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Theorem asinlem2 20165
Description: The argument to the logarithm in df-asin 20161 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 8796 . . . . 5  |-  _i  e.  CC
2 mulcl 8821 . . . . 5  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
31, 2mpan 651 . . . 4  |-  ( A  e.  CC  ->  (
_i  x.  A )  e.  CC )
4 ax-1cn 8795 . . . . . 6  |-  1  e.  CC
5 sqcl 11166 . . . . . 6  |-  ( A  e.  CC  ->  ( A ^ 2 )  e.  CC )
6 subcl 9051 . . . . . 6  |-  ( ( 1  e.  CC  /\  ( A ^ 2 )  e.  CC )  -> 
( 1  -  ( A ^ 2 ) )  e.  CC )
74, 5, 6sylancr 644 . . . . 5  |-  ( A  e.  CC  ->  (
1  -  ( A ^ 2 ) )  e.  CC )
87sqrcld 11919 . . . 4  |-  ( A  e.  CC  ->  ( sqr `  ( 1  -  ( A ^ 2 ) ) )  e.  CC )
93, 8addcomd 9014 . . 3  |-  ( A  e.  CC  ->  (
( _i  x.  A
)  +  ( sqr `  ( 1  -  ( A ^ 2 ) ) ) )  =  ( ( sqr `  (
1  -  ( A ^ 2 ) ) )  +  ( _i  x.  A ) ) )
10 mulneg2 9217 . . . . . 6  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  -u A
)  =  -u (
_i  x.  A )
)
111, 10mpan 651 . . . . 5  |-  ( A  e.  CC  ->  (
_i  x.  -u A )  =  -u ( _i  x.  A ) )
12 sqneg 11164 . . . . . . 7  |-  ( A  e.  CC  ->  ( -u A ^ 2 )  =  ( A ^
2 ) )
1312oveq2d 5874 . . . . . 6  |-  ( A  e.  CC  ->  (
1  -  ( -u A ^ 2 ) )  =  ( 1  -  ( A ^ 2 ) ) )
1413fveq2d 5529 . . . . 5  |-  ( A  e.  CC  ->  ( sqr `  ( 1  -  ( -u A ^
2 ) ) )  =  ( sqr `  (
1  -  ( A ^ 2 ) ) ) )
1511, 14oveq12d 5876 . . . 4  |-  ( A  e.  CC  ->  (
( _i  x.  -u A
)  +  ( sqr `  ( 1  -  ( -u A ^ 2 ) ) ) )  =  ( -u ( _i  x.  A )  +  ( sqr `  (
1  -  ( A ^ 2 ) ) ) ) )
163negcld 9144 . . . . 5  |-  ( A  e.  CC  ->  -u (
_i  x.  A )  e.  CC )
1716, 8addcomd 9014 . . . 4  |-  ( A  e.  CC  ->  ( -u ( _i  x.  A
)  +  ( sqr `  ( 1  -  ( A ^ 2 ) ) ) )  =  ( ( sqr `  (
1  -  ( A ^ 2 ) ) )  +  -u (
_i  x.  A )
) )
188, 3negsubd 9163 . . . 4  |-  ( A  e.  CC  ->  (
( sqr `  (
1  -  ( A ^ 2 ) ) )  +  -u (
_i  x.  A )
)  =  ( ( sqr `  ( 1  -  ( A ^
2 ) ) )  -  ( _i  x.  A ) ) )
1915, 17, 183eqtrd 2319 . . 3  |-  ( A  e.  CC  ->  (
( _i  x.  -u A
)  +  ( sqr `  ( 1  -  ( -u A ^ 2 ) ) ) )  =  ( ( sqr `  (
1  -  ( A ^ 2 ) ) )  -  ( _i  x.  A ) ) )
209, 19oveq12d 5876 . 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 11921 . . . 4  |-  ( A  e.  CC  ->  (
( sqr `  (
1  -  ( A ^ 2 ) ) ) ^ 2 )  =  ( 1  -  ( A ^ 2 ) ) )
22 sqmul 11167 . . . . . 6  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( ( _i  x.  A ) ^ 2 )  =  ( ( _i ^ 2 )  x.  ( A ^
2 ) ) )
231, 22mpan 651 . . . . 5  |-  ( A  e.  CC  ->  (
( _i  x.  A
) ^ 2 )  =  ( ( _i
^ 2 )  x.  ( A ^ 2 ) ) )
24 i2 11203 . . . . . . 7  |-  ( _i
^ 2 )  = 
-u 1
2524oveq1i 5868 . . . . . 6  |-  ( ( _i ^ 2 )  x.  ( A ^
2 ) )  =  ( -u 1  x.  ( A ^ 2 ) )
265mulm1d 9231 . . . . . 6  |-  ( A  e.  CC  ->  ( -u 1  x.  ( A ^ 2 ) )  =  -u ( A ^
2 ) )
2725, 26syl5eq 2327 . . . . 5  |-  ( A  e.  CC  ->  (
( _i ^ 2 )  x.  ( A ^ 2 ) )  =  -u ( A ^
2 ) )
2823, 27eqtrd 2315 . . . 4  |-  ( A  e.  CC  ->  (
( _i  x.  A
) ^ 2 )  =  -u ( A ^
2 ) )
2921, 28oveq12d 5876 . . 3  |-  ( A  e.  CC  ->  (
( ( sqr `  (
1  -  ( A ^ 2 ) ) ) ^ 2 )  -  ( ( _i  x.  A ) ^
2 ) )  =  ( ( 1  -  ( A ^ 2 ) )  -  -u ( A ^ 2 ) ) )
30 subsq 11210 . . . 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 642 . . 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 9164 . . 3  |-  ( A  e.  CC  ->  (
( 1  -  ( A ^ 2 ) )  -  -u ( A ^
2 ) )  =  ( ( 1  -  ( A ^ 2 ) )  +  ( A ^ 2 ) ) )
3329, 31, 323eqtr3d 2323 . 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 9060 . . 3  |-  ( ( 1  e.  CC  /\  ( A ^ 2 )  e.  CC )  -> 
( ( 1  -  ( A ^ 2 ) )  +  ( A ^ 2 ) )  =  1 )
354, 5, 34sylancr 644 . 2  |-  ( A  e.  CC  ->  (
( 1  -  ( A ^ 2 ) )  +  ( A ^
2 ) )  =  1 )
3620, 33, 353eqtrd 2319 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 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   CCcc 8735   1c1 8738   _ici 8739    + caddc 8740    x. cmul 8742    - cmin 9037   -ucneg 9038   2c2 9795   ^cexp 11104   sqrcsqr 11718
This theorem is referenced by:  asinlem3  20167  asinneg  20182
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721
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