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Theorem efiatan 20705
Description: Value of the exponential of an artcangent. (Contributed by Mario Carneiro, 2-Apr-2015.)
Assertion
Ref Expression
efiatan  |-  ( A  e.  dom arctan  ->  ( exp `  ( _i  x.  (arctan `  A ) ) )  =  ( ( sqr `  ( 1  +  ( _i  x.  A ) ) )  /  ( sqr `  ( 1  -  ( _i  x.  A
) ) ) ) )

Proof of Theorem efiatan
StepHypRef Expression
1 atanval 20677 . . . . 5  |-  ( A  e.  dom arctan  ->  (arctan `  A )  =  ( ( _i  /  2
)  x.  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) ) ) )
21oveq2d 6056 . . . 4  |-  ( A  e.  dom arctan  ->  ( _i  x.  (arctan `  A
) )  =  ( _i  x.  ( ( _i  /  2 )  x.  ( ( log `  ( 1  -  (
_i  x.  A )
) )  -  ( log `  ( 1  +  ( _i  x.  A
) ) ) ) ) ) )
3 ax-icn 9005 . . . . . 6  |-  _i  e.  CC
43a1i 11 . . . . 5  |-  ( A  e.  dom arctan  ->  _i  e.  CC )
5 halfcl 10149 . . . . . 6  |-  ( _i  e.  CC  ->  (
_i  /  2 )  e.  CC )
63, 5mp1i 12 . . . . 5  |-  ( A  e.  dom arctan  ->  ( _i 
/  2 )  e.  CC )
7 ax-1cn 9004 . . . . . . . 8  |-  1  e.  CC
8 atandm2 20670 . . . . . . . . . 10  |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  ( 1  -  ( _i  x.  A ) )  =/=  0  /\  ( 1  +  ( _i  x.  A ) )  =/=  0 ) )
98simp1bi 972 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  A  e.  CC )
10 mulcl 9030 . . . . . . . . 9  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
113, 9, 10sylancr 645 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( _i  x.  A )  e.  CC )
12 subcl 9261 . . . . . . . 8  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( 1  -  ( _i  x.  A
) )  e.  CC )
137, 11, 12sylancr 645 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( 1  -  ( _i  x.  A ) )  e.  CC )
148simp2bi 973 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( 1  -  ( _i  x.  A ) )  =/=  0 )
1513, 14logcld 20421 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( log `  ( 1  -  (
_i  x.  A )
) )  e.  CC )
16 addcl 9028 . . . . . . . 8  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( 1  +  ( _i  x.  A
) )  e.  CC )
177, 11, 16sylancr 645 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( 1  +  ( _i  x.  A ) )  e.  CC )
188simp3bi 974 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( 1  +  ( _i  x.  A ) )  =/=  0 )
1917, 18logcld 20421 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( log `  ( 1  +  ( _i  x.  A ) ) )  e.  CC )
2015, 19subcld 9367 . . . . 5  |-  ( A  e.  dom arctan  ->  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) )  e.  CC )
214, 6, 20mulassd 9067 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( _i  x.  ( _i 
/  2 ) )  x.  ( ( log `  ( 1  -  (
_i  x.  A )
) )  -  ( log `  ( 1  +  ( _i  x.  A
) ) ) ) )  =  ( _i  x.  ( ( _i 
/  2 )  x.  ( ( log `  (
1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) ) ) )
22 2cn 10026 . . . . . . . 8  |-  2  e.  CC
23 2ne0 10039 . . . . . . . 8  |-  2  =/=  0
24 divneg 9665 . . . . . . . 8  |-  ( ( 1  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  -u (
1  /  2 )  =  ( -u 1  /  2 ) )
257, 22, 23, 24mp3an 1279 . . . . . . 7  |-  -u (
1  /  2 )  =  ( -u 1  /  2 )
26 ixi 9607 . . . . . . . 8  |-  ( _i  x.  _i )  = 
-u 1
2726oveq1i 6050 . . . . . . 7  |-  ( ( _i  x.  _i )  /  2 )  =  ( -u 1  / 
2 )
283, 3, 22, 23divassi 9726 . . . . . . 7  |-  ( ( _i  x.  _i )  /  2 )  =  ( _i  x.  (
_i  /  2 ) )
2925, 27, 283eqtr2i 2430 . . . . . 6  |-  -u (
1  /  2 )  =  ( _i  x.  ( _i  /  2
) )
3029oveq1i 6050 . . . . 5  |-  ( -u ( 1  /  2
)  x.  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) ) )  =  ( ( _i  x.  ( _i  /  2
) )  x.  (
( log `  (
1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) )
3122, 23reccli 9700 . . . . . . 7  |-  ( 1  /  2 )  e.  CC
32 mulneg12 9428 . . . . . . 7  |-  ( ( ( 1  /  2
)  e.  CC  /\  ( ( log `  (
1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) )  e.  CC )  ->  ( -u ( 1  /  2
)  x.  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) ) )  =  ( ( 1  / 
2 )  x.  -u (
( log `  (
1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) ) )
3331, 20, 32sylancr 645 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( -u ( 1  /  2
)  x.  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) ) )  =  ( ( 1  / 
2 )  x.  -u (
( log `  (
1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) ) )
3415, 19negsubdi2d 9383 . . . . . . 7  |-  ( A  e.  dom arctan  ->  -u (
( log `  (
1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) )  =  ( ( log `  (
1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  (
_i  x.  A )
) ) ) )
3534oveq2d 6056 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( 1  /  2 )  x.  -u ( ( log `  ( 1  -  (
_i  x.  A )
) )  -  ( log `  ( 1  +  ( _i  x.  A
) ) ) ) )  =  ( ( 1  /  2 )  x.  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  ( _i  x.  A
) ) ) ) ) )
3631a1i 11 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( 1  /  2 )  e.  CC )
3736, 19, 15subdid 9445 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( 1  /  2 )  x.  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  ( _i  x.  A
) ) ) ) )  =  ( ( ( 1  /  2
)  x.  ( log `  ( 1  +  ( _i  x.  A ) ) ) )  -  ( ( 1  / 
2 )  x.  ( log `  ( 1  -  ( _i  x.  A
) ) ) ) ) )
3833, 35, 373eqtrd 2440 . . . . 5  |-  ( A  e.  dom arctan  ->  ( -u ( 1  /  2
)  x.  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) ) )  =  ( ( ( 1  /  2 )  x.  ( log `  (
1  +  ( _i  x.  A ) ) ) )  -  (
( 1  /  2
)  x.  ( log `  ( 1  -  (
_i  x.  A )
) ) ) ) )
3930, 38syl5eqr 2450 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( _i  x.  ( _i 
/  2 ) )  x.  ( ( log `  ( 1  -  (
_i  x.  A )
) )  -  ( log `  ( 1  +  ( _i  x.  A
) ) ) ) )  =  ( ( ( 1  /  2
)  x.  ( log `  ( 1  +  ( _i  x.  A ) ) ) )  -  ( ( 1  / 
2 )  x.  ( log `  ( 1  -  ( _i  x.  A
) ) ) ) ) )
402, 21, 393eqtr2d 2442 . . 3  |-  ( A  e.  dom arctan  ->  ( _i  x.  (arctan `  A
) )  =  ( ( ( 1  / 
2 )  x.  ( log `  ( 1  +  ( _i  x.  A
) ) ) )  -  ( ( 1  /  2 )  x.  ( log `  (
1  -  ( _i  x.  A ) ) ) ) ) )
4140fveq2d 5691 . 2  |-  ( A  e.  dom arctan  ->  ( exp `  ( _i  x.  (arctan `  A ) ) )  =  ( exp `  (
( ( 1  / 
2 )  x.  ( log `  ( 1  +  ( _i  x.  A
) ) ) )  -  ( ( 1  /  2 )  x.  ( log `  (
1  -  ( _i  x.  A ) ) ) ) ) ) )
42 mulcl 9030 . . . 4  |-  ( ( ( 1  /  2
)  e.  CC  /\  ( log `  ( 1  +  ( _i  x.  A ) ) )  e.  CC )  -> 
( ( 1  / 
2 )  x.  ( log `  ( 1  +  ( _i  x.  A
) ) ) )  e.  CC )
4331, 19, 42sylancr 645 . . 3  |-  ( A  e.  dom arctan  ->  ( ( 1  /  2 )  x.  ( log `  (
1  +  ( _i  x.  A ) ) ) )  e.  CC )
44 mulcl 9030 . . . 4  |-  ( ( ( 1  /  2
)  e.  CC  /\  ( log `  ( 1  -  ( _i  x.  A ) ) )  e.  CC )  -> 
( ( 1  / 
2 )  x.  ( log `  ( 1  -  ( _i  x.  A
) ) ) )  e.  CC )
4531, 15, 44sylancr 645 . . 3  |-  ( A  e.  dom arctan  ->  ( ( 1  /  2 )  x.  ( log `  (
1  -  ( _i  x.  A ) ) ) )  e.  CC )
46 efsub 12656 . . 3  |-  ( ( ( ( 1  / 
2 )  x.  ( log `  ( 1  +  ( _i  x.  A
) ) ) )  e.  CC  /\  (
( 1  /  2
)  x.  ( log `  ( 1  -  (
_i  x.  A )
) ) )  e.  CC )  ->  ( exp `  ( ( ( 1  /  2 )  x.  ( log `  (
1  +  ( _i  x.  A ) ) ) )  -  (
( 1  /  2
)  x.  ( log `  ( 1  -  (
_i  x.  A )
) ) ) ) )  =  ( ( exp `  ( ( 1  /  2 )  x.  ( log `  (
1  +  ( _i  x.  A ) ) ) ) )  / 
( exp `  (
( 1  /  2
)  x.  ( log `  ( 1  -  (
_i  x.  A )
) ) ) ) ) )
4743, 45, 46syl2anc 643 . 2  |-  ( A  e.  dom arctan  ->  ( exp `  ( ( ( 1  /  2 )  x.  ( log `  (
1  +  ( _i  x.  A ) ) ) )  -  (
( 1  /  2
)  x.  ( log `  ( 1  -  (
_i  x.  A )
) ) ) ) )  =  ( ( exp `  ( ( 1  /  2 )  x.  ( log `  (
1  +  ( _i  x.  A ) ) ) ) )  / 
( exp `  (
( 1  /  2
)  x.  ( log `  ( 1  -  (
_i  x.  A )
) ) ) ) ) )
4817, 18, 36cxpefd 20556 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( 1  +  ( _i  x.  A ) )  ^ c  ( 1  /  2 ) )  =  ( exp `  (
( 1  /  2
)  x.  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) ) )
49 cxpsqr 20547 . . . . 5  |-  ( ( 1  +  ( _i  x.  A ) )  e.  CC  ->  (
( 1  +  ( _i  x.  A ) )  ^ c  ( 1  /  2 ) )  =  ( sqr `  ( 1  +  ( _i  x.  A ) ) ) )
5017, 49syl 16 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( 1  +  ( _i  x.  A ) )  ^ c  ( 1  /  2 ) )  =  ( sqr `  (
1  +  ( _i  x.  A ) ) ) )
5148, 50eqtr3d 2438 . . 3  |-  ( A  e.  dom arctan  ->  ( exp `  ( ( 1  / 
2 )  x.  ( log `  ( 1  +  ( _i  x.  A
) ) ) ) )  =  ( sqr `  ( 1  +  ( _i  x.  A ) ) ) )
5213, 14, 36cxpefd 20556 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( 1  -  ( _i  x.  A ) )  ^ c  ( 1  /  2 ) )  =  ( exp `  (
( 1  /  2
)  x.  ( log `  ( 1  -  (
_i  x.  A )
) ) ) ) )
53 cxpsqr 20547 . . . . 5  |-  ( ( 1  -  ( _i  x.  A ) )  e.  CC  ->  (
( 1  -  (
_i  x.  A )
)  ^ c  ( 1  /  2 ) )  =  ( sqr `  ( 1  -  (
_i  x.  A )
) ) )
5413, 53syl 16 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( 1  -  ( _i  x.  A ) )  ^ c  ( 1  /  2 ) )  =  ( sqr `  (
1  -  ( _i  x.  A ) ) ) )
5552, 54eqtr3d 2438 . . 3  |-  ( A  e.  dom arctan  ->  ( exp `  ( ( 1  / 
2 )  x.  ( log `  ( 1  -  ( _i  x.  A
) ) ) ) )  =  ( sqr `  ( 1  -  (
_i  x.  A )
) ) )
5651, 55oveq12d 6058 . 2  |-  ( A  e.  dom arctan  ->  ( ( exp `  ( ( 1  /  2 )  x.  ( log `  (
1  +  ( _i  x.  A ) ) ) ) )  / 
( exp `  (
( 1  /  2
)  x.  ( log `  ( 1  -  (
_i  x.  A )
) ) ) ) )  =  ( ( sqr `  ( 1  +  ( _i  x.  A ) ) )  /  ( sqr `  (
1  -  ( _i  x.  A ) ) ) ) )
5741, 47, 563eqtrd 2440 1  |-  ( A  e.  dom arctan  ->  ( exp `  ( _i  x.  (arctan `  A ) ) )  =  ( ( sqr `  ( 1  +  ( _i  x.  A ) ) )  /  ( sqr `  ( 1  -  ( _i  x.  A
) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721    =/= wne 2567   dom cdm 4837   ` cfv 5413  (class class class)co 6040   CCcc 8944   0cc0 8946   1c1 8947   _ici 8948    + caddc 8949    x. cmul 8951    - cmin 9247   -ucneg 9248    / cdiv 9633   2c2 10005   sqrcsqr 11993   expce 12619   logclog 20405    ^ c ccxp 20406  arctancatan 20657
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 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ioc 10877  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-shft 11837  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-sum 12435  df-ef 12625  df-sin 12627  df-cos 12628  df-pi 12630  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-limc 19706  df-dv 19707  df-log 20407  df-cxp 20408  df-atan 20660
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