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Theorem apreim 8537
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 527 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  A  e.  RR )
21recnd 7963 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  A  e.  CC )
3 ax-icn 7884 . . . . . . 7  |-  _i  e.  CC
43a1i 9 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  _i  e.  CC )
5 simplr 528 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  B  e.  RR )
65recnd 7963 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  B  e.  CC )
74, 6mulcld 7955 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( _i  x.  B
)  e.  CC )
82, 7addcld 7954 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( A  +  ( _i  x.  B ) )  e.  CC )
9 simprl 529 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  C  e.  RR )
109recnd 7963 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  C  e.  CC )
11 simprr 531 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  D  e.  RR )
1211recnd 7963 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  D  e.  CC )
134, 12mulcld 7955 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( _i  x.  D
)  e.  CC )
1410, 13addcld 7954 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( C  +  ( _i  x.  D ) )  e.  CC )
15 eqeq1 2184 . . . . . . . . 9  |-  ( x  =  ( A  +  ( _i  x.  B
) )  ->  (
x  =  ( r  +  ( _i  x.  s ) )  <->  ( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s ) ) ) )
1615anbi1d 465 . . . . . . . 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 465 . . . . . . 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 2502 . . . . . 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 2502 . . . . 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 2184 . . . . . . . . 9  |-  ( y  =  ( C  +  ( _i  x.  D
) )  ->  (
y  =  ( t  +  ( _i  x.  u ) )  <->  ( C  +  ( _i  x.  D ) )  =  ( t  +  ( _i  x.  u ) ) ) )
2120anbi2d 464 . . . . . . . 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 465 . . . . . . 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 2502 . . . . . 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 2502 . . . . 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 8516 . . . . 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 4265 . . . 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 411 . . 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 531 . . . . . . 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 492 . . . . . . . . . 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 492 . . . . . . . . . 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 8521 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( A #  C  <->  A #  C )
)
3229, 30, 31syl2anc 411 . . . . . . . . 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 492 . . . . . . . . . 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 492 . . . . . . . . . 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 8521 . . . . . . . . . 10  |-  ( ( B  e.  RR  /\  D  e.  RR )  ->  ( B #  D  <->  B #  D )
)
3633, 34, 35syl2anc 411 . . . . . . . . 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 793 . . . . . . . 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 537 . . . . . . . . . . . 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 534 . . . . . . . . . . . . 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 8536 . . . . . . . . . . . . 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 1237 . . . . . . . . . . . 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 147 . . . . . . . . . . 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 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 ) ) )  ->  A  =  r )
44 simprlr 538 . . . . . . . . . . . 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 528 . . . . . . . . . . . . 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 8536 . . . . . . . . . . . . 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 1237 . . . . . . . . . . . 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 147 . . . . . . . . . . 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 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 ) ) )  ->  C  =  t )
5043, 49breq12d 4013 . . . . . . . . 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 114 . . . . . . . . . 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 114 . . . . . . . . . 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 4013 . . . . . . . . 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 793 . . . . . . . 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 188 . . . . . . 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 167 . . . . . 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 115 . . . . 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 2602 . . . 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 2602 . . 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 150 . 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 509 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( A #  C  <->  A #  C )
)
6235ad2ant2l 508 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( B #  D  <->  B #  D )
)
6361, 62orbi12d 793 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( A #  C  \/  B #  D )  <->  ( A #  C  \/  B #  D ) ) )
6463pm5.32i 454 . . . 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 2177 . . . . . . . . . . . 12  |-  ( A  +  ( _i  x.  B ) )  =  ( A  +  ( _i  x.  B ) )
66 eqid 2177 . . . . . . . . . . . 12  |-  ( C  +  ( _i  x.  D ) )  =  ( C  +  ( _i  x.  D ) )
6765, 66pm3.2i 272 . . . . . . . . . . 11  |-  ( ( A  +  ( _i  x.  B ) )  =  ( A  +  ( _i  x.  B
) )  /\  ( C  +  ( _i  x.  D ) )  =  ( C  +  ( _i  x.  D ) ) )
6867biantrur 303 . . . . . . . . . 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 5875 . . . . . . . . . . . . . 14  |-  ( t  =  C  ->  (
t  +  ( _i  x.  u ) )  =  ( C  +  ( _i  x.  u
) ) )
7069eqeq2d 2189 . . . . . . . . . . . . 13  |-  ( t  =  C  ->  (
( C  +  ( _i  x.  D ) )  =  ( t  +  ( _i  x.  u ) )  <->  ( C  +  ( _i  x.  D ) )  =  ( C  +  ( _i  x.  u ) ) ) )
7170anbi2d 464 . . . . . . . . . . . 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 4004 . . . . . . . . . . . . 13  |-  ( t  =  C  ->  ( A #  t 
<->  A #  C ) )
7372orbi1d 791 . . . . . . . . . . . 12  |-  ( t  =  C  ->  (
( A #  t  \/  B #  u
)  <->  ( A #  C  \/  B #  u ) ) )
7471, 73anbi12d 473 . . . . . . . . . . 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 5876 . . . . . . . . . . . . . . 15  |-  ( u  =  D  ->  (
_i  x.  u )  =  ( _i  x.  D ) )
7675oveq2d 5884 . . . . . . . . . . . . . 14  |-  ( u  =  D  ->  ( C  +  ( _i  x.  u ) )  =  ( C  +  ( _i  x.  D ) ) )
7776eqeq2d 2189 . . . . . . . . . . . . 13  |-  ( u  =  D  ->  (
( C  +  ( _i  x.  D ) )  =  ( C  +  ( _i  x.  u ) )  <->  ( C  +  ( _i  x.  D ) )  =  ( C  +  ( _i  x.  D ) ) ) )
7877anbi2d 464 . . . . . . . . . . . 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 4004 . . . . . . . . . . . . 13  |-  ( u  =  D  ->  ( B #  u 
<->  B #  D ) )
8079orbi2d 790 . . . . . . . . . . . 12  |-  ( u  =  D  ->  (
( A #  C  \/  B #  u
)  <->  ( A #  C  \/  B #  D ) ) )
8178, 80anbi12d 473 . . . . . . . . . . 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 2856 . . . . . . . . . 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 1276 . . . . . . . . 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 1203 . . . . . . . 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 5875 . . . . . . . . . . . . 13  |-  ( r  =  A  ->  (
r  +  ( _i  x.  s ) )  =  ( A  +  ( _i  x.  s
) ) )
8685eqeq2d 2189 . . . . . . . . . . . 12  |-  ( r  =  A  ->  (
( A  +  ( _i  x.  B ) )  =  ( r  +  ( _i  x.  s ) )  <->  ( A  +  ( _i  x.  B ) )  =  ( A  +  ( _i  x.  s ) ) ) )
8786anbi1d 465 . . . . . . . . . . 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 4003 . . . . . . . . . . . 12  |-  ( r  =  A  ->  (
r #  t  <->  A #  t ) )
8988orbi1d 791 . . . . . . . . . . 11  |-  ( r  =  A  ->  (
( r #  t  \/  s #  u
)  <->  ( A #  t  \/  s #  u ) ) )
9087, 89anbi12d 473 . . . . . . . . . 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 2502 . . . . . . . . 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 5876 . . . . . . . . . . . . . 14  |-  ( s  =  B  ->  (
_i  x.  s )  =  ( _i  x.  B ) )
9392oveq2d 5884 . . . . . . . . . . . . 13  |-  ( s  =  B  ->  ( A  +  ( _i  x.  s ) )  =  ( A  +  ( _i  x.  B ) ) )
9493eqeq2d 2189 . . . . . . . . . . . 12  |-  ( s  =  B  ->  (
( A  +  ( _i  x.  B ) )  =  ( A  +  ( _i  x.  s ) )  <->  ( A  +  ( _i  x.  B ) )  =  ( A  +  ( _i  x.  B ) ) ) )
9594anbi1d 465 . . . . . . . . . . 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 4003 . . . . . . . . . . . 12  |-  ( s  =  B  ->  (
s #  u  <->  B #  u ) )
9796orbi2d 790 . . . . . . . . . . 11  |-  ( s  =  B  ->  (
( A #  t  \/  s #  u
)  <->  ( A #  t  \/  B #  u ) ) )
9895, 97anbi12d 473 . . . . . . . . . 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 2502 . . . . . . . . 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 2856 . . . . . . . 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 1273 . . . . . . 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 1203 . . . . . 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 400 . . . . 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 276 . . . . 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 167 . . . 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 121 . . 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 115 . 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 129 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 104    <-> wb 105    \/ wo 708    = wceq 1353    e. wcel 2148   E.wrex 2456   class class class wbr 4000  (class class class)co 5868   CCcc 7787   RRcr 7788   _ici 7791    + caddc 7792    x. cmul 7794   # creap 8508   # cap 8515
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-cnex 7880  ax-resscn 7881  ax-1cn 7882  ax-1re 7883  ax-icn 7884  ax-addcl 7885  ax-addrcl 7886  ax-mulcl 7887  ax-mulrcl 7888  ax-addcom 7889  ax-mulcom 7890  ax-addass 7891  ax-mulass 7892  ax-distr 7893  ax-i2m1 7894  ax-0lt1 7895  ax-1rid 7896  ax-0id 7897  ax-rnegex 7898  ax-precex 7899  ax-cnre 7900  ax-pre-ltirr 7901  ax-pre-lttrn 7903  ax-pre-apti 7904  ax-pre-ltadd 7905  ax-pre-mulgt0 7906
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-iota 5173  df-fun 5213  df-fv 5219  df-riota 5824  df-ov 5871  df-oprab 5872  df-mpo 5873  df-pnf 7971  df-mnf 7972  df-ltxr 7974  df-sub 8107  df-neg 8108  df-reap 8509  df-ap 8516
This theorem is referenced by:  apirr  8539  apsym  8540  apcotr  8541  apadd1  8542  apneg  8545  mulext1  8546  apti  8556  recexaplem2  8585  crap0  8891  iap0  9118  cjap  10886  cnreim  10958  absext  11043
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