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Theorem dvlog2lem 22789
Description: Lemma for dvlog2 22790. (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 21043 . . . . . 6  |-  ( abs 
o.  -  )  e.  ( *Met `  CC )
3 ax-1cn 9550 . . . . . 6  |-  1  e.  CC
4 1re 9595 . . . . . . 7  |-  1  e.  RR
54rexri 9646 . . . . . 6  |-  1  e.  RR*
6 blssm 20684 . . . . . 6  |-  ( ( ( abs  o.  -  )  e.  ( *Met `  CC )  /\  1  e.  CC  /\  1  e.  RR* )  ->  (
1 ( ball `  ( abs  o.  -  ) ) 1 )  C_  CC )
72, 3, 5, 6mp3an 1324 . . . . 5  |-  ( 1 ( ball `  ( abs  o.  -  ) ) 1 )  C_  CC
81, 7eqsstri 3534 . . . 4  |-  S  C_  CC
98sseli 3500 . . 3  |-  ( x  e.  S  ->  x  e.  CC )
10 1m0e1 10646 . . . . . . . . 9  |-  ( 1  -  0 )  =  1
11 mnfxr 11323 . . . . . . . . . . . 12  |- -oo  e.  RR*
12 0re 9596 . . . . . . . . . . . 12  |-  0  e.  RR
13 iocssre 11604 . . . . . . . . . . . 12  |-  ( ( -oo  e.  RR*  /\  0  e.  RR )  ->  ( -oo (,] 0 )  C_  RR )
1411, 12, 13mp2an 672 . . . . . . . . . . 11  |-  ( -oo (,] 0 )  C_  RR
1514sseli 3500 . . . . . . . . . 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 11587 . . . . . . . . . . . 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 1013 . . . . . . . . . 10  |-  ( x  e.  ( -oo (,] 0 )  ->  x  <_  0 )
2115, 16, 17, 20lesub2dd 10169 . . . . . . . . 9  |-  ( x  e.  ( -oo (,] 0 )  ->  (
1  -  0 )  <_  ( 1  -  x ) )
2210, 21syl5eqbrr 4481 . . . . . . . 8  |-  ( x  e.  ( -oo (,] 0 )  ->  1  <_  ( 1  -  x
) )
23 ax-resscn 9549 . . . . . . . . . . . 12  |-  RR  C_  CC
2414, 23sstri 3513 . . . . . . . . . . 11  |-  ( -oo (,] 0 )  C_  CC
2524sseli 3500 . . . . . . . . . 10  |-  ( x  e.  ( -oo (,] 0 )  ->  x  e.  CC )
26 eqid 2467 . . . . . . . . . . 11  |-  ( abs 
o.  -  )  =  ( abs  o.  -  )
2726cnmetdval 21041 . . . . . . . . . 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 10076 . . . . . . . . . . . 12  |-  0  <_  1
3029a1i 11 . . . . . . . . . . 11  |-  ( x  e.  ( -oo (,] 0 )  ->  0  <_  1 )
3115, 16, 17, 20, 30letrd 9738 . . . . . . . . . 10  |-  ( x  e.  ( -oo (,] 0 )  ->  x  <_  1 )
3215, 17, 31abssubge0d 13226 . . . . . . . . 9  |-  ( x  e.  ( -oo (,] 0 )  ->  ( abs `  ( 1  -  x ) )  =  ( 1  -  x
) )
3328, 32eqtrd 2508 . . . . . . . 8  |-  ( x  e.  ( -oo (,] 0 )  ->  (
1 ( abs  o.  -  ) x )  =  ( 1  -  x ) )
3422, 33breqtrrd 4473 . . . . . . 7  |-  ( x  e.  ( -oo (,] 0 )  ->  1  <_  ( 1 ( abs 
o.  -  ) x
) )
35 cnmet 21042 . . . . . . . . . 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 20598 . . . . . . . . 9  |-  ( ( ( abs  o.  -  )  e.  ( Met `  CC )  /\  1  e.  CC  /\  x  e.  CC )  ->  (
1 ( abs  o.  -  ) x )  e.  RR )
3936, 37, 25, 38syl3anc 1228 . . . . . . . 8  |-  ( x  e.  ( -oo (,] 0 )  ->  (
1 ( abs  o.  -  ) x )  e.  RR )
40 lenlt 9663 . . . . . . . 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 20656 . . . . . . 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 1229 . . . . . 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 2575 . . 3  |-  ( x  e.  S  ->  -.  x  e.  ( -oo (,] 0 ) )
509, 49eldifd 3487 . 2  |-  ( x  e.  S  ->  x  e.  ( CC  \  ( -oo (,] 0 ) ) )
5150ssriv 3508 1  |-  S  C_  ( CC  \  ( -oo (,] 0 ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    /\ w3a 973    = wceq 1379    e. wcel 1767    \ cdif 3473    C_ wss 3476   class class class wbr 4447    o. ccom 5003   ` cfv 5588  (class class class)co 6284   CCcc 9490   RRcr 9491   0cc0 9492   1c1 9493   -oocmnf 9626   RR*cxr 9627    < clt 9628    <_ cle 9629    - cmin 9805   (,]cioc 11530   abscabs 13030   *Metcxmt 18202   Metcme 18203   ballcbl 18204
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-er 7311  df-map 7422  df-en 7517  df-dom 7518  df-sdom 7519  df-sup 7901  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-n0 10796  df-z 10865  df-uz 11083  df-rp 11221  df-xadd 11319  df-ioc 11534  df-seq 12076  df-exp 12135  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-psmet 18210  df-xmet 18211  df-met 18212  df-bl 18213
This theorem is referenced by:  dvlog2  22790  logtayl  22797  logtayl2  22799  efrlim  23055  lgamcvg2  28265
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