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Theorem logcnlem2 20043
Description: Lemma for logcn 20047. (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 20039 . . . . . . . . . 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 11727 . . . . . . 7  |-  ( ph  ->  ( Im `  A
)  e.  RR )
98adantr 451 . . . . . 6  |-  ( (
ph  /\  -.  A  e.  RR+ )  ->  (
Im `  A )  e.  RR )
109recnd 8906 . . . . 5  |-  ( (
ph  /\  -.  A  e.  RR+ )  ->  (
Im `  A )  e.  CC )
11 reim0b 11651 . . . . . . . . 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 2516 . . . . . 6  |-  ( ph  ->  ( -.  A  e.  RR+  ->  ( Im `  A )  =/=  0
) )
1716imp 418 . . . . 5  |-  ( (
ph  /\  -.  A  e.  RR+ )  ->  (
Im `  A )  =/=  0 )
1810, 17absrpcld 11977 . . . 4  |-  ( (
ph  /\  -.  A  e.  RR+ )  ->  ( abs `  ( Im `  A ) )  e.  RR+ )
192, 18ifclda 3626 . . 3  |-  ( ph  ->  if ( A  e.  RR+ ,  A ,  ( abs `  ( Im
`  A ) ) )  e.  RR+ )
201, 19syl5eqel 2400 . 2  |-  ( ph  ->  S  e.  RR+ )
21 logcnlem.t . . 3  |-  T  =  ( ( abs `  A
)  x.  ( R  /  ( 1  +  R ) ) )
224logdmn0 20040 . . . . . 6  |-  ( A  e.  D  ->  A  =/=  0 )
233, 22syl 15 . . . . 5  |-  ( ph  ->  A  =/=  0 )
247, 23absrpcld 11977 . . . 4  |-  ( ph  ->  ( abs `  A
)  e.  RR+ )
25 logcnlem.r . . . . 5  |-  ( ph  ->  R  e.  RR+ )
26 1rp 10405 . . . . . 6  |-  1  e.  RR+
27 rpaddcl 10421 . . . . . 6  |-  ( ( 1  e.  RR+  /\  R  e.  RR+ )  ->  (
1  +  R )  e.  RR+ )
2826, 25, 27sylancr 644 . . . . 5  |-  ( ph  ->  ( 1  +  R
)  e.  RR+ )
2925, 28rpdivcld 10454 . . . 4  |-  ( ph  ->  ( R  /  (
1  +  R ) )  e.  RR+ )
3024, 29rpmulcld 10453 . . 3  |-  ( ph  ->  ( ( abs `  A
)  x.  ( R  /  ( 1  +  R ) ) )  e.  RR+ )
3121, 30syl5eqel 2400 . 2  |-  ( ph  ->  T  e.  RR+ )
32 ifcl 3635 . 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 1633    e. wcel 1701    =/= wne 2479    \ cdif 3183   ifcif 3599   class class class wbr 4060   ` cfv 5292  (class class class)co 5900   CCcc 8780   RRcr 8781   0cc0 8782   1c1 8783    + caddc 8785    x. cmul 8787    -oocmnf 8910    <_ cle 8913    / cdiv 9468   RR+crp 10401   (,]cioc 10704   Imcim 11630   abscabs 11766
This theorem is referenced by:  logcnlem5  20046
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859  ax-pre-sup 8860
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-sup 7239  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-div 9469  df-nn 9792  df-2 9849  df-3 9850  df-n0 10013  df-z 10072  df-uz 10278  df-rp 10402  df-ioc 10708  df-seq 11094  df-exp 11152  df-cj 11631  df-re 11632  df-im 11633  df-sqr 11767  df-abs 11768
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