . . . . 6  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  D  e.  ( * Met `  X ) )
5 metnrmlem.2 . . . . . . . . 9  |-  ( ph  ->  S  e.  ( Clsd `  J ) )
65adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  S  e.  ( Clsd `  J ) )
7 eqid 2389 . . . . . . . . 9  |-  U. J  =  U. J
87cldss 17018 . . . . . . . 8  |-  ( S  e.  ( Clsd `  J
)  ->  S  C_  U. J
)
96, 8syl 16 . . . . . . 7  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  S  C_  U. J )
10 metdscn.j . . . . . . . . 9  |-  J  =  ( MetOpen `  D )
1110mopnuni 18363 . . . . . . . 8  |-  ( D  e.  ( * Met `  X )  ->  X  =  U. J )
124, 11syl 16 . . . . . . 7  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  X  =  U. J )
139, 12sseqtr4d 3330 . . . . . 6  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  S  C_  X )
14 metdscn.f . . . . . . 7  |-  F  =  ( x  e.  X  |->  sup ( ran  (
y  e.  S  |->  ( x D y ) ) ,  RR* ,  `'  <  ) )
1514metdsf 18751 . . . . . 6  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X
)  ->  F : X
--> ( 0 [,]  +oo ) )
164, 13, 15syl2anc 643 . . . . 5  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  F : X --> ( 0 [,]  +oo ) )
17 metnrmlem.3 . . . . . . . . 9  |-  ( ph  ->  T  e.  ( Clsd `  J ) )
1817adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  T  e.  ( Clsd `  J ) )
197cldss 17018 . . . . . . . 8  |-  ( T  e.  ( Clsd `  J
)  ->  T  C_  U. J
)
2018, 19syl 16 . . . . . . 7  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  T  C_  U. J )
2120, 12sseqtr4d 3330 . . . . . 6  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  T  C_  X )
22 simprr 734 . . . . . 6  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  B  e.  T )
2321, 22sseldd 3294 . . . . 5  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  B  e.  X )
2416, 23ffvelrnd 5812 . . . 4  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  -> 
( F `  B
)  e.  ( 0 [,]  +oo ) )
25 elxrge0 10942 . . . . 5  |-  ( ( F `  B )  e.  ( 0 [,] 
+oo )  <->  ( ( F `  B )  e.  RR*  /\  0  <_ 
( F `  B
) ) )
2625simplbi 447 . . . 4  |-  ( ( F `  B )  e.  ( 0 [,] 
+oo )  ->  ( F `  B )  e.  RR* )
2724, 26syl 16 . . 3  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  -> 
( F `  B
)  e.  RR* )
28 ifcl 3720 . . 3  |-  ( ( 1  e.  RR*  /\  ( F `  B )  e.  RR* )  ->  if ( 1  <_  ( F `  B ) ,  1 ,  ( F `  B ) )  e.  RR* )
292, 27, 28sylancr 645 . 2  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  if ( 1  <_  ( F `  B ) ,  1 ,  ( F `  B ) )  e.  RR* )
30 simprl 733 . . . 4  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  A  e.  S )
3113, 30sseldd 3294 . . 3  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  A  e.  X )
32 xmetcl 18272 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( A D B )  e.  RR* )
334, 31, 23, 32syl3anc 1184 . 2  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  -> 
( A D B )  e.  RR* )
34 xrmin2 10700 . . 3  |-  ( ( 1  e.  RR*  /\  ( F `  B )  e.  RR* )  ->  if ( 1  <_  ( F `  B ) ,  1 ,  ( F `  B ) )  <_  ( F `  B ) )
352, 27, 34sylancr 645 . 2  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  if ( 1  <_  ( F `  B ) ,  1 ,  ( F `  B ) )  <_  ( F `  B ) )
3614metdstri 18754 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X
)  /\  ( B  e.  X  /\  A  e.  X ) )  -> 
( F `  B
)  <_  ( ( B D A ) + e ( F `  A ) ) )
374, 13, 23, 31, 36syl22anc 1185 . . 3  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  -> 
( F `  B
)  <_  ( ( B D A ) + e ( F `  A ) ) )
38 xmetsym 18288 . . . . . 6  |-  ( ( D  e.  ( * Met `  X )  /\  B  e.  X  /\  A  e.  X
)  ->  ( B D A )  =  ( A D B ) )
394, 23, 31, 38syl3anc 1184 . . . . 5  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  -> 
( B D A )  =  ( A D B ) )
4014metds0 18753 . . . . . 6  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  S
)  ->  ( F `  A )  =  0 )
414, 13, 30, 40syl3anc 1184 . . . . 5  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  -> 
( F `  A
)  =  0 )
4239, 41oveq12d 6040 . . . 4  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  -> 
( ( B D A ) + e
( F `  A
) )  =  ( ( A D B ) + e 0 ) )
43 xaddid1 10759 . . . . 5  |-  ( ( A D B )  e.  RR*  ->  ( ( A D B ) + e 0 )  =  ( A D B ) )
4433, 43syl 16 . . . 4  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  -> 
( ( A D B ) + e
0 )  =  ( A D B ) )
4542, 44eqtrd 2421 . . 3  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  -> 
( ( B D A ) + e
( F `  A
) )  =  ( A D B ) )
4637, 45breqtrd 4179 . 2  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  -> 
( F `  B
)  <_  ( A D B ) )
4729, 27, 33, 35, 46xrletrd 10686 1  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  if ( 1  <_  ( F `  B ) ,  1 ,  ( F `  B ) )  <_  ( A D B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    i^i cin 3264    C_ wss 3265   (/)c0 3573   ifcif 3684   U.cuni 3959   class class class wbr 4155    e. cmpt 4209   `'ccnv 4819   ran crn 4821   -->wf 5392   ` cfv 5396  (class class class)co 6022   supcsup 7382   0cc0 8925   1c1 8926    +oocpnf 9052   RR*cxr 9054    < clt 9055    <_ cle 9056   + ecxad 10642   [,]cicc 10853   * Metcxmt 16614   MetOpencmopn 16619   Clsdccld 17005
This theorem is referenced by:  metnrmlem3  18764
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002  ax-pre-sup 9003
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-er 6843  df-ec 6845  df-map 6958  df-en 7048  df-dom 7049  df-sdom 7050  df-sup 7383  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-div 9612  df-nn 9935  df-2 9992  df-n0 10156  df-z 10217  df-uz 10423  df-q 10509  df-rp 10547  df-xneg 10644  df-xadd 10645  df-xmul 10646  df-icc 10857  df-topgen 13596  df-xmet 16621  df-bl 16623  df-mopn 16624  df-top 16888  df-bases 16890  df-topon 16891  df-cld 17008
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