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Theorem metnrmlem1 21231
Description: Lemma for metnrm 21234. (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 9607 . . . 4  |-  1  e.  RR
21rexri 9658 . . 3  |-  1  e.  RR*
3 metnrmlem.1 . . . . . . 7  |-  ( ph  ->  D  e.  ( *Met `  X ) )
43adantr 465 . . . . . 6  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  D  e.  ( *Met `  X ) )
5 metnrmlem.2 . . . . . . . . 9  |-  ( ph  ->  S  e.  ( Clsd `  J ) )
65adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  S  e.  ( Clsd `  J ) )
7 eqid 2467 . . . . . . . . 9  |-  U. J  =  U. J
87cldss 19398 . . . . . . . 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 20812 . . . . . . . 8  |-  ( D  e.  ( *Met `  X )  ->  X  =  U. J )
124, 11syl 16 . . . . . . 7  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  X  =  U. J )
139, 12sseqtr4d 3546 . . . . . 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 21220 . . . . . 6  |-  ( ( D  e.  ( *Met `  X )  /\  S  C_  X
)  ->  F : X
--> ( 0 [,] +oo ) )
164, 13, 15syl2anc 661 . . . . 5  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  F : X --> ( 0 [,] +oo ) )
17 metnrmlem.3 . . . . . . . . 9  |-  ( ph  ->  T  e.  ( Clsd `  J ) )
1817adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  T  e.  ( Clsd `  J ) )
197cldss 19398 . . . . . . . 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 3546 . . . . . 6  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  T  C_  X )
22 simprr 756 . . . . . 6  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  B  e.  T )
2321, 22sseldd 3510 . . . . 5  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  B  e.  X )
2416, 23ffvelrnd 6033 . . . 4  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  -> 
( F `  B
)  e.  ( 0 [,] +oo ) )
25 elxrge0 11641 . . . . 5  |-  ( ( F `  B )  e.  ( 0 [,] +oo )  <->  ( ( F `
 B )  e. 
RR*  /\  0  <_  ( F `  B ) ) )
2625simplbi 460 . . . 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 3987 . . 3  |-  ( ( 1  e.  RR*  /\  ( F `  B )  e.  RR* )  ->  if ( 1  <_  ( F `  B ) ,  1 ,  ( F `  B ) )  e.  RR* )
292, 27, 28sylancr 663 . 2  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  if ( 1  <_  ( F `  B ) ,  1 ,  ( F `  B ) )  e.  RR* )
30 simprl 755 . . . 4  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  A  e.  S )
3113, 30sseldd 3510 . . 3  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  A  e.  X )
32 xmetcl 20702 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( A D B )  e.  RR* )
334, 31, 23, 32syl3anc 1228 . 2  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  -> 
( A D B )  e.  RR* )
34 xrmin2 11391 . . 3  |-  ( ( 1  e.  RR*  /\  ( F `  B )  e.  RR* )  ->  if ( 1  <_  ( F `  B ) ,  1 ,  ( F `  B ) )  <_  ( F `  B ) )
352, 27, 34sylancr 663 . 2  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  if ( 1  <_  ( F `  B ) ,  1 ,  ( F `  B ) )  <_  ( F `  B ) )
3614metdstri 21223 . . . 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 1229 . . 3  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  -> 
( F `  B
)  <_  ( ( B D A ) +e ( F `  A ) ) )
38 xmetsym 20718 . . . . . 6  |-  ( ( D  e.  ( *Met `  X )  /\  B  e.  X  /\  A  e.  X
)  ->  ( B D A )  =  ( A D B ) )
394, 23, 31, 38syl3anc 1228 . . . . 5  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  -> 
( B D A )  =  ( A D B ) )
4014metds0 21222 . . . . . 6  |-  ( ( D  e.  ( *Met `  X )  /\  S  C_  X  /\  A  e.  S
)  ->  ( F `  A )  =  0 )
414, 13, 30, 40syl3anc 1228 . . . . 5  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  -> 
( F `  A
)  =  0 )
4239, 41oveq12d 6313 . . . 4  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  -> 
( ( B D A ) +e
( F `  A
) )  =  ( ( A D B ) +e 0 ) )
43 xaddid1 11450 . . . . 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 2508 . . 3  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  -> 
( ( B D A ) +e
( F `  A
) )  =  ( A D B ) )
4637, 45breqtrd 4477 . 2  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  -> 
( F `  B
)  <_  ( A D B ) )
4729, 27, 33, 35, 46xrletrd 11377 1  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  if ( 1  <_  ( F `  B ) ,  1 ,  ( F `  B ) )  <_  ( A D B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    i^i cin 3480    C_ wss 3481   (/)c0 3790   ifcif 3945   U.cuni 4251   class class class wbr 4453    |-> cmpt 4511   `'ccnv 5004   ran crn 5006   -->wf 5590   ` cfv 5594  (class class class)co 6295   supcsup 7912   0cc0 9504   1c1 9505   +oocpnf 9637   RR*cxr 9639    < clt 9640    <_ cle 9641   +ecxad 11328   [,]cicc 11544   *Metcxmt 18273   MetOpencmopn 18278   Clsdccld 19385
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 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-ec 7325  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-n0 10808  df-z 10877  df-uz 11095  df-q 11195  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-icc 11548  df-topgen 14716  df-psmet 18281  df-xmet 18282  df-bl 18284  df-mopn 18285  df-top 19268  df-bases 19270  df-topon 19271  df-cld 19388
This theorem is referenced by:  metnrmlem3  21233
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