ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  aprcl Unicode version

Theorem aprcl 8401
Description: Reverse closure for apartness. (Contributed by Jim Kingdon, 19-Dec-2023.)
Assertion
Ref Expression
aprcl  |-  ( A #  B  ->  ( A  e.  CC  /\  B  e.  CC ) )

Proof of Theorem aprcl
Dummy variables  r  s  t  u  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-br 3925 . . . 4  |-  ( A #  B  <->  <. A ,  B >.  e. #  )
2 eqeq1 2144 . . . . . . . . . 10  |-  ( x  =  ( 1st `  <. A ,  B >. )  ->  ( x  =  ( r  +  ( _i  x.  s ) )  <-> 
( 1st `  <. A ,  B >. )  =  ( r  +  ( _i  x.  s
) ) ) )
32anbi1d 460 . . . . . . . . 9  |-  ( x  =  ( 1st `  <. A ,  B >. )  ->  ( ( x  =  ( r  +  ( _i  x.  s ) )  /\  y  =  ( t  +  ( _i  x.  u ) ) )  <->  ( ( 1st `  <. A ,  B >. )  =  ( r  +  ( _i  x.  s ) )  /\  y  =  ( t  +  ( _i  x.  u ) ) ) ) )
43anbi1d 460 . . . . . . . 8  |-  ( x  =  ( 1st `  <. A ,  B >. )  ->  ( ( ( x  =  ( r  +  ( _i  x.  s
) )  /\  y  =  ( t  +  ( _i  x.  u
) ) )  /\  ( r #  t  \/  s #  u
) )  <->  ( (
( 1st `  <. A ,  B >. )  =  ( r  +  ( _i  x.  s
) )  /\  y  =  ( t  +  ( _i  x.  u
) ) )  /\  ( r #  t  \/  s #  u
) ) ) )
542rexbidv 2458 . . . . . . 7  |-  ( x  =  ( 1st `  <. A ,  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  ( ( ( 1st `  <. A ,  B >. )  =  ( r  +  ( _i  x.  s ) )  /\  y  =  ( t  +  ( _i  x.  u ) ) )  /\  ( r #  t  \/  s #  u ) ) ) )
652rexbidv 2458 . . . . . 6  |-  ( x  =  ( 1st `  <. A ,  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  ( ( ( 1st `  <. A ,  B >. )  =  ( r  +  ( _i  x.  s ) )  /\  y  =  ( t  +  ( _i  x.  u ) ) )  /\  ( r #  t  \/  s #  u ) ) ) )
7 eqeq1 2144 . . . . . . . . . 10  |-  ( y  =  ( 2nd `  <. A ,  B >. )  ->  ( y  =  ( t  +  ( _i  x.  u ) )  <-> 
( 2nd `  <. A ,  B >. )  =  ( t  +  ( _i  x.  u
) ) ) )
87anbi2d 459 . . . . . . . . 9  |-  ( y  =  ( 2nd `  <. A ,  B >. )  ->  ( ( ( 1st `  <. A ,  B >. )  =  ( r  +  ( _i  x.  s ) )  /\  y  =  ( t  +  ( _i  x.  u ) ) )  <-> 
( ( 1st `  <. A ,  B >. )  =  ( r  +  ( _i  x.  s
) )  /\  ( 2nd `  <. A ,  B >. )  =  ( t  +  ( _i  x.  u ) ) ) ) )
98anbi1d 460 . . . . . . . 8  |-  ( y  =  ( 2nd `  <. A ,  B >. )  ->  ( ( ( ( 1st `  <. A ,  B >. )  =  ( r  +  ( _i  x.  s ) )  /\  y  =  ( t  +  ( _i  x.  u ) ) )  /\  ( r #  t  \/  s #  u ) )  <->  ( (
( 1st `  <. A ,  B >. )  =  ( r  +  ( _i  x.  s
) )  /\  ( 2nd `  <. A ,  B >. )  =  ( t  +  ( _i  x.  u ) ) )  /\  ( r #  t  \/  s #  u ) ) ) )
1092rexbidv 2458 . . . . . . 7  |-  ( y  =  ( 2nd `  <. A ,  B >. )  ->  ( E. t  e.  RR  E. u  e.  RR  ( ( ( 1st `  <. A ,  B >. )  =  ( r  +  ( _i  x.  s ) )  /\  y  =  ( t  +  ( _i  x.  u ) ) )  /\  ( r #  t  \/  s #  u ) )  <->  E. t  e.  RR  E. u  e.  RR  ( ( ( 1st `  <. A ,  B >. )  =  ( r  +  ( _i  x.  s ) )  /\  ( 2nd `  <. A ,  B >. )  =  ( t  +  ( _i  x.  u
) ) )  /\  ( r #  t  \/  s #  u
) ) ) )
11102rexbidv 2458 . . . . . 6  |-  ( y  =  ( 2nd `  <. A ,  B >. )  ->  ( E. r  e.  RR  E. s  e.  RR  E. t  e.  RR  E. u  e.  RR  ( ( ( 1st `  <. A ,  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  ( ( ( 1st `  <. A ,  B >. )  =  ( r  +  ( _i  x.  s ) )  /\  ( 2nd `  <. A ,  B >. )  =  ( t  +  ( _i  x.  u
) ) )  /\  ( r #  t  \/  s #  u
) ) ) )
126, 11elopabi 6086 . . . . 5  |-  ( <. A ,  B >.  e. 
{ <. 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
) ) }  ->  E. r  e.  RR  E. s  e.  RR  E. t  e.  RR  E. u  e.  RR  ( ( ( 1st `  <. A ,  B >. )  =  ( r  +  ( _i  x.  s ) )  /\  ( 2nd `  <. A ,  B >. )  =  ( t  +  ( _i  x.  u
) ) )  /\  ( r #  t  \/  s #  u
) ) )
13 df-ap 8337 . . . . 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
) ) }
1412, 13eleq2s 2232 . . . 4  |-  ( <. A ,  B >.  e. # 
->  E. r  e.  RR  E. s  e.  RR  E. t  e.  RR  E. u  e.  RR  ( ( ( 1st `  <. A ,  B >. )  =  ( r  +  ( _i  x.  s ) )  /\  ( 2nd `  <. A ,  B >. )  =  ( t  +  ( _i  x.  u
) ) )  /\  ( r #  t  \/  s #  u
) ) )
151, 14sylbi 120 . . 3  |-  ( A #  B  ->  E. r  e.  RR  E. s  e.  RR  E. t  e.  RR  E. u  e.  RR  ( ( ( 1st `  <. A ,  B >. )  =  ( r  +  ( _i  x.  s ) )  /\  ( 2nd `  <. A ,  B >. )  =  ( t  +  ( _i  x.  u
) ) )  /\  ( r #  t  \/  s #  u
) ) )
16 simpl 108 . . . . . . 7  |-  ( ( ( ( 1st `  <. A ,  B >. )  =  ( r  +  ( _i  x.  s
) )  /\  ( 2nd `  <. A ,  B >. )  =  ( t  +  ( _i  x.  u ) ) )  /\  ( r #  t  \/  s #  u ) )  -> 
( ( 1st `  <. A ,  B >. )  =  ( r  +  ( _i  x.  s
) )  /\  ( 2nd `  <. A ,  B >. )  =  ( t  +  ( _i  x.  u ) ) ) )
1716reximi 2527 . . . . . 6  |-  ( E. u  e.  RR  (
( ( 1st `  <. A ,  B >. )  =  ( r  +  ( _i  x.  s
) )  /\  ( 2nd `  <. A ,  B >. )  =  ( t  +  ( _i  x.  u ) ) )  /\  ( r #  t  \/  s #  u ) )  ->  E. u  e.  RR  ( ( 1st `  <. A ,  B >. )  =  ( r  +  ( _i  x.  s
) )  /\  ( 2nd `  <. A ,  B >. )  =  ( t  +  ( _i  x.  u ) ) ) )
1817reximi 2527 . . . . 5  |-  ( E. t  e.  RR  E. u  e.  RR  (
( ( 1st `  <. A ,  B >. )  =  ( r  +  ( _i  x.  s
) )  /\  ( 2nd `  <. A ,  B >. )  =  ( t  +  ( _i  x.  u ) ) )  /\  ( r #  t  \/  s #  u ) )  ->  E. t  e.  RR  E. u  e.  RR  (
( 1st `  <. A ,  B >. )  =  ( r  +  ( _i  x.  s
) )  /\  ( 2nd `  <. A ,  B >. )  =  ( t  +  ( _i  x.  u ) ) ) )
1918reximi 2527 . . . 4  |-  ( E. s  e.  RR  E. t  e.  RR  E. u  e.  RR  ( ( ( 1st `  <. A ,  B >. )  =  ( r  +  ( _i  x.  s ) )  /\  ( 2nd `  <. A ,  B >. )  =  ( t  +  ( _i  x.  u
) ) )  /\  ( r #  t  \/  s #  u
) )  ->  E. s  e.  RR  E. t  e.  RR  E. u  e.  RR  ( ( 1st `  <. A ,  B >. )  =  ( r  +  ( _i  x.  s ) )  /\  ( 2nd `  <. A ,  B >. )  =  ( t  +  ( _i  x.  u ) ) ) )
2019reximi 2527 . . 3  |-  ( E. r  e.  RR  E. s  e.  RR  E. t  e.  RR  E. u  e.  RR  ( ( ( 1st `  <. A ,  B >. )  =  ( r  +  ( _i  x.  s ) )  /\  ( 2nd `  <. A ,  B >. )  =  ( t  +  ( _i  x.  u
) ) )  /\  ( r #  t  \/  s #  u
) )  ->  E. r  e.  RR  E. s  e.  RR  E. t  e.  RR  E. u  e.  RR  ( ( 1st `  <. A ,  B >. )  =  ( r  +  ( _i  x.  s ) )  /\  ( 2nd `  <. A ,  B >. )  =  ( t  +  ( _i  x.  u ) ) ) )
2115, 20syl 14 . 2  |-  ( A #  B  ->  E. r  e.  RR  E. s  e.  RR  E. t  e.  RR  E. u  e.  RR  ( ( 1st `  <. A ,  B >. )  =  ( r  +  ( _i  x.  s ) )  /\  ( 2nd `  <. A ,  B >. )  =  ( t  +  ( _i  x.  u ) ) ) )
2213relopabi 4660 . . . . . . . . . 10  |-  Rel #
2322brrelex1i 4577 . . . . . . . . 9  |-  ( A #  B  ->  A  e.  _V )
2423ad3antrrr 483 . . . . . . . 8  |-  ( ( ( ( A #  B  /\  ( r  e.  RR  /\  s  e.  RR ) )  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( 1st `  <. A ,  B >. )  =  ( r  +  ( _i  x.  s
) )  /\  ( 2nd `  <. A ,  B >. )  =  ( t  +  ( _i  x.  u ) ) ) )  ->  A  e.  _V )
2522brrelex2i 4578 . . . . . . . . 9  |-  ( A #  B  ->  B  e.  _V )
2625ad3antrrr 483 . . . . . . . 8  |-  ( ( ( ( A #  B  /\  ( r  e.  RR  /\  s  e.  RR ) )  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( 1st `  <. A ,  B >. )  =  ( r  +  ( _i  x.  s
) )  /\  ( 2nd `  <. A ,  B >. )  =  ( t  +  ( _i  x.  u ) ) ) )  ->  B  e.  _V )
27 op1stg 6041 . . . . . . . 8  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( 1st `  <. A ,  B >. )  =  A )
2824, 26, 27syl2anc 408 . . . . . . 7  |-  ( ( ( ( A #  B  /\  ( r  e.  RR  /\  s  e.  RR ) )  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( 1st `  <. A ,  B >. )  =  ( r  +  ( _i  x.  s
) )  /\  ( 2nd `  <. A ,  B >. )  =  ( t  +  ( _i  x.  u ) ) ) )  ->  ( 1st ` 
<. A ,  B >. )  =  A )
29 simprl 520 . . . . . . . 8  |-  ( ( ( ( A #  B  /\  ( r  e.  RR  /\  s  e.  RR ) )  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( 1st `  <. A ,  B >. )  =  ( r  +  ( _i  x.  s
) )  /\  ( 2nd `  <. A ,  B >. )  =  ( t  +  ( _i  x.  u ) ) ) )  ->  ( 1st ` 
<. A ,  B >. )  =  ( r  +  ( _i  x.  s
) ) )
30 simprl 520 . . . . . . . . . . 11  |-  ( ( A #  B  /\  (
r  e.  RR  /\  s  e.  RR )
)  ->  r  e.  RR )
3130ad2antrr 479 . . . . . . . . . 10  |-  ( ( ( ( A #  B  /\  ( r  e.  RR  /\  s  e.  RR ) )  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( 1st `  <. A ,  B >. )  =  ( r  +  ( _i  x.  s
) )  /\  ( 2nd `  <. A ,  B >. )  =  ( t  +  ( _i  x.  u ) ) ) )  ->  r  e.  RR )
3231recnd 7787 . . . . . . . . 9  |-  ( ( ( ( A #  B  /\  ( r  e.  RR  /\  s  e.  RR ) )  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( 1st `  <. A ,  B >. )  =  ( r  +  ( _i  x.  s
) )  /\  ( 2nd `  <. A ,  B >. )  =  ( t  +  ( _i  x.  u ) ) ) )  ->  r  e.  CC )
33 ax-icn 7708 . . . . . . . . . . 11  |-  _i  e.  CC
3433a1i 9 . . . . . . . . . 10  |-  ( ( ( ( A #  B  /\  ( r  e.  RR  /\  s  e.  RR ) )  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( 1st `  <. A ,  B >. )  =  ( r  +  ( _i  x.  s
) )  /\  ( 2nd `  <. A ,  B >. )  =  ( t  +  ( _i  x.  u ) ) ) )  ->  _i  e.  CC )
35 simprr 521 . . . . . . . . . . . 12  |-  ( ( A #  B  /\  (
r  e.  RR  /\  s  e.  RR )
)  ->  s  e.  RR )
3635ad2antrr 479 . . . . . . . . . . 11  |-  ( ( ( ( A #  B  /\  ( r  e.  RR  /\  s  e.  RR ) )  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( 1st `  <. A ,  B >. )  =  ( r  +  ( _i  x.  s
) )  /\  ( 2nd `  <. A ,  B >. )  =  ( t  +  ( _i  x.  u ) ) ) )  ->  s  e.  RR )
3736recnd 7787 . . . . . . . . . 10  |-  ( ( ( ( A #  B  /\  ( r  e.  RR  /\  s  e.  RR ) )  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( 1st `  <. A ,  B >. )  =  ( r  +  ( _i  x.  s
) )  /\  ( 2nd `  <. A ,  B >. )  =  ( t  +  ( _i  x.  u ) ) ) )  ->  s  e.  CC )
3834, 37mulcld 7779 . . . . . . . . 9  |-  ( ( ( ( A #  B  /\  ( r  e.  RR  /\  s  e.  RR ) )  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( 1st `  <. A ,  B >. )  =  ( r  +  ( _i  x.  s
) )  /\  ( 2nd `  <. A ,  B >. )  =  ( t  +  ( _i  x.  u ) ) ) )  ->  ( _i  x.  s )  e.  CC )
3932, 38addcld 7778 . . . . . . . 8  |-  ( ( ( ( A #  B  /\  ( r  e.  RR  /\  s  e.  RR ) )  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( 1st `  <. A ,  B >. )  =  ( r  +  ( _i  x.  s
) )  /\  ( 2nd `  <. A ,  B >. )  =  ( t  +  ( _i  x.  u ) ) ) )  ->  ( r  +  ( _i  x.  s ) )  e.  CC )
4029, 39eqeltrd 2214 . . . . . . 7  |-  ( ( ( ( A #  B  /\  ( r  e.  RR  /\  s  e.  RR ) )  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( 1st `  <. A ,  B >. )  =  ( r  +  ( _i  x.  s
) )  /\  ( 2nd `  <. A ,  B >. )  =  ( t  +  ( _i  x.  u ) ) ) )  ->  ( 1st ` 
<. A ,  B >. )  e.  CC )
4128, 40eqeltrrd 2215 . . . . . 6  |-  ( ( ( ( A #  B  /\  ( r  e.  RR  /\  s  e.  RR ) )  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( 1st `  <. A ,  B >. )  =  ( r  +  ( _i  x.  s
) )  /\  ( 2nd `  <. A ,  B >. )  =  ( t  +  ( _i  x.  u ) ) ) )  ->  A  e.  CC )
42 op2ndg 6042 . . . . . . . 8  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( 2nd `  <. A ,  B >. )  =  B )
4324, 26, 42syl2anc 408 . . . . . . 7  |-  ( ( ( ( A #  B  /\  ( r  e.  RR  /\  s  e.  RR ) )  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( 1st `  <. A ,  B >. )  =  ( r  +  ( _i  x.  s
) )  /\  ( 2nd `  <. A ,  B >. )  =  ( t  +  ( _i  x.  u ) ) ) )  ->  ( 2nd ` 
<. A ,  B >. )  =  B )
44 simprr 521 . . . . . . . 8  |-  ( ( ( ( A #  B  /\  ( r  e.  RR  /\  s  e.  RR ) )  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( 1st `  <. A ,  B >. )  =  ( r  +  ( _i  x.  s
) )  /\  ( 2nd `  <. A ,  B >. )  =  ( t  +  ( _i  x.  u ) ) ) )  ->  ( 2nd ` 
<. A ,  B >. )  =  ( t  +  ( _i  x.  u
) ) )
45 recn 7746 . . . . . . . . . . . 12  |-  ( t  e.  RR  ->  t  e.  CC )
4645adantr 274 . . . . . . . . . . 11  |-  ( ( t  e.  RR  /\  u  e.  RR )  ->  t  e.  CC )
4733a1i 9 . . . . . . . . . . . 12  |-  ( ( t  e.  RR  /\  u  e.  RR )  ->  _i  e.  CC )
48 recn 7746 . . . . . . . . . . . . 13  |-  ( u  e.  RR  ->  u  e.  CC )
4948adantl 275 . . . . . . . . . . . 12  |-  ( ( t  e.  RR  /\  u  e.  RR )  ->  u  e.  CC )
5047, 49mulcld 7779 . . . . . . . . . . 11  |-  ( ( t  e.  RR  /\  u  e.  RR )  ->  ( _i  x.  u
)  e.  CC )
5146, 50addcld 7778 . . . . . . . . . 10  |-  ( ( t  e.  RR  /\  u  e.  RR )  ->  ( t  +  ( _i  x.  u ) )  e.  CC )
5251adantl 275 . . . . . . . . 9  |-  ( ( ( A #  B  /\  ( r  e.  RR  /\  s  e.  RR ) )  /\  ( t  e.  RR  /\  u  e.  RR ) )  -> 
( t  +  ( _i  x.  u ) )  e.  CC )
5352adantr 274 . . . . . . . 8  |-  ( ( ( ( A #  B  /\  ( r  e.  RR  /\  s  e.  RR ) )  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( 1st `  <. A ,  B >. )  =  ( r  +  ( _i  x.  s
) )  /\  ( 2nd `  <. A ,  B >. )  =  ( t  +  ( _i  x.  u ) ) ) )  ->  ( t  +  ( _i  x.  u ) )  e.  CC )
5444, 53eqeltrd 2214 . . . . . . 7  |-  ( ( ( ( A #  B  /\  ( r  e.  RR  /\  s  e.  RR ) )  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( 1st `  <. A ,  B >. )  =  ( r  +  ( _i  x.  s
) )  /\  ( 2nd `  <. A ,  B >. )  =  ( t  +  ( _i  x.  u ) ) ) )  ->  ( 2nd ` 
<. A ,  B >. )  e.  CC )
5543, 54eqeltrrd 2215 . . . . . 6  |-  ( ( ( ( A #  B  /\  ( r  e.  RR  /\  s  e.  RR ) )  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( 1st `  <. A ,  B >. )  =  ( r  +  ( _i  x.  s
) )  /\  ( 2nd `  <. A ,  B >. )  =  ( t  +  ( _i  x.  u ) ) ) )  ->  B  e.  CC )
5641, 55jca 304 . . . . 5  |-  ( ( ( ( A #  B  /\  ( r  e.  RR  /\  s  e.  RR ) )  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( 1st `  <. A ,  B >. )  =  ( r  +  ( _i  x.  s
) )  /\  ( 2nd `  <. A ,  B >. )  =  ( t  +  ( _i  x.  u ) ) ) )  ->  ( A  e.  CC  /\  B  e.  CC ) )
5756ex 114 . . . 4  |-  ( ( ( A #  B  /\  ( r  e.  RR  /\  s  e.  RR ) )  /\  ( t  e.  RR  /\  u  e.  RR ) )  -> 
( ( ( 1st `  <. A ,  B >. )  =  ( r  +  ( _i  x.  s ) )  /\  ( 2nd `  <. A ,  B >. )  =  ( t  +  ( _i  x.  u ) ) )  ->  ( A  e.  CC  /\  B  e.  CC ) ) )
5857rexlimdvva 2555 . . 3  |-  ( ( A #  B  /\  (
r  e.  RR  /\  s  e.  RR )
)  ->  ( E. t  e.  RR  E. u  e.  RR  ( ( 1st `  <. A ,  B >. )  =  ( r  +  ( _i  x.  s ) )  /\  ( 2nd `  <. A ,  B >. )  =  ( t  +  ( _i  x.  u ) ) )  ->  ( A  e.  CC  /\  B  e.  CC ) ) )
5958rexlimdvva 2555 . 2  |-  ( A #  B  ->  ( E. r  e.  RR  E. s  e.  RR  E. t  e.  RR  E. u  e.  RR  ( ( 1st `  <. A ,  B >. )  =  ( r  +  ( _i  x.  s ) )  /\  ( 2nd `  <. A ,  B >. )  =  ( t  +  ( _i  x.  u ) ) )  ->  ( A  e.  CC  /\  B  e.  CC ) ) )
6021, 59mpd 13 1  |-  ( A #  B  ->  ( A  e.  CC  /\  B  e.  CC ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ wo 697    = wceq 1331    e. wcel 1480   E.wrex 2415   _Vcvv 2681   <.cop 3525   class class class wbr 3924   {copab 3983   ` cfv 5118  (class class class)co 5767   1stc1st 6029   2ndc2nd 6030   CCcc 7611   RRcr 7612   _ici 7615    + caddc 7616    x. cmul 7618   # creap 8329   # cap 8336
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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-resscn 7705  ax-icn 7708  ax-addcl 7709  ax-mulcl 7711
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-fo 5124  df-fv 5126  df-1st 6031  df-2nd 6032  df-ap 8337
This theorem is referenced by:  apsscn  8402
  Copyright terms: Public domain W3C validator