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Theorem apreim 8080
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 7516 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  A  e.  CC )
3 ax-icn 7440 . . . . . . 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 7516 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  B  e.  CC )
74, 6mulcld 7508 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( _i  x.  B
)  e.  CC )
82, 7addcld 7507 . . . 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 7516 . . . . 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 7516 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  D  e.  CC )
134, 12mulcld 7508 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( _i  x.  D
)  e.  CC )
1410, 13addcld 7507 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( C  +  ( _i  x.  D ) )  e.  CC )
15 eqeq1 2094 . . . . . . . . 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 2403 . . . . . 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 2403 . . . . 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 2094 . . . . . . . . 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 2403 . . . . . 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 2403 . . . . 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 8059 . . . . 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 4096 . . . 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 8064 . . . . . . . . . 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 8064 . . . . . . . . . 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 742 . . . . . . . 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 8079 . . . . . . . . . . . . 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 1173 . . . . . . . . . . . 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 8079 . . . . . . . . . . . . 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 1173 . . . . . . . . . . . 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 3858 . . . . . . . . 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 3858 . . . . . . . . 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 742 . . . . . . . 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 2496 . . . 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 2496 . . 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 742 . . . . 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 2088 . . . . . . . . . . . 12  |-  ( A  +  ( _i  x.  B ) )  =  ( A  +  ( _i  x.  B ) )
66 eqid 2088 . . . . . . . . . . . 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 5659 . . . . . . . . . . . . . 14  |-  ( t  =  C  ->  (
t  +  ( _i  x.  u ) )  =  ( C  +  ( _i  x.  u
) ) )
7069eqeq2d 2099 . . . . . . . . . . . . 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 3849 . . . . . . . . . . . . 13  |-  ( t  =  C  ->  ( A #  t 
<->  A #  C ) )
7372orbi1d 740 . . . . . . . . . . . 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 5660 . . . . . . . . . . . . . . 15  |-  ( u  =  D  ->  (
_i  x.  u )  =  ( _i  x.  D ) )
7675oveq2d 5668 . . . . . . . . . . . . . 14  |-  ( u  =  D  ->  ( C  +  ( _i  x.  u ) )  =  ( C  +  ( _i  x.  D ) ) )
7776eqeq2d 2099 . . . . . . . . . . . . 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 3849 . . . . . . . . . . . . 13  |-  ( u  =  D  ->  ( B #  u 
<->  B #  D ) )
8079orbi2d 739 . . . . . . . . . . . 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 2736 . . . . . . . . . 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 1212 . . . . . . . . 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 1143 . . . . . . . 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 5659 . . . . . . . . . . . . 13  |-  ( r  =  A  ->  (
r  +  ( _i  x.  s ) )  =  ( A  +  ( _i  x.  s
) ) )
8685eqeq2d 2099 . . . . . . . . . . . 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 3848 . . . . . . . . . . . 12  |-  ( r  =  A  ->  (
r #  t  <->  A #  t ) )
8988orbi1d 740 . . . . . . . . . . 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 2403 . . . . . . . . 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 5660 . . . . . . . . . . . . . 14  |-  ( s  =  B  ->  (
_i  x.  s )  =  ( _i  x.  B ) )
9392oveq2d 5668 . . . . . . . . . . . . 13  |-  ( s  =  B  ->  ( A  +  ( _i  x.  s ) )  =  ( A  +  ( _i  x.  B ) ) )
9493eqeq2d 2099 . . . . . . . . . . . 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 3848 . . . . . . . . . . . 12  |-  ( s  =  B  ->  (
s #  u  <->  B #  u ) )
9796orbi2d 739 . . . . . . . . . . 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 2403 . . . . . . . . 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 2736 . . . . . . . 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 1209 . . . . . . 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 1143 . . . . . 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 664    = wceq 1289    e. wcel 1438   E.wrex 2360   class class class wbr 3845  (class class class)co 5652   CCcc 7348   RRcr 7349   _ici 7352    + caddc 7353    x. cmul 7355   # creap 8051   # cap 8058
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-cnex 7436  ax-resscn 7437  ax-1cn 7438  ax-1re 7439  ax-icn 7440  ax-addcl 7441  ax-addrcl 7442  ax-mulcl 7443  ax-mulrcl 7444  ax-addcom 7445  ax-mulcom 7446  ax-addass 7447  ax-mulass 7448  ax-distr 7449  ax-i2m1 7450  ax-0lt1 7451  ax-1rid 7452  ax-0id 7453  ax-rnegex 7454  ax-precex 7455  ax-cnre 7456  ax-pre-ltirr 7457  ax-pre-lttrn 7459  ax-pre-apti 7460  ax-pre-ltadd 7461  ax-pre-mulgt0 7462
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-iota 4980  df-fun 5017  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-pnf 7524  df-mnf 7525  df-ltxr 7527  df-sub 7655  df-neg 7656  df-reap 8052  df-ap 8059
This theorem is referenced by:  apirr  8082  apsym  8083  apcotr  8084  apadd1  8085  apneg  8088  mulext1  8089  apti  8099  recexaplem2  8121  crap0  8418  iap0  8639  cjap  10340  absext  10496
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