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Theorem metnrmlem1 18379
Description: Lemma for metnrm 18382. (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 8853 . . . 4  |-  1  e.  RR
2 rexr 8893 . . . 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 2296 . . . . . . . . 9  |-  U. J  =  U. J
98cldss 16782 . . . . . . . 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 18003 . . . . . . . 8  |-  ( D  e.  ( * Met `  X )  ->  X  =  U. J )
135, 12syl 15 . . . . . . 7  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  X  =  U. J )
1410, 13sseqtr4d 3228 . . . . . 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 18368 . . . . . 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 16782 . . . . . . . 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 3228 . . . . . 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 3194 . . . . 5  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  B  e.  X )
25 ffvelrn 5679 . . . . 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 10763 . . . . 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 3614 . . 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 3194 . . 3  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  ->  A  e.  X )
34 xmetcl 17912 . . 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 10523 . . 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 18371 . . . 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 17928 . . . . . 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 18370 . . . . . 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 5892 . . . 4  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  -> 
( ( B D A ) + e
( F `  A
) )  =  ( ( A D B ) + e 0 ) )
45 xaddid1 10582 . . . . 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 2328 . . 3  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  -> 
( ( B D A ) + e
( F `  A
) )  =  ( A D B ) )
4839, 47breqtrd 4063 . 2  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  T ) )  -> 
( F `  B
)  <_  ( A D B ) )
4931, 29, 35, 37, 48xrletrd 10509 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 1632    e. wcel 1696    i^i cin 3164    C_ wss 3165   (/)c0 3468   ifcif 3578   U.cuni 3843   class class class wbr 4039    e. cmpt 4093   `'ccnv 4704   ran crn 4706   -->wf 5267   ` cfv 5271  (class class class)co 5874   supcsup 7209   RRcr 8752   0cc0 8753   1c1 8754    +oocpnf 8880   RR*cxr 8882    < clt 8883    <_ cle 8884   + ecxad 10466   [,]cicc 10675   * Metcxmt 16385   MetOpencmopn 16388   Clsdccld 16769
This theorem is referenced by:  metnrmlem3  18381
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-ec 6678  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-icc 10679  df-topgen 13360  df-xmet 16389  df-bl 16391  df-mopn 16392  df-top 16652  df-bases 16654  df-topon 16655  df-cld 16772
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