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Theorem apcotr 8332
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 7726 . . 3  |-  ( C  e.  CC  ->  E. u  e.  RR  E. v  e.  RR  C  =  ( u  +  ( _i  x.  v ) ) )
213ad2ant3 987 . 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 7726 . . . . . . 7  |-  ( B  e.  CC  ->  E. z  e.  RR  E. w  e.  RR  B  =  ( z  +  ( _i  x.  w ) ) )
433ad2ant2 986 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  E. z  e.  RR  E. w  e.  RR  B  =  ( z  +  ( _i  x.  w ) ) )
54ad2antrr 477 . . . . 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 7726 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  E. x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y ) ) )
763ad2ant1 985 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  E. x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y ) ) )
87adantr 272 . . . . . . . . 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 481 . . . . . . . 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 109 . . . . . . . . . . . . . . 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 506 . . . . . . . . . . . . . . 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 3910 . . . . . . . . . . . . . 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 507 . . . . . . . . . . . . . . 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 508 . . . . . . . . . . . . . . 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 503 . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . 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 504 . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . 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 8328 . . . . . . . . . . . . . . 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 1200 . . . . . . . . . . . . . 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 187 . . . . . . . . . . . . 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 503 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  RR  /\  v  e.  RR ) )  ->  u  e.  RR )
2322ad2antrr 477 . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . 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 8323 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR  /\  z  e.  RR  /\  u  e.  RR )  ->  (
x #  z  ->  (
x #  u  \/  z #  u ) ) )
2613, 16, 24, 25syl3anc 1199 . . . . . . . . . . . . . 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 504 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  RR  /\  v  e.  RR ) )  ->  v  e.  RR )
2827ad2antrr 477 . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . 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 8323 . . . . . . . . . . . . . . 15  |-  ( ( y  e.  RR  /\  w  e.  RR  /\  v  e.  RR )  ->  (
y #  w  ->  (
y #  v  \/  w #  v ) ) )
3114, 18, 29, 30syl3anc 1199 . . . . . . . . . . . . . 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 758 . . . . . . . . . . . . 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 149 . . . . . . . . . . . 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 743 . . . . . . . . . . . 12  |-  ( ( ( x #  u  \/  z #  u )  \/  ( y #  v  \/  w #  v ) )  <-> 
( ( x #  u  \/  y #  v )  \/  ( z #  u  \/  w #  v ) ) )
3533, 34syl6ib 160 . . . . . . . . . . 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 502 . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . 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 3910 . . . . . . . . . . . . 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 8328 . . . . . . . . . . . . . 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 1200 . . . . . . . . . . . . 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 187 . . . . . . . . . . . 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 3910 . . . . . . . . . . . . 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 8328 . . . . . . . . . . . . . 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 1200 . . . . . . . . . . . . 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 187 . . . . . . . . . . . 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 765 . . . . . . . . . . 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 168 . . . . . . . . . 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 114 . . . . . . . . 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 2532 . . . . . . . 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 114 . . . . . 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 2532 . . . . 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 114 . . 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 2532 . 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 103    <-> wb 104    \/ wo 680    /\ w3a 945    = wceq 1314    e. wcel 1463   E.wrex 2392   class class class wbr 3897  (class class class)co 5740   CCcc 7582   RRcr 7583   _ici 7586    + caddc 7587    x. cmul 7589   # cap 8306
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-mulrcl 7683  ax-addcom 7684  ax-mulcom 7685  ax-addass 7686  ax-mulass 7687  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-1rid 7691  ax-0id 7692  ax-rnegex 7693  ax-precex 7694  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-apti 7699  ax-pre-ltadd 7700  ax-pre-mulgt0 7701
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-iota 5056  df-fun 5093  df-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-pnf 7766  df-mnf 7767  df-ltxr 7769  df-sub 7899  df-neg 7900  df-reap 8300  df-ap 8307
This theorem is referenced by:  addext  8335  mulext  8339  mul0eqap  8394
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