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Theorem asinlem 20700
Description: The argument to the logarithm in df-asin 20697 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 9041 . . . 4  |-  _i  e.  CC
2 mulcl 9066 . . . 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 9040 . . . . 5  |-  1  e.  CC
5 sqcl 11436 . . . . 5  |-  ( A  e.  CC  ->  ( A ^ 2 )  e.  CC )
6 subcl 9297 . . . . 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 12231 . . 3  |-  ( A  e.  CC  ->  ( sqr `  ( 1  -  ( A ^ 2 ) ) )  e.  CC )
93, 8subnegd 9410 . 2  |-  ( A  e.  CC  ->  (
( _i  x.  A
)  -  -u ( sqr `  ( 1  -  ( A ^ 2 ) ) ) )  =  ( ( _i  x.  A )  +  ( sqr `  (
1  -  ( A ^ 2 ) ) ) ) )
108negcld 9390 . . 3  |-  ( A  e.  CC  ->  -u ( sqr `  ( 1  -  ( A ^ 2 ) ) )  e.  CC )
11 ax-1ne0 9051 . . . . . . 7  |-  1  =/=  0
1211necomi 2680 . . . . . 6  |-  0  =/=  1
13 0cn 9076 . . . . . . . 8  |-  0  e.  CC
1413a1i 11 . . . . . . 7  |-  ( A  e.  CC  ->  0  e.  CC )
154a1i 11 . . . . . . 7  |-  ( A  e.  CC  ->  1  e.  CC )
16 subcan2 9318 . . . . . . . 8  |-  ( ( 0  e.  CC  /\  1  e.  CC  /\  ( A ^ 2 )  e.  CC )  ->  (
( 0  -  ( A ^ 2 ) )  =  ( 1  -  ( A ^ 2 ) )  <->  0  = 
1 ) )
1716necon3bid 2633 . . . . . . 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 11437 . . . . . . . 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 11473 . . . . . . . . 9  |-  ( _i
^ 2 )  = 
-u 1
2322oveq1i 6083 . . . . . . . 8  |-  ( ( _i ^ 2 )  x.  ( A ^
2 ) )  =  ( -u 1  x.  ( A ^ 2 ) )
245mulm1d 9477 . . . . . . . 8  |-  ( A  e.  CC  ->  ( -u 1  x.  ( A ^ 2 ) )  =  -u ( A ^
2 ) )
2523, 24syl5eq 2479 . . . . . . 7  |-  ( A  e.  CC  ->  (
( _i ^ 2 )  x.  ( A ^ 2 ) )  =  -u ( A ^
2 ) )
2621, 25eqtrd 2467 . . . . . 6  |-  ( A  e.  CC  ->  (
( _i  x.  A
) ^ 2 )  =  -u ( A ^
2 ) )
27 df-neg 9286 . . . . . 6  |-  -u ( A ^ 2 )  =  ( 0  -  ( A ^ 2 ) )
2826, 27syl6eq 2483 . . . . 5  |-  ( A  e.  CC  ->  (
( _i  x.  A
) ^ 2 )  =  ( 0  -  ( A ^ 2 ) ) )
29 sqneg 11434 . . . . . . 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 12233 . . . . . 6  |-  ( A  e.  CC  ->  (
( sqr `  (
1  -  ( A ^ 2 ) ) ) ^ 2 )  =  ( 1  -  ( A ^ 2 ) ) )
3230, 31eqtrd 2467 . . . . 5  |-  ( A  e.  CC  ->  ( -u ( sqr `  (
1  -  ( A ^ 2 ) ) ) ^ 2 )  =  ( 1  -  ( A ^ 2 ) ) )
3319, 28, 323netr4d 2625 . . . 4  |-  ( A  e.  CC  ->  (
( _i  x.  A
) ^ 2 )  =/=  ( -u ( sqr `  ( 1  -  ( A ^ 2 ) ) ) ^
2 ) )
34 oveq1 6080 . . . . 5  |-  ( ( _i  x.  A )  =  -u ( sqr `  (
1  -  ( A ^ 2 ) ) )  ->  ( (
_i  x.  A ) ^ 2 )  =  ( -u ( sqr `  ( 1  -  ( A ^ 2 ) ) ) ^ 2 ) )
3534necon3i 2637 . . . 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 9412 . 2  |-  ( A  e.  CC  ->  (
( _i  x.  A
)  -  -u ( sqr `  ( 1  -  ( A ^ 2 ) ) ) )  =/=  0 )
389, 37eqnetrrd 2618 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 1652    e. wcel 1725    =/= wne 2598   ` cfv 5446  (class class class)co 6073   CCcc 8980   0cc0 8982   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  asinf  20704  asinneg  20718  efiasin  20720  asinbnd  20731  dvreasin  26270
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
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