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Theorem metnrmlem1a 18414
Description: Lemma for metnrm 18418. (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
metnrmlem1a  |-  ( (
ph  /\  A  e.  T )  ->  (
0  <  ( F `  A )  /\  if ( 1  <_  ( F `  A ) ,  1 ,  ( F `  A ) )  e.  RR+ )
)
Distinct variable groups:    x, y, A    x, D, y    y, J    x, T, y    x, S, y    x, X, y
Allowed substitution hints:    ph( x, y)    F( x, y)    J( x)

Proof of Theorem metnrmlem1a
StepHypRef Expression
1 metnrmlem.4 . . . . . 6  |-  ( ph  ->  ( S  i^i  T
)  =  (/) )
21adantr 451 . . . . 5  |-  ( (
ph  /\  A  e.  T )  ->  ( S  i^i  T )  =  (/) )
3 inelcm 3543 . . . . . . . 8  |-  ( ( A  e.  S  /\  A  e.  T )  ->  ( S  i^i  T
)  =/=  (/) )
43expcom 424 . . . . . . 7  |-  ( A  e.  T  ->  ( A  e.  S  ->  ( S  i^i  T )  =/=  (/) ) )
54adantl 452 . . . . . 6  |-  ( (
ph  /\  A  e.  T )  ->  ( A  e.  S  ->  ( S  i^i  T )  =/=  (/) ) )
65necon2bd 2528 . . . . 5  |-  ( (
ph  /\  A  e.  T )  ->  (
( S  i^i  T
)  =  (/)  ->  -.  A  e.  S )
)
72, 6mpd 14 . . . 4  |-  ( (
ph  /\  A  e.  T )  ->  -.  A  e.  S )
8 eqcom 2318 . . . . . 6  |-  ( 0  =  ( F `  A )  <->  ( F `  A )  =  0 )
9 metnrmlem.1 . . . . . . . 8  |-  ( ph  ->  D  e.  ( * Met `  X ) )
109adantr 451 . . . . . . 7  |-  ( (
ph  /\  A  e.  T )  ->  D  e.  ( * Met `  X
) )
11 metnrmlem.2 . . . . . . . . . 10  |-  ( ph  ->  S  e.  ( Clsd `  J ) )
1211adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  A  e.  T )  ->  S  e.  ( Clsd `  J
) )
13 eqid 2316 . . . . . . . . . 10  |-  U. J  =  U. J
1413cldss 16822 . . . . . . . . 9  |-  ( S  e.  ( Clsd `  J
)  ->  S  C_  U. J
)
1512, 14syl 15 . . . . . . . 8  |-  ( (
ph  /\  A  e.  T )  ->  S  C_ 
U. J )
16 metdscn.j . . . . . . . . . 10  |-  J  =  ( MetOpen `  D )
1716mopnuni 18039 . . . . . . . . 9  |-  ( D  e.  ( * Met `  X )  ->  X  =  U. J )
1810, 17syl 15 . . . . . . . 8  |-  ( (
ph  /\  A  e.  T )  ->  X  =  U. J )
1915, 18sseqtr4d 3249 . . . . . . 7  |-  ( (
ph  /\  A  e.  T )  ->  S  C_  X )
20 metnrmlem.3 . . . . . . . . . . 11  |-  ( ph  ->  T  e.  ( Clsd `  J ) )
2120adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  A  e.  T )  ->  T  e.  ( Clsd `  J
) )
2213cldss 16822 . . . . . . . . . 10  |-  ( T  e.  ( Clsd `  J
)  ->  T  C_  U. J
)
2321, 22syl 15 . . . . . . . . 9  |-  ( (
ph  /\  A  e.  T )  ->  T  C_ 
U. J )
2423, 18sseqtr4d 3249 . . . . . . . 8  |-  ( (
ph  /\  A  e.  T )  ->  T  C_  X )
25 simpr 447 . . . . . . . 8  |-  ( (
ph  /\  A  e.  T )  ->  A  e.  T )
2624, 25sseldd 3215 . . . . . . 7  |-  ( (
ph  /\  A  e.  T )  ->  A  e.  X )
27 metdscn.f . . . . . . . 8  |-  F  =  ( x  e.  X  |->  sup ( ran  (
y  e.  S  |->  ( x D y ) ) ,  RR* ,  `'  <  ) )
2827, 16metdseq0 18410 . . . . . . 7  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  X
)  ->  ( ( F `  A )  =  0  <->  A  e.  ( ( cls `  J
) `  S )
) )
2910, 19, 26, 28syl3anc 1182 . . . . . 6  |-  ( (
ph  /\  A  e.  T )  ->  (
( F `  A
)  =  0  <->  A  e.  ( ( cls `  J
) `  S )
) )
308, 29syl5bb 248 . . . . 5  |-  ( (
ph  /\  A  e.  T )  ->  (
0  =  ( F `
 A )  <->  A  e.  ( ( cls `  J
) `  S )
) )
31 cldcls 16835 . . . . . . 7  |-  ( S  e.  ( Clsd `  J
)  ->  ( ( cls `  J ) `  S )  =  S )
3212, 31syl 15 . . . . . 6  |-  ( (
ph  /\  A  e.  T )  ->  (
( cls `  J
) `  S )  =  S )
3332eleq2d 2383 . . . . 5  |-  ( (
ph  /\  A  e.  T )  ->  ( A  e.  ( ( cls `  J ) `  S )  <->  A  e.  S ) )
3430, 33bitrd 244 . . . 4  |-  ( (
ph  /\  A  e.  T )  ->  (
0  =  ( F `
 A )  <->  A  e.  S ) )
357, 34mtbird 292 . . 3  |-  ( (
ph  /\  A  e.  T )  ->  -.  0  =  ( F `  A ) )
3627metdsf 18404 . . . . . . . 8  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X
)  ->  F : X
--> ( 0 [,]  +oo ) )
3710, 19, 36syl2anc 642 . . . . . . 7  |-  ( (
ph  /\  A  e.  T )  ->  F : X --> ( 0 [,] 
+oo ) )
38 ffvelrn 5701 . . . . . . 7  |-  ( ( F : X --> ( 0 [,]  +oo )  /\  A  e.  X )  ->  ( F `  A )  e.  ( 0 [,]  +oo ) )
3937, 26, 38syl2anc 642 . . . . . 6  |-  ( (
ph  /\  A  e.  T )  ->  ( F `  A )  e.  ( 0 [,]  +oo ) )
40 elxrge0 10794 . . . . . . 7  |-  ( ( F `  A )  e.  ( 0 [,] 
+oo )  <->  ( ( F `  A )  e.  RR*  /\  0  <_ 
( F `  A
) ) )
4140simprbi 450 . . . . . 6  |-  ( ( F `  A )  e.  ( 0 [,] 
+oo )  ->  0  <_  ( F `  A
) )
4239, 41syl 15 . . . . 5  |-  ( (
ph  /\  A  e.  T )  ->  0  <_  ( F `  A
) )
43 0xr 8923 . . . . . 6  |-  0  e.  RR*
4440simplbi 446 . . . . . . 7  |-  ( ( F `  A )  e.  ( 0 [,] 
+oo )  ->  ( F `  A )  e.  RR* )
4539, 44syl 15 . . . . . 6  |-  ( (
ph  /\  A  e.  T )  ->  ( F `  A )  e.  RR* )
46 xrleloe 10525 . . . . . 6  |-  ( ( 0  e.  RR*  /\  ( F `  A )  e.  RR* )  ->  (
0  <_  ( F `  A )  <->  ( 0  <  ( F `  A )  \/  0  =  ( F `  A ) ) ) )
4743, 45, 46sylancr 644 . . . . 5  |-  ( (
ph  /\  A  e.  T )  ->  (
0  <_  ( F `  A )  <->  ( 0  <  ( F `  A )  \/  0  =  ( F `  A ) ) ) )
4842, 47mpbid 201 . . . 4  |-  ( (
ph  /\  A  e.  T )  ->  (
0  <  ( F `  A )  \/  0  =  ( F `  A ) ) )
4948ord 366 . . 3  |-  ( (
ph  /\  A  e.  T )  ->  ( -.  0  <  ( F `
 A )  -> 
0  =  ( F `
 A ) ) )
5035, 49mt3d 117 . 2  |-  ( (
ph  /\  A  e.  T )  ->  0  <  ( F `  A
) )
51 1re 8882 . . . . . 6  |-  1  e.  RR
52 rexr 8922 . . . . . 6  |-  ( 1  e.  RR  ->  1  e.  RR* )
5351, 52ax-mp 8 . . . . 5  |-  1  e.  RR*
54 ifcl 3635 . . . . 5  |-  ( ( 1  e.  RR*  /\  ( F `  A )  e.  RR* )  ->  if ( 1  <_  ( F `  A ) ,  1 ,  ( F `  A ) )  e.  RR* )
5553, 45, 54sylancr 644 . . . 4  |-  ( (
ph  /\  A  e.  T )  ->  if ( 1  <_  ( F `  A ) ,  1 ,  ( F `  A ) )  e.  RR* )
5651a1i 10 . . . 4  |-  ( (
ph  /\  A  e.  T )  ->  1  e.  RR )
57 0lt1 9341 . . . . . 6  |-  0  <  1
58 breq2 4064 . . . . . . 7  |-  ( 1  =  if ( 1  <_  ( F `  A ) ,  1 ,  ( F `  A ) )  -> 
( 0  <  1  <->  0  <  if ( 1  <_  ( F `  A ) ,  1 ,  ( F `  A ) ) ) )
59 breq2 4064 . . . . . . 7  |-  ( ( F `  A )  =  if ( 1  <_  ( F `  A ) ,  1 ,  ( F `  A ) )  -> 
( 0  <  ( F `  A )  <->  0  <  if ( 1  <_  ( F `  A ) ,  1 ,  ( F `  A ) ) ) )
6058, 59ifboth 3630 . . . . . 6  |-  ( ( 0  <  1  /\  0  <  ( F `
 A ) )  ->  0  <  if ( 1  <_  ( F `  A ) ,  1 ,  ( F `  A ) ) )
6157, 50, 60sylancr 644 . . . . 5  |-  ( (
ph  /\  A  e.  T )  ->  0  <  if ( 1  <_ 
( F `  A
) ,  1 ,  ( F `  A
) ) )
62 xrltle 10530 . . . . . 6  |-  ( ( 0  e.  RR*  /\  if ( 1  <_  ( F `  A ) ,  1 ,  ( F `  A ) )  e.  RR* )  ->  ( 0  <  if ( 1  <_  ( F `  A ) ,  1 ,  ( F `  A ) )  ->  0  <_  if ( 1  <_  ( F `  A ) ,  1 ,  ( F `  A ) ) ) )
6343, 55, 62sylancr 644 . . . . 5  |-  ( (
ph  /\  A  e.  T )  ->  (
0  <  if (
1  <_  ( F `  A ) ,  1 ,  ( F `  A ) )  -> 
0  <_  if (
1  <_  ( F `  A ) ,  1 ,  ( F `  A ) ) ) )
6461, 63mpd 14 . . . 4  |-  ( (
ph  /\  A  e.  T )  ->  0  <_  if ( 1  <_ 
( F `  A
) ,  1 ,  ( F `  A
) ) )
65 xrmin1 10553 . . . . 5  |-  ( ( 1  e.  RR*  /\  ( F `  A )  e.  RR* )  ->  if ( 1  <_  ( F `  A ) ,  1 ,  ( F `  A ) )  <_  1 )
6653, 45, 65sylancr 644 . . . 4  |-  ( (
ph  /\  A  e.  T )  ->  if ( 1  <_  ( F `  A ) ,  1 ,  ( F `  A ) )  <_  1 )
67 xrrege0 10550 . . . 4  |-  ( ( ( if ( 1  <_  ( F `  A ) ,  1 ,  ( F `  A ) )  e. 
RR*  /\  1  e.  RR )  /\  (
0  <_  if (
1  <_  ( F `  A ) ,  1 ,  ( F `  A ) )  /\  if ( 1  <_  ( F `  A ) ,  1 ,  ( F `  A ) )  <_  1 ) )  ->  if (
1  <_  ( F `  A ) ,  1 ,  ( F `  A ) )  e.  RR )
6855, 56, 64, 66, 67syl22anc 1183 . . 3  |-  ( (
ph  /\  A  e.  T )  ->  if ( 1  <_  ( F `  A ) ,  1 ,  ( F `  A ) )  e.  RR )
6968, 61elrpd 10435 . 2  |-  ( (
ph  /\  A  e.  T )  ->  if ( 1  <_  ( F `  A ) ,  1 ,  ( F `  A ) )  e.  RR+ )
7050, 69jca 518 1  |-  ( (
ph  /\  A  e.  T )  ->  (
0  <  ( F `  A )  /\  if ( 1  <_  ( F `  A ) ,  1 ,  ( F `  A ) )  e.  RR+ )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1633    e. wcel 1701    =/= wne 2479    i^i cin 3185    C_ wss 3186   (/)c0 3489   ifcif 3599   U.cuni 3864   class class class wbr 4060    e. cmpt 4114   `'ccnv 4725   ran crn 4727   -->wf 5288   ` cfv 5292  (class class class)co 5900   supcsup 7238   RRcr 8781   0cc0 8782   1c1 8783    +oocpnf 8909   RR*cxr 8911    < clt 8912    <_ cle 8913   RR+crp 10401   [,]cicc 10706   * Metcxmt 16418   MetOpencmopn 16423   Clsdccld 16809   clsccl 16811
This theorem is referenced by:  metnrmlem2  18416  metnrmlem3  18417
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859  ax-pre-sup 8860
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-iin 3945  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-er 6702  df-map 6817  df-en 6907  df-dom 6908  df-sdom 6909  df-sup 7239  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-div 9469  df-nn 9792  df-2 9849  df-n0 10013  df-z 10072  df-uz 10278  df-q 10364  df-rp 10402  df-xneg 10499  df-xadd 10500  df-xmul 10501  df-icc 10710  df-topgen 13393  df-xmet 16425  df-bl 16427  df-mopn 16428  df-top 16692  df-bases 16694  df-topon 16695  df-cld 16812  df-ntr 16813  df-cls 16814
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