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Theorem logcnlem2 20534
Description: Lemma for logcn 20538. (Contributed by Mario Carneiro, 25-Feb-2015.)
Hypotheses
Ref Expression
logcn.d  |-  D  =  ( CC  \  (  -oo (,] 0 ) )
logcnlem.s  |-  S  =  if ( A  e.  RR+ ,  A ,  ( abs `  ( Im
`  A ) ) )
logcnlem.t  |-  T  =  ( ( abs `  A
)  x.  ( R  /  ( 1  +  R ) ) )
logcnlem.a  |-  ( ph  ->  A  e.  D )
logcnlem.r  |-  ( ph  ->  R  e.  RR+ )
Assertion
Ref Expression
logcnlem2  |-  ( ph  ->  if ( S  <_  T ,  S ,  T )  e.  RR+ )

Proof of Theorem logcnlem2
StepHypRef Expression
1 logcnlem.s . . 3  |-  S  =  if ( A  e.  RR+ ,  A ,  ( abs `  ( Im
`  A ) ) )
2 simpr 448 . . . 4  |-  ( (
ph  /\  A  e.  RR+ )  ->  A  e.  RR+ )
3 logcnlem.a . . . . . . . . 9  |-  ( ph  ->  A  e.  D )
4 logcn.d . . . . . . . . . . 11  |-  D  =  ( CC  \  (  -oo (,] 0 ) )
54ellogdm 20530 . . . . . . . . . 10  |-  ( A  e.  D  <->  ( A  e.  CC  /\  ( A  e.  RR  ->  A  e.  RR+ ) ) )
65simplbi 447 . . . . . . . . 9  |-  ( A  e.  D  ->  A  e.  CC )
73, 6syl 16 . . . . . . . 8  |-  ( ph  ->  A  e.  CC )
87imcld 12000 . . . . . . 7  |-  ( ph  ->  ( Im `  A
)  e.  RR )
98adantr 452 . . . . . 6  |-  ( (
ph  /\  -.  A  e.  RR+ )  ->  (
Im `  A )  e.  RR )
109recnd 9114 . . . . 5  |-  ( (
ph  /\  -.  A  e.  RR+ )  ->  (
Im `  A )  e.  CC )
11 reim0b 11924 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( A  e.  RR  <->  ( Im `  A )  =  0 ) )
127, 11syl 16 . . . . . . . 8  |-  ( ph  ->  ( A  e.  RR  <->  ( Im `  A )  =  0 ) )
135simprbi 451 . . . . . . . . 9  |-  ( A  e.  D  ->  ( A  e.  RR  ->  A  e.  RR+ ) )
143, 13syl 16 . . . . . . . 8  |-  ( ph  ->  ( A  e.  RR  ->  A  e.  RR+ )
)
1512, 14sylbird 227 . . . . . . 7  |-  ( ph  ->  ( ( Im `  A )  =  0  ->  A  e.  RR+ ) )
1615necon3bd 2638 . . . . . 6  |-  ( ph  ->  ( -.  A  e.  RR+  ->  ( Im `  A )  =/=  0
) )
1716imp 419 . . . . 5  |-  ( (
ph  /\  -.  A  e.  RR+ )  ->  (
Im `  A )  =/=  0 )
1810, 17absrpcld 12250 . . . 4  |-  ( (
ph  /\  -.  A  e.  RR+ )  ->  ( abs `  ( Im `  A ) )  e.  RR+ )
192, 18ifclda 3766 . . 3  |-  ( ph  ->  if ( A  e.  RR+ ,  A ,  ( abs `  ( Im
`  A ) ) )  e.  RR+ )
201, 19syl5eqel 2520 . 2  |-  ( ph  ->  S  e.  RR+ )
21 logcnlem.t . . 3  |-  T  =  ( ( abs `  A
)  x.  ( R  /  ( 1  +  R ) ) )
224logdmn0 20531 . . . . . 6  |-  ( A  e.  D  ->  A  =/=  0 )
233, 22syl 16 . . . . 5  |-  ( ph  ->  A  =/=  0 )
247, 23absrpcld 12250 . . . 4  |-  ( ph  ->  ( abs `  A
)  e.  RR+ )
25 logcnlem.r . . . . 5  |-  ( ph  ->  R  e.  RR+ )
26 1rp 10616 . . . . . 6  |-  1  e.  RR+
27 rpaddcl 10632 . . . . . 6  |-  ( ( 1  e.  RR+  /\  R  e.  RR+ )  ->  (
1  +  R )  e.  RR+ )
2826, 25, 27sylancr 645 . . . . 5  |-  ( ph  ->  ( 1  +  R
)  e.  RR+ )
2925, 28rpdivcld 10665 . . . 4  |-  ( ph  ->  ( R  /  (
1  +  R ) )  e.  RR+ )
3024, 29rpmulcld 10664 . . 3  |-  ( ph  ->  ( ( abs `  A
)  x.  ( R  /  ( 1  +  R ) ) )  e.  RR+ )
3121, 30syl5eqel 2520 . 2  |-  ( ph  ->  T  e.  RR+ )
32 ifcl 3775 . 2  |-  ( ( S  e.  RR+  /\  T  e.  RR+ )  ->  if ( S  <_  T ,  S ,  T )  e.  RR+ )
3320, 31, 32syl2anc 643 1  |-  ( ph  ->  if ( S  <_  T ,  S ,  T )  e.  RR+ )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599    \ cdif 3317   ifcif 3739   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   CCcc 8988   RRcr 8989   0cc0 8990   1c1 8991    + caddc 8993    x. cmul 8995    -oocmnf 9118    <_ cle 9121    / cdiv 9677   RR+crp 10612   (,]cioc 10917   Imcim 11903   abscabs 12039
This theorem is referenced by:  logcnlem5  20537
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-ioc 10921  df-seq 11324  df-exp 11383  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041
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