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Theorem logcnlem2 19990
Description: Lemma for logcn 19994. (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 447 . . . 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 19986 . . . . . . . . . 10  |-  ( A  e.  D  <->  ( A  e.  CC  /\  ( A  e.  RR  ->  A  e.  RR+ ) ) )
65simplbi 446 . . . . . . . . 9  |-  ( A  e.  D  ->  A  e.  CC )
73, 6syl 15 . . . . . . . 8  |-  ( ph  ->  A  e.  CC )
87imcld 11680 . . . . . . 7  |-  ( ph  ->  ( Im `  A
)  e.  RR )
98adantr 451 . . . . . 6  |-  ( (
ph  /\  -.  A  e.  RR+ )  ->  (
Im `  A )  e.  RR )
109recnd 8861 . . . . 5  |-  ( (
ph  /\  -.  A  e.  RR+ )  ->  (
Im `  A )  e.  CC )
11 reim0b 11604 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( A  e.  RR  <->  ( Im `  A )  =  0 ) )
127, 11syl 15 . . . . . . . 8  |-  ( ph  ->  ( A  e.  RR  <->  ( Im `  A )  =  0 ) )
135simprbi 450 . . . . . . . . 9  |-  ( A  e.  D  ->  ( A  e.  RR  ->  A  e.  RR+ ) )
143, 13syl 15 . . . . . . . 8  |-  ( ph  ->  ( A  e.  RR  ->  A  e.  RR+ )
)
1512, 14sylbird 226 . . . . . . 7  |-  ( ph  ->  ( ( Im `  A )  =  0  ->  A  e.  RR+ ) )
1615necon3bd 2483 . . . . . 6  |-  ( ph  ->  ( -.  A  e.  RR+  ->  ( Im `  A )  =/=  0
) )
1716imp 418 . . . . 5  |-  ( (
ph  /\  -.  A  e.  RR+ )  ->  (
Im `  A )  =/=  0 )
1810, 17absrpcld 11930 . . . 4  |-  ( (
ph  /\  -.  A  e.  RR+ )  ->  ( abs `  ( Im `  A ) )  e.  RR+ )
192, 18ifclda 3592 . . 3  |-  ( ph  ->  if ( A  e.  RR+ ,  A ,  ( abs `  ( Im
`  A ) ) )  e.  RR+ )
201, 19syl5eqel 2367 . 2  |-  ( ph  ->  S  e.  RR+ )
21 logcnlem.t . . 3  |-  T  =  ( ( abs `  A
)  x.  ( R  /  ( 1  +  R ) ) )
224logdmn0 19987 . . . . . 6  |-  ( A  e.  D  ->  A  =/=  0 )
233, 22syl 15 . . . . 5  |-  ( ph  ->  A  =/=  0 )
247, 23absrpcld 11930 . . . 4  |-  ( ph  ->  ( abs `  A
)  e.  RR+ )
25 logcnlem.r . . . . 5  |-  ( ph  ->  R  e.  RR+ )
26 1rp 10358 . . . . . 6  |-  1  e.  RR+
27 rpaddcl 10374 . . . . . 6  |-  ( ( 1  e.  RR+  /\  R  e.  RR+ )  ->  (
1  +  R )  e.  RR+ )
2826, 25, 27sylancr 644 . . . . 5  |-  ( ph  ->  ( 1  +  R
)  e.  RR+ )
2925, 28rpdivcld 10407 . . . 4  |-  ( ph  ->  ( R  /  (
1  +  R ) )  e.  RR+ )
3024, 29rpmulcld 10406 . . 3  |-  ( ph  ->  ( ( abs `  A
)  x.  ( R  /  ( 1  +  R ) ) )  e.  RR+ )
3121, 30syl5eqel 2367 . 2  |-  ( ph  ->  T  e.  RR+ )
32 ifcl 3601 . 2  |-  ( ( S  e.  RR+  /\  T  e.  RR+ )  ->  if ( S  <_  T ,  S ,  T )  e.  RR+ )
3320, 31, 32syl2anc 642 1  |-  ( ph  ->  if ( S  <_  T ,  S ,  T )  e.  RR+ )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446    \ cdif 3149   ifcif 3565   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    -oocmnf 8865    <_ cle 8868    / cdiv 9423   RR+crp 10354   (,]cioc 10657   Imcim 11583   abscabs 11719
This theorem is referenced by:  logcnlem5  19993
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-ioc 10661  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721
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