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Theorem asinlem 23842
Description: The argument to the logarithm in df-asin 23839 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 9623 . . . 4  |-  _i  e.  CC
2 mulcl 9648 . . . 4  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
31, 2mpan 681 . . 3  |-  ( A  e.  CC  ->  (
_i  x.  A )  e.  CC )
4 ax-1cn 9622 . . . . 5  |-  1  e.  CC
5 sqcl 12368 . . . . 5  |-  ( A  e.  CC  ->  ( A ^ 2 )  e.  CC )
6 subcl 9899 . . . . 5  |-  ( ( 1  e.  CC  /\  ( A ^ 2 )  e.  CC )  -> 
( 1  -  ( A ^ 2 ) )  e.  CC )
74, 5, 6sylancr 674 . . . 4  |-  ( A  e.  CC  ->  (
1  -  ( A ^ 2 ) )  e.  CC )
87sqrtcld 13547 . . 3  |-  ( A  e.  CC  ->  ( sqr `  ( 1  -  ( A ^ 2 ) ) )  e.  CC )
93, 8subnegd 10018 . 2  |-  ( A  e.  CC  ->  (
( _i  x.  A
)  -  -u ( sqr `  ( 1  -  ( A ^ 2 ) ) ) )  =  ( ( _i  x.  A )  +  ( sqr `  (
1  -  ( A ^ 2 ) ) ) ) )
108negcld 9998 . . 3  |-  ( A  e.  CC  ->  -u ( sqr `  ( 1  -  ( A ^ 2 ) ) )  e.  CC )
11 0ne1 10704 . . . . . 6  |-  0  =/=  1
12 0cnd 9661 . . . . . . 7  |-  ( A  e.  CC  ->  0  e.  CC )
13 1cnd 9684 . . . . . . 7  |-  ( A  e.  CC  ->  1  e.  CC )
14 subcan2 9924 . . . . . . . 8  |-  ( ( 0  e.  CC  /\  1  e.  CC  /\  ( A ^ 2 )  e.  CC )  ->  (
( 0  -  ( A ^ 2 ) )  =  ( 1  -  ( A ^ 2 ) )  <->  0  = 
1 ) )
1514necon3bid 2679 . . . . . . 7  |-  ( ( 0  e.  CC  /\  1  e.  CC  /\  ( A ^ 2 )  e.  CC )  ->  (
( 0  -  ( A ^ 2 ) )  =/=  ( 1  -  ( A ^ 2 ) )  <->  0  =/=  1 ) )
1612, 13, 5, 15syl3anc 1276 . . . . . 6  |-  ( A  e.  CC  ->  (
( 0  -  ( A ^ 2 ) )  =/=  ( 1  -  ( A ^ 2 ) )  <->  0  =/=  1 ) )
1711, 16mpbiri 241 . . . . 5  |-  ( A  e.  CC  ->  (
0  -  ( A ^ 2 ) )  =/=  ( 1  -  ( A ^ 2 ) ) )
18 sqmul 12369 . . . . . . . 8  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( ( _i  x.  A ) ^ 2 )  =  ( ( _i ^ 2 )  x.  ( A ^
2 ) ) )
191, 18mpan 681 . . . . . . 7  |-  ( A  e.  CC  ->  (
( _i  x.  A
) ^ 2 )  =  ( ( _i
^ 2 )  x.  ( A ^ 2 ) ) )
20 i2 12406 . . . . . . . . 9  |-  ( _i
^ 2 )  = 
-u 1
2120oveq1i 6324 . . . . . . . 8  |-  ( ( _i ^ 2 )  x.  ( A ^
2 ) )  =  ( -u 1  x.  ( A ^ 2 ) )
225mulm1d 10097 . . . . . . . 8  |-  ( A  e.  CC  ->  ( -u 1  x.  ( A ^ 2 ) )  =  -u ( A ^
2 ) )
2321, 22syl5eq 2507 . . . . . . 7  |-  ( A  e.  CC  ->  (
( _i ^ 2 )  x.  ( A ^ 2 ) )  =  -u ( A ^
2 ) )
2419, 23eqtrd 2495 . . . . . 6  |-  ( A  e.  CC  ->  (
( _i  x.  A
) ^ 2 )  =  -u ( A ^
2 ) )
25 df-neg 9888 . . . . . 6  |-  -u ( A ^ 2 )  =  ( 0  -  ( A ^ 2 ) )
2624, 25syl6eq 2511 . . . . 5  |-  ( A  e.  CC  ->  (
( _i  x.  A
) ^ 2 )  =  ( 0  -  ( A ^ 2 ) ) )
27 sqneg 12366 . . . . . . 7  |-  ( ( sqr `  ( 1  -  ( A ^
2 ) ) )  e.  CC  ->  ( -u ( sqr `  (
1  -  ( A ^ 2 ) ) ) ^ 2 )  =  ( ( sqr `  ( 1  -  ( A ^ 2 ) ) ) ^ 2 ) )
288, 27syl 17 . . . . . 6  |-  ( A  e.  CC  ->  ( -u ( sqr `  (
1  -  ( A ^ 2 ) ) ) ^ 2 )  =  ( ( sqr `  ( 1  -  ( A ^ 2 ) ) ) ^ 2 ) )
297sqsqrtd 13549 . . . . . 6  |-  ( A  e.  CC  ->  (
( sqr `  (
1  -  ( A ^ 2 ) ) ) ^ 2 )  =  ( 1  -  ( A ^ 2 ) ) )
3028, 29eqtrd 2495 . . . . 5  |-  ( A  e.  CC  ->  ( -u ( sqr `  (
1  -  ( A ^ 2 ) ) ) ^ 2 )  =  ( 1  -  ( A ^ 2 ) ) )
3117, 26, 303netr4d 2712 . . . 4  |-  ( A  e.  CC  ->  (
( _i  x.  A
) ^ 2 )  =/=  ( -u ( sqr `  ( 1  -  ( A ^ 2 ) ) ) ^
2 ) )
32 oveq1 6321 . . . . 5  |-  ( ( _i  x.  A )  =  -u ( sqr `  (
1  -  ( A ^ 2 ) ) )  ->  ( (
_i  x.  A ) ^ 2 )  =  ( -u ( sqr `  ( 1  -  ( A ^ 2 ) ) ) ^ 2 ) )
3332necon3i 2667 . . . 4  |-  ( ( ( _i  x.  A
) ^ 2 )  =/=  ( -u ( sqr `  ( 1  -  ( A ^ 2 ) ) ) ^
2 )  ->  (
_i  x.  A )  =/=  -u ( sqr `  (
1  -  ( A ^ 2 ) ) ) )
3431, 33syl 17 . . 3  |-  ( A  e.  CC  ->  (
_i  x.  A )  =/=  -u ( sqr `  (
1  -  ( A ^ 2 ) ) ) )
353, 10, 34subne0d 10020 . 2  |-  ( A  e.  CC  ->  (
( _i  x.  A
)  -  -u ( sqr `  ( 1  -  ( A ^ 2 ) ) ) )  =/=  0 )
369, 35eqnetrrd 2703 1  |-  ( A  e.  CC  ->  (
( _i  x.  A
)  +  ( sqr `  ( 1  -  ( A ^ 2 ) ) ) )  =/=  0
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ w3a 991    = wceq 1454    e. wcel 1897    =/= wne 2632   ` cfv 5600  (class class class)co 6314   CCcc 9562   0cc0 9564   1c1 9565   _ici 9566    + caddc 9567    x. cmul 9569    - cmin 9885   -ucneg 9886   2c2 10686   ^cexp 12303   sqrcsqrt 13344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-cnex 9620  ax-resscn 9621  ax-1cn 9622  ax-icn 9623  ax-addcl 9624  ax-addrcl 9625  ax-mulcl 9626  ax-mulrcl 9627  ax-mulcom 9628  ax-addass 9629  ax-mulass 9630  ax-distr 9631  ax-i2m1 9632  ax-1ne0 9633  ax-1rid 9634  ax-rnegex 9635  ax-rrecex 9636  ax-cnre 9637  ax-pre-lttri 9638  ax-pre-lttrn 9639  ax-pre-ltadd 9640  ax-pre-mulgt0 9641  ax-pre-sup 9642
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-nel 2635  df-ral 2753  df-rex 2754  df-reu 2755  df-rmo 2756  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-riota 6276  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-om 6719  df-2nd 6820  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-er 7388  df-en 7595  df-dom 7596  df-sdom 7597  df-sup 7981  df-pnf 9702  df-mnf 9703  df-xr 9704  df-ltxr 9705  df-le 9706  df-sub 9887  df-neg 9888  df-div 10297  df-nn 10637  df-2 10695  df-3 10696  df-n0 10898  df-z 10966  df-uz 11188  df-rp 11331  df-seq 12245  df-exp 12304  df-cj 13210  df-re 13211  df-im 13212  df-sqrt 13346  df-abs 13347
This theorem is referenced by:  asinlem3  23845  asinf  23846  asinneg  23860  efiasin  23862  asinbnd  23873  dvasin  32072
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