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Theorem apcotr 8595
Description: Apartness is cotransitive. (Contributed by Jim Kingdon, 16-Feb-2020.)
Assertion
Ref Expression
apcotr  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A #  B  ->  ( A #  C  \/  B #  C
) ) )

Proof of Theorem apcotr
Dummy variables  u  v  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnre 7984 . . 3  |-  ( C  e.  CC  ->  E. u  e.  RR  E. v  e.  RR  C  =  ( u  +  ( _i  x.  v ) ) )
213ad2ant3 1022 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  E. u  e.  RR  E. v  e.  RR  C  =  ( u  +  ( _i  x.  v ) ) )
3 cnre 7984 . . . . . . 7  |-  ( B  e.  CC  ->  E. z  e.  RR  E. w  e.  RR  B  =  ( z  +  ( _i  x.  w ) ) )
433ad2ant2 1021 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  E. z  e.  RR  E. w  e.  RR  B  =  ( z  +  ( _i  x.  w ) ) )
54ad2antrr 488 . . . . 5  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  (
u  e.  RR  /\  v  e.  RR )
)  /\  C  =  ( u  +  (
_i  x.  v )
) )  ->  E. z  e.  RR  E. w  e.  RR  B  =  ( z  +  ( _i  x.  w ) ) )
6 cnre 7984 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  E. x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y ) ) )
763ad2ant1 1020 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  E. x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y ) ) )
87adantr 276 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  RR  /\  v  e.  RR ) )  ->  E. x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y ) ) )
98ad3antrrr 492 . . . . . . . 8  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  (
u  e.  RR  /\  v  e.  RR )
)  /\  C  =  ( u  +  (
_i  x.  v )
) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  E. x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y ) ) )
10 simpr 110 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  (
u  e.  RR  /\  v  e.  RR )
)  /\  C  =  ( u  +  (
_i  x.  v )
) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  A  =  ( x  +  ( _i  x.  y
) ) )
11 simpllr 534 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  (
u  e.  RR  /\  v  e.  RR )
)  /\  C  =  ( u  +  (
_i  x.  v )
) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  B  =  ( z  +  ( _i  x.  w
) ) )
1210, 11breq12d 4031 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  (
u  e.  RR  /\  v  e.  RR )
)  /\  C  =  ( u  +  (
_i  x.  v )
) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  ( A #  B  <->  ( x  +  ( _i  x.  y
) ) #  ( z  +  ( _i  x.  w ) ) ) )
13 simplrl 535 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  (
u  e.  RR  /\  v  e.  RR )
)  /\  C  =  ( u  +  (
_i  x.  v )
) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  x  e.  RR )
14 simplrr 536 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  (
u  e.  RR  /\  v  e.  RR )
)  /\  C  =  ( u  +  (
_i  x.  v )
) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  y  e.  RR )
15 simprl 529 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  (
u  e.  RR  /\  v  e.  RR )
)  /\  C  =  ( u  +  (
_i  x.  v )
) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  ->  z  e.  RR )
1615ad3antrrr 492 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  (
u  e.  RR  /\  v  e.  RR )
)  /\  C  =  ( u  +  (
_i  x.  v )
) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  z  e.  RR )
17 simprr 531 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  (
u  e.  RR  /\  v  e.  RR )
)  /\  C  =  ( u  +  (
_i  x.  v )
) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  ->  w  e.  RR )
1817ad3antrrr 492 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  (
u  e.  RR  /\  v  e.  RR )
)  /\  C  =  ( u  +  (
_i  x.  v )
) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  w  e.  RR )
19 apreim 8591 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  RR  /\  y  e.  RR )  /\  ( z  e.  RR  /\  w  e.  RR ) )  -> 
( ( x  +  ( _i  x.  y
) ) #  ( z  +  ( _i  x.  w ) )  <->  ( x #  z  \/  y #  w
) ) )
2013, 14, 16, 18, 19syl22anc 1250 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  (
u  e.  RR  /\  v  e.  RR )
)  /\  C  =  ( u  +  (
_i  x.  v )
) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  (
( x  +  ( _i  x.  y ) ) #  ( z  +  ( _i  x.  w
) )  <->  ( x #  z  \/  y #  w
) ) )
2112, 20bitrd 188 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  (
u  e.  RR  /\  v  e.  RR )
)  /\  C  =  ( u  +  (
_i  x.  v )
) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  ( A #  B  <->  ( x #  z  \/  y #  w ) ) )
22 simprl 529 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  RR  /\  v  e.  RR ) )  ->  u  e.  RR )
2322ad2antrr 488 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  (
u  e.  RR  /\  v  e.  RR )
)  /\  C  =  ( u  +  (
_i  x.  v )
) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  ->  u  e.  RR )
2423ad3antrrr 492 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  (
u  e.  RR  /\  v  e.  RR )
)  /\  C  =  ( u  +  (
_i  x.  v )
) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  u  e.  RR )
25 reapcotr 8586 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR  /\  z  e.  RR  /\  u  e.  RR )  ->  (
x #  z  ->  (
x #  u  \/  z #  u ) ) )
2613, 16, 24, 25syl3anc 1249 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  (
u  e.  RR  /\  v  e.  RR )
)  /\  C  =  ( u  +  (
_i  x.  v )
) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  (
x #  z  ->  (
x #  u  \/  z #  u ) ) )
27 simprr 531 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  RR  /\  v  e.  RR ) )  ->  v  e.  RR )
2827ad2antrr 488 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  (
u  e.  RR  /\  v  e.  RR )
)  /\  C  =  ( u  +  (
_i  x.  v )
) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  ->  v  e.  RR )
2928ad3antrrr 492 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  (
u  e.  RR  /\  v  e.  RR )
)  /\  C  =  ( u  +  (
_i  x.  v )
) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  v  e.  RR )
30 reapcotr 8586 . . . . . . . . . . . . . . 15  |-  ( ( y  e.  RR  /\  w  e.  RR  /\  v  e.  RR )  ->  (
y #  w  ->  (
y #  v  \/  w #  v ) ) )
3114, 18, 29, 30syl3anc 1249 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  (
u  e.  RR  /\  v  e.  RR )
)  /\  C  =  ( u  +  (
_i  x.  v )
) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  (
y #  w  ->  (
y #  v  \/  w #  v ) ) )
3226, 31orim12d 787 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  (
u  e.  RR  /\  v  e.  RR )
)  /\  C  =  ( u  +  (
_i  x.  v )
) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  (
( x #  z  \/  y #  w )  -> 
( ( x #  u  \/  z #  u )  \/  ( y #  v  \/  w #  v ) ) ) )
3321, 32sylbid 150 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  (
u  e.  RR  /\  v  e.  RR )
)  /\  C  =  ( u  +  (
_i  x.  v )
) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  ( A #  B  ->  ( ( x #  u  \/  z #  u )  \/  (
y #  v  \/  w #  v ) ) ) )
34 or4 772 . . . . . . . . . . . 12  |-  ( ( ( x #  u  \/  z #  u )  \/  ( y #  v  \/  w #  v ) )  <-> 
( ( x #  u  \/  y #  v )  \/  ( z #  u  \/  w #  v ) ) )
3533, 34imbitrdi 161 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  (
u  e.  RR  /\  v  e.  RR )
)  /\  C  =  ( u  +  (
_i  x.  v )
) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  ( A #  B  ->  ( ( x #  u  \/  y #  v )  \/  (
z #  u  \/  w #  v ) ) ) )
36 simplr 528 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  (
u  e.  RR  /\  v  e.  RR )
)  /\  C  =  ( u  +  (
_i  x.  v )
) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  ->  C  =  ( u  +  (
_i  x.  v )
) )
3736ad3antrrr 492 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  (
u  e.  RR  /\  v  e.  RR )
)  /\  C  =  ( u  +  (
_i  x.  v )
) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  C  =  ( u  +  ( _i  x.  v
) ) )
3810, 37breq12d 4031 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  (
u  e.  RR  /\  v  e.  RR )
)  /\  C  =  ( u  +  (
_i  x.  v )
) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  ( A #  C  <->  ( x  +  ( _i  x.  y
) ) #  ( u  +  ( _i  x.  v ) ) ) )
39 apreim 8591 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  RR  /\  y  e.  RR )  /\  ( u  e.  RR  /\  v  e.  RR ) )  -> 
( ( x  +  ( _i  x.  y
) ) #  ( u  +  ( _i  x.  v ) )  <->  ( x #  u  \/  y #  v
) ) )
4013, 14, 24, 29, 39syl22anc 1250 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  (
u  e.  RR  /\  v  e.  RR )
)  /\  C  =  ( u  +  (
_i  x.  v )
) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  (
( x  +  ( _i  x.  y ) ) #  ( u  +  ( _i  x.  v
) )  <->  ( x #  u  \/  y #  v
) ) )
4138, 40bitrd 188 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  (
u  e.  RR  /\  v  e.  RR )
)  /\  C  =  ( u  +  (
_i  x.  v )
) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  ( A #  C  <->  ( x #  u  \/  y #  v )
) )
4211, 37breq12d 4031 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  (
u  e.  RR  /\  v  e.  RR )
)  /\  C  =  ( u  +  (
_i  x.  v )
) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  ( B #  C  <->  ( z  +  ( _i  x.  w
) ) #  ( u  +  ( _i  x.  v ) ) ) )
43 apreim 8591 . . . . . . . . . . . . . 14  |-  ( ( ( z  e.  RR  /\  w  e.  RR )  /\  ( u  e.  RR  /\  v  e.  RR ) )  -> 
( ( z  +  ( _i  x.  w
) ) #  ( u  +  ( _i  x.  v ) )  <->  ( z #  u  \/  w #  v
) ) )
4416, 18, 24, 29, 43syl22anc 1250 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  (
u  e.  RR  /\  v  e.  RR )
)  /\  C  =  ( u  +  (
_i  x.  v )
) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  (
( z  +  ( _i  x.  w ) ) #  ( u  +  ( _i  x.  v
) )  <->  ( z #  u  \/  w #  v
) ) )
4542, 44bitrd 188 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  (
u  e.  RR  /\  v  e.  RR )
)  /\  C  =  ( u  +  (
_i  x.  v )
) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  ( B #  C  <->  ( z #  u  \/  w #  v ) ) )
4641, 45orbi12d 794 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  (
u  e.  RR  /\  v  e.  RR )
)  /\  C  =  ( u  +  (
_i  x.  v )
) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  (
( A #  C  \/  B #  C )  <->  ( (
x #  u  \/  y #  v )  \/  (
z #  u  \/  w #  v ) ) ) )
4735, 46sylibrd 169 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  (
u  e.  RR  /\  v  e.  RR )
)  /\  C  =  ( u  +  (
_i  x.  v )
) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  ( A #  B  ->  ( A #  C  \/  B #  C
) ) )
4847ex 115 . . . . . . . . 9  |-  ( ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  RR  /\  v  e.  RR ) )  /\  C  =  ( u  +  ( _i  x.  v ) ) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  ->  ( A  =  ( x  +  ( _i  x.  y
) )  ->  ( A #  B  ->  ( A #  C  \/  B #  C
) ) ) )
4948rexlimdvva 2615 . . . . . . . 8  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  (
u  e.  RR  /\  v  e.  RR )
)  /\  C  =  ( u  +  (
_i  x.  v )
) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  ( E. x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y
) )  ->  ( A #  B  ->  ( A #  C  \/  B #  C
) ) ) )
509, 49mpd 13 . . . . . . 7  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  (
u  e.  RR  /\  v  e.  RR )
)  /\  C  =  ( u  +  (
_i  x.  v )
) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  ( A #  B  ->  ( A #  C  \/  B #  C
) ) )
5150ex 115 . . . . . 6  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  (
u  e.  RR  /\  v  e.  RR )
)  /\  C  =  ( u  +  (
_i  x.  v )
) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  ->  ( B  =  ( z  +  ( _i  x.  w
) )  ->  ( A #  B  ->  ( A #  C  \/  B #  C
) ) ) )
5251rexlimdvva 2615 . . . . 5  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  (
u  e.  RR  /\  v  e.  RR )
)  /\  C  =  ( u  +  (
_i  x.  v )
) )  ->  ( E. z  e.  RR  E. w  e.  RR  B  =  ( z  +  ( _i  x.  w
) )  ->  ( A #  B  ->  ( A #  C  \/  B #  C
) ) ) )
535, 52mpd 13 . . . 4  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  (
u  e.  RR  /\  v  e.  RR )
)  /\  C  =  ( u  +  (
_i  x.  v )
) )  ->  ( A #  B  ->  ( A #  C  \/  B #  C
) ) )
5453ex 115 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  RR  /\  v  e.  RR ) )  ->  ( C  =  ( u  +  ( _i  x.  v
) )  ->  ( A #  B  ->  ( A #  C  \/  B #  C
) ) ) )
5554rexlimdvva 2615 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( E. u  e.  RR  E. v  e.  RR  C  =  ( u  +  ( _i  x.  v
) )  ->  ( A #  B  ->  ( A #  C  \/  B #  C
) ) ) )
562, 55mpd 13 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A #  B  ->  ( A #  C  \/  B #  C
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 709    /\ w3a 980    = wceq 1364    e. wcel 2160   E.wrex 2469   class class class wbr 4018  (class class class)co 5897   CCcc 7840   RRcr 7841   _ici 7844    + caddc 7845    x. cmul 7847   # cap 8569
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7933  ax-resscn 7934  ax-1cn 7935  ax-1re 7936  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-mulrcl 7941  ax-addcom 7942  ax-mulcom 7943  ax-addass 7944  ax-mulass 7945  ax-distr 7946  ax-i2m1 7947  ax-0lt1 7948  ax-1rid 7949  ax-0id 7950  ax-rnegex 7951  ax-precex 7952  ax-cnre 7953  ax-pre-ltirr 7954  ax-pre-ltwlin 7955  ax-pre-lttrn 7956  ax-pre-apti 7957  ax-pre-ltadd 7958  ax-pre-mulgt0 7959
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fv 5243  df-riota 5852  df-ov 5900  df-oprab 5901  df-mpo 5902  df-pnf 8025  df-mnf 8026  df-ltxr 8028  df-sub 8161  df-neg 8162  df-reap 8563  df-ap 8570
This theorem is referenced by:  addext  8598  mulext  8602  aptap  8638  mul0eqap  8658
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