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Theorem apreim 7770
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 496 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  A  e.  RR )
21recnd 7209 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  A  e.  CC )
3 ax-icn 7133 . . . . . . 7  |-  _i  e.  CC
43a1i 9 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  _i  e.  CC )
5 simplr 497 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  B  e.  RR )
65recnd 7209 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  B  e.  CC )
74, 6mulcld 7201 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( _i  x.  B
)  e.  CC )
82, 7addcld 7200 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( A  +  ( _i  x.  B ) )  e.  CC )
9 simprl 498 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  C  e.  RR )
109recnd 7209 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  C  e.  CC )
11 simprr 499 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  D  e.  RR )
1211recnd 7209 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  D  e.  CC )
134, 12mulcld 7201 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( _i  x.  D
)  e.  CC )
1410, 13addcld 7200 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( C  +  ( _i  x.  D ) )  e.  CC )
15 eqeq1 2088 . . . . . . . . 9  |-  ( x  =  ( A  +  ( _i  x.  B
) )  ->  (
x  =  ( r  +  ( _i  x.  s ) )  <->  ( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s ) ) ) )
1615anbi1d 453 . . . . . . . 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 453 . . . . . . 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 2392 . . . . . 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 2392 . . . . 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 2088 . . . . . . . . 9  |-  ( y  =  ( C  +  ( _i  x.  D
) )  ->  (
y  =  ( t  +  ( _i  x.  u ) )  <->  ( C  +  ( _i  x.  D ) )  =  ( t  +  ( _i  x.  u ) ) ) )
2120anbi2d 452 . . . . . . . 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 453 . . . . . . 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 2392 . . . . . 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 2392 . . . . 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 7749 . . . . 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 4032 . . . 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 403 . . 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 499 . . . . . . 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 476 . . . . . . . . . 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 476 . . . . . . . . . 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 7754 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( A #  C  <->  A #  C )
)
3229, 30, 31syl2anc 403 . . . . . . . . 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 476 . . . . . . . . . 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 476 . . . . . . . . . 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 7754 . . . . . . . . . 10  |-  ( ( B  e.  RR  /\  D  e.  RR )  ->  ( B #  D  <->  B #  D )
)
3633, 34, 35syl2anc 403 . . . . . . . . 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 740 . . . . . . . 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 504 . . . . . . . . . . . 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 501 . . . . . . . . . . . . 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 7769 . . . . . . . . . . . . 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 1169 . . . . . . . . . . . 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 145 . . . . . . . . . . 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 110 . . . . . . . . . 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 505 . . . . . . . . . . . 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 497 . . . . . . . . . . . . 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 7769 . . . . . . . . . . . . 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 1169 . . . . . . . . . . . 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 145 . . . . . . . . . . 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 110 . . . . . . . . . 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 3806 . . . . . . . . 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 112 . . . . . . . . . 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 112 . . . . . . . . . 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 3806 . . . . . . . . 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 740 . . . . . . . 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 186 . . . . . . 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 165 . . . . . 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 113 . . . . 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 2485 . . . 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 2485 . . 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 148 . 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 493 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( A #  C  <->  A #  C )
)
6235ad2ant2l 492 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( B #  D  <->  B #  D )
)
6361, 62orbi12d 740 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( A #  C  \/  B #  D )  <->  ( A #  C  \/  B #  D ) ) )
6463pm5.32i 442 . . . 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 2082 . . . . . . . . . . . 12  |-  ( A  +  ( _i  x.  B ) )  =  ( A  +  ( _i  x.  B ) )
66 eqid 2082 . . . . . . . . . . . 12  |-  ( C  +  ( _i  x.  D ) )  =  ( C  +  ( _i  x.  D ) )
6765, 66pm3.2i 266 . . . . . . . . . . 11  |-  ( ( A  +  ( _i  x.  B ) )  =  ( A  +  ( _i  x.  B
) )  /\  ( C  +  ( _i  x.  D ) )  =  ( C  +  ( _i  x.  D ) ) )
6867biantrur 297 . . . . . . . . . 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 5550 . . . . . . . . . . . . . 14  |-  ( t  =  C  ->  (
t  +  ( _i  x.  u ) )  =  ( C  +  ( _i  x.  u
) ) )
7069eqeq2d 2093 . . . . . . . . . . . . 13  |-  ( t  =  C  ->  (
( C  +  ( _i  x.  D ) )  =  ( t  +  ( _i  x.  u ) )  <->  ( C  +  ( _i  x.  D ) )  =  ( C  +  ( _i  x.  u ) ) ) )
7170anbi2d 452 . . . . . . . . . . . 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 3797 . . . . . . . . . . . . 13  |-  ( t  =  C  ->  ( A #  t 
<->  A #  C ) )
7372orbi1d 738 . . . . . . . . . . . 12  |-  ( t  =  C  ->  (
( A #  t  \/  B #  u
)  <->  ( A #  C  \/  B #  u ) ) )
7471, 73anbi12d 457 . . . . . . . . . . 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 5551 . . . . . . . . . . . . . . 15  |-  ( u  =  D  ->  (
_i  x.  u )  =  ( _i  x.  D ) )
7675oveq2d 5559 . . . . . . . . . . . . . 14  |-  ( u  =  D  ->  ( C  +  ( _i  x.  u ) )  =  ( C  +  ( _i  x.  D ) ) )
7776eqeq2d 2093 . . . . . . . . . . . . 13  |-  ( u  =  D  ->  (
( C  +  ( _i  x.  D ) )  =  ( C  +  ( _i  x.  u ) )  <->  ( C  +  ( _i  x.  D ) )  =  ( C  +  ( _i  x.  D ) ) ) )
7877anbi2d 452 . . . . . . . . . . . 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 3797 . . . . . . . . . . . . 13  |-  ( u  =  D  ->  ( B #  u 
<->  B #  D ) )
8079orbi2d 737 . . . . . . . . . . . 12  |-  ( u  =  D  ->  (
( A #  C  \/  B #  u
)  <->  ( A #  C  \/  B #  D ) ) )
8178, 80anbi12d 457 . . . . . . . . . . 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 2716 . . . . . . . . . 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 1208 . . . . . . . . 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 1139 . . . . . . . 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 5550 . . . . . . . . . . . . 13  |-  ( r  =  A  ->  (
r  +  ( _i  x.  s ) )  =  ( A  +  ( _i  x.  s
) ) )
8685eqeq2d 2093 . . . . . . . . . . . 12  |-  ( r  =  A  ->  (
( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s ) )  <->  ( A  +  ( _i  x.  B ) )  =  ( A  +  ( _i  x.  s ) ) ) )
8786anbi1d 453 . . . . . . . . . . 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 3796 . . . . . . . . . . . 12  |-  ( r  =  A  ->  (
r #  t  <->  A #  t ) )
8988orbi1d 738 . . . . . . . . . . 11  |-  ( r  =  A  ->  (
( r #  t  \/  s #  u
)  <->  ( A #  t  \/  s #  u ) ) )
9087, 89anbi12d 457 . . . . . . . . . 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 2392 . . . . . . . . 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 5551 . . . . . . . . . . . . . 14  |-  ( s  =  B  ->  (
_i  x.  s )  =  ( _i  x.  B ) )
9392oveq2d 5559 . . . . . . . . . . . . 13  |-  ( s  =  B  ->  ( A  +  ( _i  x.  s ) )  =  ( A  +  ( _i  x.  B ) ) )
9493eqeq2d 2093 . . . . . . . . . . . 12  |-  ( s  =  B  ->  (
( A  +  ( _i  x.  B ) )  =  ( A  +  ( _i  x.  s ) )  <->  ( A  +  ( _i  x.  B ) )  =  ( A  +  ( _i  x.  B ) ) ) )
9594anbi1d 453 . . . . . . . . . . 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 3796 . . . . . . . . . . . 12  |-  ( s  =  B  ->  (
s #  u  <->  B #  u ) )
9796orbi2d 737 . . . . . . . . . . 11  |-  ( s  =  B  ->  (
( A #  t  \/  s #  u
)  <->  ( A #  t  \/  B #  u ) ) )
9895, 97anbi12d 457 . . . . . . . . . 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 2392 . . . . . . . . 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 2716 . . . . . . . 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 1205 . . . . . . 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 1139 . . . . . 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 392 . . . . 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 270 . . . . 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 165 . . . 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 119 . . 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 113 . 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 127 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 102    <-> wb 103    \/ wo 662    = wceq 1285    e. wcel 1434   E.wrex 2350   class class class wbr 3793  (class class class)co 5543   CCcc 7041   RRcr 7042   _ici 7045    + caddc 7046    x. cmul 7048   # creap 7741   # cap 7748
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-mulrcl 7137  ax-addcom 7138  ax-mulcom 7139  ax-addass 7140  ax-mulass 7141  ax-distr 7142  ax-i2m1 7143  ax-0lt1 7144  ax-1rid 7145  ax-0id 7146  ax-rnegex 7147  ax-precex 7148  ax-cnre 7149  ax-pre-ltirr 7150  ax-pre-lttrn 7152  ax-pre-apti 7153  ax-pre-ltadd 7154  ax-pre-mulgt0 7155
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-iota 4897  df-fun 4934  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-pnf 7217  df-mnf 7218  df-ltxr 7220  df-sub 7348  df-neg 7349  df-reap 7742  df-ap 7749
This theorem is referenced by:  apirr  7772  apsym  7773  apcotr  7774  apadd1  7775  apneg  7778  mulext1  7779  apti  7789  recexaplem2  7809  crap0  8102  iap0  8321  cjap  9931  absext  10087
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