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Theorem apreim 8332
Description: Complex apartness in terms of real and imaginary parts. (Contributed by Jim Kingdon, 12-Feb-2020.)
Assertion
Ref Expression
apreim  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( A  +  ( _i  x.  B
) ) #  ( C  +  ( _i  x.  D ) )  <->  ( A #  C  \/  B #  D
) ) )

Proof of Theorem apreim
Dummy variables  r  s  t  u  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpll 503 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  A  e.  RR )
21recnd 7762 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  A  e.  CC )
3 ax-icn 7683 . . . . . . 7  |-  _i  e.  CC
43a1i 9 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  _i  e.  CC )
5 simplr 504 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  B  e.  RR )
65recnd 7762 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  B  e.  CC )
74, 6mulcld 7754 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( _i  x.  B
)  e.  CC )
82, 7addcld 7753 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( A  +  ( _i  x.  B ) )  e.  CC )
9 simprl 505 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  C  e.  RR )
109recnd 7762 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  C  e.  CC )
11 simprr 506 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  D  e.  RR )
1211recnd 7762 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  D  e.  CC )
134, 12mulcld 7754 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( _i  x.  D
)  e.  CC )
1410, 13addcld 7753 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( C  +  ( _i  x.  D ) )  e.  CC )
15 eqeq1 2124 . . . . . . . . 9  |-  ( x  =  ( A  +  ( _i  x.  B
) )  ->  (
x  =  ( r  +  ( _i  x.  s ) )  <->  ( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s ) ) ) )
1615anbi1d 460 . . . . . . . 8  |-  ( x  =  ( A  +  ( _i  x.  B
) )  ->  (
( x  =  ( r  +  ( _i  x.  s ) )  /\  y  =  ( t  +  ( _i  x.  u ) ) )  <->  ( ( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s ) )  /\  y  =  ( t  +  ( _i  x.  u ) ) ) ) )
1716anbi1d 460 . . . . . . 7  |-  ( x  =  ( A  +  ( _i  x.  B
) )  ->  (
( ( x  =  ( r  +  ( _i  x.  s ) )  /\  y  =  ( t  +  ( _i  x.  u ) ) )  /\  (
r #  t  \/  s #  u ) )  <->  ( ( ( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s
) )  /\  y  =  ( t  +  ( _i  x.  u
) ) )  /\  ( r #  t  \/  s #  u
) ) ) )
18172rexbidv 2437 . . . . . 6  |-  ( x  =  ( A  +  ( _i  x.  B
) )  ->  ( E. t  e.  RR  E. u  e.  RR  (
( x  =  ( r  +  ( _i  x.  s ) )  /\  y  =  ( t  +  ( _i  x.  u ) ) )  /\  ( r #  t  \/  s #  u ) )  <->  E. t  e.  RR  E. u  e.  RR  ( ( ( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s
) )  /\  y  =  ( t  +  ( _i  x.  u
) ) )  /\  ( r #  t  \/  s #  u
) ) ) )
19182rexbidv 2437 . . . . 5  |-  ( x  =  ( A  +  ( _i  x.  B
) )  ->  ( E. r  e.  RR  E. s  e.  RR  E. t  e.  RR  E. u  e.  RR  ( ( x  =  ( r  +  ( _i  x.  s
) )  /\  y  =  ( t  +  ( _i  x.  u
) ) )  /\  ( r #  t  \/  s #  u
) )  <->  E. r  e.  RR  E. s  e.  RR  E. t  e.  RR  E. u  e.  RR  ( ( ( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s
) )  /\  y  =  ( t  +  ( _i  x.  u
) ) )  /\  ( r #  t  \/  s #  u
) ) ) )
20 eqeq1 2124 . . . . . . . . 9  |-  ( y  =  ( C  +  ( _i  x.  D
) )  ->  (
y  =  ( t  +  ( _i  x.  u ) )  <->  ( C  +  ( _i  x.  D ) )  =  ( t  +  ( _i  x.  u ) ) ) )
2120anbi2d 459 . . . . . . . 8  |-  ( y  =  ( C  +  ( _i  x.  D
) )  ->  (
( ( A  +  ( _i  x.  B
) )  =  ( r  +  ( _i  x.  s ) )  /\  y  =  ( t  +  ( _i  x.  u ) ) )  <->  ( ( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s ) )  /\  ( C  +  ( _i  x.  D ) )  =  ( t  +  ( _i  x.  u ) ) ) ) )
2221anbi1d 460 . . . . . . 7  |-  ( y  =  ( C  +  ( _i  x.  D
) )  ->  (
( ( ( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s ) )  /\  y  =  ( t  +  ( _i  x.  u ) ) )  /\  (
r #  t  \/  s #  u ) )  <->  ( ( ( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s
) )  /\  ( C  +  ( _i  x.  D ) )  =  ( t  +  ( _i  x.  u ) ) )  /\  (
r #  t  \/  s #  u ) ) ) )
23222rexbidv 2437 . . . . . 6  |-  ( y  =  ( C  +  ( _i  x.  D
) )  ->  ( E. t  e.  RR  E. u  e.  RR  (
( ( A  +  ( _i  x.  B
) )  =  ( r  +  ( _i  x.  s ) )  /\  y  =  ( t  +  ( _i  x.  u ) ) )  /\  ( r #  t  \/  s #  u ) )  <->  E. t  e.  RR  E. u  e.  RR  ( ( ( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s
) )  /\  ( C  +  ( _i  x.  D ) )  =  ( t  +  ( _i  x.  u ) ) )  /\  (
r #  t  \/  s #  u ) ) ) )
24232rexbidv 2437 . . . . 5  |-  ( y  =  ( C  +  ( _i  x.  D
) )  ->  ( E. r  e.  RR  E. s  e.  RR  E. t  e.  RR  E. u  e.  RR  ( ( ( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s
) )  /\  y  =  ( t  +  ( _i  x.  u
) ) )  /\  ( r #  t  \/  s #  u
) )  <->  E. r  e.  RR  E. s  e.  RR  E. t  e.  RR  E. u  e.  RR  ( ( ( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s
) )  /\  ( C  +  ( _i  x.  D ) )  =  ( t  +  ( _i  x.  u ) ) )  /\  (
r #  t  \/  s #  u ) ) ) )
25 df-ap 8311 . . . . 5  |- #  =  { <. x ,  y >.  |  E. r  e.  RR  E. s  e.  RR  E. t  e.  RR  E. u  e.  RR  ( ( x  =  ( r  +  ( _i  x.  s
) )  /\  y  =  ( t  +  ( _i  x.  u
) ) )  /\  ( r #  t  \/  s #  u
) ) }
2619, 24, 25brabg 4161 . . . 4  |-  ( ( ( A  +  ( _i  x.  B ) )  e.  CC  /\  ( C  +  (
_i  x.  D )
)  e.  CC )  ->  ( ( A  +  ( _i  x.  B ) ) #  ( C  +  ( _i  x.  D ) )  <->  E. r  e.  RR  E. s  e.  RR  E. t  e.  RR  E. u  e.  RR  ( ( ( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s
) )  /\  ( C  +  ( _i  x.  D ) )  =  ( t  +  ( _i  x.  u ) ) )  /\  (
r #  t  \/  s #  u ) ) ) )
278, 14, 26syl2anc 408 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( A  +  ( _i  x.  B
) ) #  ( C  +  ( _i  x.  D ) )  <->  E. r  e.  RR  E. s  e.  RR  E. t  e.  RR  E. u  e.  RR  ( ( ( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s
) )  /\  ( C  +  ( _i  x.  D ) )  =  ( t  +  ( _i  x.  u ) ) )  /\  (
r #  t  \/  s #  u ) ) ) )
28 simprr 506 . . . . . . 7  |-  ( ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( r  e.  RR  /\  s  e.  RR ) )  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( ( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s ) )  /\  ( C  +  (
_i  x.  D )
)  =  ( t  +  ( _i  x.  u ) ) )  /\  ( r #  t  \/  s #  u ) ) )  ->  ( r #  t  \/  s #  u ) )
291ad3antrrr 483 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( r  e.  RR  /\  s  e.  RR ) )  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( ( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s ) )  /\  ( C  +  (
_i  x.  D )
)  =  ( t  +  ( _i  x.  u ) ) )  /\  ( r #  t  \/  s #  u ) ) )  ->  A  e.  RR )
309ad3antrrr 483 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( r  e.  RR  /\  s  e.  RR ) )  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( ( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s ) )  /\  ( C  +  (
_i  x.  D )
)  =  ( t  +  ( _i  x.  u ) ) )  /\  ( r #  t  \/  s #  u ) ) )  ->  C  e.  RR )
31 apreap 8316 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( A #  C  <->  A #  C )
)
3229, 30, 31syl2anc 408 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( r  e.  RR  /\  s  e.  RR ) )  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( ( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s ) )  /\  ( C  +  (
_i  x.  D )
)  =  ( t  +  ( _i  x.  u ) ) )  /\  ( r #  t  \/  s #  u ) ) )  ->  ( A #  C  <->  A #  C ) )
335ad3antrrr 483 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( r  e.  RR  /\  s  e.  RR ) )  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( ( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s ) )  /\  ( C  +  (
_i  x.  D )
)  =  ( t  +  ( _i  x.  u ) ) )  /\  ( r #  t  \/  s #  u ) ) )  ->  B  e.  RR )
3411ad3antrrr 483 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( r  e.  RR  /\  s  e.  RR ) )  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( ( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s ) )  /\  ( C  +  (
_i  x.  D )
)  =  ( t  +  ( _i  x.  u ) ) )  /\  ( r #  t  \/  s #  u ) ) )  ->  D  e.  RR )
35 apreap 8316 . . . . . . . . . 10  |-  ( ( B  e.  RR  /\  D  e.  RR )  ->  ( B #  D  <->  B #  D )
)
3633, 34, 35syl2anc 408 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( r  e.  RR  /\  s  e.  RR ) )  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( ( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s ) )  /\  ( C  +  (
_i  x.  D )
)  =  ( t  +  ( _i  x.  u ) ) )  /\  ( r #  t  \/  s #  u ) ) )  ->  ( B #  D  <->  B #  D ) )
3732, 36orbi12d 767 . . . . . . . 8  |-  ( ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( r  e.  RR  /\  s  e.  RR ) )  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( ( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s ) )  /\  ( C  +  (
_i  x.  D )
)  =  ( t  +  ( _i  x.  u ) ) )  /\  ( r #  t  \/  s #  u ) ) )  ->  ( ( A #  C  \/  B #  D
)  <->  ( A #  C  \/  B #  D ) ) )
38 simprll 511 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( r  e.  RR  /\  s  e.  RR ) )  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( ( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s ) )  /\  ( C  +  (
_i  x.  D )
)  =  ( t  +  ( _i  x.  u ) ) )  /\  ( r #  t  \/  s #  u ) ) )  ->  ( A  +  ( _i  x.  B
) )  =  ( r  +  ( _i  x.  s ) ) )
39 simpllr 508 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( r  e.  RR  /\  s  e.  RR ) )  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( ( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s ) )  /\  ( C  +  (
_i  x.  D )
)  =  ( t  +  ( _i  x.  u ) ) )  /\  ( r #  t  \/  s #  u ) ) )  ->  ( r  e.  RR  /\  s  e.  RR ) )
40 cru 8331 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( r  e.  RR  /\  s  e.  RR ) )  -> 
( ( A  +  ( _i  x.  B
) )  =  ( r  +  ( _i  x.  s ) )  <-> 
( A  =  r  /\  B  =  s ) ) )
4129, 33, 39, 40syl21anc 1200 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( r  e.  RR  /\  s  e.  RR ) )  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( ( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s ) )  /\  ( C  +  (
_i  x.  D )
)  =  ( t  +  ( _i  x.  u ) ) )  /\  ( r #  t  \/  s #  u ) ) )  ->  ( ( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s ) )  <->  ( A  =  r  /\  B  =  s ) ) )
4238, 41mpbid 146 . . . . . . . . . . 11  |-  ( ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( r  e.  RR  /\  s  e.  RR ) )  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( ( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s ) )  /\  ( C  +  (
_i  x.  D )
)  =  ( t  +  ( _i  x.  u ) ) )  /\  ( r #  t  \/  s #  u ) ) )  ->  ( A  =  r  /\  B  =  s ) )
4342simpld 111 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( r  e.  RR  /\  s  e.  RR ) )  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( ( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s ) )  /\  ( C  +  (
_i  x.  D )
)  =  ( t  +  ( _i  x.  u ) ) )  /\  ( r #  t  \/  s #  u ) ) )  ->  A  =  r )
44 simprlr 512 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( r  e.  RR  /\  s  e.  RR ) )  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( ( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s ) )  /\  ( C  +  (
_i  x.  D )
)  =  ( t  +  ( _i  x.  u ) ) )  /\  ( r #  t  \/  s #  u ) ) )  ->  ( C  +  ( _i  x.  D
) )  =  ( t  +  ( _i  x.  u ) ) )
45 simplr 504 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( r  e.  RR  /\  s  e.  RR ) )  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( ( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s ) )  /\  ( C  +  (
_i  x.  D )
)  =  ( t  +  ( _i  x.  u ) ) )  /\  ( r #  t  \/  s #  u ) ) )  ->  ( t  e.  RR  /\  u  e.  RR ) )
46 cru 8331 . . . . . . . . . . . . 13  |-  ( ( ( C  e.  RR  /\  D  e.  RR )  /\  ( t  e.  RR  /\  u  e.  RR ) )  -> 
( ( C  +  ( _i  x.  D
) )  =  ( t  +  ( _i  x.  u ) )  <-> 
( C  =  t  /\  D  =  u ) ) )
4730, 34, 45, 46syl21anc 1200 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( r  e.  RR  /\  s  e.  RR ) )  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( ( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s ) )  /\  ( C  +  (
_i  x.  D )
)  =  ( t  +  ( _i  x.  u ) ) )  /\  ( r #  t  \/  s #  u ) ) )  ->  ( ( C  +  ( _i  x.  D ) )  =  ( t  +  ( _i  x.  u ) )  <->  ( C  =  t  /\  D  =  u ) ) )
4844, 47mpbid 146 . . . . . . . . . . 11  |-  ( ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( r  e.  RR  /\  s  e.  RR ) )  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( ( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s ) )  /\  ( C  +  (
_i  x.  D )
)  =  ( t  +  ( _i  x.  u ) ) )  /\  ( r #  t  \/  s #  u ) ) )  ->  ( C  =  t  /\  D  =  u ) )
4948simpld 111 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( r  e.  RR  /\  s  e.  RR ) )  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( ( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s ) )  /\  ( C  +  (
_i  x.  D )
)  =  ( t  +  ( _i  x.  u ) ) )  /\  ( r #  t  \/  s #  u ) ) )  ->  C  =  t )
5043, 49breq12d 3912 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( r  e.  RR  /\  s  e.  RR ) )  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( ( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s ) )  /\  ( C  +  (
_i  x.  D )
)  =  ( t  +  ( _i  x.  u ) ) )  /\  ( r #  t  \/  s #  u ) ) )  ->  ( A #  C  <->  r #  t )
)
5142simprd 113 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( r  e.  RR  /\  s  e.  RR ) )  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( ( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s ) )  /\  ( C  +  (
_i  x.  D )
)  =  ( t  +  ( _i  x.  u ) ) )  /\  ( r #  t  \/  s #  u ) ) )  ->  B  =  s )
5248simprd 113 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( r  e.  RR  /\  s  e.  RR ) )  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( ( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s ) )  /\  ( C  +  (
_i  x.  D )
)  =  ( t  +  ( _i  x.  u ) ) )  /\  ( r #  t  \/  s #  u ) ) )  ->  D  =  u )
5351, 52breq12d 3912 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( r  e.  RR  /\  s  e.  RR ) )  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( ( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s ) )  /\  ( C  +  (
_i  x.  D )
)  =  ( t  +  ( _i  x.  u ) ) )  /\  ( r #  t  \/  s #  u ) ) )  ->  ( B #  D  <->  s #  u )
)
5450, 53orbi12d 767 . . . . . . . 8  |-  ( ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( r  e.  RR  /\  s  e.  RR ) )  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( ( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s ) )  /\  ( C  +  (
_i  x.  D )
)  =  ( t  +  ( _i  x.  u ) ) )  /\  ( r #  t  \/  s #  u ) ) )  ->  ( ( A #  C  \/  B #  D )  <->  ( r #  t  \/  s #  u ) ) )
5537, 54bitrd 187 . . . . . . 7  |-  ( ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( r  e.  RR  /\  s  e.  RR ) )  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( ( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s ) )  /\  ( C  +  (
_i  x.  D )
)  =  ( t  +  ( _i  x.  u ) ) )  /\  ( r #  t  \/  s #  u ) ) )  ->  ( ( A #  C  \/  B #  D
)  <->  ( r #  t  \/  s #  u ) ) )
5628, 55mpbird 166 . . . . . 6  |-  ( ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( r  e.  RR  /\  s  e.  RR ) )  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( ( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s ) )  /\  ( C  +  (
_i  x.  D )
)  =  ( t  +  ( _i  x.  u ) ) )  /\  ( r #  t  \/  s #  u ) ) )  ->  ( A #  C  \/  B #  D )
)
5756ex 114 . . . . 5  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( r  e.  RR  /\  s  e.  RR ) )  /\  ( t  e.  RR  /\  u  e.  RR ) )  -> 
( ( ( ( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s
) )  /\  ( C  +  ( _i  x.  D ) )  =  ( t  +  ( _i  x.  u ) ) )  /\  (
r #  t  \/  s #  u ) )  ->  ( A #  C  \/  B #  D
) ) )
5857rexlimdvva 2534 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( r  e.  RR  /\  s  e.  RR ) )  ->  ( E. t  e.  RR  E. u  e.  RR  ( ( ( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s
) )  /\  ( C  +  ( _i  x.  D ) )  =  ( t  +  ( _i  x.  u ) ) )  /\  (
r #  t  \/  s #  u ) )  ->  ( A #  C  \/  B #  D
) ) )
5958rexlimdvva 2534 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( E. r  e.  RR  E. s  e.  RR  E. t  e.  RR  E. u  e.  RR  ( ( ( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s
) )  /\  ( C  +  ( _i  x.  D ) )  =  ( t  +  ( _i  x.  u ) ) )  /\  (
r #  t  \/  s #  u ) )  ->  ( A #  C  \/  B #  D
) ) )
6027, 59sylbid 149 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( A  +  ( _i  x.  B
) ) #  ( C  +  ( _i  x.  D ) )  -> 
( A #  C  \/  B #  D ) ) )
6131ad2ant2r 500 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( A #  C  <->  A #  C )
)
6235ad2ant2l 499 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( B #  D  <->  B #  D )
)
6361, 62orbi12d 767 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( A #  C  \/  B #  D )  <->  ( A #  C  \/  B #  D ) ) )
6463pm5.32i 449 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( A #  C  \/  B #  D ) )  <->  ( (
( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( A #  C  \/  B #  D
) ) )
65 eqid 2117 . . . . . . . . . . . 12  |-  ( A  +  ( _i  x.  B ) )  =  ( A  +  ( _i  x.  B ) )
66 eqid 2117 . . . . . . . . . . . 12  |-  ( C  +  ( _i  x.  D ) )  =  ( C  +  ( _i  x.  D ) )
6765, 66pm3.2i 270 . . . . . . . . . . 11  |-  ( ( A  +  ( _i  x.  B ) )  =  ( A  +  ( _i  x.  B
) )  /\  ( C  +  ( _i  x.  D ) )  =  ( C  +  ( _i  x.  D ) ) )
6867biantrur 301 . . . . . . . . . 10  |-  ( ( A #  C  \/  B #  D )  <-> 
( ( ( A  +  ( _i  x.  B ) )  =  ( A  +  ( _i  x.  B ) )  /\  ( C  +  ( _i  x.  D ) )  =  ( C  +  ( _i  x.  D ) ) )  /\  ( A #  C  \/  B #  D )
) )
69 oveq1 5749 . . . . . . . . . . . . . 14  |-  ( t  =  C  ->  (
t  +  ( _i  x.  u ) )  =  ( C  +  ( _i  x.  u
) ) )
7069eqeq2d 2129 . . . . . . . . . . . . 13  |-  ( t  =  C  ->  (
( C  +  ( _i  x.  D ) )  =  ( t  +  ( _i  x.  u ) )  <->  ( C  +  ( _i  x.  D ) )  =  ( C  +  ( _i  x.  u ) ) ) )
7170anbi2d 459 . . . . . . . . . . . 12  |-  ( t  =  C  ->  (
( ( A  +  ( _i  x.  B
) )  =  ( A  +  ( _i  x.  B ) )  /\  ( C  +  ( _i  x.  D
) )  =  ( t  +  ( _i  x.  u ) ) )  <->  ( ( A  +  ( _i  x.  B ) )  =  ( A  +  ( _i  x.  B ) )  /\  ( C  +  ( _i  x.  D ) )  =  ( C  +  ( _i  x.  u ) ) ) ) )
72 breq2 3903 . . . . . . . . . . . . 13  |-  ( t  =  C  ->  ( A #  t 
<->  A #  C ) )
7372orbi1d 765 . . . . . . . . . . . 12  |-  ( t  =  C  ->  (
( A #  t  \/  B #  u
)  <->  ( A #  C  \/  B #  u ) ) )
7471, 73anbi12d 464 . . . . . . . . . . 11  |-  ( t  =  C  ->  (
( ( ( A  +  ( _i  x.  B ) )  =  ( A  +  ( _i  x.  B ) )  /\  ( C  +  ( _i  x.  D ) )  =  ( t  +  ( _i  x.  u ) ) )  /\  ( A #  t  \/  B #  u )
)  <->  ( ( ( A  +  ( _i  x.  B ) )  =  ( A  +  ( _i  x.  B
) )  /\  ( C  +  ( _i  x.  D ) )  =  ( C  +  ( _i  x.  u ) ) )  /\  ( A #  C  \/  B #  u )
) ) )
75 oveq2 5750 . . . . . . . . . . . . . . 15  |-  ( u  =  D  ->  (
_i  x.  u )  =  ( _i  x.  D ) )
7675oveq2d 5758 . . . . . . . . . . . . . 14  |-  ( u  =  D  ->  ( C  +  ( _i  x.  u ) )  =  ( C  +  ( _i  x.  D ) ) )
7776eqeq2d 2129 . . . . . . . . . . . . 13  |-  ( u  =  D  ->  (
( C  +  ( _i  x.  D ) )  =  ( C  +  ( _i  x.  u ) )  <->  ( C  +  ( _i  x.  D ) )  =  ( C  +  ( _i  x.  D ) ) ) )
7877anbi2d 459 . . . . . . . . . . . 12  |-  ( u  =  D  ->  (
( ( A  +  ( _i  x.  B
) )  =  ( A  +  ( _i  x.  B ) )  /\  ( C  +  ( _i  x.  D
) )  =  ( C  +  ( _i  x.  u ) ) )  <->  ( ( A  +  ( _i  x.  B ) )  =  ( A  +  ( _i  x.  B ) )  /\  ( C  +  ( _i  x.  D ) )  =  ( C  +  ( _i  x.  D ) ) ) ) )
79 breq2 3903 . . . . . . . . . . . . 13  |-  ( u  =  D  ->  ( B #  u 
<->  B #  D ) )
8079orbi2d 764 . . . . . . . . . . . 12  |-  ( u  =  D  ->  (
( A #  C  \/  B #  u
)  <->  ( A #  C  \/  B #  D ) ) )
8178, 80anbi12d 464 . . . . . . . . . . 11  |-  ( u  =  D  ->  (
( ( ( A  +  ( _i  x.  B ) )  =  ( A  +  ( _i  x.  B ) )  /\  ( C  +  ( _i  x.  D ) )  =  ( C  +  ( _i  x.  u ) ) )  /\  ( A #  C  \/  B #  u )
)  <->  ( ( ( A  +  ( _i  x.  B ) )  =  ( A  +  ( _i  x.  B
) )  /\  ( C  +  ( _i  x.  D ) )  =  ( C  +  ( _i  x.  D ) ) )  /\  ( A #  C  \/  B #  D )
) ) )
8274, 81rspc2ev 2778 . . . . . . . . . 10  |-  ( ( C  e.  RR  /\  D  e.  RR  /\  (
( ( A  +  ( _i  x.  B
) )  =  ( A  +  ( _i  x.  B ) )  /\  ( C  +  ( _i  x.  D
) )  =  ( C  +  ( _i  x.  D ) ) )  /\  ( A #  C  \/  B #  D ) ) )  ->  E. t  e.  RR  E. u  e.  RR  (
( ( A  +  ( _i  x.  B
) )  =  ( A  +  ( _i  x.  B ) )  /\  ( C  +  ( _i  x.  D
) )  =  ( t  +  ( _i  x.  u ) ) )  /\  ( A #  t  \/  B #  u ) ) )
8368, 82syl3an3b 1239 . . . . . . . . 9  |-  ( ( C  e.  RR  /\  D  e.  RR  /\  ( A #  C  \/  B #  D )
)  ->  E. t  e.  RR  E. u  e.  RR  ( ( ( A  +  ( _i  x.  B ) )  =  ( A  +  ( _i  x.  B
) )  /\  ( C  +  ( _i  x.  D ) )  =  ( t  +  ( _i  x.  u ) ) )  /\  ( A #  t  \/  B #  u )
) )
84833expa 1166 . . . . . . . 8  |-  ( ( ( C  e.  RR  /\  D  e.  RR )  /\  ( A #  C  \/  B #  D ) )  ->  E. t  e.  RR  E. u  e.  RR  (
( ( A  +  ( _i  x.  B
) )  =  ( A  +  ( _i  x.  B ) )  /\  ( C  +  ( _i  x.  D
) )  =  ( t  +  ( _i  x.  u ) ) )  /\  ( A #  t  \/  B #  u ) ) )
85 oveq1 5749 . . . . . . . . . . . . 13  |-  ( r  =  A  ->  (
r  +  ( _i  x.  s ) )  =  ( A  +  ( _i  x.  s
) ) )
8685eqeq2d 2129 . . . . . . . . . . . 12  |-  ( r  =  A  ->  (
( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s ) )  <->  ( A  +  ( _i  x.  B ) )  =  ( A  +  ( _i  x.  s ) ) ) )
8786anbi1d 460 . . . . . . . . . . 11  |-  ( r  =  A  ->  (
( ( A  +  ( _i  x.  B
) )  =  ( r  +  ( _i  x.  s ) )  /\  ( C  +  ( _i  x.  D
) )  =  ( t  +  ( _i  x.  u ) ) )  <->  ( ( A  +  ( _i  x.  B ) )  =  ( A  +  ( _i  x.  s ) )  /\  ( C  +  ( _i  x.  D ) )  =  ( t  +  ( _i  x.  u ) ) ) ) )
88 breq1 3902 . . . . . . . . . . . 12  |-  ( r  =  A  ->  (
r #  t  <->  A #  t ) )
8988orbi1d 765 . . . . . . . . . . 11  |-  ( r  =  A  ->  (
( r #  t  \/  s #  u
)  <->  ( A #  t  \/  s #  u ) ) )
9087, 89anbi12d 464 . . . . . . . . . 10  |-  ( r  =  A  ->  (
( ( ( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s ) )  /\  ( C  +  ( _i  x.  D ) )  =  ( t  +  ( _i  x.  u ) ) )  /\  (
r #  t  \/  s #  u ) )  <->  ( ( ( A  +  ( _i  x.  B ) )  =  ( A  +  ( _i  x.  s
) )  /\  ( C  +  ( _i  x.  D ) )  =  ( t  +  ( _i  x.  u ) ) )  /\  ( A #  t  \/  s #  u )
) ) )
91902rexbidv 2437 . . . . . . . . 9  |-  ( r  =  A  ->  ( E. t  e.  RR  E. u  e.  RR  (
( ( A  +  ( _i  x.  B
) )  =  ( r  +  ( _i  x.  s ) )  /\  ( C  +  ( _i  x.  D
) )  =  ( t  +  ( _i  x.  u ) ) )  /\  ( r #  t  \/  s #  u ) )  <->  E. t  e.  RR  E. u  e.  RR  ( ( ( A  +  ( _i  x.  B ) )  =  ( A  +  ( _i  x.  s
) )  /\  ( C  +  ( _i  x.  D ) )  =  ( t  +  ( _i  x.  u ) ) )  /\  ( A #  t  \/  s #  u )
) ) )
92 oveq2 5750 . . . . . . . . . . . . . 14  |-  ( s  =  B  ->  (
_i  x.  s )  =  ( _i  x.  B ) )
9392oveq2d 5758 . . . . . . . . . . . . 13  |-  ( s  =  B  ->  ( A  +  ( _i  x.  s ) )  =  ( A  +  ( _i  x.  B ) ) )
9493eqeq2d 2129 . . . . . . . . . . . 12  |-  ( s  =  B  ->  (
( A  +  ( _i  x.  B ) )  =  ( A  +  ( _i  x.  s ) )  <->  ( A  +  ( _i  x.  B ) )  =  ( A  +  ( _i  x.  B ) ) ) )
9594anbi1d 460 . . . . . . . . . . 11  |-  ( s  =  B  ->  (
( ( A  +  ( _i  x.  B
) )  =  ( A  +  ( _i  x.  s ) )  /\  ( C  +  ( _i  x.  D
) )  =  ( t  +  ( _i  x.  u ) ) )  <->  ( ( A  +  ( _i  x.  B ) )  =  ( A  +  ( _i  x.  B ) )  /\  ( C  +  ( _i  x.  D ) )  =  ( t  +  ( _i  x.  u ) ) ) ) )
96 breq1 3902 . . . . . . . . . . . 12  |-  ( s  =  B  ->  (
s #  u  <->  B #  u ) )
9796orbi2d 764 . . . . . . . . . . 11  |-  ( s  =  B  ->  (
( A #  t  \/  s #  u
)  <->  ( A #  t  \/  B #  u ) ) )
9895, 97anbi12d 464 . . . . . . . . . 10  |-  ( s  =  B  ->  (
( ( ( A  +  ( _i  x.  B ) )  =  ( A  +  ( _i  x.  s ) )  /\  ( C  +  ( _i  x.  D ) )  =  ( t  +  ( _i  x.  u ) ) )  /\  ( A #  t  \/  s #  u )
)  <->  ( ( ( A  +  ( _i  x.  B ) )  =  ( A  +  ( _i  x.  B
) )  /\  ( C  +  ( _i  x.  D ) )  =  ( t  +  ( _i  x.  u ) ) )  /\  ( A #  t  \/  B #  u )
) ) )
99982rexbidv 2437 . . . . . . . . 9  |-  ( s  =  B  ->  ( E. t  e.  RR  E. u  e.  RR  (
( ( A  +  ( _i  x.  B
) )  =  ( A  +  ( _i  x.  s ) )  /\  ( C  +  ( _i  x.  D
) )  =  ( t  +  ( _i  x.  u ) ) )  /\  ( A #  t  \/  s #  u ) )  <->  E. t  e.  RR  E. u  e.  RR  ( ( ( A  +  ( _i  x.  B ) )  =  ( A  +  ( _i  x.  B
) )  /\  ( C  +  ( _i  x.  D ) )  =  ( t  +  ( _i  x.  u ) ) )  /\  ( A #  t  \/  B #  u )
) ) )
10091, 99rspc2ev 2778 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  E. t  e.  RR  E. u  e.  RR  ( ( ( A  +  ( _i  x.  B ) )  =  ( A  +  ( _i  x.  B
) )  /\  ( C  +  ( _i  x.  D ) )  =  ( t  +  ( _i  x.  u ) ) )  /\  ( A #  t  \/  B #  u )
) )  ->  E. r  e.  RR  E. s  e.  RR  E. t  e.  RR  E. u  e.  RR  ( ( ( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s
) )  /\  ( C  +  ( _i  x.  D ) )  =  ( t  +  ( _i  x.  u ) ) )  /\  (
r #  t  \/  s #  u ) ) )
10184, 100syl3an3 1236 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  (
( C  e.  RR  /\  D  e.  RR )  /\  ( A #  C  \/  B #  D ) ) )  ->  E. r  e.  RR  E. s  e.  RR  E. t  e.  RR  E. u  e.  RR  ( ( ( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s
) )  /\  ( C  +  ( _i  x.  D ) )  =  ( t  +  ( _i  x.  u ) ) )  /\  (
r #  t  \/  s #  u ) ) )
1021013expa 1166 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( ( C  e.  RR  /\  D  e.  RR )  /\  ( A #  C  \/  B #  D )
) )  ->  E. r  e.  RR  E. s  e.  RR  E. t  e.  RR  E. u  e.  RR  ( ( ( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s
) )  /\  ( C  +  ( _i  x.  D ) )  =  ( t  +  ( _i  x.  u ) ) )  /\  (
r #  t  \/  s #  u ) ) )
103102anassrs 397 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( A #  C  \/  B #  D
) )  ->  E. r  e.  RR  E. s  e.  RR  E. t  e.  RR  E. u  e.  RR  ( ( ( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s
) )  /\  ( C  +  ( _i  x.  D ) )  =  ( t  +  ( _i  x.  u ) ) )  /\  (
r #  t  \/  s #  u ) ) )
10427adantr 274 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( A #  C  \/  B #  D
) )  ->  (
( A  +  ( _i  x.  B ) ) #  ( C  +  ( _i  x.  D
) )  <->  E. r  e.  RR  E. s  e.  RR  E. t  e.  RR  E. u  e.  RR  ( ( ( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s
) )  /\  ( C  +  ( _i  x.  D ) )  =  ( t  +  ( _i  x.  u ) ) )  /\  (
r #  t  \/  s #  u ) ) ) )
105103, 104mpbird 166 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( A #  C  \/  B #  D
) )  ->  ( A  +  ( _i  x.  B ) ) #  ( C  +  ( _i  x.  D ) ) )
10664, 105sylbi 120 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( A #  C  \/  B #  D ) )  -> 
( A  +  ( _i  x.  B ) ) #  ( C  +  ( _i  x.  D
) ) )
107106ex 114 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( A #  C  \/  B #  D )  ->  ( A  +  ( _i  x.  B ) ) #  ( C  +  ( _i  x.  D
) ) ) )
10860, 107impbid 128 1  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( A  +  ( _i  x.  B
) ) #  ( C  +  ( _i  x.  D ) )  <->  ( A #  C  \/  B #  D
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 682    = wceq 1316    e. wcel 1465   E.wrex 2394   class class class wbr 3899  (class class class)co 5742   CCcc 7586   RRcr 7587   _ici 7590    + caddc 7591    x. cmul 7593   # creap 8303   # cap 8310
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-pnf 7770  df-mnf 7771  df-ltxr 7773  df-sub 7903  df-neg 7904  df-reap 8304  df-ap 8311
This theorem is referenced by:  apirr  8334  apsym  8335  apcotr  8336  apadd1  8337  apneg  8340  mulext1  8341  apti  8351  recexaplem2  8380  crap0  8680  iap0  8901  cjap  10633  cnreim  10705  absext  10790
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