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Theorem efiatan 23917
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 23889 . . . . 5  |-  ( A  e.  dom arctan  ->  (arctan `  A )  =  ( ( _i  /  2
)  x.  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) ) ) )
21oveq2d 6324 . . . 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 9616 . . . . . 6  |-  _i  e.  CC
43a1i 11 . . . . 5  |-  ( A  e.  dom arctan  ->  _i  e.  CC )
5 halfcl 10861 . . . . . 6  |-  ( _i  e.  CC  ->  (
_i  /  2 )  e.  CC )
63, 5mp1i 13 . . . . 5  |-  ( A  e.  dom arctan  ->  ( _i 
/  2 )  e.  CC )
7 ax-1cn 9615 . . . . . . . 8  |-  1  e.  CC
8 atandm2 23882 . . . . . . . . . 10  |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  ( 1  -  ( _i  x.  A ) )  =/=  0  /\  ( 1  +  ( _i  x.  A ) )  =/=  0 ) )
98simp1bi 1045 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  A  e.  CC )
10 mulcl 9641 . . . . . . . . 9  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
113, 9, 10sylancr 676 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( _i  x.  A )  e.  CC )
12 subcl 9894 . . . . . . . 8  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( 1  -  ( _i  x.  A
) )  e.  CC )
137, 11, 12sylancr 676 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( 1  -  ( _i  x.  A ) )  e.  CC )
148simp2bi 1046 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( 1  -  ( _i  x.  A ) )  =/=  0 )
1513, 14logcld 23599 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( log `  ( 1  -  (
_i  x.  A )
) )  e.  CC )
16 addcl 9639 . . . . . . . 8  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( 1  +  ( _i  x.  A
) )  e.  CC )
177, 11, 16sylancr 676 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( 1  +  ( _i  x.  A ) )  e.  CC )
188simp3bi 1047 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( 1  +  ( _i  x.  A ) )  =/=  0 )
1917, 18logcld 23599 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( log `  ( 1  +  ( _i  x.  A ) ) )  e.  CC )
2015, 19subcld 10005 . . . . 5  |-  ( A  e.  dom arctan  ->  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) )  e.  CC )
214, 6, 20mulassd 9684 . . . 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 10702 . . . . . . . 8  |-  2  e.  CC
23 2ne0 10724 . . . . . . . 8  |-  2  =/=  0
24 divneg 10324 . . . . . . . 8  |-  ( ( 1  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  -u (
1  /  2 )  =  ( -u 1  /  2 ) )
257, 22, 23, 24mp3an 1390 . . . . . . 7  |-  -u (
1  /  2 )  =  ( -u 1  /  2 )
26 ixi 10263 . . . . . . . 8  |-  ( _i  x.  _i )  = 
-u 1
2726oveq1i 6318 . . . . . . 7  |-  ( ( _i  x.  _i )  /  2 )  =  ( -u 1  / 
2 )
283, 3, 22, 23divassi 10385 . . . . . . 7  |-  ( ( _i  x.  _i )  /  2 )  =  ( _i  x.  (
_i  /  2 ) )
2925, 27, 283eqtr2i 2499 . . . . . 6  |-  -u (
1  /  2 )  =  ( _i  x.  ( _i  /  2
) )
3029oveq1i 6318 . . . . 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 10852 . . . . . . 7  |-  ( 1  /  2 )  e.  CC
32 mulneg12 10078 . . . . . . 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 676 . . . . . 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 10021 . . . . . . 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 6324 . . . . . 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 10095 . . . . . 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 2509 . . . . 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 2519 . . . 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 2511 . . 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 5883 . 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 9641 . . . 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 676 . . 3  |-  ( A  e.  dom arctan  ->  ( ( 1  /  2 )  x.  ( log `  (
1  +  ( _i  x.  A ) ) ) )  e.  CC )
44 mulcl 9641 . . . 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 676 . . 3  |-  ( A  e.  dom arctan  ->  ( ( 1  /  2 )  x.  ( log `  (
1  -  ( _i  x.  A ) ) ) )  e.  CC )
46 efsub 14231 . . 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 673 . 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 23736 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( 1  +  ( _i  x.  A ) )  ^c  ( 1  /  2 ) )  =  ( exp `  (
( 1  /  2
)  x.  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) ) )
49 cxpsqrt 23727 . . . . 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 2507 . . 3  |-  ( A  e.  dom arctan  ->  ( exp `  ( ( 1  / 
2 )  x.  ( log `  ( 1  +  ( _i  x.  A
) ) ) ) )  =  ( sqr `  ( 1  +  ( _i  x.  A ) ) ) )
5213, 14, 36cxpefd 23736 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( 1  -  ( _i  x.  A ) )  ^c  ( 1  /  2 ) )  =  ( exp `  (
( 1  /  2
)  x.  ( log `  ( 1  -  (
_i  x.  A )
) ) ) ) )
53 cxpsqrt 23727 . . . . 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 2507 . . 3  |-  ( A  e.  dom arctan  ->  ( exp `  ( ( 1  / 
2 )  x.  ( log `  ( 1  -  ( _i  x.  A
) ) ) ) )  =  ( sqr `  ( 1  -  (
_i  x.  A )
) ) )
5651, 55oveq12d 6326 . 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 2509 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 1452    e. wcel 1904    =/= wne 2641   dom cdm 4839   ` cfv 5589  (class class class)co 6308   CCcc 9555   0cc0 9557   1c1 9558   _ici 9559    + caddc 9560    x. cmul 9562    - cmin 9880   -ucneg 9881    / cdiv 10291   2c2 10681   sqrcsqrt 13373   expce 14191   logclog 23583    ^c ccxp 23584  arctancatan 23869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ioc 11665  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-fac 12498  df-bc 12526  df-hash 12554  df-shft 13207  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-limsup 13603  df-clim 13629  df-rlim 13630  df-sum 13830  df-ef 14198  df-sin 14200  df-cos 14201  df-pi 14203  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cn 20320  df-cnp 20321  df-haus 20408  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-xms 21413  df-ms 21414  df-tms 21415  df-cncf 21988  df-limc 22900  df-dv 22901  df-log 23585  df-cxp 23586  df-atan 23872
This theorem is referenced by: (None)
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