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Theorem recex 10080
Description: Existence of reciprocal of nonzero complex number. (Contributed by Eric Schmidt, 22-May-2007.)
Assertion
Ref Expression
recex  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  E. x  e.  CC  ( A  x.  x
)  =  1 )
Distinct variable group:    x, A

Proof of Theorem recex
Dummy variables  y 
a  b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnre 9494 . . 3  |-  ( A  e.  CC  ->  E. a  e.  RR  E. b  e.  RR  A  =  ( a  +  ( _i  x.  b ) ) )
2 recextlem2 10079 . . . . . . . . 9  |-  ( ( a  e.  RR  /\  b  e.  RR  /\  (
a  +  ( _i  x.  b ) )  =/=  0 )  -> 
( ( a  x.  a )  +  ( b  x.  b ) )  =/=  0 )
323expia 1190 . . . . . . . 8  |-  ( ( a  e.  RR  /\  b  e.  RR )  ->  ( ( a  +  ( _i  x.  b
) )  =/=  0  ->  ( ( a  x.  a )  +  ( b  x.  b ) )  =/=  0 ) )
4 remulcl 9479 . . . . . . . . . . . . 13  |-  ( ( a  e.  RR  /\  a  e.  RR )  ->  ( a  x.  a
)  e.  RR )
54anidms 645 . . . . . . . . . . . 12  |-  ( a  e.  RR  ->  (
a  x.  a )  e.  RR )
6 remulcl 9479 . . . . . . . . . . . . 13  |-  ( ( b  e.  RR  /\  b  e.  RR )  ->  ( b  x.  b
)  e.  RR )
76anidms 645 . . . . . . . . . . . 12  |-  ( b  e.  RR  ->  (
b  x.  b )  e.  RR )
8 readdcl 9477 . . . . . . . . . . . 12  |-  ( ( ( a  x.  a
)  e.  RR  /\  ( b  x.  b
)  e.  RR )  ->  ( ( a  x.  a )  +  ( b  x.  b
) )  e.  RR )
95, 7, 8syl2an 477 . . . . . . . . . . 11  |-  ( ( a  e.  RR  /\  b  e.  RR )  ->  ( ( a  x.  a )  +  ( b  x.  b ) )  e.  RR )
10 ax-rrecex 9466 . . . . . . . . . . 11  |-  ( ( ( ( a  x.  a )  +  ( b  x.  b ) )  e.  RR  /\  ( ( a  x.  a )  +  ( b  x.  b ) )  =/=  0 )  ->  E. y  e.  RR  ( ( ( a  x.  a )  +  ( b  x.  b
) )  x.  y
)  =  1 )
119, 10sylan 471 . . . . . . . . . 10  |-  ( ( ( a  e.  RR  /\  b  e.  RR )  /\  ( ( a  x.  a )  +  ( b  x.  b
) )  =/=  0
)  ->  E. y  e.  RR  ( ( ( a  x.  a )  +  ( b  x.  b ) )  x.  y )  =  1 )
12 recn 9484 . . . . . . . . . . . 12  |-  ( a  e.  RR  ->  a  e.  CC )
13 recn 9484 . . . . . . . . . . . 12  |-  ( b  e.  RR  ->  b  e.  CC )
14 recn 9484 . . . . . . . . . . . . . 14  |-  ( y  e.  RR  ->  y  e.  CC )
15 ax-icn 9453 . . . . . . . . . . . . . . . . . . . 20  |-  _i  e.  CC
16 mulcl 9478 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( _i  e.  CC  /\  b  e.  CC )  ->  ( _i  x.  b
)  e.  CC )
1715, 16mpan 670 . . . . . . . . . . . . . . . . . . 19  |-  ( b  e.  CC  ->  (
_i  x.  b )  e.  CC )
18 subcl 9721 . . . . . . . . . . . . . . . . . . 19  |-  ( ( a  e.  CC  /\  ( _i  x.  b
)  e.  CC )  ->  ( a  -  ( _i  x.  b
) )  e.  CC )
1917, 18sylan2 474 . . . . . . . . . . . . . . . . . 18  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( a  -  (
_i  x.  b )
)  e.  CC )
20 mulcl 9478 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( a  -  (
_i  x.  b )
)  e.  CC  /\  y  e.  CC )  ->  ( ( a  -  ( _i  x.  b
) )  x.  y
)  e.  CC )
2119, 20sylan 471 . . . . . . . . . . . . . . . . 17  |-  ( ( ( a  e.  CC  /\  b  e.  CC )  /\  y  e.  CC )  ->  ( ( a  -  ( _i  x.  b ) )  x.  y )  e.  CC )
2221adantr 465 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( a  e.  CC  /\  b  e.  CC )  /\  y  e.  CC )  /\  (
( ( a  x.  a )  +  ( b  x.  b ) )  x.  y )  =  1 )  -> 
( ( a  -  ( _i  x.  b
) )  x.  y
)  e.  CC )
23 addcl 9476 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( a  e.  CC  /\  ( _i  x.  b
)  e.  CC )  ->  ( a  +  ( _i  x.  b
) )  e.  CC )
2417, 23sylan2 474 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( a  +  ( _i  x.  b ) )  e.  CC )
2524adantr 465 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( a  e.  CC  /\  b  e.  CC )  /\  y  e.  CC )  ->  ( a  +  ( _i  x.  b
) )  e.  CC )
2619adantr 465 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( a  e.  CC  /\  b  e.  CC )  /\  y  e.  CC )  ->  ( a  -  ( _i  x.  b
) )  e.  CC )
27 simpr 461 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( a  e.  CC  /\  b  e.  CC )  /\  y  e.  CC )  ->  y  e.  CC )
2825, 26, 27mulassd 9521 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( a  e.  CC  /\  b  e.  CC )  /\  y  e.  CC )  ->  ( ( ( a  +  ( _i  x.  b ) )  x.  ( a  -  ( _i  x.  b
) ) )  x.  y )  =  ( ( a  +  ( _i  x.  b ) )  x.  ( ( a  -  ( _i  x.  b ) )  x.  y ) ) )
29 recextlem1 10078 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( ( a  +  ( _i  x.  b
) )  x.  (
a  -  ( _i  x.  b ) ) )  =  ( ( a  x.  a )  +  ( b  x.  b ) ) )
3029adantr 465 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( a  e.  CC  /\  b  e.  CC )  /\  y  e.  CC )  ->  ( ( a  +  ( _i  x.  b ) )  x.  ( a  -  (
_i  x.  b )
) )  =  ( ( a  x.  a
)  +  ( b  x.  b ) ) )
3130oveq1d 6216 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( a  e.  CC  /\  b  e.  CC )  /\  y  e.  CC )  ->  ( ( ( a  +  ( _i  x.  b ) )  x.  ( a  -  ( _i  x.  b
) ) )  x.  y )  =  ( ( ( a  x.  a )  +  ( b  x.  b ) )  x.  y ) )
3228, 31eqtr3d 2497 . . . . . . . . . . . . . . . . 17  |-  ( ( ( a  e.  CC  /\  b  e.  CC )  /\  y  e.  CC )  ->  ( ( a  +  ( _i  x.  b ) )  x.  ( ( a  -  ( _i  x.  b
) )  x.  y
) )  =  ( ( ( a  x.  a )  +  ( b  x.  b ) )  x.  y ) )
33 id 22 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( a  x.  a )  +  ( b  x.  b ) )  x.  y )  =  1  ->  (
( ( a  x.  a )  +  ( b  x.  b ) )  x.  y )  =  1 )
3432, 33sylan9eq 2515 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( a  e.  CC  /\  b  e.  CC )  /\  y  e.  CC )  /\  (
( ( a  x.  a )  +  ( b  x.  b ) )  x.  y )  =  1 )  -> 
( ( a  +  ( _i  x.  b
) )  x.  (
( a  -  (
_i  x.  b )
)  x.  y ) )  =  1 )
35 oveq2 6209 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  ( ( a  -  ( _i  x.  b ) )  x.  y )  ->  (
( a  +  ( _i  x.  b ) )  x.  x )  =  ( ( a  +  ( _i  x.  b ) )  x.  ( ( a  -  ( _i  x.  b
) )  x.  y
) ) )
3635eqeq1d 2456 . . . . . . . . . . . . . . . . 17  |-  ( x  =  ( ( a  -  ( _i  x.  b ) )  x.  y )  ->  (
( ( a  +  ( _i  x.  b
) )  x.  x
)  =  1  <->  (
( a  +  ( _i  x.  b ) )  x.  ( ( a  -  ( _i  x.  b ) )  x.  y ) )  =  1 ) )
3736rspcev 3179 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( a  -  ( _i  x.  b
) )  x.  y
)  e.  CC  /\  ( ( a  +  ( _i  x.  b
) )  x.  (
( a  -  (
_i  x.  b )
)  x.  y ) )  =  1 )  ->  E. x  e.  CC  ( ( a  +  ( _i  x.  b
) )  x.  x
)  =  1 )
3822, 34, 37syl2anc 661 . . . . . . . . . . . . . . 15  |-  ( ( ( ( a  e.  CC  /\  b  e.  CC )  /\  y  e.  CC )  /\  (
( ( a  x.  a )  +  ( b  x.  b ) )  x.  y )  =  1 )  ->  E. x  e.  CC  ( ( a  +  ( _i  x.  b
) )  x.  x
)  =  1 )
3938exp31 604 . . . . . . . . . . . . . 14  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( y  e.  CC  ->  ( ( ( ( a  x.  a )  +  ( b  x.  b ) )  x.  y )  =  1  ->  E. x  e.  CC  ( ( a  +  ( _i  x.  b
) )  x.  x
)  =  1 ) ) )
4014, 39syl5 32 . . . . . . . . . . . . 13  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( y  e.  RR  ->  ( ( ( ( a  x.  a )  +  ( b  x.  b ) )  x.  y )  =  1  ->  E. x  e.  CC  ( ( a  +  ( _i  x.  b
) )  x.  x
)  =  1 ) ) )
4140rexlimdv 2946 . . . . . . . . . . . 12  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( E. y  e.  RR  ( ( ( a  x.  a )  +  ( b  x.  b ) )  x.  y )  =  1  ->  E. x  e.  CC  ( ( a  +  ( _i  x.  b
) )  x.  x
)  =  1 ) )
4212, 13, 41syl2an 477 . . . . . . . . . . 11  |-  ( ( a  e.  RR  /\  b  e.  RR )  ->  ( E. y  e.  RR  ( ( ( a  x.  a )  +  ( b  x.  b ) )  x.  y )  =  1  ->  E. x  e.  CC  ( ( a  +  ( _i  x.  b
) )  x.  x
)  =  1 ) )
4342adantr 465 . . . . . . . . . 10  |-  ( ( ( a  e.  RR  /\  b  e.  RR )  /\  ( ( a  x.  a )  +  ( b  x.  b
) )  =/=  0
)  ->  ( E. y  e.  RR  (
( ( a  x.  a )  +  ( b  x.  b ) )  x.  y )  =  1  ->  E. x  e.  CC  ( ( a  +  ( _i  x.  b ) )  x.  x )  =  1 ) )
4411, 43mpd 15 . . . . . . . . 9  |-  ( ( ( a  e.  RR  /\  b  e.  RR )  /\  ( ( a  x.  a )  +  ( b  x.  b
) )  =/=  0
)  ->  E. x  e.  CC  ( ( a  +  ( _i  x.  b ) )  x.  x )  =  1 )
4544ex 434 . . . . . . . 8  |-  ( ( a  e.  RR  /\  b  e.  RR )  ->  ( ( ( a  x.  a )  +  ( b  x.  b
) )  =/=  0  ->  E. x  e.  CC  ( ( a  +  ( _i  x.  b
) )  x.  x
)  =  1 ) )
463, 45syld 44 . . . . . . 7  |-  ( ( a  e.  RR  /\  b  e.  RR )  ->  ( ( a  +  ( _i  x.  b
) )  =/=  0  ->  E. x  e.  CC  ( ( a  +  ( _i  x.  b
) )  x.  x
)  =  1 ) )
4746adantr 465 . . . . . 6  |-  ( ( ( a  e.  RR  /\  b  e.  RR )  /\  A  =  ( a  +  ( _i  x.  b ) ) )  ->  ( (
a  +  ( _i  x.  b ) )  =/=  0  ->  E. x  e.  CC  ( ( a  +  ( _i  x.  b ) )  x.  x )  =  1 ) )
48 neeq1 2733 . . . . . . 7  |-  ( A  =  ( a  +  ( _i  x.  b
) )  ->  ( A  =/=  0  <->  ( a  +  ( _i  x.  b ) )  =/=  0 ) )
4948adantl 466 . . . . . 6  |-  ( ( ( a  e.  RR  /\  b  e.  RR )  /\  A  =  ( a  +  ( _i  x.  b ) ) )  ->  ( A  =/=  0  <->  ( a  +  ( _i  x.  b
) )  =/=  0
) )
50 oveq1 6208 . . . . . . . . 9  |-  ( A  =  ( a  +  ( _i  x.  b
) )  ->  ( A  x.  x )  =  ( ( a  +  ( _i  x.  b ) )  x.  x ) )
5150eqeq1d 2456 . . . . . . . 8  |-  ( A  =  ( a  +  ( _i  x.  b
) )  ->  (
( A  x.  x
)  =  1  <->  (
( a  +  ( _i  x.  b ) )  x.  x )  =  1 ) )
5251rexbidv 2868 . . . . . . 7  |-  ( A  =  ( a  +  ( _i  x.  b
) )  ->  ( E. x  e.  CC  ( A  x.  x
)  =  1  <->  E. x  e.  CC  (
( a  +  ( _i  x.  b ) )  x.  x )  =  1 ) )
5352adantl 466 . . . . . 6  |-  ( ( ( a  e.  RR  /\  b  e.  RR )  /\  A  =  ( a  +  ( _i  x.  b ) ) )  ->  ( E. x  e.  CC  ( A  x.  x )  =  1  <->  E. x  e.  CC  ( ( a  +  ( _i  x.  b ) )  x.  x )  =  1 ) )
5447, 49, 533imtr4d 268 . . . . 5  |-  ( ( ( a  e.  RR  /\  b  e.  RR )  /\  A  =  ( a  +  ( _i  x.  b ) ) )  ->  ( A  =/=  0  ->  E. x  e.  CC  ( A  x.  x )  =  1 ) )
5554ex 434 . . . 4  |-  ( ( a  e.  RR  /\  b  e.  RR )  ->  ( A  =  ( a  +  ( _i  x.  b ) )  ->  ( A  =/=  0  ->  E. x  e.  CC  ( A  x.  x )  =  1 ) ) )
5655rexlimivv 2952 . . 3  |-  ( E. a  e.  RR  E. b  e.  RR  A  =  ( a  +  ( _i  x.  b
) )  ->  ( A  =/=  0  ->  E. x  e.  CC  ( A  x.  x )  =  1 ) )
571, 56syl 16 . 2  |-  ( A  e.  CC  ->  ( A  =/=  0  ->  E. x  e.  CC  ( A  x.  x )  =  1 ) )
5857imp 429 1  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  E. x  e.  CC  ( A  x.  x
)  =  1 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2648   E.wrex 2800  (class class class)co 6201   CCcc 9392   RRcr 9393   0cc0 9394   1c1 9395   _ici 9396    + caddc 9397    x. cmul 9399    - cmin 9707
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-po 4750  df-so 4751  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710
This theorem is referenced by:  mulcand  10081  receu  10093
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