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Theorem metnrmlem1 21862
Description: Lemma for metnrm 21880. (Contributed by Mario Carneiro, 14-Jan-2014.) (Revised 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 )
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 9642 . . . 4  |-  1  e.  RR
21rexri 9693 . . 3  |-  1  e.  RR*
3 metnrmlem.1 . . . . . . 7  |-  ( ph  ->  D  e.  ( *Met `  X ) )
43adantr 466 . . . . . 6  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  D  e.  ( *Met `  X ) )
5 metnrmlem.2 . . . . . . . . 9  |-  ( ph  ->  S  e.  ( Clsd `  J ) )
65adantr 466 . . . . . . . 8  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  S  e.  ( Clsd `  J ) )
7 eqid 2422 . . . . . . . . 9  |-  U. J  =  U. J
87cldss 20030 . . . . . . . 8  |-  ( S  e.  ( Clsd `  J
)  ->  S  C_  U. J
)
96, 8syl 17 . . . . . . 7  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  S  C_  U. J )
10 metdscn.j . . . . . . . . 9  |-  J  =  ( MetOpen `  D )
1110mopnuni 21442 . . . . . . . 8  |-  ( D  e.  ( *Met `  X )  ->  X  =  U. J )
124, 11syl 17 . . . . . . 7  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  X  =  U. J )
139, 12sseqtr4d 3501 . . . . . 6  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  S  C_  X )
14 metdscn.f . . . . . . 7  |-  F  =  ( x  e.  X  |-> inf ( ran  ( y  e.  S  |->  ( x D y ) ) ,  RR* ,  <  )
)
1514metdsf 21851 . . . . . 6  |-  ( ( D  e.  ( *Met `  X )  /\  S  C_  X
)  ->  F : X
--> ( 0 [,] +oo ) )
164, 13, 15syl2anc 665 . . . . 5  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  F : X --> ( 0 [,] +oo ) )
17 metnrmlem.3 . . . . . . . . 9  |-  ( ph  ->  T  e.  ( Clsd `  J ) )
1817adantr 466 . . . . . . . 8  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  T  e.  ( Clsd `  J ) )
197cldss 20030 . . . . . . . 8  |-  ( T  e.  ( Clsd `  J
)  ->  T  C_  U. J
)
2018, 19syl 17 . . . . . . 7  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  T  C_  U. J )
2120, 12sseqtr4d 3501 . . . . . 6  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  T  C_  X )
22 simprr 764 . . . . . 6  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  B  e.  T )
2321, 22sseldd 3465 . . . . 5  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  B  e.  X )
2416, 23ffvelrnd 6034 . . . 4  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  -> 
( F `  B
)  e.  ( 0 [,] +oo ) )
25 elxrge0 11741 . . . . 5  |-  ( ( F `  B )  e.  ( 0 [,] +oo )  <->  ( ( F `
 B )  e. 
RR*  /\  0  <_  ( F `  B ) ) )
2625simplbi 461 . . . 4  |-  ( ( F `  B )  e.  ( 0 [,] +oo )  ->  ( F `
 B )  e. 
RR* )
2724, 26syl 17 . . 3  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  -> 
( F `  B
)  e.  RR* )
28 ifcl 3951 . . 3  |-  ( ( 1  e.  RR*  /\  ( F `  B )  e.  RR* )  ->  if ( 1  <_  ( F `  B ) ,  1 ,  ( F `  B ) )  e.  RR* )
292, 27, 28sylancr 667 . 2  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  if ( 1  <_  ( F `  B ) ,  1 ,  ( F `  B ) )  e.  RR* )
30 simprl 762 . . . 4  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  A  e.  S )
3113, 30sseldd 3465 . . 3  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  A  e.  X )
32 xmetcl 21332 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( A D B )  e.  RR* )
334, 31, 23, 32syl3anc 1264 . 2  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  -> 
( A D B )  e.  RR* )
34 xrmin2 11473 . . 3  |-  ( ( 1  e.  RR*  /\  ( F `  B )  e.  RR* )  ->  if ( 1  <_  ( F `  B ) ,  1 ,  ( F `  B ) )  <_  ( F `  B ) )
352, 27, 34sylancr 667 . 2  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  if ( 1  <_  ( F `  B ) ,  1 ,  ( F `  B ) )  <_  ( F `  B ) )
3614metdstri 21854 . . . 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 1265 . . 3  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  -> 
( F `  B
)  <_  ( ( B D A ) +e ( F `  A ) ) )
38 xmetsym 21348 . . . . . 6  |-  ( ( D  e.  ( *Met `  X )  /\  B  e.  X  /\  A  e.  X
)  ->  ( B D A )  =  ( A D B ) )
394, 23, 31, 38syl3anc 1264 . . . . 5  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  -> 
( B D A )  =  ( A D B ) )
4014metds0 21853 . . . . . 6  |-  ( ( D  e.  ( *Met `  X )  /\  S  C_  X  /\  A  e.  S
)  ->  ( F `  A )  =  0 )
414, 13, 30, 40syl3anc 1264 . . . . 5  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  -> 
( F `  A
)  =  0 )
4239, 41oveq12d 6319 . . . 4  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  -> 
( ( B D A ) +e
( F `  A
) )  =  ( ( A D B ) +e 0 ) )
43 xaddid1 11532 . . . . 5  |-  ( ( A D B )  e.  RR*  ->  ( ( A D B ) +e 0 )  =  ( A D B ) )
4433, 43syl 17 . . . 4  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  -> 
( ( A D B ) +e 0 )  =  ( A D B ) )
4542, 44eqtrd 2463 . . 3  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  -> 
( ( B D A ) +e
( F `  A
) )  =  ( A D B ) )
4637, 45breqtrd 4445 . 2  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  -> 
( F `  B
)  <_  ( A D B ) )
4729, 27, 33, 35, 46xrletrd 11459 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 370    = wceq 1437    e. wcel 1868    i^i cin 3435    C_ wss 3436   (/)c0 3761   ifcif 3909   U.cuni 4216   class class class wbr 4420    |-> cmpt 4479   ran crn 4850   -->wf 5593   ` cfv 5597  (class class class)co 6301  infcinf 7957   0cc0 9539   1c1 9540   +oocpnf 9672   RR*cxr 9674    < clt 9675    <_ cle 9676   +ecxad 11407   [,]cicc 11638   *Metcxmt 18942   MetOpencmopn 18947   Clsdccld 20017
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-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-we 4810  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-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  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-om 6703  df-1st 6803  df-2nd 6804  df-wrecs 7032  df-recs 7094  df-rdg 7132  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-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-icc 11642  df-topgen 15329  df-psmet 18949  df-xmet 18950  df-bl 18952  df-mopn 18953  df-top 19907  df-bases 19908  df-topon 19909  df-cld 20020
This theorem is referenced by:  metnrmlem3  21864
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