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Theorem apreap 7652
Description: Complex apartness and real apartness agree on the real numbers. (Contributed by Jim Kingdon, 31-Jan-2020.)
Assertion
Ref Expression
apreap  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A #  B  <->  A #  B )
)

Proof of Theorem apreap
Dummy variables  r  s  t  u  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2062 . . . . . . . 8  |-  ( x  =  A  ->  (
x  =  ( r  +  ( _i  x.  s ) )  <->  A  =  ( r  +  ( _i  x.  s ) ) ) )
21anbi1d 446 . . . . . . 7  |-  ( x  =  A  ->  (
( x  =  ( r  +  ( _i  x.  s ) )  /\  y  =  ( t  +  ( _i  x.  u ) ) )  <->  ( A  =  ( r  +  ( _i  x.  s ) )  /\  y  =  ( t  +  ( _i  x.  u ) ) ) ) )
32anbi1d 446 . . . . . 6  |-  ( x  =  A  ->  (
( ( x  =  ( r  +  ( _i  x.  s ) )  /\  y  =  ( t  +  ( _i  x.  u ) ) )  /\  (
r #  t  \/  s #  u ) )  <->  ( ( A  =  ( r  +  ( _i  x.  s
) )  /\  y  =  ( t  +  ( _i  x.  u
) ) )  /\  ( r #  t  \/  s #  u
) ) ) )
432rexbidv 2366 . . . . 5  |-  ( x  =  A  ->  ( 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  =  ( r  +  ( _i  x.  s
) )  /\  y  =  ( t  +  ( _i  x.  u
) ) )  /\  ( r #  t  \/  s #  u
) ) ) )
542rexbidv 2366 . . . 4  |-  ( x  =  A  ->  ( 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  =  ( r  +  ( _i  x.  s
) )  /\  y  =  ( t  +  ( _i  x.  u
) ) )  /\  ( r #  t  \/  s #  u
) ) ) )
6 eqeq1 2062 . . . . . . . 8  |-  ( y  =  B  ->  (
y  =  ( t  +  ( _i  x.  u ) )  <->  B  =  ( t  +  ( _i  x.  u ) ) ) )
76anbi2d 445 . . . . . . 7  |-  ( y  =  B  ->  (
( A  =  ( r  +  ( _i  x.  s ) )  /\  y  =  ( t  +  ( _i  x.  u ) ) )  <->  ( A  =  ( r  +  ( _i  x.  s ) )  /\  B  =  ( t  +  ( _i  x.  u ) ) ) ) )
87anbi1d 446 . . . . . 6  |-  ( y  =  B  ->  (
( ( A  =  ( r  +  ( _i  x.  s ) )  /\  y  =  ( t  +  ( _i  x.  u ) ) )  /\  (
r #  t  \/  s #  u ) )  <->  ( ( A  =  ( r  +  ( _i  x.  s
) )  /\  B  =  ( t  +  ( _i  x.  u
) ) )  /\  ( r #  t  \/  s #  u
) ) ) )
982rexbidv 2366 . . . . 5  |-  ( y  =  B  ->  ( E. t  e.  RR  E. u  e.  RR  (
( A  =  ( r  +  ( _i  x.  s ) )  /\  y  =  ( t  +  ( _i  x.  u ) ) )  /\  ( r #  t  \/  s #  u ) )  <->  E. t  e.  RR  E. u  e.  RR  ( ( A  =  ( r  +  ( _i  x.  s
) )  /\  B  =  ( t  +  ( _i  x.  u
) ) )  /\  ( r #  t  \/  s #  u
) ) ) )
1092rexbidv 2366 . . . 4  |-  ( y  =  B  ->  ( E. r  e.  RR  E. s  e.  RR  E. t  e.  RR  E. u  e.  RR  ( ( A  =  ( 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  =  ( r  +  ( _i  x.  s
) )  /\  B  =  ( t  +  ( _i  x.  u
) ) )  /\  ( r #  t  \/  s #  u
) ) ) )
11 df-ap 7647 . . . 4  |- #  =  { <. 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
) ) }
125, 10, 11brabg 4034 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A #  B  <->  E. r  e.  RR  E. s  e.  RR  E. t  e.  RR  E. u  e.  RR  ( ( A  =  ( r  +  ( _i  x.  s
) )  /\  B  =  ( t  +  ( _i  x.  u
) ) )  /\  ( r #  t  \/  s #  u
) ) ) )
13 simplll 493 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
r  e.  RR  /\  s  e.  RR )
)  /\  ( t  e.  RR  /\  u  e.  RR ) )  ->  A  e.  RR )
1413adantr 265 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
r  e.  RR  /\  s  e.  RR )
)  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( A  =  ( r  +  ( _i  x.  s ) )  /\  B  =  ( t  +  ( _i  x.  u ) ) )  /\  (
r #  t  \/  s #  u ) ) )  ->  A  e.  RR )
15 simplrl 495 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
r  e.  RR  /\  s  e.  RR )
)  /\  ( t  e.  RR  /\  u  e.  RR ) )  -> 
r  e.  RR )
1615adantr 265 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
r  e.  RR  /\  s  e.  RR )
)  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( A  =  ( r  +  ( _i  x.  s ) )  /\  B  =  ( t  +  ( _i  x.  u ) ) )  /\  (
r #  t  \/  s #  u ) ) )  ->  r  e.  RR )
17 simplrr 496 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
r  e.  RR  /\  s  e.  RR )
)  /\  ( t  e.  RR  /\  u  e.  RR ) )  -> 
s  e.  RR )
1817adantr 265 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
r  e.  RR  /\  s  e.  RR )
)  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( A  =  ( r  +  ( _i  x.  s ) )  /\  B  =  ( t  +  ( _i  x.  u ) ) )  /\  (
r #  t  \/  s #  u ) ) )  ->  s  e.  RR )
19 simprll 497 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
r  e.  RR  /\  s  e.  RR )
)  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( A  =  ( r  +  ( _i  x.  s ) )  /\  B  =  ( t  +  ( _i  x.  u ) ) )  /\  (
r #  t  \/  s #  u ) ) )  ->  A  =  ( r  +  ( _i  x.  s
) ) )
20 rereim 7651 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  r  e.  RR )  /\  ( s  e.  RR  /\  A  =  ( r  +  ( _i  x.  s ) ) ) )  -> 
( r  =  A  /\  s  =  0 ) )
2114, 16, 18, 19, 20syl22anc 1147 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
r  e.  RR  /\  s  e.  RR )
)  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( A  =  ( r  +  ( _i  x.  s ) )  /\  B  =  ( t  +  ( _i  x.  u ) ) )  /\  (
r #  t  \/  s #  u ) ) )  ->  (
r  =  A  /\  s  =  0 ) )
2221simprd 111 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
r  e.  RR  /\  s  e.  RR )
)  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( A  =  ( r  +  ( _i  x.  s ) )  /\  B  =  ( t  +  ( _i  x.  u ) ) )  /\  (
r #  t  \/  s #  u ) ) )  ->  s  =  0 )
23 simpllr 494 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
r  e.  RR  /\  s  e.  RR )
)  /\  ( t  e.  RR  /\  u  e.  RR ) )  ->  B  e.  RR )
2423adantr 265 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
r  e.  RR  /\  s  e.  RR )
)  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( A  =  ( r  +  ( _i  x.  s ) )  /\  B  =  ( t  +  ( _i  x.  u ) ) )  /\  (
r #  t  \/  s #  u ) ) )  ->  B  e.  RR )
25 simplrl 495 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
r  e.  RR  /\  s  e.  RR )
)  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( A  =  ( r  +  ( _i  x.  s ) )  /\  B  =  ( t  +  ( _i  x.  u ) ) )  /\  (
r #  t  \/  s #  u ) ) )  ->  t  e.  RR )
26 simplrr 496 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
r  e.  RR  /\  s  e.  RR )
)  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( A  =  ( r  +  ( _i  x.  s ) )  /\  B  =  ( t  +  ( _i  x.  u ) ) )  /\  (
r #  t  \/  s #  u ) ) )  ->  u  e.  RR )
27 simprlr 498 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
r  e.  RR  /\  s  e.  RR )
)  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( A  =  ( r  +  ( _i  x.  s ) )  /\  B  =  ( t  +  ( _i  x.  u ) ) )  /\  (
r #  t  \/  s #  u ) ) )  ->  B  =  ( t  +  ( _i  x.  u
) ) )
28 rereim 7651 . . . . . . . . . . . 12  |-  ( ( ( B  e.  RR  /\  t  e.  RR )  /\  ( u  e.  RR  /\  B  =  ( t  +  ( _i  x.  u ) ) ) )  -> 
( t  =  B  /\  u  =  0 ) )
2924, 25, 26, 27, 28syl22anc 1147 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
r  e.  RR  /\  s  e.  RR )
)  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( A  =  ( r  +  ( _i  x.  s ) )  /\  B  =  ( t  +  ( _i  x.  u ) ) )  /\  (
r #  t  \/  s #  u ) ) )  ->  (
t  =  B  /\  u  =  0 ) )
3029simprd 111 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
r  e.  RR  /\  s  e.  RR )
)  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( A  =  ( r  +  ( _i  x.  s ) )  /\  B  =  ( t  +  ( _i  x.  u ) ) )  /\  (
r #  t  \/  s #  u ) ) )  ->  u  =  0 )
3122, 30eqtr4d 2091 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
r  e.  RR  /\  s  e.  RR )
)  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( A  =  ( r  +  ( _i  x.  s ) )  /\  B  =  ( t  +  ( _i  x.  u ) ) )  /\  (
r #  t  \/  s #  u ) ) )  ->  s  =  u )
32 reapti 7644 . . . . . . . . . 10  |-  ( ( s  e.  RR  /\  u  e.  RR )  ->  ( s  =  u  <->  -.  s #  u ) )
3318, 26, 32syl2anc 397 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
r  e.  RR  /\  s  e.  RR )
)  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( A  =  ( r  +  ( _i  x.  s ) )  /\  B  =  ( t  +  ( _i  x.  u ) ) )  /\  (
r #  t  \/  s #  u ) ) )  ->  (
s  =  u  <->  -.  s #  u
) )
3431, 33mpbid 139 . . . . . . . 8  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
r  e.  RR  /\  s  e.  RR )
)  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( A  =  ( r  +  ( _i  x.  s ) )  /\  B  =  ( t  +  ( _i  x.  u ) ) )  /\  (
r #  t  \/  s #  u ) ) )  ->  -.  s #  u )
35 simprr 492 . . . . . . . 8  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
r  e.  RR  /\  s  e.  RR )
)  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( A  =  ( r  +  ( _i  x.  s ) )  /\  B  =  ( t  +  ( _i  x.  u ) ) )  /\  (
r #  t  \/  s #  u ) ) )  ->  (
r #  t  \/  s #  u ) )
3634, 35ecased 1255 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
r  e.  RR  /\  s  e.  RR )
)  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( A  =  ( r  +  ( _i  x.  s ) )  /\  B  =  ( t  +  ( _i  x.  u ) ) )  /\  (
r #  t  \/  s #  u ) ) )  ->  r #  t
)
3721simpld 109 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
r  e.  RR  /\  s  e.  RR )
)  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( A  =  ( r  +  ( _i  x.  s ) )  /\  B  =  ( t  +  ( _i  x.  u ) ) )  /\  (
r #  t  \/  s #  u ) ) )  ->  r  =  A )
3829simpld 109 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
r  e.  RR  /\  s  e.  RR )
)  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( A  =  ( r  +  ( _i  x.  s ) )  /\  B  =  ( t  +  ( _i  x.  u ) ) )  /\  (
r #  t  \/  s #  u ) ) )  ->  t  =  B )
3936, 37, 383brtr3d 3821 . . . . . 6  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
r  e.  RR  /\  s  e.  RR )
)  /\  ( t  e.  RR  /\  u  e.  RR ) )  /\  ( ( A  =  ( r  +  ( _i  x.  s ) )  /\  B  =  ( t  +  ( _i  x.  u ) ) )  /\  (
r #  t  \/  s #  u ) ) )  ->  A #  B
)
4039ex 112 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
r  e.  RR  /\  s  e.  RR )
)  /\  ( t  e.  RR  /\  u  e.  RR ) )  -> 
( ( ( A  =  ( r  +  ( _i  x.  s
) )  /\  B  =  ( t  +  ( _i  x.  u
) ) )  /\  ( r #  t  \/  s #  u
) )  ->  A #  B
) )
4140rexlimdvva 2457 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( r  e.  RR  /\  s  e.  RR ) )  -> 
( E. t  e.  RR  E. u  e.  RR  ( ( A  =  ( r  +  ( _i  x.  s
) )  /\  B  =  ( t  +  ( _i  x.  u
) ) )  /\  ( r #  t  \/  s #  u
) )  ->  A #  B
) )
4241rexlimdvva 2457 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( E. r  e.  RR  E. s  e.  RR  E. t  e.  RR  E. u  e.  RR  ( ( A  =  ( r  +  ( _i  x.  s
) )  /\  B  =  ( t  +  ( _i  x.  u
) ) )  /\  ( r #  t  \/  s #  u
) )  ->  A #  B
) )
4312, 42sylbid 143 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A #  B  ->  A #  B ) )
44 ax-icn 7037 . . . . . . . 8  |-  _i  e.  CC
4544mul01i 7460 . . . . . . 7  |-  ( _i  x.  0 )  =  0
4645oveq2i 5551 . . . . . 6  |-  ( A  +  ( _i  x.  0 ) )  =  ( A  +  0 )
47 simp1 915 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B
)  ->  A  e.  RR )
4847recnd 7113 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B
)  ->  A  e.  CC )
4948addid1d 7223 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B
)  ->  ( A  +  0 )  =  A )
5046, 49syl5req 2101 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B
)  ->  A  =  ( A  +  (
_i  x.  0 ) ) )
5145oveq2i 5551 . . . . . 6  |-  ( B  +  ( _i  x.  0 ) )  =  ( B  +  0 )
52 simp2 916 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B
)  ->  B  e.  RR )
5352recnd 7113 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B
)  ->  B  e.  CC )
5453addid1d 7223 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B
)  ->  ( B  +  0 )  =  B )
5551, 54syl5req 2101 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B
)  ->  B  =  ( B  +  (
_i  x.  0 ) ) )
56 olc 642 . . . . . . 7  |-  ( A #  B  ->  ( 0 #  0  \/  A #  B ) )
57563ad2ant3 938 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B
)  ->  ( 0 #  0  \/  A #  B ) )
5857orcomd 658 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B
)  ->  ( A #  B  \/  0 #  0 ) )
5950, 55, 58jca31 296 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B
)  ->  ( ( A  =  ( A  +  ( _i  x.  0 ) )  /\  B  =  ( B  +  ( _i  x.  0 ) ) )  /\  ( A #  B  \/  0 #  0 ) ) )
60 0red 7086 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B
)  ->  0  e.  RR )
61 simpr 107 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  u  =  0 )  ->  u  =  0 )
6261oveq2d 5556 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  u  =  0 )  -> 
( _i  x.  u
)  =  ( _i  x.  0 ) )
6362oveq2d 5556 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  u  =  0 )  -> 
( B  +  ( _i  x.  u ) )  =  ( B  +  ( _i  x.  0 ) ) )
6463eqeq2d 2067 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  u  =  0 )  -> 
( B  =  ( B  +  ( _i  x.  u ) )  <-> 
B  =  ( B  +  ( _i  x.  0 ) ) ) )
6564anbi2d 445 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  u  =  0 )  -> 
( ( A  =  ( A  +  ( _i  x.  0 ) )  /\  B  =  ( B  +  ( _i  x.  u ) ) )  <->  ( A  =  ( A  +  ( _i  x.  0
) )  /\  B  =  ( B  +  ( _i  x.  0
) ) ) ) )
6661breq2d 3804 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  u  =  0 )  -> 
( 0 #  u  <->  0 #  0 ) )
6766orbi2d 714 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  u  =  0 )  -> 
( ( A #  B  \/  0 #  u )  <->  ( A #  B  \/  0 #  0 ) ) )
6865, 67anbi12d 450 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  u  =  0 )  -> 
( ( ( A  =  ( A  +  ( _i  x.  0
) )  /\  B  =  ( B  +  ( _i  x.  u
) ) )  /\  ( A #  B  \/  0 #  u
) )  <->  ( ( A  =  ( A  +  ( _i  x.  0 ) )  /\  B  =  ( B  +  ( _i  x.  0 ) ) )  /\  ( A #  B  \/  0 #  0 ) ) ) )
6960, 68rspcedv 2677 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B
)  ->  ( (
( A  =  ( A  +  ( _i  x.  0 ) )  /\  B  =  ( B  +  ( _i  x.  0 ) ) )  /\  ( A #  B  \/  0 #  0 ) )  ->  E. u  e.  RR  ( ( A  =  ( A  +  ( _i  x.  0 ) )  /\  B  =  ( B  +  ( _i  x.  u ) ) )  /\  ( A #  B  \/  0 #  u ) ) ) )
70 simpr 107 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  t  =  B )  ->  t  =  B )
7170oveq1d 5555 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  t  =  B )  ->  (
t  +  ( _i  x.  u ) )  =  ( B  +  ( _i  x.  u
) ) )
7271eqeq2d 2067 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  t  =  B )  ->  ( B  =  ( t  +  ( _i  x.  u ) )  <->  B  =  ( B  +  (
_i  x.  u )
) ) )
7372anbi2d 445 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  t  =  B )  ->  (
( A  =  ( A  +  ( _i  x.  0 ) )  /\  B  =  ( t  +  ( _i  x.  u ) ) )  <->  ( A  =  ( A  +  ( _i  x.  0 ) )  /\  B  =  ( B  +  ( _i  x.  u ) ) ) ) )
7470breq2d 3804 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  t  =  B )  ->  ( A #  t 
<->  A #  B ) )
7574orbi1d 715 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  t  =  B )  ->  (
( A #  t  \/  0 #  u
)  <->  ( A #  B  \/  0 #  u ) ) )
7673, 75anbi12d 450 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  t  =  B )  ->  (
( ( A  =  ( A  +  ( _i  x.  0 ) )  /\  B  =  ( t  +  ( _i  x.  u ) ) )  /\  ( A #  t  \/  0 #  u ) )  <->  ( ( A  =  ( A  +  ( _i  x.  0
) )  /\  B  =  ( B  +  ( _i  x.  u
) ) )  /\  ( A #  B  \/  0 #  u
) ) ) )
7776rexbidv 2344 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  t  =  B )  ->  ( E. u  e.  RR  ( ( A  =  ( A  +  ( _i  x.  0 ) )  /\  B  =  ( t  +  ( _i  x.  u ) ) )  /\  ( A #  t  \/  0 #  u ) )  <->  E. u  e.  RR  ( ( A  =  ( A  +  ( _i  x.  0 ) )  /\  B  =  ( B  +  ( _i  x.  u ) ) )  /\  ( A #  B  \/  0 #  u ) ) ) )
7852, 77rspcedv 2677 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B
)  ->  ( E. u  e.  RR  (
( A  =  ( A  +  ( _i  x.  0 ) )  /\  B  =  ( B  +  ( _i  x.  u ) ) )  /\  ( A #  B  \/  0 #  u ) )  ->  E. t  e.  RR  E. u  e.  RR  (
( A  =  ( A  +  ( _i  x.  0 ) )  /\  B  =  ( t  +  ( _i  x.  u ) ) )  /\  ( A #  t  \/  0 #  u ) ) ) )
7969, 78syld 44 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B
)  ->  ( (
( A  =  ( A  +  ( _i  x.  0 ) )  /\  B  =  ( B  +  ( _i  x.  0 ) ) )  /\  ( A #  B  \/  0 #  0 ) )  ->  E. t  e.  RR  E. u  e.  RR  (
( A  =  ( A  +  ( _i  x.  0 ) )  /\  B  =  ( t  +  ( _i  x.  u ) ) )  /\  ( A #  t  \/  0 #  u ) ) ) )
80 simpr 107 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  s  =  0 )  -> 
s  =  0 )
8180oveq2d 5556 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  s  =  0 )  -> 
( _i  x.  s
)  =  ( _i  x.  0 ) )
8281oveq2d 5556 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  s  =  0 )  -> 
( A  +  ( _i  x.  s ) )  =  ( A  +  ( _i  x.  0 ) ) )
8382eqeq2d 2067 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  s  =  0 )  -> 
( A  =  ( A  +  ( _i  x.  s ) )  <-> 
A  =  ( A  +  ( _i  x.  0 ) ) ) )
8483anbi1d 446 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  s  =  0 )  -> 
( ( A  =  ( A  +  ( _i  x.  s ) )  /\  B  =  ( t  +  ( _i  x.  u ) ) )  <->  ( A  =  ( A  +  ( _i  x.  0
) )  /\  B  =  ( t  +  ( _i  x.  u
) ) ) ) )
8580breq1d 3802 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  s  =  0 )  -> 
( s #  u  <->  0 #  u ) )
8685orbi2d 714 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  s  =  0 )  -> 
( ( A #  t  \/  s #  u )  <->  ( A #  t  \/  0 #  u ) ) )
8784, 86anbi12d 450 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  s  =  0 )  -> 
( ( ( A  =  ( A  +  ( _i  x.  s
) )  /\  B  =  ( t  +  ( _i  x.  u
) ) )  /\  ( A #  t  \/  s #  u
) )  <->  ( ( A  =  ( A  +  ( _i  x.  0 ) )  /\  B  =  ( t  +  ( _i  x.  u ) ) )  /\  ( A #  t  \/  0 #  u ) ) ) )
88872rexbidv 2366 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  s  =  0 )  -> 
( E. t  e.  RR  E. u  e.  RR  ( ( A  =  ( A  +  ( _i  x.  s
) )  /\  B  =  ( t  +  ( _i  x.  u
) ) )  /\  ( A #  t  \/  s #  u
) )  <->  E. t  e.  RR  E. u  e.  RR  ( ( A  =  ( A  +  ( _i  x.  0
) )  /\  B  =  ( t  +  ( _i  x.  u
) ) )  /\  ( A #  t  \/  0 #  u
) ) ) )
8960, 88rspcedv 2677 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B
)  ->  ( E. t  e.  RR  E. u  e.  RR  ( ( A  =  ( A  +  ( _i  x.  0
) )  /\  B  =  ( t  +  ( _i  x.  u
) ) )  /\  ( A #  t  \/  0 #  u
) )  ->  E. s  e.  RR  E. t  e.  RR  E. u  e.  RR  ( ( A  =  ( A  +  ( _i  x.  s
) )  /\  B  =  ( t  +  ( _i  x.  u
) ) )  /\  ( A #  t  \/  s #  u
) ) ) )
90 simpr 107 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  r  =  A )  ->  r  =  A )
9190oveq1d 5555 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  r  =  A )  ->  (
r  +  ( _i  x.  s ) )  =  ( A  +  ( _i  x.  s
) ) )
9291eqeq2d 2067 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  r  =  A )  ->  ( A  =  ( r  +  ( _i  x.  s ) )  <->  A  =  ( A  +  (
_i  x.  s )
) ) )
9392anbi1d 446 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  r  =  A )  ->  (
( A  =  ( r  +  ( _i  x.  s ) )  /\  B  =  ( t  +  ( _i  x.  u ) ) )  <->  ( A  =  ( A  +  ( _i  x.  s ) )  /\  B  =  ( t  +  ( _i  x.  u ) ) ) ) )
9490breq1d 3802 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  r  =  A )  ->  (
r #  t  <->  A #  t ) )
9594orbi1d 715 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  r  =  A )  ->  (
( r #  t  \/  s #  u
)  <->  ( A #  t  \/  s #  u ) ) )
9693, 95anbi12d 450 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  r  =  A )  ->  (
( ( A  =  ( r  +  ( _i  x.  s ) )  /\  B  =  ( t  +  ( _i  x.  u ) ) )  /\  (
r #  t  \/  s #  u ) )  <->  ( ( A  =  ( A  +  ( _i  x.  s
) )  /\  B  =  ( t  +  ( _i  x.  u
) ) )  /\  ( A #  t  \/  s #  u
) ) ) )
9796rexbidv 2344 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  r  =  A )  ->  ( E. u  e.  RR  ( ( A  =  ( r  +  ( _i  x.  s ) )  /\  B  =  ( t  +  ( _i  x.  u ) ) )  /\  (
r #  t  \/  s #  u ) )  <->  E. u  e.  RR  ( ( A  =  ( A  +  ( _i  x.  s ) )  /\  B  =  ( t  +  ( _i  x.  u ) ) )  /\  ( A #  t  \/  s #  u )
) ) )
98972rexbidv 2366 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  r  =  A )  ->  ( E. s  e.  RR  E. t  e.  RR  E. u  e.  RR  (
( A  =  ( r  +  ( _i  x.  s ) )  /\  B  =  ( t  +  ( _i  x.  u ) ) )  /\  ( r #  t  \/  s #  u ) )  <->  E. s  e.  RR  E. t  e.  RR  E. u  e.  RR  ( ( A  =  ( A  +  ( _i  x.  s
) )  /\  B  =  ( t  +  ( _i  x.  u
) ) )  /\  ( A #  t  \/  s #  u
) ) ) )
9947, 98rspcedv 2677 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B
)  ->  ( E. s  e.  RR  E. t  e.  RR  E. u  e.  RR  ( ( A  =  ( A  +  ( _i  x.  s
) )  /\  B  =  ( t  +  ( _i  x.  u
) ) )  /\  ( A #  t  \/  s #  u
) )  ->  E. r  e.  RR  E. s  e.  RR  E. t  e.  RR  E. u  e.  RR  ( ( A  =  ( r  +  ( _i  x.  s
) )  /\  B  =  ( t  +  ( _i  x.  u
) ) )  /\  ( r #  t  \/  s #  u
) ) ) )
10079, 89, 993syld 55 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B
)  ->  ( (
( A  =  ( A  +  ( _i  x.  0 ) )  /\  B  =  ( B  +  ( _i  x.  0 ) ) )  /\  ( A #  B  \/  0 #  0 ) )  ->  E. r  e.  RR  E. s  e.  RR  E. t  e.  RR  E. u  e.  RR  ( ( A  =  ( r  +  ( _i  x.  s
) )  /\  B  =  ( t  +  ( _i  x.  u
) ) )  /\  ( r #  t  \/  s #  u
) ) ) )
101123adant3 935 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B
)  ->  ( A #  B 
<->  E. r  e.  RR  E. s  e.  RR  E. t  e.  RR  E. u  e.  RR  ( ( A  =  ( r  +  ( _i  x.  s
) )  /\  B  =  ( t  +  ( _i  x.  u
) ) )  /\  ( r #  t  \/  s #  u
) ) ) )
102100, 101sylibrd 162 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B
)  ->  ( (
( A  =  ( A  +  ( _i  x.  0 ) )  /\  B  =  ( B  +  ( _i  x.  0 ) ) )  /\  ( A #  B  \/  0 #  0 ) )  ->  A #  B ) )
10359, 102mpd 13 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B
)  ->  A #  B
)
1041033expia 1117 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A #  B  ->  A #  B
) )
10543, 104impbid 124 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A #  B  <->  A #  B )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 101    <-> wb 102    \/ wo 639    /\ w3a 896    = wceq 1259    e. wcel 1409   E.wrex 2324   class class class wbr 3792  (class class class)co 5540   RRcr 6946   0cc0 6947   _ici 6949    + caddc 6950    x. cmul 6952   # creap 7639   # cap 7646
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339  ax-cnex 7033  ax-resscn 7034  ax-1cn 7035  ax-1re 7036  ax-icn 7037  ax-addcl 7038  ax-addrcl 7039  ax-mulcl 7040  ax-mulrcl 7041  ax-addcom 7042  ax-mulcom 7043  ax-addass 7044  ax-mulass 7045  ax-distr 7046  ax-i2m1 7047  ax-1rid 7049  ax-0id 7050  ax-rnegex 7051  ax-precex 7052  ax-cnre 7053  ax-pre-ltirr 7054  ax-pre-lttrn 7056  ax-pre-apti 7057  ax-pre-ltadd 7058  ax-pre-mulgt0 7059
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-riota 5496  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-1o 6032  df-2o 6033  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-enq0 6580  df-nq0 6581  df-0nq0 6582  df-plq0 6583  df-mq0 6584  df-inp 6622  df-i1p 6623  df-iplp 6624  df-iltp 6626  df-enr 6869  df-nr 6870  df-ltr 6873  df-0r 6874  df-1r 6875  df-0 6954  df-1 6955  df-r 6957  df-lt 6960  df-pnf 7121  df-mnf 7122  df-ltxr 7124  df-sub 7247  df-neg 7248  df-reap 7640  df-ap 7647
This theorem is referenced by:  reaplt  7653  apreim  7668  apirr  7670  apti  7687  recexap  7708  rerecclap  7781
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