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Theorem asinlem2 20577
Description: The argument to the logarithm in df-asin 20573 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 8983 . . . . 5  |-  _i  e.  CC
2 mulcl 9008 . . . . 5  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
31, 2mpan 652 . . . 4  |-  ( A  e.  CC  ->  (
_i  x.  A )  e.  CC )
4 ax-1cn 8982 . . . . . 6  |-  1  e.  CC
5 sqcl 11372 . . . . . 6  |-  ( A  e.  CC  ->  ( A ^ 2 )  e.  CC )
6 subcl 9238 . . . . . 6  |-  ( ( 1  e.  CC  /\  ( A ^ 2 )  e.  CC )  -> 
( 1  -  ( A ^ 2 ) )  e.  CC )
74, 5, 6sylancr 645 . . . . 5  |-  ( A  e.  CC  ->  (
1  -  ( A ^ 2 ) )  e.  CC )
87sqrcld 12167 . . . 4  |-  ( A  e.  CC  ->  ( sqr `  ( 1  -  ( A ^ 2 ) ) )  e.  CC )
93, 8addcomd 9201 . . 3  |-  ( A  e.  CC  ->  (
( _i  x.  A
)  +  ( sqr `  ( 1  -  ( A ^ 2 ) ) ) )  =  ( ( sqr `  (
1  -  ( A ^ 2 ) ) )  +  ( _i  x.  A ) ) )
10 mulneg2 9404 . . . . . 6  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  -u A
)  =  -u (
_i  x.  A )
)
111, 10mpan 652 . . . . 5  |-  ( A  e.  CC  ->  (
_i  x.  -u A )  =  -u ( _i  x.  A ) )
12 sqneg 11370 . . . . . . 7  |-  ( A  e.  CC  ->  ( -u A ^ 2 )  =  ( A ^
2 ) )
1312oveq2d 6037 . . . . . 6  |-  ( A  e.  CC  ->  (
1  -  ( -u A ^ 2 ) )  =  ( 1  -  ( A ^ 2 ) ) )
1413fveq2d 5673 . . . . 5  |-  ( A  e.  CC  ->  ( sqr `  ( 1  -  ( -u A ^
2 ) ) )  =  ( sqr `  (
1  -  ( A ^ 2 ) ) ) )
1511, 14oveq12d 6039 . . . 4  |-  ( A  e.  CC  ->  (
( _i  x.  -u A
)  +  ( sqr `  ( 1  -  ( -u A ^ 2 ) ) ) )  =  ( -u ( _i  x.  A )  +  ( sqr `  (
1  -  ( A ^ 2 ) ) ) ) )
163negcld 9331 . . . . 5  |-  ( A  e.  CC  ->  -u (
_i  x.  A )  e.  CC )
1716, 8addcomd 9201 . . . 4  |-  ( A  e.  CC  ->  ( -u ( _i  x.  A
)  +  ( sqr `  ( 1  -  ( A ^ 2 ) ) ) )  =  ( ( sqr `  (
1  -  ( A ^ 2 ) ) )  +  -u (
_i  x.  A )
) )
188, 3negsubd 9350 . . . 4  |-  ( A  e.  CC  ->  (
( sqr `  (
1  -  ( A ^ 2 ) ) )  +  -u (
_i  x.  A )
)  =  ( ( sqr `  ( 1  -  ( A ^
2 ) ) )  -  ( _i  x.  A ) ) )
1915, 17, 183eqtrd 2424 . . 3  |-  ( A  e.  CC  ->  (
( _i  x.  -u A
)  +  ( sqr `  ( 1  -  ( -u A ^ 2 ) ) ) )  =  ( ( sqr `  (
1  -  ( A ^ 2 ) ) )  -  ( _i  x.  A ) ) )
209, 19oveq12d 6039 . 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 12169 . . . 4  |-  ( A  e.  CC  ->  (
( sqr `  (
1  -  ( A ^ 2 ) ) ) ^ 2 )  =  ( 1  -  ( A ^ 2 ) ) )
22 sqmul 11373 . . . . . 6  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( ( _i  x.  A ) ^ 2 )  =  ( ( _i ^ 2 )  x.  ( A ^
2 ) ) )
231, 22mpan 652 . . . . 5  |-  ( A  e.  CC  ->  (
( _i  x.  A
) ^ 2 )  =  ( ( _i
^ 2 )  x.  ( A ^ 2 ) ) )
24 i2 11409 . . . . . . 7  |-  ( _i
^ 2 )  = 
-u 1
2524oveq1i 6031 . . . . . 6  |-  ( ( _i ^ 2 )  x.  ( A ^
2 ) )  =  ( -u 1  x.  ( A ^ 2 ) )
265mulm1d 9418 . . . . . 6  |-  ( A  e.  CC  ->  ( -u 1  x.  ( A ^ 2 ) )  =  -u ( A ^
2 ) )
2725, 26syl5eq 2432 . . . . 5  |-  ( A  e.  CC  ->  (
( _i ^ 2 )  x.  ( A ^ 2 ) )  =  -u ( A ^
2 ) )
2823, 27eqtrd 2420 . . . 4  |-  ( A  e.  CC  ->  (
( _i  x.  A
) ^ 2 )  =  -u ( A ^
2 ) )
2921, 28oveq12d 6039 . . 3  |-  ( A  e.  CC  ->  (
( ( sqr `  (
1  -  ( A ^ 2 ) ) ) ^ 2 )  -  ( ( _i  x.  A ) ^
2 ) )  =  ( ( 1  -  ( A ^ 2 ) )  -  -u ( A ^ 2 ) ) )
30 subsq 11416 . . . 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 643 . . 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 9351 . . 3  |-  ( A  e.  CC  ->  (
( 1  -  ( A ^ 2 ) )  -  -u ( A ^
2 ) )  =  ( ( 1  -  ( A ^ 2 ) )  +  ( A ^ 2 ) ) )
3329, 31, 323eqtr3d 2428 . 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 9247 . . 3  |-  ( ( 1  e.  CC  /\  ( A ^ 2 )  e.  CC )  -> 
( ( 1  -  ( A ^ 2 ) )  +  ( A ^ 2 ) )  =  1 )
354, 5, 34sylancr 645 . 2  |-  ( A  e.  CC  ->  (
( 1  -  ( A ^ 2 ) )  +  ( A ^
2 ) )  =  1 )
3620, 33, 353eqtrd 2424 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 set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   ` cfv 5395  (class class class)co 6021   CCcc 8922   1c1 8925   _ici 8926    + caddc 8927    x. cmul 8929    - cmin 9224   -ucneg 9225   2c2 9982   ^cexp 11310   sqrcsqr 11966
This theorem is referenced by:  asinlem3  20579  asinneg  20594
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-sup 7382  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-n0 10155  df-z 10216  df-uz 10422  df-rp 10546  df-seq 11252  df-exp 11311  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-abs 11969
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