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Theorem logrec 20622
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 9649 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 1  /  A
)  e.  CC )
2 recne0 9655 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 1  /  A
)  =/=  0 )
3 eflog 20435 . . . . . . . 8  |-  ( ( ( 1  /  A
)  e.  CC  /\  ( 1  /  A
)  =/=  0 )  ->  ( exp `  ( log `  ( 1  /  A ) ) )  =  ( 1  /  A ) )
41, 2, 3syl2anc 643 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( exp `  ( log `  ( 1  /  A ) ) )  =  ( 1  /  A ) )
54eqcomd 2417 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 1  /  A
)  =  ( exp `  ( log `  (
1  /  A ) ) ) )
65oveq2d 6064 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 1  /  (
1  /  A ) )  =  ( 1  /  ( exp `  ( log `  ( 1  /  A ) ) ) ) )
7 eflog 20435 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( exp `  ( log `  A ) )  =  A )
8 recrec 9675 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 1  /  (
1  /  A ) )  =  A )
97, 8eqtr4d 2447 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( exp `  ( log `  A ) )  =  ( 1  / 
( 1  /  A
) ) )
101, 2logcld 20429 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( log `  (
1  /  A ) )  e.  CC )
11 efneg 12662 . . . . . 6  |-  ( ( log `  ( 1  /  A ) )  e.  CC  ->  ( exp `  -u ( log `  (
1  /  A ) ) )  =  ( 1  /  ( exp `  ( log `  (
1  /  A ) ) ) ) )
1210, 11syl 16 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( exp `  -u ( log `  ( 1  /  A ) ) )  =  ( 1  / 
( exp `  ( log `  ( 1  /  A ) ) ) ) )
136, 9, 123eqtr4d 2454 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( exp `  ( log `  A ) )  =  ( exp `  -u ( log `  ( 1  /  A ) ) ) )
14133adant3 977 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  (
Im `  ( log `  A ) )  =/= 
pi )  ->  ( exp `  ( log `  A
) )  =  ( exp `  -u ( log `  ( 1  /  A ) ) ) )
1514fveq2d 5699 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  (
Im `  ( log `  A ) )  =/= 
pi )  ->  ( log `  ( exp `  ( log `  A ) ) )  =  ( log `  ( exp `  -u ( log `  ( 1  /  A ) ) ) ) )
16 logrncl 20426 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( log `  A
)  e.  ran  log )
17163adant3 977 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  (
Im `  ( log `  A ) )  =/= 
pi )  ->  ( log `  A )  e. 
ran  log )
18 logef 20437 . . 3  |-  ( ( log `  A )  e.  ran  log  ->  ( log `  ( exp `  ( log `  A
) ) )  =  ( log `  A
) )
1917, 18syl 16 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  (
Im `  ( log `  A ) )  =/= 
pi )  ->  ( log `  ( exp `  ( log `  A ) ) )  =  ( log `  A ) )
20 df-ne 2577 . . . . 5  |-  ( ( Im `  ( log `  A ) )  =/= 
pi 
<->  -.  ( Im `  ( log `  A ) )  =  pi )
21 lognegb 20445 . . . . . . . . . . . 12  |-  ( ( ( 1  /  A
)  e.  CC  /\  ( 1  /  A
)  =/=  0 )  ->  ( -u (
1  /  A )  e.  RR+  <->  ( Im `  ( log `  ( 1  /  A ) ) )  =  pi ) )
221, 2, 21syl2anc 643 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( -u ( 1  /  A )  e.  RR+  <->  (
Im `  ( log `  ( 1  /  A
) ) )  =  pi ) )
2322biimprd 215 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( Im `  ( log `  ( 1  /  A ) ) )  =  pi  ->  -u ( 1  /  A
)  e.  RR+ )
)
24 ax-1cn 9012 . . . . . . . . . . . 12  |-  1  e.  CC
25 divneg2 9702 . . . . . . . . . . . 12  |-  ( ( 1  e.  CC  /\  A  e.  CC  /\  A  =/=  0 )  ->  -u (
1  /  A )  =  ( 1  /  -u A ) )
2624, 25mp3an1 1266 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  -u ( 1  /  A
)  =  ( 1  /  -u A ) )
2726eleq1d 2478 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( -u ( 1  /  A )  e.  RR+  <->  (
1  /  -u A
)  e.  RR+ )
)
2823, 27sylibd 206 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( Im `  ( log `  ( 1  /  A ) ) )  =  pi  ->  ( 1  /  -u A
)  e.  RR+ )
)
29 negcl 9270 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  -u A  e.  CC )
3029adantr 452 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  -u A  e.  CC )
31 negeq0 9319 . . . . . . . . . . . 12  |-  ( A  e.  CC  ->  ( A  =  0  <->  -u A  =  0 ) )
3231necon3bid 2610 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  ( A  =/=  0  <->  -u A  =/=  0 ) )
3332biimpa 471 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  -u A  =/=  0 )
34 rpreccl 10599 . . . . . . . . . . 11  |-  ( ( 1  /  -u A
)  e.  RR+  ->  ( 1  /  ( 1  /  -u A ) )  e.  RR+ )
35 recrec 9675 . . . . . . . . . . . 12  |-  ( (
-u A  e.  CC  /\  -u A  =/=  0
)  ->  ( 1  /  ( 1  /  -u A ) )  = 
-u A )
3635eleq1d 2478 . . . . . . . . . . 11  |-  ( (
-u A  e.  CC  /\  -u A  =/=  0
)  ->  ( (
1  /  ( 1  /  -u A ) )  e.  RR+  <->  -u A  e.  RR+ ) )
3734, 36syl5ib 211 . . . . . . . . . 10  |-  ( (
-u A  e.  CC  /\  -u A  =/=  0
)  ->  ( (
1  /  -u A
)  e.  RR+  ->  -u A  e.  RR+ ) )
3830, 33, 37syl2anc 643 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( 1  /  -u A )  e.  RR+  -> 
-u A  e.  RR+ ) )
3928, 38syld 42 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( Im `  ( log `  ( 1  /  A ) ) )  =  pi  ->  -u A  e.  RR+ ) )
40 lognegb 20445 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( -u A  e.  RR+  <->  (
Im `  ( log `  A ) )  =  pi ) )
4139, 40sylibd 206 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( Im `  ( log `  ( 1  /  A ) ) )  =  pi  ->  ( Im `  ( log `  A ) )  =  pi ) )
4241con3d 127 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( -.  ( Im
`  ( log `  A
) )  =  pi 
->  -.  ( Im `  ( log `  ( 1  /  A ) ) )  =  pi ) )
43423impia 1150 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  -.  ( Im `  ( log `  A ) )  =  pi )  ->  -.  ( Im `  ( log `  ( 1  /  A
) ) )  =  pi )
4420, 43syl3an3b 1222 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  (
Im `  ( log `  A ) )  =/= 
pi )  ->  -.  ( Im `  ( log `  ( 1  /  A
) ) )  =  pi )
45 logrncl 20426 . . . . . . 7  |-  ( ( ( 1  /  A
)  e.  CC  /\  ( 1  /  A
)  =/=  0 )  ->  ( log `  (
1  /  A ) )  e.  ran  log )
461, 2, 45syl2anc 643 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( log `  (
1  /  A ) )  e.  ran  log )
47 logreclem 20621 . . . . . 6  |-  ( ( ( log `  (
1  /  A ) )  e.  ran  log  /\ 
-.  ( Im `  ( log `  ( 1  /  A ) ) )  =  pi )  ->  -u ( log `  (
1  /  A ) )  e.  ran  log )
4846, 47sylan 458 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  -.  ( Im
`  ( log `  (
1  /  A ) ) )  =  pi )  ->  -u ( log `  ( 1  /  A
) )  e.  ran  log )
49483impa 1148 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  -.  ( Im `  ( log `  ( 1  /  A
) ) )  =  pi )  ->  -u ( log `  ( 1  /  A ) )  e. 
ran  log )
5044, 49syld3an3 1229 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  (
Im `  ( log `  A ) )  =/= 
pi )  ->  -u ( log `  ( 1  /  A ) )  e. 
ran  log )
51 logef 20437 . . 3  |-  ( -u ( log `  ( 1  /  A ) )  e.  ran  log  ->  ( log `  ( exp `  -u ( log `  (
1  /  A ) ) ) )  = 
-u ( log `  (
1  /  A ) ) )
5250, 51syl 16 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  (
Im `  ( log `  A ) )  =/= 
pi )  ->  ( log `  ( exp `  -u ( log `  ( 1  /  A ) ) ) )  =  -u ( log `  ( 1  /  A ) ) )
5315, 19, 523eqtr3d 2452 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2575   ran crn 4846   ` cfv 5421  (class class class)co 6048   CCcc 8952   0cc0 8954   1c1 8955   -ucneg 9256    / cdiv 9641   RR+crp 10576   Imcim 11866   expce 12627   picpi 12632   logclog 20413
This theorem is referenced by:  isosctrlem2  20624  logbrec  24366
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-inf2 7560  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032  ax-addf 9033  ax-mulf 9034
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-iin 4064  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-of 6272  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-2o 6692  df-oadd 6695  df-er 6872  df-map 6987  df-pm 6988  df-ixp 7031  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-fi 7382  df-sup 7412  df-oi 7443  df-card 7790  df-cda 8012  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-3 10023  df-4 10024  df-5 10025  df-6 10026  df-7 10027  df-8 10028  df-9 10029  df-10 10030  df-n0 10186  df-z 10247  df-dec 10347  df-uz 10453  df-q 10539  df-rp 10577  df-xneg 10674  df-xadd 10675  df-xmul 10676  df-ioo 10884  df-ioc 10885  df-ico 10886  df-icc 10887  df-fz 11008  df-fzo 11099  df-fl 11165  df-mod 11214  df-seq 11287  df-exp 11346  df-fac 11530  df-bc 11557  df-hash 11582  df-shft 11845  df-cj 11867  df-re 11868  df-im 11869  df-sqr 12003  df-abs 12004  df-limsup 12228  df-clim 12245  df-rlim 12246  df-sum 12443  df-ef 12633  df-sin 12635  df-cos 12636  df-pi 12638  df-struct 13434  df-ndx 13435  df-slot 13436  df-base 13437  df-sets 13438  df-ress 13439  df-plusg 13505  df-mulr 13506  df-starv 13507  df-sca 13508  df-vsca 13509  df-tset 13511  df-ple 13512  df-ds 13514  df-unif 13515  df-hom 13516  df-cco 13517  df-rest 13613  df-topn 13614  df-topgen 13630  df-pt 13631  df-prds 13634  df-xrs 13689  df-0g 13690  df-gsum 13691  df-qtop 13696  df-imas 13697  df-xps 13699  df-mre 13774  df-mrc 13775  df-acs 13777  df-mnd 14653  df-submnd 14702  df-mulg 14778  df-cntz 15079  df-cmn 15377  df-psmet 16657  df-xmet 16658  df-met 16659  df-bl 16660  df-mopn 16661  df-fbas 16662  df-fg 16663  df-cnfld 16667  df-top 16926  df-bases 16928  df-topon 16929  df-topsp 16930  df-cld 17046  df-ntr 17047  df-cls 17048  df-nei 17125  df-lp 17163  df-perf 17164  df-cn 17253  df-cnp 17254  df-haus 17341  df-tx 17555  df-hmeo 17748  df-fil 17839  df-fm 17931  df-flim 17932  df-flf 17933  df-xms 18311  df-ms 18312  df-tms 18313  df-cncf 18869  df-limc 19714  df-dv 19715  df-log 20415
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