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Theorem asinlem2 22264
Description: The argument to the logarithm in df-asin 22260 has the property that replacing  A with  -u A in the expression gives the reciprocal. (Contributed by Mario Carneiro, 1-Apr-2015.)
Assertion
Ref Expression
asinlem2  |-  ( A  e.  CC  ->  (
( ( _i  x.  A )  +  ( sqr `  ( 1  -  ( A ^
2 ) ) ) )  x.  ( ( _i  x.  -u A
)  +  ( sqr `  ( 1  -  ( -u A ^ 2 ) ) ) ) )  =  1 )

Proof of Theorem asinlem2
StepHypRef Expression
1 ax-icn 9341 . . . . 5  |-  _i  e.  CC
2 mulcl 9366 . . . . 5  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
31, 2mpan 670 . . . 4  |-  ( A  e.  CC  ->  (
_i  x.  A )  e.  CC )
4 ax-1cn 9340 . . . . . 6  |-  1  e.  CC
5 sqcl 11928 . . . . . 6  |-  ( A  e.  CC  ->  ( A ^ 2 )  e.  CC )
6 subcl 9609 . . . . . 6  |-  ( ( 1  e.  CC  /\  ( A ^ 2 )  e.  CC )  -> 
( 1  -  ( A ^ 2 ) )  e.  CC )
74, 5, 6sylancr 663 . . . . 5  |-  ( A  e.  CC  ->  (
1  -  ( A ^ 2 ) )  e.  CC )
87sqrcld 12923 . . . 4  |-  ( A  e.  CC  ->  ( sqr `  ( 1  -  ( A ^ 2 ) ) )  e.  CC )
93, 8addcomd 9571 . . 3  |-  ( A  e.  CC  ->  (
( _i  x.  A
)  +  ( sqr `  ( 1  -  ( A ^ 2 ) ) ) )  =  ( ( sqr `  (
1  -  ( A ^ 2 ) ) )  +  ( _i  x.  A ) ) )
10 mulneg2 9782 . . . . . 6  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  -u A
)  =  -u (
_i  x.  A )
)
111, 10mpan 670 . . . . 5  |-  ( A  e.  CC  ->  (
_i  x.  -u A )  =  -u ( _i  x.  A ) )
12 sqneg 11926 . . . . . . 7  |-  ( A  e.  CC  ->  ( -u A ^ 2 )  =  ( A ^
2 ) )
1312oveq2d 6107 . . . . . 6  |-  ( A  e.  CC  ->  (
1  -  ( -u A ^ 2 ) )  =  ( 1  -  ( A ^ 2 ) ) )
1413fveq2d 5695 . . . . 5  |-  ( A  e.  CC  ->  ( sqr `  ( 1  -  ( -u A ^
2 ) ) )  =  ( sqr `  (
1  -  ( A ^ 2 ) ) ) )
1511, 14oveq12d 6109 . . . 4  |-  ( A  e.  CC  ->  (
( _i  x.  -u A
)  +  ( sqr `  ( 1  -  ( -u A ^ 2 ) ) ) )  =  ( -u ( _i  x.  A )  +  ( sqr `  (
1  -  ( A ^ 2 ) ) ) ) )
163negcld 9706 . . . . 5  |-  ( A  e.  CC  ->  -u (
_i  x.  A )  e.  CC )
1716, 8addcomd 9571 . . . 4  |-  ( A  e.  CC  ->  ( -u ( _i  x.  A
)  +  ( sqr `  ( 1  -  ( A ^ 2 ) ) ) )  =  ( ( sqr `  (
1  -  ( A ^ 2 ) ) )  +  -u (
_i  x.  A )
) )
188, 3negsubd 9725 . . . 4  |-  ( A  e.  CC  ->  (
( sqr `  (
1  -  ( A ^ 2 ) ) )  +  -u (
_i  x.  A )
)  =  ( ( sqr `  ( 1  -  ( A ^
2 ) ) )  -  ( _i  x.  A ) ) )
1915, 17, 183eqtrd 2479 . . 3  |-  ( A  e.  CC  ->  (
( _i  x.  -u A
)  +  ( sqr `  ( 1  -  ( -u A ^ 2 ) ) ) )  =  ( ( sqr `  (
1  -  ( A ^ 2 ) ) )  -  ( _i  x.  A ) ) )
209, 19oveq12d 6109 . 2  |-  ( A  e.  CC  ->  (
( ( _i  x.  A )  +  ( sqr `  ( 1  -  ( A ^
2 ) ) ) )  x.  ( ( _i  x.  -u A
)  +  ( sqr `  ( 1  -  ( -u A ^ 2 ) ) ) ) )  =  ( ( ( sqr `  ( 1  -  ( A ^
2 ) ) )  +  ( _i  x.  A ) )  x.  ( ( sqr `  (
1  -  ( A ^ 2 ) ) )  -  ( _i  x.  A ) ) ) )
217sqsqrd 12925 . . . 4  |-  ( A  e.  CC  ->  (
( sqr `  (
1  -  ( A ^ 2 ) ) ) ^ 2 )  =  ( 1  -  ( A ^ 2 ) ) )
22 sqmul 11929 . . . . . 6  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( ( _i  x.  A ) ^ 2 )  =  ( ( _i ^ 2 )  x.  ( A ^
2 ) ) )
231, 22mpan 670 . . . . 5  |-  ( A  e.  CC  ->  (
( _i  x.  A
) ^ 2 )  =  ( ( _i
^ 2 )  x.  ( A ^ 2 ) ) )
24 i2 11966 . . . . . . 7  |-  ( _i
^ 2 )  = 
-u 1
2524oveq1i 6101 . . . . . 6  |-  ( ( _i ^ 2 )  x.  ( A ^
2 ) )  =  ( -u 1  x.  ( A ^ 2 ) )
265mulm1d 9796 . . . . . 6  |-  ( A  e.  CC  ->  ( -u 1  x.  ( A ^ 2 ) )  =  -u ( A ^
2 ) )
2725, 26syl5eq 2487 . . . . 5  |-  ( A  e.  CC  ->  (
( _i ^ 2 )  x.  ( A ^ 2 ) )  =  -u ( A ^
2 ) )
2823, 27eqtrd 2475 . . . 4  |-  ( A  e.  CC  ->  (
( _i  x.  A
) ^ 2 )  =  -u ( A ^
2 ) )
2921, 28oveq12d 6109 . . 3  |-  ( A  e.  CC  ->  (
( ( sqr `  (
1  -  ( A ^ 2 ) ) ) ^ 2 )  -  ( ( _i  x.  A ) ^
2 ) )  =  ( ( 1  -  ( A ^ 2 ) )  -  -u ( A ^ 2 ) ) )
30 subsq 11973 . . . 4  |-  ( ( ( sqr `  (
1  -  ( A ^ 2 ) ) )  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( ( ( sqr `  ( 1  -  ( A ^
2 ) ) ) ^ 2 )  -  ( ( _i  x.  A ) ^ 2 ) )  =  ( ( ( sqr `  (
1  -  ( A ^ 2 ) ) )  +  ( _i  x.  A ) )  x.  ( ( sqr `  ( 1  -  ( A ^ 2 ) ) )  -  ( _i  x.  A ) ) ) )
318, 3, 30syl2anc 661 . . 3  |-  ( A  e.  CC  ->  (
( ( sqr `  (
1  -  ( A ^ 2 ) ) ) ^ 2 )  -  ( ( _i  x.  A ) ^
2 ) )  =  ( ( ( sqr `  ( 1  -  ( A ^ 2 ) ) )  +  ( _i  x.  A ) )  x.  ( ( sqr `  ( 1  -  ( A ^ 2 ) ) )  -  ( _i  x.  A ) ) ) )
327, 5subnegd 9726 . . 3  |-  ( A  e.  CC  ->  (
( 1  -  ( A ^ 2 ) )  -  -u ( A ^
2 ) )  =  ( ( 1  -  ( A ^ 2 ) )  +  ( A ^ 2 ) ) )
3329, 31, 323eqtr3d 2483 . 2  |-  ( A  e.  CC  ->  (
( ( sqr `  (
1  -  ( A ^ 2 ) ) )  +  ( _i  x.  A ) )  x.  ( ( sqr `  ( 1  -  ( A ^ 2 ) ) )  -  ( _i  x.  A ) ) )  =  ( ( 1  -  ( A ^ 2 ) )  +  ( A ^
2 ) ) )
34 npcan 9619 . . 3  |-  ( ( 1  e.  CC  /\  ( A ^ 2 )  e.  CC )  -> 
( ( 1  -  ( A ^ 2 ) )  +  ( A ^ 2 ) )  =  1 )
354, 5, 34sylancr 663 . 2  |-  ( A  e.  CC  ->  (
( 1  -  ( A ^ 2 ) )  +  ( A ^
2 ) )  =  1 )
3620, 33, 353eqtrd 2479 1  |-  ( A  e.  CC  ->  (
( ( _i  x.  A )  +  ( sqr `  ( 1  -  ( A ^
2 ) ) ) )  x.  ( ( _i  x.  -u A
)  +  ( sqr `  ( 1  -  ( -u A ^ 2 ) ) ) ) )  =  1 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   ` cfv 5418  (class class class)co 6091   CCcc 9280   1c1 9283   _ici 9284    + caddc 9285    x. cmul 9287    - cmin 9595   -ucneg 9596   2c2 10371   ^cexp 11865   sqrcsqr 12722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-2nd 6578  df-recs 6832  df-rdg 6866  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-sup 7691  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-n0 10580  df-z 10647  df-uz 10862  df-rp 10992  df-seq 11807  df-exp 11866  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725
This theorem is referenced by:  asinlem3  22266  asinneg  22281
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