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Theorem metnrmlem1a 21488
Description: Lemma for metnrm 21492. (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 465 . . . . 5  |-  ( (
ph  /\  A  e.  T )  ->  ( S  i^i  T )  =  (/) )
3 inelcm 3884 . . . . . . . 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 466 . . . . . 6  |-  ( (
ph  /\  A  e.  T )  ->  ( A  e.  S  ->  ( S  i^i  T )  =/=  (/) ) )
65necon2bd 2672 . . . . 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 2466 . . . . . 6  |-  ( 0  =  ( F `  A )  <->  ( F `  A )  =  0 )
9 metnrmlem.1 . . . . . . . 8  |-  ( ph  ->  D  e.  ( *Met `  X ) )
109adantr 465 . . . . . . 7  |-  ( (
ph  /\  A  e.  T )  ->  D  e.  ( *Met `  X ) )
11 metnrmlem.2 . . . . . . . . . 10  |-  ( ph  ->  S  e.  ( Clsd `  J ) )
1211adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  A  e.  T )  ->  S  e.  ( Clsd `  J
) )
13 eqid 2457 . . . . . . . . . 10  |-  U. J  =  U. J
1413cldss 19657 . . . . . . . . 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 21070 . . . . . . . . 9  |-  ( D  e.  ( *Met `  X )  ->  X  =  U. J )
1810, 17syl 16 . . . . . . . 8  |-  ( (
ph  /\  A  e.  T )  ->  X  =  U. J )
1915, 18sseqtr4d 3536 . . . . . . 7  |-  ( (
ph  /\  A  e.  T )  ->  S  C_  X )
20 metnrmlem.3 . . . . . . . . . . 11  |-  ( ph  ->  T  e.  ( Clsd `  J ) )
2120adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  A  e.  T )  ->  T  e.  ( Clsd `  J
) )
2213cldss 19657 . . . . . . . . . 10  |-  ( T  e.  ( Clsd `  J
)  ->  T  C_  U. J
)
2321, 22syl 16 . . . . . . . . 9  |-  ( (
ph  /\  A  e.  T )  ->  T  C_ 
U. J )
2423, 18sseqtr4d 3536 . . . . . . . 8  |-  ( (
ph  /\  A  e.  T )  ->  T  C_  X )
25 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  A  e.  T )  ->  A  e.  T )
2624, 25sseldd 3500 . . . . . . 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 21484 . . . . . . 7  |-  ( ( D  e.  ( *Met `  X )  /\  S  C_  X  /\  A  e.  X
)  ->  ( ( F `  A )  =  0  <->  A  e.  ( ( cls `  J
) `  S )
) )
2910, 19, 26, 28syl3anc 1228 . . . . . 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 19670 . . . . . . 7  |-  ( S  e.  ( Clsd `  J
)  ->  ( ( cls `  J ) `  S )  =  S )
3212, 31syl 16 . . . . . 6  |-  ( (
ph  /\  A  e.  T )  ->  (
( cls `  J
) `  S )  =  S )
3332eleq2d 2527 . . . . 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 21478 . . . . . . . 8  |-  ( ( D  e.  ( *Met `  X )  /\  S  C_  X
)  ->  F : X
--> ( 0 [,] +oo ) )
3710, 19, 36syl2anc 661 . . . . . . 7  |-  ( (
ph  /\  A  e.  T )  ->  F : X --> ( 0 [,] +oo ) )
3837, 26ffvelrnd 6033 . . . . . 6  |-  ( (
ph  /\  A  e.  T )  ->  ( F `  A )  e.  ( 0 [,] +oo ) )
39 elxrge0 11654 . . . . . . 7  |-  ( ( F `  A )  e.  ( 0 [,] +oo )  <->  ( ( F `
 A )  e. 
RR*  /\  0  <_  ( F `  A ) ) )
4039simprbi 464 . . . . . 6  |-  ( ( F `  A )  e.  ( 0 [,] +oo )  ->  0  <_ 
( F `  A
) )
4138, 40syl 16 . . . . 5  |-  ( (
ph  /\  A  e.  T )  ->  0  <_  ( F `  A
) )
42 0xr 9657 . . . . . 6  |-  0  e.  RR*
4339simplbi 460 . . . . . . 7  |-  ( ( F `  A )  e.  ( 0 [,] +oo )  ->  ( F `
 A )  e. 
RR* )
4438, 43syl 16 . . . . . 6  |-  ( (
ph  /\  A  e.  T )  ->  ( F `  A )  e.  RR* )
45 xrleloe 11375 . . . . . 6  |-  ( ( 0  e.  RR*  /\  ( F `  A )  e.  RR* )  ->  (
0  <_  ( F `  A )  <->  ( 0  <  ( F `  A )  \/  0  =  ( F `  A ) ) ) )
4642, 44, 45sylancr 663 . . . . 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 9612 . . . . . 6  |-  1  e.  RR
5150rexri 9663 . . . . 5  |-  1  e.  RR*
52 ifcl 3986 . . . . 5  |-  ( ( 1  e.  RR*  /\  ( F `  A )  e.  RR* )  ->  if ( 1  <_  ( F `  A ) ,  1 ,  ( F `  A ) )  e.  RR* )
5351, 44, 52sylancr 663 . . . 4  |-  ( (
ph  /\  A  e.  T )  ->  if ( 1  <_  ( F `  A ) ,  1 ,  ( F `  A ) )  e.  RR* )
54 1red 9628 . . . 4  |-  ( (
ph  /\  A  e.  T )  ->  1  e.  RR )
55 0lt1 10096 . . . . . 6  |-  0  <  1
56 breq2 4460 . . . . . . 7  |-  ( 1  =  if ( 1  <_  ( F `  A ) ,  1 ,  ( F `  A ) )  -> 
( 0  <  1  <->  0  <  if ( 1  <_  ( F `  A ) ,  1 ,  ( F `  A ) ) ) )
57 breq2 4460 . . . . . . 7  |-  ( ( F `  A )  =  if ( 1  <_  ( F `  A ) ,  1 ,  ( F `  A ) )  -> 
( 0  <  ( F `  A )  <->  0  <  if ( 1  <_  ( F `  A ) ,  1 ,  ( F `  A ) ) ) )
5856, 57ifboth 3980 . . . . . 6  |-  ( ( 0  <  1  /\  0  <  ( F `
 A ) )  ->  0  <  if ( 1  <_  ( F `  A ) ,  1 ,  ( F `  A ) ) )
5955, 49, 58sylancr 663 . . . . 5  |-  ( (
ph  /\  A  e.  T )  ->  0  <  if ( 1  <_ 
( F `  A
) ,  1 ,  ( F `  A
) ) )
60 xrltle 11380 . . . . . 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 663 . . . . 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 11403 . . . . 5  |-  ( ( 1  e.  RR*  /\  ( F `  A )  e.  RR* )  ->  if ( 1  <_  ( F `  A ) ,  1 ,  ( F `  A ) )  <_  1 )
6451, 44, 63sylancr 663 . . . 4  |-  ( (
ph  /\  A  e.  T )  ->  if ( 1  <_  ( F `  A ) ,  1 ,  ( F `  A ) )  <_  1 )
65 xrrege0 11400 . . . 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 1229 . . 3  |-  ( (
ph  /\  A  e.  T )  ->  if ( 1  <_  ( F `  A ) ,  1 ,  ( F `  A ) )  e.  RR )
6766, 59elrpd 11279 . 2  |-  ( (
ph  /\  A  e.  T )  ->  if ( 1  <_  ( F `  A ) ,  1 ,  ( F `  A ) )  e.  RR+ )
6849, 67jca 532 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 1395    e. wcel 1819    =/= wne 2652    i^i cin 3470    C_ wss 3471   (/)c0 3793   ifcif 3944   U.cuni 4251   class class class wbr 4456    |-> cmpt 4515   `'ccnv 5007   ran crn 5009   -->wf 5590   ` cfv 5594  (class class class)co 6296   supcsup 7918   RRcr 9508   0cc0 9509   1c1 9510   +oocpnf 9642   RR*cxr 9644    < clt 9645    <_ cle 9646   RR+crp 11245   [,]cicc 11557   *Metcxmt 18530   MetOpencmopn 18535   Clsdccld 19644   clsccl 19646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-icc 11561  df-topgen 14861  df-psmet 18538  df-xmet 18539  df-bl 18541  df-mopn 18542  df-top 19526  df-bases 19528  df-topon 19529  df-cld 19647  df-ntr 19648  df-cls 19649
This theorem is referenced by:  metnrmlem2  21490  metnrmlem3  21491
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