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Theorem dvlog2lem 23676
Description: Lemma for dvlog2 23677. (Contributed by Mario Carneiro, 1-Mar-2015.)
Hypothesis
Ref Expression
dvlog2.s  |-  S  =  ( 1 ( ball `  ( abs  o.  -  ) ) 1 )
Assertion
Ref Expression
dvlog2lem  |-  S  C_  ( CC  \  ( -oo (,] 0 ) )

Proof of Theorem dvlog2lem
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 dvlog2.s . . . . 5  |-  S  =  ( 1 ( ball `  ( abs  o.  -  ) ) 1 )
2 cnxmet 21871 . . . . . 6  |-  ( abs 
o.  -  )  e.  ( *Met `  CC )
3 ax-1cn 9615 . . . . . 6  |-  1  e.  CC
4 1re 9660 . . . . . . 7  |-  1  e.  RR
54rexri 9711 . . . . . 6  |-  1  e.  RR*
6 blssm 21511 . . . . . 6  |-  ( ( ( abs  o.  -  )  e.  ( *Met `  CC )  /\  1  e.  CC  /\  1  e.  RR* )  ->  (
1 ( ball `  ( abs  o.  -  ) ) 1 )  C_  CC )
72, 3, 5, 6mp3an 1390 . . . . 5  |-  ( 1 ( ball `  ( abs  o.  -  ) ) 1 )  C_  CC
81, 7eqsstri 3448 . . . 4  |-  S  C_  CC
98sseli 3414 . . 3  |-  ( x  e.  S  ->  x  e.  CC )
10 1m0e1 10742 . . . . . . . . 9  |-  ( 1  -  0 )  =  1
11 mnfxr 11437 . . . . . . . . . . . 12  |- -oo  e.  RR*
12 0re 9661 . . . . . . . . . . . 12  |-  0  e.  RR
13 iocssre 11739 . . . . . . . . . . . 12  |-  ( ( -oo  e.  RR*  /\  0  e.  RR )  ->  ( -oo (,] 0 )  C_  RR )
1411, 12, 13mp2an 686 . . . . . . . . . . 11  |-  ( -oo (,] 0 )  C_  RR
1514sseli 3414 . . . . . . . . . 10  |-  ( x  e.  ( -oo (,] 0 )  ->  x  e.  RR )
1612a1i 11 . . . . . . . . . 10  |-  ( x  e.  ( -oo (,] 0 )  ->  0  e.  RR )
174a1i 11 . . . . . . . . . 10  |-  ( x  e.  ( -oo (,] 0 )  ->  1  e.  RR )
18 elioc2 11722 . . . . . . . . . . . 12  |-  ( ( -oo  e.  RR*  /\  0  e.  RR )  ->  (
x  e.  ( -oo (,] 0 )  <->  ( x  e.  RR  /\ -oo  <  x  /\  x  <_  0
) ) )
1911, 12, 18mp2an 686 . . . . . . . . . . 11  |-  ( x  e.  ( -oo (,] 0 )  <->  ( x  e.  RR  /\ -oo  <  x  /\  x  <_  0
) )
2019simp3bi 1047 . . . . . . . . . 10  |-  ( x  e.  ( -oo (,] 0 )  ->  x  <_  0 )
2115, 16, 17, 20lesub2dd 10251 . . . . . . . . 9  |-  ( x  e.  ( -oo (,] 0 )  ->  (
1  -  0 )  <_  ( 1  -  x ) )
2210, 21syl5eqbrr 4430 . . . . . . . 8  |-  ( x  e.  ( -oo (,] 0 )  ->  1  <_  ( 1  -  x
) )
23 ax-resscn 9614 . . . . . . . . . . . 12  |-  RR  C_  CC
2414, 23sstri 3427 . . . . . . . . . . 11  |-  ( -oo (,] 0 )  C_  CC
2524sseli 3414 . . . . . . . . . 10  |-  ( x  e.  ( -oo (,] 0 )  ->  x  e.  CC )
26 eqid 2471 . . . . . . . . . . 11  |-  ( abs 
o.  -  )  =  ( abs  o.  -  )
2726cnmetdval 21869 . . . . . . . . . 10  |-  ( ( 1  e.  CC  /\  x  e.  CC )  ->  ( 1 ( abs 
o.  -  ) x
)  =  ( abs `  ( 1  -  x
) ) )
283, 25, 27sylancr 676 . . . . . . . . 9  |-  ( x  e.  ( -oo (,] 0 )  ->  (
1 ( abs  o.  -  ) x )  =  ( abs `  (
1  -  x ) ) )
29 0le1 10158 . . . . . . . . . . . 12  |-  0  <_  1
3029a1i 11 . . . . . . . . . . 11  |-  ( x  e.  ( -oo (,] 0 )  ->  0  <_  1 )
3115, 16, 17, 20, 30letrd 9809 . . . . . . . . . 10  |-  ( x  e.  ( -oo (,] 0 )  ->  x  <_  1 )
3215, 17, 31abssubge0d 13570 . . . . . . . . 9  |-  ( x  e.  ( -oo (,] 0 )  ->  ( abs `  ( 1  -  x ) )  =  ( 1  -  x
) )
3328, 32eqtrd 2505 . . . . . . . 8  |-  ( x  e.  ( -oo (,] 0 )  ->  (
1 ( abs  o.  -  ) x )  =  ( 1  -  x ) )
3422, 33breqtrrd 4422 . . . . . . 7  |-  ( x  e.  ( -oo (,] 0 )  ->  1  <_  ( 1 ( abs 
o.  -  ) x
) )
35 cnmet 21870 . . . . . . . . . 10  |-  ( abs 
o.  -  )  e.  ( Met `  CC )
3635a1i 11 . . . . . . . . 9  |-  ( x  e.  ( -oo (,] 0 )  ->  ( abs  o.  -  )  e.  ( Met `  CC ) )
373a1i 11 . . . . . . . . 9  |-  ( x  e.  ( -oo (,] 0 )  ->  1  e.  CC )
38 metcl 21425 . . . . . . . . 9  |-  ( ( ( abs  o.  -  )  e.  ( Met `  CC )  /\  1  e.  CC  /\  x  e.  CC )  ->  (
1 ( abs  o.  -  ) x )  e.  RR )
3936, 37, 25, 38syl3anc 1292 . . . . . . . 8  |-  ( x  e.  ( -oo (,] 0 )  ->  (
1 ( abs  o.  -  ) x )  e.  RR )
40 lenlt 9730 . . . . . . . 8  |-  ( ( 1  e.  RR  /\  ( 1 ( abs 
o.  -  ) x
)  e.  RR )  ->  ( 1  <_ 
( 1 ( abs 
o.  -  ) x
)  <->  -.  ( 1 ( abs  o.  -  ) x )  <  1 ) )
414, 39, 40sylancr 676 . . . . . . 7  |-  ( x  e.  ( -oo (,] 0 )  ->  (
1  <_  ( 1 ( abs  o.  -  ) x )  <->  -.  (
1 ( abs  o.  -  ) x )  <  1 ) )
4234, 41mpbid 215 . . . . . 6  |-  ( x  e.  ( -oo (,] 0 )  ->  -.  ( 1 ( abs 
o.  -  ) x
)  <  1 )
432a1i 11 . . . . . . 7  |-  ( x  e.  ( -oo (,] 0 )  ->  ( abs  o.  -  )  e.  ( *Met `  CC ) )
445a1i 11 . . . . . . 7  |-  ( x  e.  ( -oo (,] 0 )  ->  1  e.  RR* )
45 elbl2 21483 . . . . . . 7  |-  ( ( ( ( abs  o.  -  )  e.  ( *Met `  CC )  /\  1  e.  RR* )  /\  ( 1  e.  CC  /\  x  e.  CC ) )  -> 
( x  e.  ( 1 ( ball `  ( abs  o.  -  ) ) 1 )  <->  ( 1 ( abs  o.  -  ) x )  <  1 ) )
4643, 44, 37, 25, 45syl22anc 1293 . . . . . 6  |-  ( x  e.  ( -oo (,] 0 )  ->  (
x  e.  ( 1 ( ball `  ( abs  o.  -  ) ) 1 )  <->  ( 1 ( abs  o.  -  ) x )  <  1 ) )
4742, 46mtbird 308 . . . . 5  |-  ( x  e.  ( -oo (,] 0 )  ->  -.  x  e.  ( 1 ( ball `  ( abs  o.  -  ) ) 1 ) )
4847con2i 124 . . . 4  |-  ( x  e.  ( 1 (
ball `  ( abs  o. 
-  ) ) 1 )  ->  -.  x  e.  ( -oo (,] 0
) )
4948, 1eleq2s 2567 . . 3  |-  ( x  e.  S  ->  -.  x  e.  ( -oo (,] 0 ) )
509, 49eldifd 3401 . 2  |-  ( x  e.  S  ->  x  e.  ( CC  \  ( -oo (,] 0 ) ) )
5150ssriv 3422 1  |-  S  C_  ( CC  \  ( -oo (,] 0 ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 189    /\ w3a 1007    = wceq 1452    e. wcel 1904    \ cdif 3387    C_ wss 3390   class class class wbr 4395    o. ccom 4843   ` cfv 5589  (class class class)co 6308   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558   -oocmnf 9691   RR*cxr 9692    < clt 9693    <_ cle 9694    - cmin 9880   (,]cioc 11661   abscabs 13374   *Metcxmt 19032   Metcme 19033   ballcbl 19034
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-sup 7974  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-xadd 11433  df-ioc 11665  df-seq 12252  df-exp 12311  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042
This theorem is referenced by:  dvlog2  23677  logtayl  23684  logtayl2  23686  efrlim  23974  lgamcvg2  24059
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