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Theorem recexap 8583
Description: Existence of reciprocal of nonzero complex number. (Contributed by Jim Kingdon, 20-Feb-2020.)
Assertion
Ref Expression
recexap  |-  ( ( A  e.  CC  /\  A #  0 )  ->  E. x  e.  CC  ( A  x.  x )  =  1 )
Distinct variable group:    x, A

Proof of Theorem recexap
Dummy variables  y  a  b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnre 7928 . . 3  |-  ( A  e.  CC  ->  E. a  e.  RR  E. b  e.  RR  A  =  ( a  +  ( _i  x.  b ) ) )
2 recexaplem2 8582 . . . . . . . . 9  |-  ( ( a  e.  RR  /\  b  e.  RR  /\  (
a  +  ( _i  x.  b ) ) #  0 )  ->  (
( a  x.  a
)  +  ( b  x.  b ) ) #  0 )
323expia 1205 . . . . . . . 8  |-  ( ( a  e.  RR  /\  b  e.  RR )  ->  ( ( a  +  ( _i  x.  b
) ) #  0  -> 
( ( a  x.  a )  +  ( b  x.  b ) ) #  0 ) )
4 remulcl 7914 . . . . . . . . . . . 12  |-  ( ( a  e.  RR  /\  a  e.  RR )  ->  ( a  x.  a
)  e.  RR )
54anidms 397 . . . . . . . . . . 11  |-  ( a  e.  RR  ->  (
a  x.  a )  e.  RR )
6 remulcl 7914 . . . . . . . . . . . 12  |-  ( ( b  e.  RR  /\  b  e.  RR )  ->  ( b  x.  b
)  e.  RR )
76anidms 397 . . . . . . . . . . 11  |-  ( b  e.  RR  ->  (
b  x.  b )  e.  RR )
8 readdcl 7912 . . . . . . . . . . 11  |-  ( ( ( a  x.  a
)  e.  RR  /\  ( b  x.  b
)  e.  RR )  ->  ( ( a  x.  a )  +  ( b  x.  b
) )  e.  RR )
95, 7, 8syl2an 289 . . . . . . . . . 10  |-  ( ( a  e.  RR  /\  b  e.  RR )  ->  ( ( a  x.  a )  +  ( b  x.  b ) )  e.  RR )
10 0re 7932 . . . . . . . . . 10  |-  0  e.  RR
11 apreap 8518 . . . . . . . . . 10  |-  ( ( ( ( a  x.  a )  +  ( b  x.  b ) )  e.  RR  /\  0  e.  RR )  ->  ( ( ( a  x.  a )  +  ( b  x.  b
) ) #  0  <->  (
( a  x.  a
)  +  ( b  x.  b ) ) #  0 ) )
129, 10, 11sylancl 413 . . . . . . . . 9  |-  ( ( a  e.  RR  /\  b  e.  RR )  ->  ( ( ( a  x.  a )  +  ( b  x.  b
) ) #  0  <->  (
( a  x.  a
)  +  ( b  x.  b ) ) #  0 ) )
13 recexre 8509 . . . . . . . . . . . 12  |-  ( ( ( ( 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 )
149, 13sylan 283 . . . . . . . . . . 11  |-  ( ( ( 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 )
15 recn 7919 . . . . . . . . . . . . 13  |-  ( a  e.  RR  ->  a  e.  CC )
16 recn 7919 . . . . . . . . . . . . 13  |-  ( b  e.  RR  ->  b  e.  CC )
17 recn 7919 . . . . . . . . . . . . . . 15  |-  ( y  e.  RR  ->  y  e.  CC )
18 ax-icn 7881 . . . . . . . . . . . . . . . . . . . . 21  |-  _i  e.  CC
19 mulcl 7913 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( _i  e.  CC  /\  b  e.  CC )  ->  ( _i  x.  b
)  e.  CC )
2018, 19mpan 424 . . . . . . . . . . . . . . . . . . . 20  |-  ( b  e.  CC  ->  (
_i  x.  b )  e.  CC )
21 subcl 8130 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( a  e.  CC  /\  ( _i  x.  b
)  e.  CC )  ->  ( a  -  ( _i  x.  b
) )  e.  CC )
2220, 21sylan2 286 . . . . . . . . . . . . . . . . . . 19  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( a  -  (
_i  x.  b )
)  e.  CC )
23 mulcl 7913 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( a  -  (
_i  x.  b )
)  e.  CC  /\  y  e.  CC )  ->  ( ( a  -  ( _i  x.  b
) )  x.  y
)  e.  CC )
2422, 23sylan 283 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( a  e.  CC  /\  b  e.  CC )  /\  y  e.  CC )  ->  ( ( a  -  ( _i  x.  b ) )  x.  y )  e.  CC )
2524adantr 276 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( 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 )
26 addcl 7911 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( a  e.  CC  /\  ( _i  x.  b
)  e.  CC )  ->  ( a  +  ( _i  x.  b
) )  e.  CC )
2720, 26sylan2 286 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( a  +  ( _i  x.  b ) )  e.  CC )
2827adantr 276 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( a  e.  CC  /\  b  e.  CC )  /\  y  e.  CC )  ->  ( a  +  ( _i  x.  b
) )  e.  CC )
2922adantr 276 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( a  e.  CC  /\  b  e.  CC )  /\  y  e.  CC )  ->  ( a  -  ( _i  x.  b
) )  e.  CC )
30 simpr 110 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( a  e.  CC  /\  b  e.  CC )  /\  y  e.  CC )  ->  y  e.  CC )
3128, 29, 30mulassd 7955 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( 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 ) ) )
32 recextlem1 8581 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( ( a  +  ( _i  x.  b
) )  x.  (
a  -  ( _i  x.  b ) ) )  =  ( ( a  x.  a )  +  ( b  x.  b ) ) )
3332adantr 276 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( a  e.  CC  /\  b  e.  CC )  /\  y  e.  CC )  ->  ( ( a  +  ( _i  x.  b ) )  x.  ( a  -  (
_i  x.  b )
) )  =  ( ( a  x.  a
)  +  ( b  x.  b ) ) )
3433oveq1d 5880 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( 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 ) )
3531, 34eqtr3d 2210 . . . . . . . . . . . . . . . . . 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 ) )
36 id 19 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( a  x.  a )  +  ( b  x.  b ) )  x.  y )  =  1  ->  (
( ( a  x.  a )  +  ( b  x.  b ) )  x.  y )  =  1 )
3735, 36sylan9eq 2228 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( 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 )
38 oveq2 5873 . . . . . . . . . . . . . . . . . . 19  |-  ( x  =  ( ( a  -  ( _i  x.  b ) )  x.  y )  ->  (
( a  +  ( _i  x.  b ) )  x.  x )  =  ( ( a  +  ( _i  x.  b ) )  x.  ( ( a  -  ( _i  x.  b
) )  x.  y
) ) )
3938eqeq1d 2184 . . . . . . . . . . . . . . . . . 18  |-  ( 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 ) )
4039rspcev 2839 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( 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 )
4125, 37, 40syl2anc 411 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( 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 )
4241exp31 364 . . . . . . . . . . . . . . 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 ) ) )
4317, 42syl5 32 . . . . . . . . . . . . . 14  |-  ( ( 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 ) ) )
4443rexlimdv 2591 . . . . . . . . . . . . 13  |-  ( ( 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 ) )
4515, 16, 44syl2an 289 . . . . . . . . . . . 12  |-  ( ( 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 ) )
4645adantr 276 . . . . . . . . . . 11  |-  ( ( ( 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 ) )
4714, 46mpd 13 . . . . . . . . . 10  |-  ( ( ( a  e.  RR  /\  b  e.  RR )  /\  ( ( a  x.  a )  +  ( b  x.  b
) ) #  0 )  ->  E. x  e.  CC  ( ( a  +  ( _i  x.  b
) )  x.  x
)  =  1 )
4847ex 115 . . . . . . . . 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 ) )
4912, 48sylbid 150 . . . . . . . 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 ) )
503, 49syld 45 . . . . . . 7  |-  ( ( a  e.  RR  /\  b  e.  RR )  ->  ( ( a  +  ( _i  x.  b
) ) #  0  ->  E. x  e.  CC  ( ( a  +  ( _i  x.  b
) )  x.  x
)  =  1 ) )
5150adantr 276 . . . . . 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 ) )
52 breq1 4001 . . . . . . 7  |-  ( A  =  ( a  +  ( _i  x.  b
) )  ->  ( A #  0  <->  ( a  +  ( _i  x.  b
) ) #  0 ) )
5352adantl 277 . . . . . 6  |-  ( ( ( a  e.  RR  /\  b  e.  RR )  /\  A  =  ( a  +  ( _i  x.  b ) ) )  ->  ( A #  0 
<->  ( a  +  ( _i  x.  b ) ) #  0 ) )
54 oveq1 5872 . . . . . . . . 9  |-  ( A  =  ( a  +  ( _i  x.  b
) )  ->  ( A  x.  x )  =  ( ( a  +  ( _i  x.  b ) )  x.  x ) )
5554eqeq1d 2184 . . . . . . . 8  |-  ( A  =  ( a  +  ( _i  x.  b
) )  ->  (
( A  x.  x
)  =  1  <->  (
( a  +  ( _i  x.  b ) )  x.  x )  =  1 ) )
5655rexbidv 2476 . . . . . . 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 ) )
5756adantl 277 . . . . . 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 ) )
5851, 53, 573imtr4d 203 . . . . 5  |-  ( ( ( a  e.  RR  /\  b  e.  RR )  /\  A  =  ( a  +  ( _i  x.  b ) ) )  ->  ( A #  0  ->  E. x  e.  CC  ( A  x.  x
)  =  1 ) )
5958ex 115 . . . 4  |-  ( ( a  e.  RR  /\  b  e.  RR )  ->  ( A  =  ( a  +  ( _i  x.  b ) )  ->  ( A #  0  ->  E. x  e.  CC  ( A  x.  x
)  =  1 ) ) )
6059rexlimivv 2598 . . 3  |-  ( E. a  e.  RR  E. b  e.  RR  A  =  ( a  +  ( _i  x.  b
) )  ->  ( A #  0  ->  E. x  e.  CC  ( A  x.  x )  =  1 ) )
611, 60syl 14 . 2  |-  ( A  e.  CC  ->  ( A #  0  ->  E. x  e.  CC  ( A  x.  x )  =  1 ) )
6261imp 124 1  |-  ( ( A  e.  CC  /\  A #  0 )  ->  E. x  e.  CC  ( A  x.  x )  =  1 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2146   E.wrex 2454   class class class wbr 3998  (class class class)co 5865   CCcc 7784   RRcr 7785   0cc0 7786   1c1 7787   _ici 7788    + caddc 7789    x. cmul 7791    - cmin 8102   # creap 8505   # cap 8512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903  ax-pre-mulext 7904
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-id 4287  df-po 4290  df-iso 4291  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-iota 5170  df-fun 5210  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-reap 8506  df-ap 8513
This theorem is referenced by:  mulap0  8584  mulcanapd  8591  receuap  8599
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