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Theorem cjap 9506
Description: Complex conjugate and apartness. (Contributed by Jim Kingdon, 14-Jun-2020.)
Assertion
Ref Expression
cjap  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( * `  A ) #  ( * `  B )  <->  A #  B
) )

Proof of Theorem cjap
Dummy variables  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnre 7023 . . 3  |-  ( A  e.  CC  ->  E. x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y ) ) )
21adantr 261 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  E. x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y
) ) )
3 cnre 7023 . . . . . 6  |-  ( B  e.  CC  ->  E. z  e.  RR  E. w  e.  RR  B  =  ( z  +  ( _i  x.  w ) ) )
43ad3antlr 462 . . . . 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 ) ) )
5 simplrr 488 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  y  e.  RR )
65ad2antrr 457 . . . . . . . . . . 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 )
76recnd 7054 . . . . . . . . . 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  e.  CC )
8 simplrr 488 . . . . . . . . . . 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 )
98recnd 7054 . . . . . . . . . 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 ) ) )  ->  w  e.  CC )
10 apneg 7602 . . . . . . . . . 10  |-  ( ( y  e.  CC  /\  w  e.  CC )  ->  ( y #  w  <->  -u y #  -u w ) )
117, 9, 10syl2anc 391 . . . . . . . . 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 ) ) )  ->  (
y #  w  <->  -u y #  -u w ) )
1211orbi2d 704 . . . . . . . 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 #  z  \/  y #  w )  <->  ( x #  z  \/  -u y #  -u w ) ) )
13 simpllr 486 . . . . . . . . . 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  =  ( x  +  ( _i  x.  y
) ) )
14 simpr 103 . . . . . . . . . 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 ) ) )  ->  B  =  ( z  +  ( _i  x.  w
) ) )
1513, 14breq12d 3777 . . . . . . . . 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 ) ) ) )
16 simplrl 487 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  x  e.  RR )
1716ad2antrr 457 . . . . . . . . . 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  e.  RR )
18 simplrl 487 . . . . . . . . . 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 ) ) )  ->  z  e.  RR )
19 apreim 7594 . . . . . . . . . 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
) ) )
2017, 6, 18, 8, 19syl22anc 1136 . . . . . . . . 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
) ) )
2115, 20bitrd 177 . . . . . . . 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 ) ) )
2213fveq2d 5182 . . . . . . . . . . 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 ) ) )  ->  (
* `  A )  =  ( * `  ( x  +  (
_i  x.  y )
) ) )
23 cjreim 9503 . . . . . . . . . . . 12  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( * `  (
x  +  ( _i  x.  y ) ) )  =  ( x  -  ( _i  x.  y ) ) )
2417, 6, 23syl2anc 391 . . . . . . . . . . 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  +  ( _i  x.  y ) ) )  =  ( x  -  ( _i  x.  y
) ) )
2522, 24eqtrd 2072 . . . . . . . . . 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 )  =  ( x  -  ( _i  x.  y
) ) )
2614fveq2d 5182 . . . . . . . . . . 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 ) ) )  ->  (
* `  B )  =  ( * `  ( z  +  ( _i  x.  w ) ) ) )
27 cjreim 9503 . . . . . . . . . . . 12  |-  ( ( z  e.  RR  /\  w  e.  RR )  ->  ( * `  (
z  +  ( _i  x.  w ) ) )  =  ( z  -  ( _i  x.  w ) ) )
2818, 8, 27syl2anc 391 . . . . . . . . . . 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  +  ( _i  x.  w ) ) )  =  ( z  -  ( _i  x.  w
) ) )
2926, 28eqtrd 2072 . . . . . . . . . 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 ) ) )  ->  (
* `  B )  =  ( z  -  ( _i  x.  w
) ) )
3025, 29breq12d 3777 . . . . . . . . 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 ) ) ) )
3117recnd 7054 . . . . . . . . . . 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.  CC )
32 ax-icn 6979 . . . . . . . . . . . 12  |-  _i  e.  CC
3332a1i 9 . . . . . . . . . . 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 ) ) )  ->  _i  e.  CC )
34 submul2 7396 . . . . . . . . . . 11  |-  ( ( x  e.  CC  /\  _i  e.  CC  /\  y  e.  CC )  ->  (
x  -  ( _i  x.  y ) )  =  ( x  +  ( _i  x.  -u y
) ) )
3531, 33, 7, 34syl3anc 1135 . . . . . . . . . 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  -  ( _i  x.  y ) )  =  ( x  +  ( _i  x.  -u y
) ) )
3618recnd 7054 . . . . . . . . . . 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.  CC )
37 submul2 7396 . . . . . . . . . . 11  |-  ( ( z  e.  CC  /\  _i  e.  CC  /\  w  e.  CC )  ->  (
z  -  ( _i  x.  w ) )  =  ( z  +  ( _i  x.  -u w
) ) )
3836, 33, 9, 37syl3anc 1135 . . . . . . . . . 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 ) ) )  ->  (
z  -  ( _i  x.  w ) )  =  ( z  +  ( _i  x.  -u w
) ) )
3935, 38breq12d 3777 . . . . . . . . 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  +  ( _i  x.  -u y ) ) #  ( z  +  ( _i  x.  -u w ) ) ) )
406renegcld 7378 . . . . . . . . . 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 ) ) )  ->  -u y  e.  RR )
418renegcld 7378 . . . . . . . . . 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 ) ) )  ->  -u w  e.  RR )
42 apreim 7594 . . . . . . . . . 10  |-  ( ( ( x  e.  RR  /\  -u y  e.  RR )  /\  ( z  e.  RR  /\  -u w  e.  RR ) )  -> 
( ( x  +  ( _i  x.  -u y
) ) #  ( z  +  ( _i  x.  -u w ) )  <->  ( x #  z  \/  -u y #  -u w ) ) )
4317, 40, 18, 41, 42syl22anc 1136 . . . . . . . . 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.  -u y
) ) #  ( z  +  ( _i  x.  -u w ) )  <->  ( x #  z  \/  -u y #  -u w ) ) )
4430, 39, 433bitrd 203 . . . . . . . 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  \/  -u y #  -u w ) ) )
4512, 21, 443bitr4rd 210 . . . . . . 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
) )
4645ex 108 . . . . . 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
) ) )
4746rexlimdvva 2440 . . . . 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
) ) )
484, 47mpd 13 . . . 4  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  (
( * `  A
) #  ( * `  B )  <->  A #  B
) )
4948ex 108 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( A  =  ( x  +  ( _i  x.  y ) )  ->  ( ( * `
 A ) #  ( * `  B )  <-> 
A #  B ) ) )
5049rexlimdvva 2440 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( E. x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y ) )  ->  ( ( * `
 A ) #  ( * `  B )  <-> 
A #  B ) ) )
512, 50mpd 13 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( * `  A ) #  ( * `  B )  <->  A #  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    <-> wb 98    \/ wo 629    = wceq 1243    e. wcel 1393   E.wrex 2307   class class class wbr 3764   ` cfv 4902  (class class class)co 5512   CCcc 6887   RRcr 6888   _ici 6891    + caddc 6892    x. cmul 6894    - cmin 7182   -ucneg 7183   # cap 7572   *ccj 9439
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311  ax-cnex 6975  ax-resscn 6976  ax-1cn 6977  ax-1re 6978  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-mulrcl 6983  ax-addcom 6984  ax-mulcom 6985  ax-addass 6986  ax-mulass 6987  ax-distr 6988  ax-i2m1 6989  ax-1rid 6991  ax-0id 6992  ax-rnegex 6993  ax-precex 6994  ax-cnre 6995  ax-pre-ltirr 6996  ax-pre-ltwlin 6997  ax-pre-lttrn 6998  ax-pre-apti 6999  ax-pre-ltadd 7000  ax-pre-mulgt0 7001  ax-pre-mulext 7002
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-reu 2313  df-rmo 2314  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-i1p 6565  df-iplp 6566  df-iltp 6568  df-enr 6811  df-nr 6812  df-ltr 6815  df-0r 6816  df-1r 6817  df-0 6896  df-1 6897  df-r 6899  df-lt 6902  df-pnf 7062  df-mnf 7063  df-xr 7064  df-ltxr 7065  df-le 7066  df-sub 7184  df-neg 7185  df-reap 7566  df-ap 7573  df-div 7652  df-2 7973  df-cj 9442  df-re 9443  df-im 9444
This theorem is referenced by:  cjap0  9507
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