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Theorem apreim 8385
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 519 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  A  e.  RR )
21recnd 7814 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  A  e.  CC )
3 ax-icn 7735 . . . . . . 7  |-  _i  e.  CC
43a1i 9 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  _i  e.  CC )
5 simplr 520 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  B  e.  RR )
65recnd 7814 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  B  e.  CC )
74, 6mulcld 7806 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( _i  x.  B
)  e.  CC )
82, 7addcld 7805 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( A  +  ( _i  x.  B ) )  e.  CC )
9 simprl 521 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  C  e.  RR )
109recnd 7814 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  C  e.  CC )
11 simprr 522 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  D  e.  RR )
1211recnd 7814 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  D  e.  CC )
134, 12mulcld 7806 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( _i  x.  D
)  e.  CC )
1410, 13addcld 7805 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( C  +  ( _i  x.  D ) )  e.  CC )
15 eqeq1 2147 . . . . . . . . 9  |-  ( x  =  ( A  +  ( _i  x.  B
) )  ->  (
x  =  ( r  +  ( _i  x.  s ) )  <->  ( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s ) ) ) )
1615anbi1d 461 . . . . . . . 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 461 . . . . . . 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 2461 . . . . . 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 2461 . . . . 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 2147 . . . . . . . . 9  |-  ( y  =  ( C  +  ( _i  x.  D
) )  ->  (
y  =  ( t  +  ( _i  x.  u ) )  <->  ( C  +  ( _i  x.  D ) )  =  ( t  +  ( _i  x.  u ) ) ) )
2120anbi2d 460 . . . . . . . 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 461 . . . . . . 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 2461 . . . . . 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 2461 . . . . 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 8364 . . . . 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 4195 . . . 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 409 . . 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 522 . . . . . . 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 484 . . . . . . . . . 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 484 . . . . . . . . . 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 8369 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( A #  C  <->  A #  C )
)
3229, 30, 31syl2anc 409 . . . . . . . . 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 484 . . . . . . . . . 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 484 . . . . . . . . . 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 8369 . . . . . . . . . 10  |-  ( ( B  e.  RR  /\  D  e.  RR )  ->  ( B #  D  <->  B #  D )
)
3633, 34, 35syl2anc 409 . . . . . . . . 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 783 . . . . . . . 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 527 . . . . . . . . . . . 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 524 . . . . . . . . . . . . 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 8384 . . . . . . . . . . . . 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 1216 . . . . . . . . . . . 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 528 . . . . . . . . . . . 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 520 . . . . . . . . . . . . 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 8384 . . . . . . . . . . . . 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 1216 . . . . . . . . . . . 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 3946 . . . . . . . . 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 3946 . . . . . . . . 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 783 . . . . . . . 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 2558 . . . 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 2558 . . 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 501 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( A #  C  <->  A #  C )
)
6235ad2ant2l 500 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( B #  D  <->  B #  D )
)
6361, 62orbi12d 783 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( A #  C  \/  B #  D )  <->  ( A #  C  \/  B #  D ) ) )
6463pm5.32i 450 . . . 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 2140 . . . . . . . . . . . 12  |-  ( A  +  ( _i  x.  B ) )  =  ( A  +  ( _i  x.  B ) )
66 eqid 2140 . . . . . . . . . . . 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 5785 . . . . . . . . . . . . . 14  |-  ( t  =  C  ->  (
t  +  ( _i  x.  u ) )  =  ( C  +  ( _i  x.  u
) ) )
7069eqeq2d 2152 . . . . . . . . . . . . 13  |-  ( t  =  C  ->  (
( C  +  ( _i  x.  D ) )  =  ( t  +  ( _i  x.  u ) )  <->  ( C  +  ( _i  x.  D ) )  =  ( C  +  ( _i  x.  u ) ) ) )
7170anbi2d 460 . . . . . . . . . . . 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 3937 . . . . . . . . . . . . 13  |-  ( t  =  C  ->  ( A #  t 
<->  A #  C ) )
7372orbi1d 781 . . . . . . . . . . . 12  |-  ( t  =  C  ->  (
( A #  t  \/  B #  u
)  <->  ( A #  C  \/  B #  u ) ) )
7471, 73anbi12d 465 . . . . . . . . . . 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 5786 . . . . . . . . . . . . . . 15  |-  ( u  =  D  ->  (
_i  x.  u )  =  ( _i  x.  D ) )
7675oveq2d 5794 . . . . . . . . . . . . . 14  |-  ( u  =  D  ->  ( C  +  ( _i  x.  u ) )  =  ( C  +  ( _i  x.  D ) ) )
7776eqeq2d 2152 . . . . . . . . . . . . 13  |-  ( u  =  D  ->  (
( C  +  ( _i  x.  D ) )  =  ( C  +  ( _i  x.  u ) )  <->  ( C  +  ( _i  x.  D ) )  =  ( C  +  ( _i  x.  D ) ) ) )
7877anbi2d 460 . . . . . . . . . . . 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 3937 . . . . . . . . . . . . 13  |-  ( u  =  D  ->  ( B #  u 
<->  B #  D ) )
8079orbi2d 780 . . . . . . . . . . . 12  |-  ( u  =  D  ->  (
( A #  C  \/  B #  u
)  <->  ( A #  C  \/  B #  D ) ) )
8178, 80anbi12d 465 . . . . . . . . . . 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 2805 . . . . . . . . . 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 1255 . . . . . . . . 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 1182 . . . . . . . 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 5785 . . . . . . . . . . . . 13  |-  ( r  =  A  ->  (
r  +  ( _i  x.  s ) )  =  ( A  +  ( _i  x.  s
) ) )
8685eqeq2d 2152 . . . . . . . . . . . 12  |-  ( r  =  A  ->  (
( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s ) )  <->  ( A  +  ( _i  x.  B ) )  =  ( A  +  ( _i  x.  s ) ) ) )
8786anbi1d 461 . . . . . . . . . . 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 3936 . . . . . . . . . . . 12  |-  ( r  =  A  ->  (
r #  t  <->  A #  t ) )
8988orbi1d 781 . . . . . . . . . . 11  |-  ( r  =  A  ->  (
( r #  t  \/  s #  u
)  <->  ( A #  t  \/  s #  u ) ) )
9087, 89anbi12d 465 . . . . . . . . . 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 2461 . . . . . . . . 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 5786 . . . . . . . . . . . . . 14  |-  ( s  =  B  ->  (
_i  x.  s )  =  ( _i  x.  B ) )
9392oveq2d 5794 . . . . . . . . . . . . 13  |-  ( s  =  B  ->  ( A  +  ( _i  x.  s ) )  =  ( A  +  ( _i  x.  B ) ) )
9493eqeq2d 2152 . . . . . . . . . . . 12  |-  ( s  =  B  ->  (
( A  +  ( _i  x.  B ) )  =  ( A  +  ( _i  x.  s ) )  <->  ( A  +  ( _i  x.  B ) )  =  ( A  +  ( _i  x.  B ) ) ) )
9594anbi1d 461 . . . . . . . . . . 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 3936 . . . . . . . . . . . 12  |-  ( s  =  B  ->  (
s #  u  <->  B #  u ) )
9796orbi2d 780 . . . . . . . . . . 11  |-  ( s  =  B  ->  (
( A #  t  \/  s #  u
)  <->  ( A #  t  \/  B #  u ) ) )
9895, 97anbi12d 465 . . . . . . . . . 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 2461 . . . . . . . . 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 2805 . . . . . . . 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 1252 . . . . . . 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 1182 . . . . . 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 398 . . . . 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 698    = wceq 1332    e. wcel 1481   E.wrex 2418   class class class wbr 3933  (class class class)co 5778   CCcc 7638   RRcr 7639   _ici 7642    + caddc 7643    x. cmul 7645   # creap 8356   # cap 8363
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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4050  ax-pow 4102  ax-pr 4135  ax-un 4359  ax-setind 4456  ax-cnex 7731  ax-resscn 7732  ax-1cn 7733  ax-1re 7734  ax-icn 7735  ax-addcl 7736  ax-addrcl 7737  ax-mulcl 7738  ax-mulrcl 7739  ax-addcom 7740  ax-mulcom 7741  ax-addass 7742  ax-mulass 7743  ax-distr 7744  ax-i2m1 7745  ax-0lt1 7746  ax-1rid 7747  ax-0id 7748  ax-rnegex 7749  ax-precex 7750  ax-cnre 7751  ax-pre-ltirr 7752  ax-pre-lttrn 7754  ax-pre-apti 7755  ax-pre-ltadd 7756  ax-pre-mulgt0 7757
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2689  df-sbc 2911  df-dif 3074  df-un 3076  df-in 3078  df-ss 3085  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-uni 3741  df-br 3934  df-opab 3994  df-id 4219  df-xp 4549  df-rel 4550  df-cnv 4551  df-co 4552  df-dm 4553  df-iota 5092  df-fun 5129  df-fv 5135  df-riota 5734  df-ov 5781  df-oprab 5782  df-mpo 5783  df-pnf 7822  df-mnf 7823  df-ltxr 7825  df-sub 7955  df-neg 7956  df-reap 8357  df-ap 8364
This theorem is referenced by:  apirr  8387  apsym  8388  apcotr  8389  apadd1  8390  apneg  8393  mulext1  8394  apti  8404  recexaplem2  8433  crap0  8736  iap0  8963  cjap  10706  cnreim  10778  absext  10863
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