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Theorem apti 8609
Description: Complex apartness is tight. (Contributed by Jim Kingdon, 21-Feb-2020.)
Assertion
Ref Expression
apti  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  =  B  <->  -.  A #  B )
)

Proof of Theorem apti
Dummy variables  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnre 7983 . . 3  |-  ( A  e.  CC  ->  E. x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y ) ) )
21adantr 276 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  E. x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y
) ) )
3 cnre 7983 . . . . . . 7  |-  ( B  e.  CC  ->  E. z  e.  RR  E. w  e.  RR  B  =  ( z  +  ( _i  x.  w ) ) )
43adantl 277 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  E. z  e.  RR  E. w  e.  RR  B  =  ( z  +  ( _i  x.  w
) ) )
54ad2antrr 488 . . . . 5  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  E. z  e.  RR  E. w  e.  RR  B  =  ( z  +  ( _i  x.  w ) ) )
6 simpr 110 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  e.  RR  /\  y  e.  RR ) )
76ad3antrrr 492 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  A  =  ( x  +  ( _i  x.  y ) ) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  (
x  e.  RR  /\  y  e.  RR )
)
8 simplr 528 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  A  =  ( x  +  ( _i  x.  y ) ) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  (
z  e.  RR  /\  w  e.  RR )
)
9 cru 8589 . . . . . . . . 9  |-  ( ( ( x  e.  RR  /\  y  e.  RR )  /\  ( z  e.  RR  /\  w  e.  RR ) )  -> 
( ( x  +  ( _i  x.  y
) )  =  ( z  +  ( _i  x.  w ) )  <-> 
( x  =  z  /\  y  =  w ) ) )
107, 8, 9syl2anc 411 . . . . . . . 8  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  A  =  ( x  +  ( _i  x.  y ) ) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  (
( x  +  ( _i  x.  y ) )  =  ( z  +  ( _i  x.  w ) )  <->  ( x  =  z  /\  y  =  w ) ) )
11 simpllr 534 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  A  =  ( x  +  ( _i  x.  y ) ) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  A  =  ( x  +  ( _i  x.  y
) ) )
12 simpr 110 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  A  =  ( x  +  ( _i  x.  y ) ) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  B  =  ( z  +  ( _i  x.  w
) ) )
1311, 12eqeq12d 2204 . . . . . . . 8  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  A  =  ( x  +  ( _i  x.  y ) ) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  ( A  =  B  <->  ( x  +  ( _i  x.  y ) )  =  ( z  +  ( _i  x.  w ) ) ) )
14 apreim 8590 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR  /\  y  e.  RR )  /\  ( z  e.  RR  /\  w  e.  RR ) )  -> 
( ( x  +  ( _i  x.  y
) ) #  ( z  +  ( _i  x.  w ) )  <->  ( x #  z  \/  y #  w
) ) )
1514notbid 668 . . . . . . . . . . 11  |-  ( ( ( x  e.  RR  /\  y  e.  RR )  /\  ( z  e.  RR  /\  w  e.  RR ) )  -> 
( -.  ( x  +  ( _i  x.  y ) ) #  ( z  +  ( _i  x.  w ) )  <->  -.  ( x #  z  \/  y #  w ) ) )
16 ioran 753 . . . . . . . . . . 11  |-  ( -.  ( x #  z  \/  y #  w )  <->  ( -.  x #  z  /\  -.  y #  w ) )
1715, 16bitrdi 196 . . . . . . . . . 10  |-  ( ( ( x  e.  RR  /\  y  e.  RR )  /\  ( z  e.  RR  /\  w  e.  RR ) )  -> 
( -.  ( x  +  ( _i  x.  y ) ) #  ( z  +  ( _i  x.  w ) )  <-> 
( -.  x #  z  /\  -.  y #  w ) ) )
187, 8, 17syl2anc 411 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  A  =  ( x  +  ( _i  x.  y ) ) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  ( -.  ( x  +  ( _i  x.  y ) ) #  ( z  +  ( _i  x.  w
) )  <->  ( -.  x #  z  /\  -.  y #  w ) ) )
1911, 12breq12d 4031 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  A  =  ( x  +  ( _i  x.  y ) ) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  ( A #  B  <->  ( x  +  ( _i  x.  y
) ) #  ( z  +  ( _i  x.  w ) ) ) )
2019notbid 668 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  A  =  ( x  +  ( _i  x.  y ) ) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  ( -.  A #  B  <->  -.  (
x  +  ( _i  x.  y ) ) #  ( z  +  ( _i  x.  w ) ) ) )
217simpld 112 . . . . . . . . . . 11  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  A  =  ( x  +  ( _i  x.  y ) ) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  x  e.  RR )
228simpld 112 . . . . . . . . . . 11  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  A  =  ( x  +  ( _i  x.  y ) ) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  z  e.  RR )
23 reapti 8566 . . . . . . . . . . . 12  |-  ( ( x  e.  RR  /\  z  e.  RR )  ->  ( x  =  z  <->  -.  x #  z ) )
24 apreap 8574 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR  /\  z  e.  RR )  ->  ( x #  z  <->  x #  z )
)
2524notbid 668 . . . . . . . . . . . 12  |-  ( ( x  e.  RR  /\  z  e.  RR )  ->  ( -.  x #  z  <->  -.  x #  z ) )
2623, 25bitr4d 191 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  z  e.  RR )  ->  ( x  =  z  <->  -.  x #  z )
)
2721, 22, 26syl2anc 411 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  A  =  ( x  +  ( _i  x.  y ) ) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  (
x  =  z  <->  -.  x #  z ) )
287simprd 114 . . . . . . . . . . 11  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  A  =  ( x  +  ( _i  x.  y ) ) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  y  e.  RR )
298simprd 114 . . . . . . . . . . 11  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  A  =  ( x  +  ( _i  x.  y ) ) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  w  e.  RR )
30 reapti 8566 . . . . . . . . . . . 12  |-  ( ( y  e.  RR  /\  w  e.  RR )  ->  ( y  =  w  <->  -.  y #  w ) )
31 apreap 8574 . . . . . . . . . . . . 13  |-  ( ( y  e.  RR  /\  w  e.  RR )  ->  ( y #  w  <->  y #  w )
)
3231notbid 668 . . . . . . . . . . . 12  |-  ( ( y  e.  RR  /\  w  e.  RR )  ->  ( -.  y #  w  <->  -.  y #  w ) )
3330, 32bitr4d 191 . . . . . . . . . . 11  |-  ( ( y  e.  RR  /\  w  e.  RR )  ->  ( y  =  w  <->  -.  y #  w )
)
3428, 29, 33syl2anc 411 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  A  =  ( x  +  ( _i  x.  y ) ) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  (
y  =  w  <->  -.  y #  w ) )
3527, 34anbi12d 473 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  A  =  ( x  +  ( _i  x.  y ) ) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  (
( x  =  z  /\  y  =  w )  <->  ( -.  x #  z  /\  -.  y #  w ) ) )
3618, 20, 353bitr4d 220 . . . . . . . 8  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  A  =  ( x  +  ( _i  x.  y ) ) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  ( -.  A #  B  <->  ( x  =  z  /\  y  =  w ) ) )
3710, 13, 363bitr4d 220 . . . . . . 7  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  A  =  ( x  +  ( _i  x.  y ) ) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  ( A  =  B  <->  -.  A #  B ) )
3837ex 115 . . . . . 6  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  ->  ( B  =  ( z  +  ( _i  x.  w
) )  ->  ( A  =  B  <->  -.  A #  B ) ) )
3938rexlimdvva 2615 . . . . 5  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  ( E. z  e.  RR  E. w  e.  RR  B  =  ( z  +  ( _i  x.  w
) )  ->  ( A  =  B  <->  -.  A #  B ) ) )
405, 39mpd 13 . . . 4  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  ( A  =  B  <->  -.  A #  B ) )
4140ex 115 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( A  =  ( x  +  ( _i  x.  y ) )  ->  ( A  =  B  <->  -.  A #  B
) ) )
4241rexlimdvva 2615 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( E. x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y ) )  ->  ( A  =  B  <->  -.  A #  B
) ) )
432, 42mpd 13 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  =  B  <->  -.  A #  B )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 709    = wceq 1364    e. wcel 2160   E.wrex 2469   class class class wbr 4018  (class class class)co 5896   CCcc 7839   RRcr 7840   _ici 7843    + caddc 7844    x. cmul 7846   # creap 8561   # cap 8568
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7932  ax-resscn 7933  ax-1cn 7934  ax-1re 7935  ax-icn 7936  ax-addcl 7937  ax-addrcl 7938  ax-mulcl 7939  ax-mulrcl 7940  ax-addcom 7941  ax-mulcom 7942  ax-addass 7943  ax-mulass 7944  ax-distr 7945  ax-i2m1 7946  ax-0lt1 7947  ax-1rid 7948  ax-0id 7949  ax-rnegex 7950  ax-precex 7951  ax-cnre 7952  ax-pre-ltirr 7953  ax-pre-lttrn 7955  ax-pre-apti 7956  ax-pre-ltadd 7957  ax-pre-mulgt0 7958
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fv 5243  df-riota 5852  df-ov 5899  df-oprab 5900  df-mpo 5901  df-pnf 8024  df-mnf 8025  df-ltxr 8027  df-sub 8160  df-neg 8161  df-reap 8562  df-ap 8569
This theorem is referenced by:  apne  8610  apcon4bid  8611  cnstab  8632  aptap  8637  qapne  9669  expeq0  10582  nn0opthd  10734  recvguniq  11036  climuni  11333  dedekindeu  14558  dedekindicclemicc  14567  ivthinc  14578  limcimo  14591  cnplimclemle  14594  coseq0q4123  14712  cos11  14731  refeq  15235  triap  15236
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