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Theorem efiatan 23824
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 23796 . . . . 5  |-  ( A  e.  dom arctan  ->  (arctan `  A )  =  ( ( _i  /  2
)  x.  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) ) ) )
21oveq2d 6317 . . . 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 9598 . . . . . 6  |-  _i  e.  CC
43a1i 11 . . . . 5  |-  ( A  e.  dom arctan  ->  _i  e.  CC )
5 halfcl 10838 . . . . . 6  |-  ( _i  e.  CC  ->  (
_i  /  2 )  e.  CC )
63, 5mp1i 13 . . . . 5  |-  ( A  e.  dom arctan  ->  ( _i 
/  2 )  e.  CC )
7 ax-1cn 9597 . . . . . . . 8  |-  1  e.  CC
8 atandm2 23789 . . . . . . . . . 10  |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  ( 1  -  ( _i  x.  A ) )  =/=  0  /\  ( 1  +  ( _i  x.  A ) )  =/=  0 ) )
98simp1bi 1020 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  A  e.  CC )
10 mulcl 9623 . . . . . . . . 9  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
113, 9, 10sylancr 667 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( _i  x.  A )  e.  CC )
12 subcl 9874 . . . . . . . 8  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( 1  -  ( _i  x.  A
) )  e.  CC )
137, 11, 12sylancr 667 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( 1  -  ( _i  x.  A ) )  e.  CC )
148simp2bi 1021 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( 1  -  ( _i  x.  A ) )  =/=  0 )
1513, 14logcld 23506 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( log `  ( 1  -  (
_i  x.  A )
) )  e.  CC )
16 addcl 9621 . . . . . . . 8  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( 1  +  ( _i  x.  A
) )  e.  CC )
177, 11, 16sylancr 667 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( 1  +  ( _i  x.  A ) )  e.  CC )
188simp3bi 1022 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( 1  +  ( _i  x.  A ) )  =/=  0 )
1917, 18logcld 23506 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( log `  ( 1  +  ( _i  x.  A ) ) )  e.  CC )
2015, 19subcld 9986 . . . . 5  |-  ( A  e.  dom arctan  ->  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) )  e.  CC )
214, 6, 20mulassd 9666 . . . 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 10680 . . . . . . . 8  |-  2  e.  CC
23 2ne0 10702 . . . . . . . 8  |-  2  =/=  0
24 divneg 10302 . . . . . . . 8  |-  ( ( 1  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  -u (
1  /  2 )  =  ( -u 1  /  2 ) )
257, 22, 23, 24mp3an 1360 . . . . . . 7  |-  -u (
1  /  2 )  =  ( -u 1  /  2 )
26 ixi 10241 . . . . . . . 8  |-  ( _i  x.  _i )  = 
-u 1
2726oveq1i 6311 . . . . . . 7  |-  ( ( _i  x.  _i )  /  2 )  =  ( -u 1  / 
2 )
283, 3, 22, 23divassi 10363 . . . . . . 7  |-  ( ( _i  x.  _i )  /  2 )  =  ( _i  x.  (
_i  /  2 ) )
2925, 27, 283eqtr2i 2457 . . . . . 6  |-  -u (
1  /  2 )  =  ( _i  x.  ( _i  /  2
) )
3029oveq1i 6311 . . . . 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 ) ) ) ) )
31 halfcn 10829 . . . . . . 7  |-  ( 1  /  2 )  e.  CC
32 mulneg12 10057 . . . . . . 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 667 . . . . . 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 10002 . . . . . . 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 6317 . . . . . 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 10074 . . . . . 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 2467 . . . . 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 2477 . . . 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 2469 . . 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 5881 . 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 9623 . . . 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 667 . . 3  |-  ( A  e.  dom arctan  ->  ( ( 1  /  2 )  x.  ( log `  (
1  +  ( _i  x.  A ) ) ) )  e.  CC )
44 mulcl 9623 . . . 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 667 . . 3  |-  ( A  e.  dom arctan  ->  ( ( 1  /  2 )  x.  ( log `  (
1  -  ( _i  x.  A ) ) ) )  e.  CC )
46 efsub 14141 . . 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 665 . 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 23643 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( 1  +  ( _i  x.  A ) )  ^c  ( 1  /  2 ) )  =  ( exp `  (
( 1  /  2
)  x.  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) ) )
49 cxpsqrt 23634 . . . . 5  |-  ( ( 1  +  ( _i  x.  A ) )  e.  CC  ->  (
( 1  +  ( _i  x.  A ) )  ^c  ( 1  /  2 ) )  =  ( sqr `  ( 1  +  ( _i  x.  A ) ) ) )
5017, 49syl 17 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( 1  +  ( _i  x.  A ) )  ^c  ( 1  /  2 ) )  =  ( sqr `  (
1  +  ( _i  x.  A ) ) ) )
5148, 50eqtr3d 2465 . . 3  |-  ( A  e.  dom arctan  ->  ( exp `  ( ( 1  / 
2 )  x.  ( log `  ( 1  +  ( _i  x.  A
) ) ) ) )  =  ( sqr `  ( 1  +  ( _i  x.  A ) ) ) )
5213, 14, 36cxpefd 23643 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( 1  -  ( _i  x.  A ) )  ^c  ( 1  /  2 ) )  =  ( exp `  (
( 1  /  2
)  x.  ( log `  ( 1  -  (
_i  x.  A )
) ) ) ) )
53 cxpsqrt 23634 . . . . 5  |-  ( ( 1  -  ( _i  x.  A ) )  e.  CC  ->  (
( 1  -  (
_i  x.  A )
)  ^c  ( 1  /  2 ) )  =  ( sqr `  ( 1  -  (
_i  x.  A )
) ) )
5413, 53syl 17 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( 1  -  ( _i  x.  A ) )  ^c  ( 1  /  2 ) )  =  ( sqr `  (
1  -  ( _i  x.  A ) ) ) )
5552, 54eqtr3d 2465 . . 3  |-  ( A  e.  dom arctan  ->  ( exp `  ( ( 1  / 
2 )  x.  ( log `  ( 1  -  ( _i  x.  A
) ) ) ) )  =  ( sqr `  ( 1  -  (
_i  x.  A )
) ) )
5651, 55oveq12d 6319 . 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 2467 1  |-  ( A  e.  dom arctan  ->  ( exp `  ( _i  x.  (arctan `  A ) ) )  =  ( ( sqr `  ( 1  +  ( _i  x.  A ) ) )  /  ( sqr `  ( 1  -  ( _i  x.  A
) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1868    =/= wne 2618   dom cdm 4849   ` cfv 5597  (class class class)co 6301   CCcc 9537   0cc0 9539   1c1 9540   _ici 9541    + caddc 9542    x. cmul 9544    - cmin 9860   -ucneg 9861    / cdiv 10269   2c2 10659   sqrcsqrt 13284   expce 14101   logclog 23490    ^c ccxp 23491  arctancatan 23776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-inf2 8148  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-iin 4299  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-se 4809  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-isom 5606  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-of 6541  df-om 6703  df-1st 6803  df-2nd 6804  df-supp 6922  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-1o 7186  df-2o 7187  df-oadd 7190  df-er 7367  df-map 7478  df-pm 7479  df-ixp 7527  df-en 7574  df-dom 7575  df-sdom 7576  df-fin 7577  df-fsupp 7886  df-fi 7927  df-sup 7958  df-inf 7959  df-oi 8027  df-card 8374  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ioc 11640  df-ico 11641  df-icc 11642  df-fz 11785  df-fzo 11916  df-fl 12027  df-mod 12096  df-seq 12213  df-exp 12272  df-fac 12459  df-bc 12487  df-hash 12515  df-shft 13118  df-cj 13150  df-re 13151  df-im 13152  df-sqrt 13286  df-abs 13287  df-limsup 13513  df-clim 13539  df-rlim 13540  df-sum 13740  df-ef 14108  df-sin 14110  df-cos 14111  df-pi 14113  df-struct 15110  df-ndx 15111  df-slot 15112  df-base 15113  df-sets 15114  df-ress 15115  df-plusg 15190  df-mulr 15191  df-starv 15192  df-sca 15193  df-vsca 15194  df-ip 15195  df-tset 15196  df-ple 15197  df-ds 15199  df-unif 15200  df-hom 15201  df-cco 15202  df-rest 15308  df-topn 15309  df-0g 15327  df-gsum 15328  df-topgen 15329  df-pt 15330  df-prds 15333  df-xrs 15387  df-qtop 15393  df-imas 15394  df-xps 15397  df-mre 15479  df-mrc 15480  df-acs 15482  df-mgm 16475  df-sgrp 16514  df-mnd 16524  df-submnd 16570  df-mulg 16663  df-cntz 16958  df-cmn 17419  df-psmet 18949  df-xmet 18950  df-met 18951  df-bl 18952  df-mopn 18953  df-fbas 18954  df-fg 18955  df-cnfld 18958  df-top 19907  df-bases 19908  df-topon 19909  df-topsp 19910  df-cld 20020  df-ntr 20021  df-cls 20022  df-nei 20100  df-lp 20138  df-perf 20139  df-cn 20229  df-cnp 20230  df-haus 20317  df-tx 20563  df-hmeo 20756  df-fil 20847  df-fm 20939  df-flim 20940  df-flf 20941  df-xms 21321  df-ms 21322  df-tms 21323  df-cncf 21896  df-limc 22807  df-dv 22808  df-log 23492  df-cxp 23493  df-atan 23779
This theorem is referenced by: (None)
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