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Theorem dvlog2lem 22095
Description: Lemma for dvlog2 22096. (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 20350 . . . . . 6  |-  ( abs 
o.  -  )  e.  ( *Met `  CC )
3 ax-1cn 9338 . . . . . 6  |-  1  e.  CC
4 1re 9383 . . . . . . 7  |-  1  e.  RR
54rexri 9434 . . . . . 6  |-  1  e.  RR*
6 blssm 19991 . . . . . 6  |-  ( ( ( abs  o.  -  )  e.  ( *Met `  CC )  /\  1  e.  CC  /\  1  e.  RR* )  ->  (
1 ( ball `  ( abs  o.  -  ) ) 1 )  C_  CC )
72, 3, 5, 6mp3an 1314 . . . . 5  |-  ( 1 ( ball `  ( abs  o.  -  ) ) 1 )  C_  CC
81, 7eqsstri 3384 . . . 4  |-  S  C_  CC
98sseli 3350 . . 3  |-  ( x  e.  S  ->  x  e.  CC )
10 1m0e1 10430 . . . . . . . . 9  |-  ( 1  -  0 )  =  1
11 mnfxr 11092 . . . . . . . . . . . 12  |- -oo  e.  RR*
12 0re 9384 . . . . . . . . . . . 12  |-  0  e.  RR
13 iocssre 11373 . . . . . . . . . . . 12  |-  ( ( -oo  e.  RR*  /\  0  e.  RR )  ->  ( -oo (,] 0 )  C_  RR )
1411, 12, 13mp2an 672 . . . . . . . . . . 11  |-  ( -oo (,] 0 )  C_  RR
1514sseli 3350 . . . . . . . . . 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 11356 . . . . . . . . . . . 12  |-  ( ( -oo  e.  RR*  /\  0  e.  RR )  ->  (
x  e.  ( -oo (,] 0 )  <->  ( x  e.  RR  /\ -oo  <  x  /\  x  <_  0
) ) )
1911, 12, 18mp2an 672 . . . . . . . . . . 11  |-  ( x  e.  ( -oo (,] 0 )  <->  ( x  e.  RR  /\ -oo  <  x  /\  x  <_  0
) )
2019simp3bi 1005 . . . . . . . . . 10  |-  ( x  e.  ( -oo (,] 0 )  ->  x  <_  0 )
2115, 16, 17, 20lesub2dd 9954 . . . . . . . . 9  |-  ( x  e.  ( -oo (,] 0 )  ->  (
1  -  0 )  <_  ( 1  -  x ) )
2210, 21syl5eqbrr 4324 . . . . . . . 8  |-  ( x  e.  ( -oo (,] 0 )  ->  1  <_  ( 1  -  x
) )
23 ax-resscn 9337 . . . . . . . . . . . 12  |-  RR  C_  CC
2414, 23sstri 3363 . . . . . . . . . . 11  |-  ( -oo (,] 0 )  C_  CC
2524sseli 3350 . . . . . . . . . 10  |-  ( x  e.  ( -oo (,] 0 )  ->  x  e.  CC )
26 eqid 2441 . . . . . . . . . . 11  |-  ( abs 
o.  -  )  =  ( abs  o.  -  )
2726cnmetdval 20348 . . . . . . . . . 10  |-  ( ( 1  e.  CC  /\  x  e.  CC )  ->  ( 1 ( abs 
o.  -  ) x
)  =  ( abs `  ( 1  -  x
) ) )
283, 25, 27sylancr 663 . . . . . . . . 9  |-  ( x  e.  ( -oo (,] 0 )  ->  (
1 ( abs  o.  -  ) x )  =  ( abs `  (
1  -  x ) ) )
29 0le1 9861 . . . . . . . . . . . 12  |-  0  <_  1
3029a1i 11 . . . . . . . . . . 11  |-  ( x  e.  ( -oo (,] 0 )  ->  0  <_  1 )
3115, 16, 17, 20, 30letrd 9526 . . . . . . . . . 10  |-  ( x  e.  ( -oo (,] 0 )  ->  x  <_  1 )
3215, 17, 31abssubge0d 12916 . . . . . . . . 9  |-  ( x  e.  ( -oo (,] 0 )  ->  ( abs `  ( 1  -  x ) )  =  ( 1  -  x
) )
3328, 32eqtrd 2473 . . . . . . . 8  |-  ( x  e.  ( -oo (,] 0 )  ->  (
1 ( abs  o.  -  ) x )  =  ( 1  -  x ) )
3422, 33breqtrrd 4316 . . . . . . 7  |-  ( x  e.  ( -oo (,] 0 )  ->  1  <_  ( 1 ( abs 
o.  -  ) x
) )
35 cnmet 20349 . . . . . . . . . 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 19905 . . . . . . . . 9  |-  ( ( ( abs  o.  -  )  e.  ( Met `  CC )  /\  1  e.  CC  /\  x  e.  CC )  ->  (
1 ( abs  o.  -  ) x )  e.  RR )
3936, 37, 25, 38syl3anc 1218 . . . . . . . 8  |-  ( x  e.  ( -oo (,] 0 )  ->  (
1 ( abs  o.  -  ) x )  e.  RR )
40 lenlt 9451 . . . . . . . 8  |-  ( ( 1  e.  RR  /\  ( 1 ( abs 
o.  -  ) x
)  e.  RR )  ->  ( 1  <_ 
( 1 ( abs 
o.  -  ) x
)  <->  -.  ( 1 ( abs  o.  -  ) x )  <  1 ) )
414, 39, 40sylancr 663 . . . . . . 7  |-  ( x  e.  ( -oo (,] 0 )  ->  (
1  <_  ( 1 ( abs  o.  -  ) x )  <->  -.  (
1 ( abs  o.  -  ) x )  <  1 ) )
4234, 41mpbid 210 . . . . . 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 19963 . . . . . . 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 1219 . . . . . 6  |-  ( x  e.  ( -oo (,] 0 )  ->  (
x  e.  ( 1 ( ball `  ( abs  o.  -  ) ) 1 )  <->  ( 1 ( abs  o.  -  ) x )  <  1 ) )
4742, 46mtbird 301 . . . . 5  |-  ( x  e.  ( -oo (,] 0 )  ->  -.  x  e.  ( 1 ( ball `  ( abs  o.  -  ) ) 1 ) )
4847con2i 120 . . . 4  |-  ( x  e.  ( 1 (
ball `  ( abs  o. 
-  ) ) 1 )  ->  -.  x  e.  ( -oo (,] 0
) )
4948, 1eleq2s 2533 . . 3  |-  ( x  e.  S  ->  -.  x  e.  ( -oo (,] 0 ) )
509, 49eldifd 3337 . 2  |-  ( x  e.  S  ->  x  e.  ( CC  \  ( -oo (,] 0 ) ) )
5150ssriv 3358 1  |-  S  C_  ( CC  \  ( -oo (,] 0 ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    /\ w3a 965    = wceq 1369    e. wcel 1756    \ cdif 3323    C_ wss 3326   class class class wbr 4290    o. ccom 4842   ` cfv 5416  (class class class)co 6089   CCcc 9278   RRcr 9279   0cc0 9280   1c1 9281   -oocmnf 9414   RR*cxr 9415    < clt 9416    <_ cle 9417    - cmin 9593   (,]cioc 11299   abscabs 12721   *Metcxmt 17799   Metcme 17800   ballcbl 17801
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357  ax-pre-sup 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-recs 6830  df-rdg 6864  df-er 7099  df-map 7214  df-en 7309  df-dom 7310  df-sdom 7311  df-sup 7689  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-div 9992  df-nn 10321  df-2 10378  df-3 10379  df-n0 10578  df-z 10645  df-uz 10860  df-rp 10990  df-xadd 11088  df-ioc 11303  df-seq 11805  df-exp 11864  df-cj 12586  df-re 12587  df-im 12588  df-sqr 12722  df-abs 12723  df-psmet 17807  df-xmet 17808  df-met 17809  df-bl 17810
This theorem is referenced by:  dvlog2  22096  logtayl  22103  logtayl2  22105  efrlim  22361  lgamcvg2  27039
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