MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  asinlem2 Structured version   Unicode version

Theorem asinlem2 20701
Description: The argument to the logarithm in df-asin 20697 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 9041 . . . . 5  |-  _i  e.  CC
2 mulcl 9066 . . . . 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 9040 . . . . . 6  |-  1  e.  CC
5 sqcl 11436 . . . . . 6  |-  ( A  e.  CC  ->  ( A ^ 2 )  e.  CC )
6 subcl 9297 . . . . . 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 12231 . . . 4  |-  ( A  e.  CC  ->  ( sqr `  ( 1  -  ( A ^ 2 ) ) )  e.  CC )
93, 8addcomd 9260 . . 3  |-  ( A  e.  CC  ->  (
( _i  x.  A
)  +  ( sqr `  ( 1  -  ( A ^ 2 ) ) ) )  =  ( ( sqr `  (
1  -  ( A ^ 2 ) ) )  +  ( _i  x.  A ) ) )
10 mulneg2 9463 . . . . . 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 11434 . . . . . . 7  |-  ( A  e.  CC  ->  ( -u A ^ 2 )  =  ( A ^
2 ) )
1312oveq2d 6089 . . . . . 6  |-  ( A  e.  CC  ->  (
1  -  ( -u A ^ 2 ) )  =  ( 1  -  ( A ^ 2 ) ) )
1413fveq2d 5724 . . . . 5  |-  ( A  e.  CC  ->  ( sqr `  ( 1  -  ( -u A ^
2 ) ) )  =  ( sqr `  (
1  -  ( A ^ 2 ) ) ) )
1511, 14oveq12d 6091 . . . 4  |-  ( A  e.  CC  ->  (
( _i  x.  -u A
)  +  ( sqr `  ( 1  -  ( -u A ^ 2 ) ) ) )  =  ( -u ( _i  x.  A )  +  ( sqr `  (
1  -  ( A ^ 2 ) ) ) ) )
163negcld 9390 . . . . 5  |-  ( A  e.  CC  ->  -u (
_i  x.  A )  e.  CC )
1716, 8addcomd 9260 . . . 4  |-  ( A  e.  CC  ->  ( -u ( _i  x.  A
)  +  ( sqr `  ( 1  -  ( A ^ 2 ) ) ) )  =  ( ( sqr `  (
1  -  ( A ^ 2 ) ) )  +  -u (
_i  x.  A )
) )
188, 3negsubd 9409 . . . 4  |-  ( A  e.  CC  ->  (
( sqr `  (
1  -  ( A ^ 2 ) ) )  +  -u (
_i  x.  A )
)  =  ( ( sqr `  ( 1  -  ( A ^
2 ) ) )  -  ( _i  x.  A ) ) )
1915, 17, 183eqtrd 2471 . . 3  |-  ( A  e.  CC  ->  (
( _i  x.  -u A
)  +  ( sqr `  ( 1  -  ( -u A ^ 2 ) ) ) )  =  ( ( sqr `  (
1  -  ( A ^ 2 ) ) )  -  ( _i  x.  A ) ) )
209, 19oveq12d 6091 . 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 12233 . . . 4  |-  ( A  e.  CC  ->  (
( sqr `  (
1  -  ( A ^ 2 ) ) ) ^ 2 )  =  ( 1  -  ( A ^ 2 ) ) )
22 sqmul 11437 . . . . . 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 11473 . . . . . . 7  |-  ( _i
^ 2 )  = 
-u 1
2524oveq1i 6083 . . . . . 6  |-  ( ( _i ^ 2 )  x.  ( A ^
2 ) )  =  ( -u 1  x.  ( A ^ 2 ) )
265mulm1d 9477 . . . . . 6  |-  ( A  e.  CC  ->  ( -u 1  x.  ( A ^ 2 ) )  =  -u ( A ^
2 ) )
2725, 26syl5eq 2479 . . . . 5  |-  ( A  e.  CC  ->  (
( _i ^ 2 )  x.  ( A ^ 2 ) )  =  -u ( A ^
2 ) )
2823, 27eqtrd 2467 . . . 4  |-  ( A  e.  CC  ->  (
( _i  x.  A
) ^ 2 )  =  -u ( A ^
2 ) )
2921, 28oveq12d 6091 . . 3  |-  ( A  e.  CC  ->  (
( ( sqr `  (
1  -  ( A ^ 2 ) ) ) ^ 2 )  -  ( ( _i  x.  A ) ^
2 ) )  =  ( ( 1  -  ( A ^ 2 ) )  -  -u ( A ^ 2 ) ) )
30 subsq 11480 . . . 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 9410 . . 3  |-  ( A  e.  CC  ->  (
( 1  -  ( A ^ 2 ) )  -  -u ( A ^
2 ) )  =  ( ( 1  -  ( A ^ 2 ) )  +  ( A ^ 2 ) ) )
3329, 31, 323eqtr3d 2475 . 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 9306 . . 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 2471 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 1652    e. wcel 1725   ` cfv 5446  (class class class)co 6073   CCcc 8980   1c1 8983   _ici 8984    + caddc 8985    x. cmul 8987    - cmin 9283   -ucneg 9284   2c2 10041   ^cexp 11374   sqrcsqr 12030
This theorem is referenced by:  asinlem3  20703  asinneg  20718
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033
  Copyright terms: Public domain W3C validator