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Theorem apreap 8316
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 2124 . . . . . . . 8  |-  ( x  =  A  ->  (
x  =  ( r  +  ( _i  x.  s ) )  <->  A  =  ( r  +  ( _i  x.  s ) ) ) )
21anbi1d 460 . . . . . . 7  |-  ( x  =  A  ->  (
( x  =  ( r  +  ( _i  x.  s ) )  /\  y  =  ( t  +  ( _i  x.  u ) ) )  <->  ( A  =  ( r  +  ( _i  x.  s ) )  /\  y  =  ( t  +  ( _i  x.  u ) ) ) ) )
32anbi1d 460 . . . . . 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 2437 . . . . 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 2437 . . . 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 2124 . . . . . . . 8  |-  ( y  =  B  ->  (
y  =  ( t  +  ( _i  x.  u ) )  <->  B  =  ( t  +  ( _i  x.  u ) ) ) )
76anbi2d 459 . . . . . . 7  |-  ( y  =  B  ->  (
( A  =  ( r  +  ( _i  x.  s ) )  /\  y  =  ( t  +  ( _i  x.  u ) ) )  <->  ( A  =  ( r  +  ( _i  x.  s ) )  /\  B  =  ( t  +  ( _i  x.  u ) ) ) ) )
87anbi1d 460 . . . . . 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 2437 . . . . 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 2437 . . . 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 8311 . . . 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 4161 . . 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 507 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
r  e.  RR  /\  s  e.  RR )
)  /\  ( t  e.  RR  /\  u  e.  RR ) )  ->  A  e.  RR )
1413adantr 274 . . . . . . . . . . . 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 509 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
r  e.  RR  /\  s  e.  RR )
)  /\  ( t  e.  RR  /\  u  e.  RR ) )  -> 
r  e.  RR )
1615adantr 274 . . . . . . . . . . . 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 510 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
r  e.  RR  /\  s  e.  RR )
)  /\  ( t  e.  RR  /\  u  e.  RR ) )  -> 
s  e.  RR )
1817adantr 274 . . . . . . . . . . . 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 511 . . . . . . . . . . . 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 8315 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  r  e.  RR )  /\  ( s  e.  RR  /\  A  =  ( r  +  ( _i  x.  s ) ) ) )  -> 
( r  =  A  /\  s  =  0 ) )
2114, 16, 18, 19, 20syl22anc 1202 . . . . . . . . . . 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 113 . . . . . . . . . 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 508 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
r  e.  RR  /\  s  e.  RR )
)  /\  ( t  e.  RR  /\  u  e.  RR ) )  ->  B  e.  RR )
2423adantr 274 . . . . . . . . . . . 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 509 . . . . . . . . . . . 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 510 . . . . . . . . . . . 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 512 . . . . . . . . . . . 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 8315 . . . . . . . . . . . 12  |-  ( ( ( B  e.  RR  /\  t  e.  RR )  /\  ( u  e.  RR  /\  B  =  ( t  +  ( _i  x.  u ) ) ) )  -> 
( t  =  B  /\  u  =  0 ) )
2924, 25, 26, 27, 28syl22anc 1202 . . . . . . . . . . 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 113 . . . . . . . . . 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 2153 . . . . . . . . 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 8308 . . . . . . . . . 10  |-  ( ( s  e.  RR  /\  u  e.  RR )  ->  ( s  =  u  <->  -.  s #  u ) )
3318, 26, 32syl2anc 408 . . . . . . . . 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 146 . . . . . . . 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 506 . . . . . . . 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 1312 . . . . . . 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 111 . . . . . . 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 111 . . . . . . 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 3929 . . . . . 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 114 . . . . 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 2534 . . . 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 2534 . . 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 149 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A #  B  ->  A #  B ) )
44 ax-icn 7683 . . . . . . . 8  |-  _i  e.  CC
4544mul01i 8121 . . . . . . 7  |-  ( _i  x.  0 )  =  0
4645oveq2i 5753 . . . . . 6  |-  ( A  +  ( _i  x.  0 ) )  =  ( A  +  0 )
47 simp1 966 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B
)  ->  A  e.  RR )
4847recnd 7762 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B
)  ->  A  e.  CC )
4948addid1d 7879 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B
)  ->  ( A  +  0 )  =  A )
5046, 49syl5req 2163 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B
)  ->  A  =  ( A  +  (
_i  x.  0 ) ) )
5145oveq2i 5753 . . . . . 6  |-  ( B  +  ( _i  x.  0 ) )  =  ( B  +  0 )
52 simp2 967 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B
)  ->  B  e.  RR )
5352recnd 7762 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B
)  ->  B  e.  CC )
5453addid1d 7879 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B
)  ->  ( B  +  0 )  =  B )
5551, 54syl5req 2163 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B
)  ->  B  =  ( B  +  (
_i  x.  0 ) ) )
56 olc 685 . . . . . . 7  |-  ( A #  B  ->  ( 0 #  0  \/  A #  B ) )
57563ad2ant3 989 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B
)  ->  ( 0 #  0  \/  A #  B ) )
5857orcomd 703 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B
)  ->  ( A #  B  \/  0 #  0 ) )
5950, 55, 58jca31 307 . . . 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 7735 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B
)  ->  0  e.  RR )
61 simpr 109 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  u  =  0 )  ->  u  =  0 )
6261oveq2d 5758 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  u  =  0 )  -> 
( _i  x.  u
)  =  ( _i  x.  0 ) )
6362oveq2d 5758 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  u  =  0 )  -> 
( B  +  ( _i  x.  u ) )  =  ( B  +  ( _i  x.  0 ) ) )
6463eqeq2d 2129 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  u  =  0 )  -> 
( B  =  ( B  +  ( _i  x.  u ) )  <-> 
B  =  ( B  +  ( _i  x.  0 ) ) ) )
6564anbi2d 459 . . . . . . . . 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 3911 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  u  =  0 )  -> 
( 0 #  u  <->  0 #  0 ) )
6766orbi2d 764 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  u  =  0 )  -> 
( ( A #  B  \/  0 #  u )  <->  ( A #  B  \/  0 #  0 ) ) )
6865, 67anbi12d 464 . . . . . . . 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 2767 . . . . . . 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 109 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  t  =  B )  ->  t  =  B )
7170oveq1d 5757 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  t  =  B )  ->  (
t  +  ( _i  x.  u ) )  =  ( B  +  ( _i  x.  u
) ) )
7271eqeq2d 2129 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  t  =  B )  ->  ( B  =  ( t  +  ( _i  x.  u ) )  <->  B  =  ( B  +  (
_i  x.  u )
) ) )
7372anbi2d 459 . . . . . . . . . 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 3911 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  t  =  B )  ->  ( A #  t 
<->  A #  B ) )
7574orbi1d 765 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  t  =  B )  ->  (
( A #  t  \/  0 #  u
)  <->  ( A #  B  \/  0 #  u ) ) )
7673, 75anbi12d 464 . . . . . . . . 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 2415 . . . . . . . 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 2767 . . . . . . 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 45 . . . . . 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 109 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  s  =  0 )  -> 
s  =  0 )
8180oveq2d 5758 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  s  =  0 )  -> 
( _i  x.  s
)  =  ( _i  x.  0 ) )
8281oveq2d 5758 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  s  =  0 )  -> 
( A  +  ( _i  x.  s ) )  =  ( A  +  ( _i  x.  0 ) ) )
8382eqeq2d 2129 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  s  =  0 )  -> 
( A  =  ( A  +  ( _i  x.  s ) )  <-> 
A  =  ( A  +  ( _i  x.  0 ) ) ) )
8483anbi1d 460 . . . . . . . . 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 3909 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  s  =  0 )  -> 
( s #  u  <->  0 #  u ) )
8685orbi2d 764 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  s  =  0 )  -> 
( ( A #  t  \/  s #  u )  <->  ( A #  t  \/  0 #  u ) ) )
8784, 86anbi12d 464 . . . . . . . 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 2437 . . . . . . 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 2767 . . . . . 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 109 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  r  =  A )  ->  r  =  A )
9190oveq1d 5757 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  r  =  A )  ->  (
r  +  ( _i  x.  s ) )  =  ( A  +  ( _i  x.  s
) ) )
9291eqeq2d 2129 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  r  =  A )  ->  ( A  =  ( r  +  ( _i  x.  s ) )  <->  A  =  ( A  +  (
_i  x.  s )
) ) )
9392anbi1d 460 . . . . . . . . . 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 3909 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  r  =  A )  ->  (
r #  t  <->  A #  t ) )
9594orbi1d 765 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A #  B )  /\  r  =  A )  ->  (
( r #  t  \/  s #  u
)  <->  ( A #  t  \/  s #  u ) ) )
9693, 95anbi12d 464 . . . . . . . . 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 2415 . . . . . . . 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 2437 . . . . . . 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 2767 . . . . . 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 57 . . . . 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 986 . . . . 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 168 . . . 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 1168 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A #  B  ->  A #  B
) )
10543, 104impbid 128 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A #  B  <->  A #  B )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 682    /\ w3a 947    = wceq 1316    e. wcel 1465   E.wrex 2394   class class class wbr 3899  (class class class)co 5742   RRcr 7587   0cc0 7588   _ici 7590    + caddc 7591    x. cmul 7593   # creap 8303   # cap 8310
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-pnf 7770  df-mnf 7771  df-ltxr 7773  df-sub 7903  df-neg 7904  df-reap 8304  df-ap 8311
This theorem is referenced by:  reaplt  8317  apreim  8332  apirr  8334  apti  8351  recexap  8381  rerecclap  8457
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