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Theorem metnrmlem1a 20276
Description: Lemma for metnrm 20280. (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 462 . . . . 5  |-  ( (
ph  /\  A  e.  T )  ->  ( S  i^i  T )  =  (/) )
3 inelcm 3721 . . . . . . . 8  |-  ( ( A  e.  S  /\  A  e.  T )  ->  ( S  i^i  T
)  =/=  (/) )
43expcom 435 . . . . . . 7  |-  ( A  e.  T  ->  ( A  e.  S  ->  ( S  i^i  T )  =/=  (/) ) )
54adantl 463 . . . . . 6  |-  ( (
ph  /\  A  e.  T )  ->  ( A  e.  S  ->  ( S  i^i  T )  =/=  (/) ) )
65necon2bd 2650 . . . . 5  |-  ( (
ph  /\  A  e.  T )  ->  (
( S  i^i  T
)  =  (/)  ->  -.  A  e.  S )
)
72, 6mpd 15 . . . 4  |-  ( (
ph  /\  A  e.  T )  ->  -.  A  e.  S )
8 eqcom 2435 . . . . . 6  |-  ( 0  =  ( F `  A )  <->  ( F `  A )  =  0 )
9 metnrmlem.1 . . . . . . . 8  |-  ( ph  ->  D  e.  ( *Met `  X ) )
109adantr 462 . . . . . . 7  |-  ( (
ph  /\  A  e.  T )  ->  D  e.  ( *Met `  X ) )
11 metnrmlem.2 . . . . . . . . . 10  |-  ( ph  ->  S  e.  ( Clsd `  J ) )
1211adantr 462 . . . . . . . . 9  |-  ( (
ph  /\  A  e.  T )  ->  S  e.  ( Clsd `  J
) )
13 eqid 2433 . . . . . . . . . 10  |-  U. J  =  U. J
1413cldss 18475 . . . . . . . . 9  |-  ( S  e.  ( Clsd `  J
)  ->  S  C_  U. J
)
1512, 14syl 16 . . . . . . . 8  |-  ( (
ph  /\  A  e.  T )  ->  S  C_ 
U. J )
16 metdscn.j . . . . . . . . . 10  |-  J  =  ( MetOpen `  D )
1716mopnuni 19858 . . . . . . . . 9  |-  ( D  e.  ( *Met `  X )  ->  X  =  U. J )
1810, 17syl 16 . . . . . . . 8  |-  ( (
ph  /\  A  e.  T )  ->  X  =  U. J )
1915, 18sseqtr4d 3381 . . . . . . 7  |-  ( (
ph  /\  A  e.  T )  ->  S  C_  X )
20 metnrmlem.3 . . . . . . . . . . 11  |-  ( ph  ->  T  e.  ( Clsd `  J ) )
2120adantr 462 . . . . . . . . . 10  |-  ( (
ph  /\  A  e.  T )  ->  T  e.  ( Clsd `  J
) )
2213cldss 18475 . . . . . . . . . 10  |-  ( T  e.  ( Clsd `  J
)  ->  T  C_  U. J
)
2321, 22syl 16 . . . . . . . . 9  |-  ( (
ph  /\  A  e.  T )  ->  T  C_ 
U. J )
2423, 18sseqtr4d 3381 . . . . . . . 8  |-  ( (
ph  /\  A  e.  T )  ->  T  C_  X )
25 simpr 458 . . . . . . . 8  |-  ( (
ph  /\  A  e.  T )  ->  A  e.  T )
2624, 25sseldd 3345 . . . . . . 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 20272 . . . . . . 7  |-  ( ( D  e.  ( *Met `  X )  /\  S  C_  X  /\  A  e.  X
)  ->  ( ( F `  A )  =  0  <->  A  e.  ( ( cls `  J
) `  S )
) )
2910, 19, 26, 28syl3anc 1211 . . . . . 6  |-  ( (
ph  /\  A  e.  T )  ->  (
( F `  A
)  =  0  <->  A  e.  ( ( cls `  J
) `  S )
) )
308, 29syl5bb 257 . . . . 5  |-  ( (
ph  /\  A  e.  T )  ->  (
0  =  ( F `
 A )  <->  A  e.  ( ( cls `  J
) `  S )
) )
31 cldcls 18488 . . . . . . 7  |-  ( S  e.  ( Clsd `  J
)  ->  ( ( cls `  J ) `  S )  =  S )
3212, 31syl 16 . . . . . 6  |-  ( (
ph  /\  A  e.  T )  ->  (
( cls `  J
) `  S )  =  S )
3332eleq2d 2500 . . . . 5  |-  ( (
ph  /\  A  e.  T )  ->  ( A  e.  ( ( cls `  J ) `  S )  <->  A  e.  S ) )
3430, 33bitrd 253 . . . 4  |-  ( (
ph  /\  A  e.  T )  ->  (
0  =  ( F `
 A )  <->  A  e.  S ) )
357, 34mtbird 301 . . 3  |-  ( (
ph  /\  A  e.  T )  ->  -.  0  =  ( F `  A ) )
3627metdsf 20266 . . . . . . . 8  |-  ( ( D  e.  ( *Met `  X )  /\  S  C_  X
)  ->  F : X
--> ( 0 [,] +oo ) )
3710, 19, 36syl2anc 654 . . . . . . 7  |-  ( (
ph  /\  A  e.  T )  ->  F : X --> ( 0 [,] +oo ) )
3837, 26ffvelrnd 5832 . . . . . 6  |-  ( (
ph  /\  A  e.  T )  ->  ( F `  A )  e.  ( 0 [,] +oo ) )
39 elxrge0 11381 . . . . . . 7  |-  ( ( F `  A )  e.  ( 0 [,] +oo )  <->  ( ( F `
 A )  e. 
RR*  /\  0  <_  ( F `  A ) ) )
4039simprbi 461 . . . . . 6  |-  ( ( F `  A )  e.  ( 0 [,] +oo )  ->  0  <_ 
( F `  A
) )
4138, 40syl 16 . . . . 5  |-  ( (
ph  /\  A  e.  T )  ->  0  <_  ( F `  A
) )
42 0xr 9418 . . . . . 6  |-  0  e.  RR*
4339simplbi 457 . . . . . . 7  |-  ( ( F `  A )  e.  ( 0 [,] +oo )  ->  ( F `
 A )  e. 
RR* )
4438, 43syl 16 . . . . . 6  |-  ( (
ph  /\  A  e.  T )  ->  ( F `  A )  e.  RR* )
45 xrleloe 11109 . . . . . 6  |-  ( ( 0  e.  RR*  /\  ( F `  A )  e.  RR* )  ->  (
0  <_  ( F `  A )  <->  ( 0  <  ( F `  A )  \/  0  =  ( F `  A ) ) ) )
4642, 44, 45sylancr 656 . . . . 5  |-  ( (
ph  /\  A  e.  T )  ->  (
0  <_  ( F `  A )  <->  ( 0  <  ( F `  A )  \/  0  =  ( F `  A ) ) ) )
4741, 46mpbid 210 . . . 4  |-  ( (
ph  /\  A  e.  T )  ->  (
0  <  ( F `  A )  \/  0  =  ( F `  A ) ) )
4847ord 377 . . 3  |-  ( (
ph  /\  A  e.  T )  ->  ( -.  0  <  ( F `
 A )  -> 
0  =  ( F `
 A ) ) )
4935, 48mt3d 125 . 2  |-  ( (
ph  /\  A  e.  T )  ->  0  <  ( F `  A
) )
50 1re 9373 . . . . . 6  |-  1  e.  RR
5150rexri 9424 . . . . 5  |-  1  e.  RR*
52 ifcl 3819 . . . . 5  |-  ( ( 1  e.  RR*  /\  ( F `  A )  e.  RR* )  ->  if ( 1  <_  ( F `  A ) ,  1 ,  ( F `  A ) )  e.  RR* )
5351, 44, 52sylancr 656 . . . 4  |-  ( (
ph  /\  A  e.  T )  ->  if ( 1  <_  ( F `  A ) ,  1 ,  ( F `  A ) )  e.  RR* )
54 1red 9389 . . . 4  |-  ( (
ph  /\  A  e.  T )  ->  1  e.  RR )
55 0lt1 9850 . . . . . 6  |-  0  <  1
56 breq2 4284 . . . . . . 7  |-  ( 1  =  if ( 1  <_  ( F `  A ) ,  1 ,  ( F `  A ) )  -> 
( 0  <  1  <->  0  <  if ( 1  <_  ( F `  A ) ,  1 ,  ( F `  A ) ) ) )
57 breq2 4284 . . . . . . 7  |-  ( ( F `  A )  =  if ( 1  <_  ( F `  A ) ,  1 ,  ( F `  A ) )  -> 
( 0  <  ( F `  A )  <->  0  <  if ( 1  <_  ( F `  A ) ,  1 ,  ( F `  A ) ) ) )
5856, 57ifboth 3813 . . . . . 6  |-  ( ( 0  <  1  /\  0  <  ( F `
 A ) )  ->  0  <  if ( 1  <_  ( F `  A ) ,  1 ,  ( F `  A ) ) )
5955, 49, 58sylancr 656 . . . . 5  |-  ( (
ph  /\  A  e.  T )  ->  0  <  if ( 1  <_ 
( F `  A
) ,  1 ,  ( F `  A
) ) )
60 xrltle 11114 . . . . . 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 ) ) ) )
6142, 53, 60sylancr 656 . . . . 5  |-  ( (
ph  /\  A  e.  T )  ->  (
0  <  if (
1  <_  ( F `  A ) ,  1 ,  ( F `  A ) )  -> 
0  <_  if (
1  <_  ( F `  A ) ,  1 ,  ( F `  A ) ) ) )
6259, 61mpd 15 . . . 4  |-  ( (
ph  /\  A  e.  T )  ->  0  <_  if ( 1  <_ 
( F `  A
) ,  1 ,  ( F `  A
) ) )
63 xrmin1 11137 . . . . 5  |-  ( ( 1  e.  RR*  /\  ( F `  A )  e.  RR* )  ->  if ( 1  <_  ( F `  A ) ,  1 ,  ( F `  A ) )  <_  1 )
6451, 44, 63sylancr 656 . . . 4  |-  ( (
ph  /\  A  e.  T )  ->  if ( 1  <_  ( F `  A ) ,  1 ,  ( F `  A ) )  <_  1 )
65 xrrege0 11134 . . . 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 )
6653, 54, 62, 64, 65syl22anc 1212 . . 3  |-  ( (
ph  /\  A  e.  T )  ->  if ( 1  <_  ( F `  A ) ,  1 ,  ( F `  A ) )  e.  RR )
6766, 59elrpd 11013 . 2  |-  ( (
ph  /\  A  e.  T )  ->  if ( 1  <_  ( F `  A ) ,  1 ,  ( F `  A ) )  e.  RR+ )
6849, 67jca 529 1  |-  ( (
ph  /\  A  e.  T )  ->  (
0  <  ( F `  A )  /\  if ( 1  <_  ( F `  A ) ,  1 ,  ( F `  A ) )  e.  RR+ )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1362    e. wcel 1755    =/= wne 2596    i^i cin 3315    C_ wss 3316   (/)c0 3625   ifcif 3779   U.cuni 4079   class class class wbr 4280    e. cmpt 4338   `'ccnv 4826   ran crn 4828   -->wf 5402   ` cfv 5406  (class class class)co 6080   supcsup 7678   RRcr 9269   0cc0 9270   1c1 9271   +oocpnf 9403   RR*cxr 9405    < clt 9406    <_ cle 9407   RR+crp 10979   [,]cicc 11291   *Metcxmt 17645   MetOpencmopn 17650   Clsdccld 18462   clsccl 18464
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-iin 4162  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-er 7089  df-map 7204  df-en 7299  df-dom 7300  df-sdom 7301  df-sup 7679  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-2 10368  df-n0 10568  df-z 10635  df-uz 10850  df-q 10942  df-rp 10980  df-xneg 11077  df-xadd 11078  df-xmul 11079  df-icc 11295  df-topgen 14365  df-psmet 17653  df-xmet 17654  df-bl 17656  df-mopn 17657  df-top 18345  df-bases 18347  df-topon 18348  df-cld 18465  df-ntr 18466  df-cls 18467
This theorem is referenced by:  metnrmlem2  20278  metnrmlem3  20279
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