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Theorem metnrmlem1 18363
Description: Lemma for metnrm 18366. (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 8837 . . . 4  |-  1  e.  RR
2 rexr 8877 . . . 4  |-  ( 1  e.  RR  ->  1  e.  RR* )
31, 2ax-mp 8 . . 3  |-  1  e.  RR*
4 metnrmlem.1 . . . . . . 7  |-  ( ph  ->  D  e.  ( * Met `  X ) )
54adantr 451 . . . . . 6  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  D  e.  ( * Met `  X ) )
6 metnrmlem.2 . . . . . . . . 9  |-  ( ph  ->  S  e.  ( Clsd `  J ) )
76adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  S  e.  ( Clsd `  J ) )
8 eqid 2283 . . . . . . . . 9  |-  U. J  =  U. J
98cldss 16766 . . . . . . . 8  |-  ( S  e.  ( Clsd `  J
)  ->  S  C_  U. J
)
107, 9syl 15 . . . . . . 7  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  S  C_  U. J )
11 metdscn.j . . . . . . . . 9  |-  J  =  ( MetOpen `  D )
1211mopnuni 17987 . . . . . . . 8  |-  ( D  e.  ( * Met `  X )  ->  X  =  U. J )
135, 12syl 15 . . . . . . 7  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  X  =  U. J )
1410, 13sseqtr4d 3215 . . . . . 6  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  S  C_  X )
15 metdscn.f . . . . . . 7  |-  F  =  ( x  e.  X  |->  sup ( ran  (
y  e.  S  |->  ( x D y ) ) ,  RR* ,  `'  <  ) )
1615metdsf 18352 . . . . . 6  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X
)  ->  F : X
--> ( 0 [,]  +oo ) )
175, 14, 16syl2anc 642 . . . . 5  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  F : X --> ( 0 [,]  +oo ) )
18 metnrmlem.3 . . . . . . . . 9  |-  ( ph  ->  T  e.  ( Clsd `  J ) )
1918adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  T  e.  ( Clsd `  J ) )
208cldss 16766 . . . . . . . 8  |-  ( T  e.  ( Clsd `  J
)  ->  T  C_  U. J
)
2119, 20syl 15 . . . . . . 7  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  T  C_  U. J )
2221, 13sseqtr4d 3215 . . . . . 6  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  T  C_  X )
23 simprr 733 . . . . . 6  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  B  e.  T )
2422, 23sseldd 3181 . . . . 5  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  B  e.  X )
25 ffvelrn 5663 . . . . 5  |-  ( ( F : X --> ( 0 [,]  +oo )  /\  B  e.  X )  ->  ( F `  B )  e.  ( 0 [,]  +oo ) )
2617, 24, 25syl2anc 642 . . . 4  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  -> 
( F `  B
)  e.  ( 0 [,]  +oo ) )
27 elxrge0 10747 . . . . 5  |-  ( ( F `  B )  e.  ( 0 [,] 
+oo )  <->  ( ( F `  B )  e.  RR*  /\  0  <_ 
( F `  B
) ) )
2827simplbi 446 . . . 4  |-  ( ( F `  B )  e.  ( 0 [,] 
+oo )  ->  ( F `  B )  e.  RR* )
2926, 28syl 15 . . 3  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  -> 
( F `  B
)  e.  RR* )
30 ifcl 3601 . . 3  |-  ( ( 1  e.  RR*  /\  ( F `  B )  e.  RR* )  ->  if ( 1  <_  ( F `  B ) ,  1 ,  ( F `  B ) )  e.  RR* )
313, 29, 30sylancr 644 . 2  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  if ( 1  <_  ( F `  B ) ,  1 ,  ( F `  B ) )  e.  RR* )
32 simprl 732 . . . 4  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  A  e.  S )
3314, 32sseldd 3181 . . 3  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  A  e.  X )
34 xmetcl 17896 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( A D B )  e.  RR* )
355, 33, 24, 34syl3anc 1182 . 2  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  -> 
( A D B )  e.  RR* )
36 xrmin2 10507 . . 3  |-  ( ( 1  e.  RR*  /\  ( F `  B )  e.  RR* )  ->  if ( 1  <_  ( F `  B ) ,  1 ,  ( F `  B ) )  <_  ( F `  B ) )
373, 29, 36sylancr 644 . 2  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  if ( 1  <_  ( F `  B ) ,  1 ,  ( F `  B ) )  <_  ( F `  B ) )
3815metdstri 18355 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X
)  /\  ( B  e.  X  /\  A  e.  X ) )  -> 
( F `  B
)  <_  ( ( B D A ) + e ( F `  A ) ) )
395, 14, 24, 33, 38syl22anc 1183 . . 3  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  -> 
( F `  B
)  <_  ( ( B D A ) + e ( F `  A ) ) )
40 xmetsym 17912 . . . . . 6  |-  ( ( D  e.  ( * Met `  X )  /\  B  e.  X  /\  A  e.  X
)  ->  ( B D A )  =  ( A D B ) )
415, 24, 33, 40syl3anc 1182 . . . . 5  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  -> 
( B D A )  =  ( A D B ) )
4215metds0 18354 . . . . . 6  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  S
)  ->  ( F `  A )  =  0 )
435, 14, 32, 42syl3anc 1182 . . . . 5  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  -> 
( F `  A
)  =  0 )
4441, 43oveq12d 5876 . . . 4  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  -> 
( ( B D A ) + e
( F `  A
) )  =  ( ( A D B ) + e 0 ) )
45 xaddid1 10566 . . . . 5  |-  ( ( A D B )  e.  RR*  ->  ( ( A D B ) + e 0 )  =  ( A D B ) )
4635, 45syl 15 . . . 4  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  -> 
( ( A D B ) + e
0 )  =  ( A D B ) )
4744, 46eqtrd 2315 . . 3  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  -> 
( ( B D A ) + e
( F `  A
) )  =  ( A D B ) )
4839, 47breqtrd 4047 . 2  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  -> 
( F `  B
)  <_  ( A D B ) )
4931, 29, 35, 37, 48xrletrd 10493 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 358    = wceq 1623    e. wcel 1684    i^i cin 3151    C_ wss 3152   (/)c0 3455   ifcif 3565   U.cuni 3827   class class class wbr 4023    e. cmpt 4077   `'ccnv 4688   ran crn 4690   -->wf 5251   ` cfv 5255  (class class class)co 5858   supcsup 7193   RRcr 8736   0cc0 8737   1c1 8738    +oocpnf 8864   RR*cxr 8866    < clt 8867    <_ cle 8868   + ecxad 10450   [,]cicc 10659   * Metcxmt 16369   MetOpencmopn 16372   Clsdccld 16753
This theorem is referenced by:  metnrmlem3  18365
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-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  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-recs 6388  df-rdg 6423  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-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-icc 10663  df-topgen 13344  df-xmet 16373  df-bl 16375  df-mopn 16376  df-top 16636  df-bases 16638  df-topon 16639  df-cld 16756
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