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Theorem recapb 8628
Description: A complex number has a multiplicative inverse if and only if it is apart from zero. Theorem 11.2.4 of [HoTT], p. (varies), generalized from real to complex numbers. (Contributed by Jim Kingdon, 18-Jan-2025.)
Assertion
Ref Expression
recapb  |-  ( A  e.  CC  ->  ( A #  0  <->  E. x  e.  CC  ( A  x.  x
)  =  1 ) )
Distinct variable group:    x, A

Proof of Theorem recapb
StepHypRef Expression
1 recexap 8610 . . 3  |-  ( ( A  e.  CC  /\  A #  0 )  ->  E. x  e.  CC  ( A  x.  x )  =  1 )
21ex 115 . 2  |-  ( A  e.  CC  ->  ( A #  0  ->  E. x  e.  CC  ( A  x.  x )  =  1 ) )
3 simpl 109 . . . 4  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  x.  x
)  =  1 ) )  ->  A  e.  CC )
4 simprl 529 . . . 4  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  x.  x
)  =  1 ) )  ->  x  e.  CC )
5 simprr 531 . . . . 5  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  x.  x
)  =  1 ) )  ->  ( A  x.  x )  =  1 )
6 1ap0 8547 . . . . 5  |-  1 #  0
75, 6eqbrtrdi 4043 . . . 4  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  x.  x
)  =  1 ) )  ->  ( A  x.  x ) #  0 )
83, 4, 7mulap0bad 8616 . . 3  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  x.  x
)  =  1 ) )  ->  A #  0
)
98rexlimdvaa 2595 . 2  |-  ( A  e.  CC  ->  ( E. x  e.  CC  ( A  x.  x
)  =  1  ->  A #  0 ) )
102, 9impbid 129 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 2148   E.wrex 2456   class class class wbr 4004  (class class class)co 5875   CCcc 7809   0cc0 7811   1c1 7812    x. cmul 7816   # cap 8538
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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-mulrcl 7910  ax-addcom 7911  ax-mulcom 7912  ax-addass 7913  ax-mulass 7914  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-1rid 7918  ax-0id 7919  ax-rnegex 7920  ax-precex 7921  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925  ax-pre-apti 7926  ax-pre-ltadd 7927  ax-pre-mulgt0 7928  ax-pre-mulext 7929
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-id 4294  df-po 4297  df-iso 4298  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-iota 5179  df-fun 5219  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-sub 8130  df-neg 8131  df-reap 8532  df-ap 8539
This theorem is referenced by:  rerecapb  8800
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