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Theorem logrec 20117
Description: Logarithm of a reciprocal changes sign. (Contributed by Saveliy Skresanov, 28-Dec-2016.)
Assertion
Ref Expression
logrec  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  (
Im `  ( log `  A ) )  =/= 
pi )  ->  ( log `  A )  = 
-u ( log `  (
1  /  A ) ) )

Proof of Theorem logrec
StepHypRef Expression
1 reccl 9431 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 1  /  A
)  e.  CC )
2 recne0 9437 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 1  /  A
)  =/=  0 )
3 eflog 19933 . . . . . . . 8  |-  ( ( ( 1  /  A
)  e.  CC  /\  ( 1  /  A
)  =/=  0 )  ->  ( exp `  ( log `  ( 1  /  A ) ) )  =  ( 1  /  A ) )
41, 2, 3syl2anc 642 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( exp `  ( log `  ( 1  /  A ) ) )  =  ( 1  /  A ) )
54eqcomd 2288 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 1  /  A
)  =  ( exp `  ( log `  (
1  /  A ) ) ) )
65oveq2d 5874 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 1  /  (
1  /  A ) )  =  ( 1  /  ( exp `  ( log `  ( 1  /  A ) ) ) ) )
7 eflog 19933 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( exp `  ( log `  A ) )  =  A )
8 recrec 9457 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 1  /  (
1  /  A ) )  =  A )
97, 8eqtr4d 2318 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( exp `  ( log `  A ) )  =  ( 1  / 
( 1  /  A
) ) )
10 logcl 19926 . . . . . . 7  |-  ( ( ( 1  /  A
)  e.  CC  /\  ( 1  /  A
)  =/=  0 )  ->  ( log `  (
1  /  A ) )  e.  CC )
111, 2, 10syl2anc 642 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( log `  (
1  /  A ) )  e.  CC )
12 efneg 12378 . . . . . 6  |-  ( ( log `  ( 1  /  A ) )  e.  CC  ->  ( exp `  -u ( log `  (
1  /  A ) ) )  =  ( 1  /  ( exp `  ( log `  (
1  /  A ) ) ) ) )
1311, 12syl 15 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( exp `  -u ( log `  ( 1  /  A ) ) )  =  ( 1  / 
( exp `  ( log `  ( 1  /  A ) ) ) ) )
146, 9, 133eqtr4d 2325 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( exp `  ( log `  A ) )  =  ( exp `  -u ( log `  ( 1  /  A ) ) ) )
15143adant3 975 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  (
Im `  ( log `  A ) )  =/= 
pi )  ->  ( exp `  ( log `  A
) )  =  ( exp `  -u ( log `  ( 1  /  A ) ) ) )
1615fveq2d 5529 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  (
Im `  ( log `  A ) )  =/= 
pi )  ->  ( log `  ( exp `  ( log `  A ) ) )  =  ( log `  ( exp `  -u ( log `  ( 1  /  A ) ) ) ) )
17 logrncl 19925 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( log `  A
)  e.  ran  log )
18173adant3 975 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  (
Im `  ( log `  A ) )  =/= 
pi )  ->  ( log `  A )  e. 
ran  log )
19 logef 19935 . . 3  |-  ( ( log `  A )  e.  ran  log  ->  ( log `  ( exp `  ( log `  A
) ) )  =  ( log `  A
) )
2018, 19syl 15 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  (
Im `  ( log `  A ) )  =/= 
pi )  ->  ( log `  ( exp `  ( log `  A ) ) )  =  ( log `  A ) )
21 df-ne 2448 . . . . 5  |-  ( ( Im `  ( log `  A ) )  =/= 
pi 
<->  -.  ( Im `  ( log `  A ) )  =  pi )
22 lognegb 19943 . . . . . . . . . . . 12  |-  ( ( ( 1  /  A
)  e.  CC  /\  ( 1  /  A
)  =/=  0 )  ->  ( -u (
1  /  A )  e.  RR+  <->  ( Im `  ( log `  ( 1  /  A ) ) )  =  pi ) )
231, 2, 22syl2anc 642 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( -u ( 1  /  A )  e.  RR+  <->  (
Im `  ( log `  ( 1  /  A
) ) )  =  pi ) )
2423biimprd 214 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( Im `  ( log `  ( 1  /  A ) ) )  =  pi  ->  -u ( 1  /  A
)  e.  RR+ )
)
25 ax-1cn 8795 . . . . . . . . . . . 12  |-  1  e.  CC
26 divneg2 9484 . . . . . . . . . . . 12  |-  ( ( 1  e.  CC  /\  A  e.  CC  /\  A  =/=  0 )  ->  -u (
1  /  A )  =  ( 1  /  -u A ) )
2725, 26mp3an1 1264 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  -u ( 1  /  A
)  =  ( 1  /  -u A ) )
2827eleq1d 2349 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( -u ( 1  /  A )  e.  RR+  <->  (
1  /  -u A
)  e.  RR+ )
)
2924, 28sylibd 205 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( Im `  ( log `  ( 1  /  A ) ) )  =  pi  ->  ( 1  /  -u A
)  e.  RR+ )
)
30 negcl 9052 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  -u A  e.  CC )
3130adantr 451 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  -u A  e.  CC )
32 negeq0 9101 . . . . . . . . . . . 12  |-  ( A  e.  CC  ->  ( A  =  0  <->  -u A  =  0 ) )
3332necon3bid 2481 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  ( A  =/=  0  <->  -u A  =/=  0 ) )
3433biimpa 470 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  -u A  =/=  0 )
35 rpreccl 10377 . . . . . . . . . . 11  |-  ( ( 1  /  -u A
)  e.  RR+  ->  ( 1  /  ( 1  /  -u A ) )  e.  RR+ )
36 recrec 9457 . . . . . . . . . . . 12  |-  ( (
-u A  e.  CC  /\  -u A  =/=  0
)  ->  ( 1  /  ( 1  /  -u A ) )  = 
-u A )
3736eleq1d 2349 . . . . . . . . . . 11  |-  ( (
-u A  e.  CC  /\  -u A  =/=  0
)  ->  ( (
1  /  ( 1  /  -u A ) )  e.  RR+  <->  -u A  e.  RR+ ) )
3835, 37syl5ib 210 . . . . . . . . . 10  |-  ( (
-u A  e.  CC  /\  -u A  =/=  0
)  ->  ( (
1  /  -u A
)  e.  RR+  ->  -u A  e.  RR+ ) )
3931, 34, 38syl2anc 642 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( 1  /  -u A )  e.  RR+  -> 
-u A  e.  RR+ ) )
4029, 39syld 40 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( Im `  ( log `  ( 1  /  A ) ) )  =  pi  ->  -u A  e.  RR+ ) )
41 lognegb 19943 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( -u A  e.  RR+  <->  (
Im `  ( log `  A ) )  =  pi ) )
4240, 41sylibd 205 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( Im `  ( log `  ( 1  /  A ) ) )  =  pi  ->  ( Im `  ( log `  A ) )  =  pi ) )
4342con3d 125 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( -.  ( Im
`  ( log `  A
) )  =  pi 
->  -.  ( Im `  ( log `  ( 1  /  A ) ) )  =  pi ) )
44433impia 1148 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  -.  ( Im `  ( log `  A ) )  =  pi )  ->  -.  ( Im `  ( log `  ( 1  /  A
) ) )  =  pi )
4521, 44syl3an3b 1220 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  (
Im `  ( log `  A ) )  =/= 
pi )  ->  -.  ( Im `  ( log `  ( 1  /  A
) ) )  =  pi )
46 logrncl 19925 . . . . . . 7  |-  ( ( ( 1  /  A
)  e.  CC  /\  ( 1  /  A
)  =/=  0 )  ->  ( log `  (
1  /  A ) )  e.  ran  log )
471, 2, 46syl2anc 642 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( log `  (
1  /  A ) )  e.  ran  log )
48 logreclem 20116 . . . . . 6  |-  ( ( ( log `  (
1  /  A ) )  e.  ran  log  /\ 
-.  ( Im `  ( log `  ( 1  /  A ) ) )  =  pi )  ->  -u ( log `  (
1  /  A ) )  e.  ran  log )
4947, 48sylan 457 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  -.  ( Im
`  ( log `  (
1  /  A ) ) )  =  pi )  ->  -u ( log `  ( 1  /  A
) )  e.  ran  log )
50493impa 1146 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  -.  ( Im `  ( log `  ( 1  /  A
) ) )  =  pi )  ->  -u ( log `  ( 1  /  A ) )  e. 
ran  log )
5145, 50syld3an3 1227 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  (
Im `  ( log `  A ) )  =/= 
pi )  ->  -u ( log `  ( 1  /  A ) )  e. 
ran  log )
52 logef 19935 . . 3  |-  ( -u ( log `  ( 1  /  A ) )  e.  ran  log  ->  ( log `  ( exp `  -u ( log `  (
1  /  A ) ) ) )  = 
-u ( log `  (
1  /  A ) ) )
5351, 52syl 15 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  (
Im `  ( log `  A ) )  =/= 
pi )  ->  ( log `  ( exp `  -u ( log `  ( 1  /  A ) ) ) )  =  -u ( log `  ( 1  /  A ) ) )
5416, 20, 533eqtr3d 2323 1  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  (
Im `  ( log `  A ) )  =/= 
pi )  ->  ( log `  A )  = 
-u ( log `  (
1  /  A ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   ran crn 4690   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738   -ucneg 9038    / cdiv 9423   RR+crp 10354   Imcim 11583   expce 12343   picpi 12348   logclog 19912
This theorem is referenced by:  isosctrlem2  20119  logbrec  23407
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-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  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  ax-addf 8816  ax-mulf 8817
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-int 3863  df-iun 3907  df-iin 3908  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-se 4353  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-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  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-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-sin 12351  df-cos 12352  df-pi 12354  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217  df-log 19914
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