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Theorem apadd1 8539
Description: Addition respects apartness. Analogue of addcan 8111 for apartness. (Contributed by Jim Kingdon, 13-Feb-2020.)
Assertion
Ref Expression
apadd1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A #  B  <->  ( A  +  C ) #  ( B  +  C ) ) )

Proof of Theorem apadd1
Dummy variables  u  v  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnre 7928 . . 3  |-  ( C  e.  CC  ->  E. u  e.  RR  E. v  e.  RR  C  =  ( u  +  ( _i  x.  v ) ) )
213ad2ant3 1020 . 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 7928 . . . . . . 7  |-  ( B  e.  CC  ->  E. z  e.  RR  E. w  e.  RR  B  =  ( z  +  ( _i  x.  w ) ) )
433ad2ant2 1019 . . . . . 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 7928 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  E. x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y ) ) )
763ad2ant1 1018 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  E. x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y ) ) )
87ad2antrr 488 . . . . . . . . 9  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  (
u  e.  RR  /\  v  e.  RR )
)  /\  C  =  ( u  +  (
_i  x.  v )
) )  ->  E. x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y ) ) )
98ad2antrr 488 . . . . . . . 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 simplrl 535 . . . . . . . . . . . 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 )
) )  ->  x  e.  RR )
11 simplrr 536 . . . . . . . . . . . 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 )
) )  ->  y  e.  RR )
12 simprl 529 . . . . . . . . . . . . 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 )
)  ->  z  e.  RR )
1312ad3antrrr 492 . . . . . . . . . . . 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 )
) )  ->  z  e.  RR )
14 simprr 531 . . . . . . . . . . . . 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 )
)  ->  w  e.  RR )
1514ad3antrrr 492 . . . . . . . . . . . 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 )
) )  ->  w  e.  RR )
16 apreim 8534 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR  /\  y  e.  RR )  /\  ( z  e.  RR  /\  w  e.  RR ) )  -> 
( ( x  +  ( _i  x.  y
) ) #  ( z  +  ( _i  x.  w ) )  <->  ( x #  z  \/  y #  w
) ) )
1710, 11, 13, 15, 16syl22anc 1239 . . . . . . . . . . 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 )
) )  ->  (
( x  +  ( _i  x.  y ) ) #  ( z  +  ( _i  x.  w
) )  <->  ( x #  z  \/  y #  w
) ) )
18 simpr 110 . . . . . . . . . . . 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  =  ( x  +  ( _i  x.  y
) ) )
19 simpllr 534 . . . . . . . . . . . 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  =  ( z  +  ( _i  x.  w
) ) )
2018, 19breq12d 4011 . . . . . . . . . . 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  +  ( _i  x.  y
) ) #  ( z  +  ( _i  x.  w ) ) ) )
21 simprl 529 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  RR  /\  v  e.  RR ) )  ->  u  e.  RR )
2221ad2antrr 488 . . . . . . . . . . . . . . 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 )
)  ->  u  e.  RR )
2322ad3antrrr 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 )
) )  ->  u  e.  RR )
2410, 23readdcld 7961 . . . . . . . . . . . . 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  +  u )  e.  RR )
25 simprr 531 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  RR  /\  v  e.  RR ) )  ->  v  e.  RR )
2625ad2antrr 488 . . . . . . . . . . . . . . 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 )
)  ->  v  e.  RR )
2726ad3antrrr 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 )
) )  ->  v  e.  RR )
2811, 27readdcld 7961 . . . . . . . . . . . . 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 )
) )  ->  (
y  +  v )  e.  RR )
2913, 23readdcld 7961 . . . . . . . . . . . . 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  +  u )  e.  RR )
3015, 27readdcld 7961 . . . . . . . . . . . . 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 )
) )  ->  (
w  +  v )  e.  RR )
31 apreim 8534 . . . . . . . . . . . . 13  |-  ( ( ( ( x  +  u )  e.  RR  /\  ( y  +  v )  e.  RR )  /\  ( ( z  +  u )  e.  RR  /\  ( w  +  v )  e.  RR ) )  -> 
( ( ( x  +  u )  +  ( _i  x.  (
y  +  v ) ) ) #  ( ( z  +  u )  +  ( _i  x.  ( w  +  v
) ) )  <->  ( (
x  +  u ) #  ( z  +  u
)  \/  ( y  +  v ) #  ( w  +  v ) ) ) )
3224, 28, 29, 30, 31syl22anc 1239 . . . . . . . . . . . 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 )
) )  ->  (
( ( x  +  u )  +  ( _i  x.  ( y  +  v ) ) ) #  ( ( z  +  u )  +  ( _i  x.  (
w  +  v ) ) )  <->  ( (
x  +  u ) #  ( z  +  u
)  \/  ( y  +  v ) #  ( w  +  v ) ) ) )
3310recnd 7960 . . . . . . . . . . . . . . 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.  CC )
34 ax-icn 7881 . . . . . . . . . . . . . . . . 17  |-  _i  e.  CC
3534a1i 9 . . . . . . . . . . . . . . . 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 )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  _i  e.  CC )
3611recnd 7960 . . . . . . . . . . . . . . . 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 )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  y  e.  CC )
3735, 36mulcld 7952 . . . . . . . . . . . . . . 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 )
) )  ->  (
_i  x.  y )  e.  CC )
3823recnd 7960 . . . . . . . . . . . . . . 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.  CC )
3927recnd 7960 . . . . . . . . . . . . . . . 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 )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  v  e.  CC )
4035, 39mulcld 7952 . . . . . . . . . . . . . . 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 )
) )  ->  (
_i  x.  v )  e.  CC )
4133, 37, 38, 40add4d 8100 . . . . . . . . . . . . . 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 ) )  +  ( u  +  ( _i  x.  v ) ) )  =  ( ( x  +  u )  +  ( ( _i  x.  y )  +  ( _i  x.  v ) ) ) )
42 simplr 528 . . . . . . . . . . . . . . . 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 )
)  ->  C  =  ( u  +  (
_i  x.  v )
) )
4342ad3antrrr 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 )
) )  ->  C  =  ( u  +  ( _i  x.  v
) ) )
4418, 43oveq12d 5883 . . . . . . . . . . . . . 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  +  C )  =  ( ( x  +  ( _i  x.  y ) )  +  ( u  +  ( _i  x.  v ) ) ) )
4535, 36, 39adddid 7956 . . . . . . . . . . . . . . 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 )
) )  ->  (
_i  x.  ( y  +  v ) )  =  ( ( _i  x.  y )  +  ( _i  x.  v
) ) )
4645oveq2d 5881 . . . . . . . . . . . . . 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  +  u
)  +  ( _i  x.  ( y  +  v ) ) )  =  ( ( x  +  u )  +  ( ( _i  x.  y )  +  ( _i  x.  v ) ) ) )
4741, 44, 463eqtr4d 2218 . . . . . . . . . . . . 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  +  u )  +  ( _i  x.  (
y  +  v ) ) ) )
4813recnd 7960 . . . . . . . . . . . . . . 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.  CC )
4915recnd 7960 . . . . . . . . . . . . . . . 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 )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  w  e.  CC )
5035, 49mulcld 7952 . . . . . . . . . . . . . . 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 )
) )  ->  (
_i  x.  w )  e.  CC )
5148, 50, 38, 40add4d 8100 . . . . . . . . . . . . . 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 )
) )  ->  (
( z  +  ( _i  x.  w ) )  +  ( u  +  ( _i  x.  v ) ) )  =  ( ( z  +  u )  +  ( ( _i  x.  w )  +  ( _i  x.  v ) ) ) )
5219, 43oveq12d 5883 . . . . . . . . . . . . . 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 )
) )  ->  ( B  +  C )  =  ( ( z  +  ( _i  x.  w ) )  +  ( u  +  ( _i  x.  v ) ) ) )
5335, 49, 39adddid 7956 . . . . . . . . . . . . . . 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 )
) )  ->  (
_i  x.  ( w  +  v ) )  =  ( ( _i  x.  w )  +  ( _i  x.  v
) ) )
5453oveq2d 5881 . . . . . . . . . . . . . 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 )
) )  ->  (
( z  +  u
)  +  ( _i  x.  ( w  +  v ) ) )  =  ( ( z  +  u )  +  ( ( _i  x.  w )  +  ( _i  x.  v ) ) ) )
5551, 52, 543eqtr4d 2218 . . . . . . . . . . . . 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  +  u )  +  ( _i  x.  (
w  +  v ) ) ) )
5647, 55breq12d 4011 . . . . . . . . . . . 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
) #  ( B  +  C )  <->  ( (
x  +  u )  +  ( _i  x.  ( y  +  v ) ) ) #  ( ( z  +  u
)  +  ( _i  x.  ( w  +  v ) ) ) ) )
57 reapadd1 8527 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR  /\  z  e.  RR  /\  u  e.  RR )  ->  (
x #  z  <->  ( x  +  u ) #  ( z  +  u ) ) )
5810, 13, 23, 57syl3anc 1238 . . . . . . . . . . . . 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  <->  ( x  +  u ) #  ( z  +  u ) ) )
59 reapadd1 8527 . . . . . . . . . . . . . 14  |-  ( ( y  e.  RR  /\  w  e.  RR  /\  v  e.  RR )  ->  (
y #  w  <->  ( y  +  v ) #  ( w  +  v ) ) )
6011, 15, 27, 59syl3anc 1238 . . . . . . . . . . . . 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 )
) )  ->  (
y #  w  <->  ( y  +  v ) #  ( w  +  v ) ) )
6158, 60orbi12d 793 . . . . . . . . . . . 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 )
) )  ->  (
( x #  z  \/  y #  w )  <->  ( (
x  +  u ) #  ( z  +  u
)  \/  ( y  +  v ) #  ( w  +  v ) ) ) )
6232, 56, 613bitr4d 220 . . . . . . . . . . 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 #  z  \/  y #  w
) ) )
6317, 20, 623bitr4d 220 . . . . . . . . . 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 ) ) )
6463ex 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 ) ) ) )
6564rexlimdvva 2600 . . . . . . . 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 ) ) ) )
669, 65mpd 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 ) ) )
6766ex 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 ) ) ) )
6867rexlimdvva 2600 . . . . 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 ) ) ) )
695, 68mpd 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 ) ) )
7069ex 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 ) ) ) )
7170rexlimdvva 2600 . 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 ) ) ) )
722, 71mpd 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 708    /\ w3a 978    = wceq 1353    e. wcel 2146   E.wrex 2454   class class class wbr 3998  (class class class)co 5865   CCcc 7784   RRcr 7785   _ici 7788    + caddc 7789    x. cmul 7791   # cap 8512
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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-iota 5170  df-fun 5210  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-pnf 7968  df-mnf 7969  df-ltxr 7971  df-sub 8104  df-neg 8105  df-reap 8506  df-ap 8513
This theorem is referenced by:  apadd2  8540  addext  8541  apsub1  8573  subap0  8574
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