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Theorem dvlog2lem 23011
Description: Lemma for dvlog2 23012. (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 21258 . . . . . 6  |-  ( abs 
o.  -  )  e.  ( *Met `  CC )
3 ax-1cn 9553 . . . . . 6  |-  1  e.  CC
4 1re 9598 . . . . . . 7  |-  1  e.  RR
54rexri 9649 . . . . . 6  |-  1  e.  RR*
6 blssm 20899 . . . . . 6  |-  ( ( ( abs  o.  -  )  e.  ( *Met `  CC )  /\  1  e.  CC  /\  1  e.  RR* )  ->  (
1 ( ball `  ( abs  o.  -  ) ) 1 )  C_  CC )
72, 3, 5, 6mp3an 1325 . . . . 5  |-  ( 1 ( ball `  ( abs  o.  -  ) ) 1 )  C_  CC
81, 7eqsstri 3519 . . . 4  |-  S  C_  CC
98sseli 3485 . . 3  |-  ( x  e.  S  ->  x  e.  CC )
10 1m0e1 10653 . . . . . . . . 9  |-  ( 1  -  0 )  =  1
11 mnfxr 11334 . . . . . . . . . . . 12  |- -oo  e.  RR*
12 0re 9599 . . . . . . . . . . . 12  |-  0  e.  RR
13 iocssre 11615 . . . . . . . . . . . 12  |-  ( ( -oo  e.  RR*  /\  0  e.  RR )  ->  ( -oo (,] 0 )  C_  RR )
1411, 12, 13mp2an 672 . . . . . . . . . . 11  |-  ( -oo (,] 0 )  C_  RR
1514sseli 3485 . . . . . . . . . 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 11598 . . . . . . . . . . . 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 1014 . . . . . . . . . 10  |-  ( x  e.  ( -oo (,] 0 )  ->  x  <_  0 )
2115, 16, 17, 20lesub2dd 10176 . . . . . . . . 9  |-  ( x  e.  ( -oo (,] 0 )  ->  (
1  -  0 )  <_  ( 1  -  x ) )
2210, 21syl5eqbrr 4471 . . . . . . . 8  |-  ( x  e.  ( -oo (,] 0 )  ->  1  <_  ( 1  -  x
) )
23 ax-resscn 9552 . . . . . . . . . . . 12  |-  RR  C_  CC
2414, 23sstri 3498 . . . . . . . . . . 11  |-  ( -oo (,] 0 )  C_  CC
2524sseli 3485 . . . . . . . . . 10  |-  ( x  e.  ( -oo (,] 0 )  ->  x  e.  CC )
26 eqid 2443 . . . . . . . . . . 11  |-  ( abs 
o.  -  )  =  ( abs  o.  -  )
2726cnmetdval 21256 . . . . . . . . . 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 10083 . . . . . . . . . . . 12  |-  0  <_  1
3029a1i 11 . . . . . . . . . . 11  |-  ( x  e.  ( -oo (,] 0 )  ->  0  <_  1 )
3115, 16, 17, 20, 30letrd 9742 . . . . . . . . . 10  |-  ( x  e.  ( -oo (,] 0 )  ->  x  <_  1 )
3215, 17, 31abssubge0d 13245 . . . . . . . . 9  |-  ( x  e.  ( -oo (,] 0 )  ->  ( abs `  ( 1  -  x ) )  =  ( 1  -  x
) )
3328, 32eqtrd 2484 . . . . . . . 8  |-  ( x  e.  ( -oo (,] 0 )  ->  (
1 ( abs  o.  -  ) x )  =  ( 1  -  x ) )
3422, 33breqtrrd 4463 . . . . . . 7  |-  ( x  e.  ( -oo (,] 0 )  ->  1  <_  ( 1 ( abs 
o.  -  ) x
) )
35 cnmet 21257 . . . . . . . . . 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 20813 . . . . . . . . 9  |-  ( ( ( abs  o.  -  )  e.  ( Met `  CC )  /\  1  e.  CC  /\  x  e.  CC )  ->  (
1 ( abs  o.  -  ) x )  e.  RR )
3936, 37, 25, 38syl3anc 1229 . . . . . . . 8  |-  ( x  e.  ( -oo (,] 0 )  ->  (
1 ( abs  o.  -  ) x )  e.  RR )
40 lenlt 9666 . . . . . . . 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 20871 . . . . . . 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 1230 . . . . . 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 2551 . . 3  |-  ( x  e.  S  ->  -.  x  e.  ( -oo (,] 0 ) )
509, 49eldifd 3472 . 2  |-  ( x  e.  S  ->  x  e.  ( CC  \  ( -oo (,] 0 ) ) )
5150ssriv 3493 1  |-  S  C_  ( CC  \  ( -oo (,] 0 ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    /\ w3a 974    = wceq 1383    e. wcel 1804    \ cdif 3458    C_ wss 3461   class class class wbr 4437    o. ccom 4993   ` cfv 5578  (class class class)co 6281   CCcc 9493   RRcr 9494   0cc0 9495   1c1 9496   -oocmnf 9629   RR*cxr 9630    < clt 9631    <_ cle 9632    - cmin 9810   (,]cioc 11541   abscabs 13049   *Metcxmt 18382   Metcme 18383   ballcbl 18384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-sup 7903  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-nn 10544  df-2 10601  df-3 10602  df-n0 10803  df-z 10872  df-uz 11093  df-rp 11232  df-xadd 11330  df-ioc 11545  df-seq 12090  df-exp 12149  df-cj 12914  df-re 12915  df-im 12916  df-sqrt 13050  df-abs 13051  df-psmet 18390  df-xmet 18391  df-met 18392  df-bl 18393
This theorem is referenced by:  dvlog2  23012  logtayl  23019  logtayl2  23021  efrlim  23277  lgamcvg2  28575
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