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Theorem asinlem 20575
Description: The argument to the logarithm in df-asin 20572 is always nonzero. (Contributed by Mario Carneiro, 31-Mar-2015.)
Assertion
Ref Expression
asinlem  |-  ( A  e.  CC  ->  (
( _i  x.  A
)  +  ( sqr `  ( 1  -  ( A ^ 2 ) ) ) )  =/=  0
)

Proof of Theorem asinlem
StepHypRef Expression
1 ax-icn 8982 . . . 4  |-  _i  e.  CC
2 mulcl 9007 . . . 4  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
31, 2mpan 652 . . 3  |-  ( A  e.  CC  ->  (
_i  x.  A )  e.  CC )
4 ax-1cn 8981 . . . . 5  |-  1  e.  CC
5 sqcl 11371 . . . . 5  |-  ( A  e.  CC  ->  ( A ^ 2 )  e.  CC )
6 subcl 9237 . . . . 5  |-  ( ( 1  e.  CC  /\  ( A ^ 2 )  e.  CC )  -> 
( 1  -  ( A ^ 2 ) )  e.  CC )
74, 5, 6sylancr 645 . . . 4  |-  ( A  e.  CC  ->  (
1  -  ( A ^ 2 ) )  e.  CC )
87sqrcld 12166 . . 3  |-  ( A  e.  CC  ->  ( sqr `  ( 1  -  ( A ^ 2 ) ) )  e.  CC )
93, 8subnegd 9350 . 2  |-  ( A  e.  CC  ->  (
( _i  x.  A
)  -  -u ( sqr `  ( 1  -  ( A ^ 2 ) ) ) )  =  ( ( _i  x.  A )  +  ( sqr `  (
1  -  ( A ^ 2 ) ) ) ) )
108negcld 9330 . . 3  |-  ( A  e.  CC  ->  -u ( sqr `  ( 1  -  ( A ^ 2 ) ) )  e.  CC )
11 ax-1ne0 8992 . . . . . . 7  |-  1  =/=  0
1211necomi 2632 . . . . . 6  |-  0  =/=  1
13 0cn 9017 . . . . . . . 8  |-  0  e.  CC
1413a1i 11 . . . . . . 7  |-  ( A  e.  CC  ->  0  e.  CC )
154a1i 11 . . . . . . 7  |-  ( A  e.  CC  ->  1  e.  CC )
16 subcan2 9258 . . . . . . . 8  |-  ( ( 0  e.  CC  /\  1  e.  CC  /\  ( A ^ 2 )  e.  CC )  ->  (
( 0  -  ( A ^ 2 ) )  =  ( 1  -  ( A ^ 2 ) )  <->  0  = 
1 ) )
1716necon3bid 2585 . . . . . . 7  |-  ( ( 0  e.  CC  /\  1  e.  CC  /\  ( A ^ 2 )  e.  CC )  ->  (
( 0  -  ( A ^ 2 ) )  =/=  ( 1  -  ( A ^ 2 ) )  <->  0  =/=  1 ) )
1814, 15, 5, 17syl3anc 1184 . . . . . 6  |-  ( A  e.  CC  ->  (
( 0  -  ( A ^ 2 ) )  =/=  ( 1  -  ( A ^ 2 ) )  <->  0  =/=  1 ) )
1912, 18mpbiri 225 . . . . 5  |-  ( A  e.  CC  ->  (
0  -  ( A ^ 2 ) )  =/=  ( 1  -  ( A ^ 2 ) ) )
20 sqmul 11372 . . . . . . . 8  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( ( _i  x.  A ) ^ 2 )  =  ( ( _i ^ 2 )  x.  ( A ^
2 ) ) )
211, 20mpan 652 . . . . . . 7  |-  ( A  e.  CC  ->  (
( _i  x.  A
) ^ 2 )  =  ( ( _i
^ 2 )  x.  ( A ^ 2 ) ) )
22 i2 11408 . . . . . . . . 9  |-  ( _i
^ 2 )  = 
-u 1
2322oveq1i 6030 . . . . . . . 8  |-  ( ( _i ^ 2 )  x.  ( A ^
2 ) )  =  ( -u 1  x.  ( A ^ 2 ) )
245mulm1d 9417 . . . . . . . 8  |-  ( A  e.  CC  ->  ( -u 1  x.  ( A ^ 2 ) )  =  -u ( A ^
2 ) )
2523, 24syl5eq 2431 . . . . . . 7  |-  ( A  e.  CC  ->  (
( _i ^ 2 )  x.  ( A ^ 2 ) )  =  -u ( A ^
2 ) )
2621, 25eqtrd 2419 . . . . . 6  |-  ( A  e.  CC  ->  (
( _i  x.  A
) ^ 2 )  =  -u ( A ^
2 ) )
27 df-neg 9226 . . . . . 6  |-  -u ( A ^ 2 )  =  ( 0  -  ( A ^ 2 ) )
2826, 27syl6eq 2435 . . . . 5  |-  ( A  e.  CC  ->  (
( _i  x.  A
) ^ 2 )  =  ( 0  -  ( A ^ 2 ) ) )
29 sqneg 11369 . . . . . . 7  |-  ( ( sqr `  ( 1  -  ( A ^
2 ) ) )  e.  CC  ->  ( -u ( sqr `  (
1  -  ( A ^ 2 ) ) ) ^ 2 )  =  ( ( sqr `  ( 1  -  ( A ^ 2 ) ) ) ^ 2 ) )
308, 29syl 16 . . . . . 6  |-  ( A  e.  CC  ->  ( -u ( sqr `  (
1  -  ( A ^ 2 ) ) ) ^ 2 )  =  ( ( sqr `  ( 1  -  ( A ^ 2 ) ) ) ^ 2 ) )
317sqsqrd 12168 . . . . . 6  |-  ( A  e.  CC  ->  (
( sqr `  (
1  -  ( A ^ 2 ) ) ) ^ 2 )  =  ( 1  -  ( A ^ 2 ) ) )
3230, 31eqtrd 2419 . . . . 5  |-  ( A  e.  CC  ->  ( -u ( sqr `  (
1  -  ( A ^ 2 ) ) ) ^ 2 )  =  ( 1  -  ( A ^ 2 ) ) )
3319, 28, 323netr4d 2577 . . . 4  |-  ( A  e.  CC  ->  (
( _i  x.  A
) ^ 2 )  =/=  ( -u ( sqr `  ( 1  -  ( A ^ 2 ) ) ) ^
2 ) )
34 oveq1 6027 . . . . 5  |-  ( ( _i  x.  A )  =  -u ( sqr `  (
1  -  ( A ^ 2 ) ) )  ->  ( (
_i  x.  A ) ^ 2 )  =  ( -u ( sqr `  ( 1  -  ( A ^ 2 ) ) ) ^ 2 ) )
3534necon3i 2589 . . . 4  |-  ( ( ( _i  x.  A
) ^ 2 )  =/=  ( -u ( sqr `  ( 1  -  ( A ^ 2 ) ) ) ^
2 )  ->  (
_i  x.  A )  =/=  -u ( sqr `  (
1  -  ( A ^ 2 ) ) ) )
3633, 35syl 16 . . 3  |-  ( A  e.  CC  ->  (
_i  x.  A )  =/=  -u ( sqr `  (
1  -  ( A ^ 2 ) ) ) )
373, 10, 36subne0d 9352 . 2  |-  ( A  e.  CC  ->  (
( _i  x.  A
)  -  -u ( sqr `  ( 1  -  ( A ^ 2 ) ) ) )  =/=  0 )
389, 37eqnetrrd 2570 1  |-  ( A  e.  CC  ->  (
( _i  x.  A
)  +  ( sqr `  ( 1  -  ( A ^ 2 ) ) ) )  =/=  0
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2550   ` cfv 5394  (class class class)co 6020   CCcc 8921   0cc0 8923   1c1 8924   _ici 8925    + caddc 8926    x. cmul 8928    - cmin 9223   -ucneg 9224   2c2 9981   ^cexp 11309   sqrcsqr 11965
This theorem is referenced by:  asinlem3  20578  asinf  20579  asinneg  20593  efiasin  20595  asinbnd  20606  dvreasin  25980
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 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-sup 7381  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-n0 10154  df-z 10215  df-uz 10421  df-rp 10545  df-seq 11251  df-exp 11310  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968
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