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Theorem efiatan 23831
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 23803 . . . . 5  |-  ( A  e.  dom arctan  ->  (arctan `  A )  =  ( ( _i  /  2
)  x.  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) ) ) )
21oveq2d 6304 . . . 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 9595 . . . . . 6  |-  _i  e.  CC
43a1i 11 . . . . 5  |-  ( A  e.  dom arctan  ->  _i  e.  CC )
5 halfcl 10835 . . . . . 6  |-  ( _i  e.  CC  ->  (
_i  /  2 )  e.  CC )
63, 5mp1i 13 . . . . 5  |-  ( A  e.  dom arctan  ->  ( _i 
/  2 )  e.  CC )
7 ax-1cn 9594 . . . . . . . 8  |-  1  e.  CC
8 atandm2 23796 . . . . . . . . . 10  |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  ( 1  -  ( _i  x.  A ) )  =/=  0  /\  ( 1  +  ( _i  x.  A ) )  =/=  0 ) )
98simp1bi 1022 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  A  e.  CC )
10 mulcl 9620 . . . . . . . . 9  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
113, 9, 10sylancr 668 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( _i  x.  A )  e.  CC )
12 subcl 9871 . . . . . . . 8  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( 1  -  ( _i  x.  A
) )  e.  CC )
137, 11, 12sylancr 668 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( 1  -  ( _i  x.  A ) )  e.  CC )
148simp2bi 1023 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( 1  -  ( _i  x.  A ) )  =/=  0 )
1513, 14logcld 23513 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( log `  ( 1  -  (
_i  x.  A )
) )  e.  CC )
16 addcl 9618 . . . . . . . 8  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( 1  +  ( _i  x.  A
) )  e.  CC )
177, 11, 16sylancr 668 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( 1  +  ( _i  x.  A ) )  e.  CC )
188simp3bi 1024 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( 1  +  ( _i  x.  A ) )  =/=  0 )
1917, 18logcld 23513 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( log `  ( 1  +  ( _i  x.  A ) ) )  e.  CC )
2015, 19subcld 9983 . . . . 5  |-  ( A  e.  dom arctan  ->  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) )  e.  CC )
214, 6, 20mulassd 9663 . . . 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 10677 . . . . . . . 8  |-  2  e.  CC
23 2ne0 10699 . . . . . . . 8  |-  2  =/=  0
24 divneg 10299 . . . . . . . 8  |-  ( ( 1  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  -u (
1  /  2 )  =  ( -u 1  /  2 ) )
257, 22, 23, 24mp3an 1363 . . . . . . 7  |-  -u (
1  /  2 )  =  ( -u 1  /  2 )
26 ixi 10238 . . . . . . . 8  |-  ( _i  x.  _i )  = 
-u 1
2726oveq1i 6298 . . . . . . 7  |-  ( ( _i  x.  _i )  /  2 )  =  ( -u 1  / 
2 )
283, 3, 22, 23divassi 10360 . . . . . . 7  |-  ( ( _i  x.  _i )  /  2 )  =  ( _i  x.  (
_i  /  2 ) )
2925, 27, 283eqtr2i 2478 . . . . . 6  |-  -u (
1  /  2 )  =  ( _i  x.  ( _i  /  2
) )
3029oveq1i 6298 . . . . 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 10826 . . . . . . 7  |-  ( 1  /  2 )  e.  CC
32 mulneg12 10054 . . . . . . 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 668 . . . . . 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 9999 . . . . . . 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 6304 . . . . . 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 10071 . . . . . 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 2488 . . . . 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 2498 . . . 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 2490 . . 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 5867 . 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 9620 . . . 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 668 . . 3  |-  ( A  e.  dom arctan  ->  ( ( 1  /  2 )  x.  ( log `  (
1  +  ( _i  x.  A ) ) ) )  e.  CC )
44 mulcl 9620 . . . 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 668 . . 3  |-  ( A  e.  dom arctan  ->  ( ( 1  /  2 )  x.  ( log `  (
1  -  ( _i  x.  A ) ) ) )  e.  CC )
46 efsub 14147 . . 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 666 . 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 23650 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( 1  +  ( _i  x.  A ) )  ^c  ( 1  /  2 ) )  =  ( exp `  (
( 1  /  2
)  x.  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) ) )
49 cxpsqrt 23641 . . . . 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 2486 . . 3  |-  ( A  e.  dom arctan  ->  ( exp `  ( ( 1  / 
2 )  x.  ( log `  ( 1  +  ( _i  x.  A
) ) ) ) )  =  ( sqr `  ( 1  +  ( _i  x.  A ) ) ) )
5213, 14, 36cxpefd 23650 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( 1  -  ( _i  x.  A ) )  ^c  ( 1  /  2 ) )  =  ( exp `  (
( 1  /  2
)  x.  ( log `  ( 1  -  (
_i  x.  A )
) ) ) ) )
53 cxpsqrt 23641 . . . . 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 2486 . . 3  |-  ( A  e.  dom arctan  ->  ( exp `  ( ( 1  / 
2 )  x.  ( log `  ( 1  -  ( _i  x.  A
) ) ) ) )  =  ( sqr `  ( 1  -  (
_i  x.  A )
) ) )
5651, 55oveq12d 6306 . 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 2488 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 1443    e. wcel 1886    =/= wne 2621   dom cdm 4833   ` cfv 5581  (class class class)co 6288   CCcc 9534   0cc0 9536   1c1 9537   _ici 9538    + caddc 9539    x. cmul 9541    - cmin 9857   -ucneg 9858    / cdiv 10266   2c2 10656   sqrcsqrt 13289   expce 14107   logclog 23497    ^c ccxp 23498  arctancatan 23783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-inf2 8143  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614  ax-addf 9615  ax-mulf 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-fal 1449  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-iin 4280  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-of 6528  df-om 6690  df-1st 6790  df-2nd 6791  df-supp 6912  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-2o 7180  df-oadd 7183  df-er 7360  df-map 7471  df-pm 7472  df-ixp 7520  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-fsupp 7881  df-fi 7922  df-sup 7953  df-inf 7954  df-oi 8022  df-card 8370  df-cda 8595  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-3 10666  df-4 10667  df-5 10668  df-6 10669  df-7 10670  df-8 10671  df-9 10672  df-10 10673  df-n0 10867  df-z 10935  df-dec 11049  df-uz 11157  df-q 11262  df-rp 11300  df-xneg 11406  df-xadd 11407  df-xmul 11408  df-ioo 11636  df-ioc 11637  df-ico 11638  df-icc 11639  df-fz 11782  df-fzo 11913  df-fl 12025  df-mod 12094  df-seq 12211  df-exp 12270  df-fac 12457  df-bc 12485  df-hash 12513  df-shft 13123  df-cj 13155  df-re 13156  df-im 13157  df-sqrt 13291  df-abs 13292  df-limsup 13519  df-clim 13545  df-rlim 13546  df-sum 13746  df-ef 14114  df-sin 14116  df-cos 14117  df-pi 14119  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-sets 15120  df-ress 15121  df-plusg 15196  df-mulr 15197  df-starv 15198  df-sca 15199  df-vsca 15200  df-ip 15201  df-tset 15202  df-ple 15203  df-ds 15205  df-unif 15206  df-hom 15207  df-cco 15208  df-rest 15314  df-topn 15315  df-0g 15333  df-gsum 15334  df-topgen 15335  df-pt 15336  df-prds 15339  df-xrs 15393  df-qtop 15399  df-imas 15400  df-xps 15403  df-mre 15485  df-mrc 15486  df-acs 15488  df-mgm 16481  df-sgrp 16520  df-mnd 16530  df-submnd 16576  df-mulg 16669  df-cntz 16964  df-cmn 17425  df-psmet 18955  df-xmet 18956  df-met 18957  df-bl 18958  df-mopn 18959  df-fbas 18960  df-fg 18961  df-cnfld 18964  df-top 19914  df-bases 19915  df-topon 19916  df-topsp 19917  df-cld 20027  df-ntr 20028  df-cls 20029  df-nei 20107  df-lp 20145  df-perf 20146  df-cn 20236  df-cnp 20237  df-haus 20324  df-tx 20570  df-hmeo 20763  df-fil 20854  df-fm 20946  df-flim 20947  df-flf 20948  df-xms 21328  df-ms 21329  df-tms 21330  df-cncf 21903  df-limc 22814  df-dv 22815  df-log 23499  df-cxp 23500  df-atan 23786
This theorem is referenced by: (None)
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