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Theorem apreap 8546
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 #ℝ 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 2184 . . . . . . . 8 |- ( x = A -> ( x = ( r + ( _i x. s ) ) <-> A = ( r + ( _i x. s ) ) ) )
21anbi1d 465 . . . . . . 7 |- ( x = A -> ( ( x = ( r + ( _i x. s ) ) /\ y = ( t + ( _i x. u ) ) ) <-> ( A = ( r + ( _i x. s ) ) /\ y = ( t + ( _i x. u ) ) ) ) )
32anbi1d 465 . . . . . 6 |- ( x = A -> ( ( ( x = ( r + ( _i x. s ) ) /\ y = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) <-> ( ( A = ( r + ( _i x. s ) ) /\ y = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) ) )
432rexbidv 2502 . . . . 5 |- ( x = A -> ( E. t e. RR E. u e. RR ( ( x = ( r + ( _i x. s ) ) /\ y = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) <-> E. t e. RR E. u e. RR ( ( A = ( r + ( _i x. s ) ) /\ y = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) ) )
542rexbidv 2502 . . . 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 #ℝ t \/ s #ℝ 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 #ℝ t \/ s #ℝ u ) ) ) )
6 eqeq1 2184 . . . . . . . 8 |- ( y = B -> ( y = ( t + ( _i x. u ) ) <-> B = ( t + ( _i x. u ) ) ) )
76anbi2d 464 . . . . . . 7 |- ( y = B -> ( ( A = ( r + ( _i x. s ) ) /\ y = ( t + ( _i x. u ) ) ) <-> ( A = ( r + ( _i x. s ) ) /\ B = ( t + ( _i x. u ) ) ) ) )
87anbi1d 465 . . . . . 6 |- ( y = B -> ( ( ( A = ( r + ( _i x. s ) ) /\ y = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) <-> ( ( A = ( r + ( _i x. s ) ) /\ B = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) ) )
982rexbidv 2502 . . . . 5 |- ( y = B -> ( E. t e. RR E. u e. RR ( ( A = ( r + ( _i x. s ) ) /\ y = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) <-> E. t e. RR E. u e. RR ( ( A = ( r + ( _i x. s ) ) /\ B = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) ) )
1092rexbidv 2502 . . . 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 #ℝ t \/ s #ℝ 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 #ℝ t \/ s #ℝ u ) ) ) )
11 df-ap 8541 . . . 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 #ℝ t \/ s #ℝ u ) ) }
125, 10, 11brabg 4271 . . 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 #ℝ t \/ s #ℝ u ) ) ) )
13 simplll 533 . . . . . . . . . . . . 13 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( r e. RR /\ s e. RR ) ) /\ ( t e. RR /\ u e. RR ) ) -> A e. RR )
1413adantr 276 . . . . . . . . . . . 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 #ℝ t \/ s #ℝ u ) ) ) -> A e. RR )
15 simplrl 535 . . . . . . . . . . . . 13 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( r e. RR /\ s e. RR ) ) /\ ( t e. RR /\ u e. RR ) ) -> r e. RR )
1615adantr 276 . . . . . . . . . . . 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 #ℝ t \/ s #ℝ u ) ) ) -> r e. RR )
17 simplrr 536 . . . . . . . . . . . . 13 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( r e. RR /\ s e. RR ) ) /\ ( t e. RR /\ u e. RR ) ) -> s e. RR )
1817adantr 276 . . . . . . . . . . . 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 #ℝ t \/ s #ℝ u ) ) ) -> s e. RR )
19 simprll 537 . . . . . . . . . . . 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 #ℝ t \/ s #ℝ u ) ) ) -> A = ( r + ( _i x. s ) ) )
20 rereim 8545 . . . . . . . . . . . 12 |- ( ( ( A e. RR /\ r e. RR ) /\ ( s e. RR /\ A = ( r + ( _i x. s ) ) ) ) -> ( r = A /\ s = 0 ) )
2114, 16, 18, 19, 20syl22anc 1239 . . . . . . . . . . 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 #ℝ t \/ s #ℝ u ) ) ) -> ( r = A /\ s = 0 ) )
2221simprd 114 . . . . . . . . . 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 #ℝ t \/ s #ℝ u ) ) ) -> s = 0 )
23 simpllr 534 . . . . . . . . . . . . 13 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( r e. RR /\ s e. RR ) ) /\ ( t e. RR /\ u e. RR ) ) -> B e. RR )
2423adantr 276 . . . . . . . . . . . 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 #ℝ t \/ s #ℝ u ) ) ) -> B e. RR )
25 simplrl 535 . . . . . . . . . . . 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 #ℝ t \/ s #ℝ u ) ) ) -> t e. RR )
26 simplrr 536 . . . . . . . . . . . 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 #ℝ t \/ s #ℝ u ) ) ) -> u e. RR )
27 simprlr 538 . . . . . . . . . . . 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 #ℝ t \/ s #ℝ u ) ) ) -> B = ( t + ( _i x. u ) ) )
28 rereim 8545 . . . . . . . . . . . 12 |- ( ( ( B e. RR /\ t e. RR ) /\ ( u e. RR /\ B = ( t + ( _i x. u ) ) ) ) -> ( t = B /\ u = 0 ) )
2924, 25, 26, 27, 28syl22anc 1239 . . . . . . . . . . 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 #ℝ t \/ s #ℝ u ) ) ) -> ( t = B /\ u = 0 ) )
3029simprd 114 . . . . . . . . . 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 #ℝ t \/ s #ℝ u ) ) ) -> u = 0 )
3122, 30eqtr4d 2213 . . . . . . . . 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 #ℝ t \/ s #ℝ u ) ) ) -> s = u )
32 reapti 8538 . . . . . . . . . 10 |- ( ( s e. RR /\ u e. RR ) -> ( s = u <-> -. s #ℝ u ) )
3318, 26, 32syl2anc 411 . . . . . . . . 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 #ℝ t \/ s #ℝ u ) ) ) -> ( s = u <-> -. s #ℝ u ) )
3431, 33mpbid 147 . . . . . . . 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 #ℝ t \/ s #ℝ u ) ) ) -> -. s #ℝ u )
35 simprr 531 . . . . . . . 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 #ℝ t \/ s #ℝ u ) ) ) -> ( r #ℝ t \/ s #ℝ u ) )
3634, 35ecased 1349 . . . . . . 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 #ℝ t \/ s #ℝ u ) ) ) -> r #ℝ t )
3721simpld 112 . . . . . . 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 #ℝ t \/ s #ℝ u ) ) ) -> r = A )
3829simpld 112 . . . . . . 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 #ℝ t \/ s #ℝ u ) ) ) -> t = B )
3936, 37, 383brtr3d 4036 . . . . . 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 #ℝ t \/ s #ℝ u ) ) ) -> A #ℝ B )
4039ex 115 . . . . 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 #ℝ t \/ s #ℝ u ) ) -> A #ℝ B ) )
4140rexlimdvva 2602 . . . 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 #ℝ t \/ s #ℝ u ) ) -> A #ℝ B ) )
4241rexlimdvva 2602 . . 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 #ℝ t \/ s #ℝ u ) ) -> A #ℝ B ) )
4312, 42sylbid 150 . 2 |- ( ( A e. RR /\ B e. RR ) -> ( A # B -> A #ℝ B ) )
44 ax-icn 7908 . . . . . . . 8 |- _i e. CC
4544mul01i 8350 . . . . . . 7 |- ( _i x. 0 ) = 0
4645oveq2i 5888 . . . . . 6 |- ( A + ( _i x. 0 ) ) = ( A + 0 )
47 simp1 997 . . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> A e. RR )
4847recnd 7988 . . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> A e. CC )
4948addid1d 8108 . . . . . 6 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> ( A + 0 ) = A )
5046, 49eqtr2id 2223 . . . . 5 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> A = ( A + ( _i x. 0 ) ) )
5145oveq2i 5888 . . . . . 6 |- ( B + ( _i x. 0 ) ) = ( B + 0 )
52 simp2 998 . . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> B e. RR )
5352recnd 7988 . . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> B e. CC )
5453addid1d 8108 . . . . . 6 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> ( B + 0 ) = B )
5551, 54eqtr2id 2223 . . . . 5 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> B = ( B + ( _i x. 0 ) ) )
56 olc 711 . . . . . . 7 |- ( A #ℝ B -> ( 0 #ℝ 0 \/ A #ℝ B ) )
57563ad2ant3 1020 . . . . . 6 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> ( 0 #ℝ 0 \/ A #ℝ B ) )
5857orcomd 729 . . . . 5 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> ( A #ℝ B \/ 0 #ℝ 0 ) )
5950, 55, 58jca31 309 . . . 4 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> ( ( A = ( A + ( _i x. 0 ) ) /\ B = ( B + ( _i x. 0 ) ) ) /\ ( A #ℝ B \/ 0 #ℝ 0 ) ) )
60 0red 7960 . . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> 0 e. RR )
61 simpr 110 . . . . . . . . . . . . 13 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ u = 0 ) -> u = 0 )
6261oveq2d 5893 . . . . . . . . . . . 12 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ u = 0 ) -> ( _i x. u ) = ( _i x. 0 ) )
6362oveq2d 5893 . . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ u = 0 ) -> ( B + ( _i x. u ) ) = ( B + ( _i x. 0 ) ) )
6463eqeq2d 2189 . . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ u = 0 ) -> ( B = ( B + ( _i x. u ) ) <-> B = ( B + ( _i x. 0 ) ) ) )
6564anbi2d 464 . . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ u = 0 ) -> ( ( A = ( A + ( _i x. 0 ) ) /\ B = ( B + ( _i x. u ) ) ) <-> ( A = ( A + ( _i x. 0 ) ) /\ B = ( B + ( _i x. 0 ) ) ) ) )
6661breq2d 4017 . . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ u = 0 ) -> ( 0 #ℝ u <-> 0 #ℝ 0 ) )
6766orbi2d 790 . . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ u = 0 ) -> ( ( A #ℝ B \/ 0 #ℝ u ) <-> ( A #ℝ B \/ 0 #ℝ 0 ) ) )
6865, 67anbi12d 473 . . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ u = 0 ) -> ( ( ( A = ( A + ( _i x. 0 ) ) /\ B = ( B + ( _i x. u ) ) ) /\ ( A #ℝ B \/ 0 #ℝ u ) ) <-> ( ( A = ( A + ( _i x. 0 ) ) /\ B = ( B + ( _i x. 0 ) ) ) /\ ( A #ℝ B \/ 0 #ℝ 0 ) ) ) )
6960, 68rspcedv 2847 . . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> ( ( ( A = ( A + ( _i x. 0 ) ) /\ B = ( B + ( _i x. 0 ) ) ) /\ ( A #ℝ B \/ 0 #ℝ 0 ) ) -> E. u e. RR ( ( A = ( A + ( _i x. 0 ) ) /\ B = ( B + ( _i x. u ) ) ) /\ ( A #ℝ B \/ 0 #ℝ u ) ) ) )
70 simpr 110 . . . . . . . . . . . . 13 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ t = B ) -> t = B )
7170oveq1d 5892 . . . . . . . . . . . 12 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ t = B ) -> ( t + ( _i x. u ) ) = ( B + ( _i x. u ) ) )
7271eqeq2d 2189 . . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ t = B ) -> ( B = ( t + ( _i x. u ) ) <-> B = ( B + ( _i x. u ) ) ) )
7372anbi2d 464 . . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ t = B ) -> ( ( A = ( A + ( _i x. 0 ) ) /\ B = ( t + ( _i x. u ) ) ) <-> ( A = ( A + ( _i x. 0 ) ) /\ B = ( B + ( _i x. u ) ) ) ) )
7470breq2d 4017 . . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ t = B ) -> ( A #ℝ t <-> A #ℝ B ) )
7574orbi1d 791 . . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ t = B ) -> ( ( A #ℝ t \/ 0 #ℝ u ) <-> ( A #ℝ B \/ 0 #ℝ u ) ) )
7673, 75anbi12d 473 . . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ t = B ) -> ( ( ( A = ( A + ( _i x. 0 ) ) /\ B = ( t + ( _i x. u ) ) ) /\ ( A #ℝ t \/ 0 #ℝ u ) ) <-> ( ( A = ( A + ( _i x. 0 ) ) /\ B = ( B + ( _i x. u ) ) ) /\ ( A #ℝ B \/ 0 #ℝ u ) ) ) )
7776rexbidv 2478 . . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ t = B ) -> ( E. u e. RR ( ( A = ( A + ( _i x. 0 ) ) /\ B = ( t + ( _i x. u ) ) ) /\ ( A #ℝ t \/ 0 #ℝ u ) ) <-> E. u e. RR ( ( A = ( A + ( _i x. 0 ) ) /\ B = ( B + ( _i x. u ) ) ) /\ ( A #ℝ B \/ 0 #ℝ u ) ) ) )
7852, 77rspcedv 2847 . . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> ( E. u e. RR ( ( A = ( A + ( _i x. 0 ) ) /\ B = ( B + ( _i x. u ) ) ) /\ ( A #ℝ B \/ 0 #ℝ u ) ) -> E. t e. RR E. u e. RR ( ( A = ( A + ( _i x. 0 ) ) /\ B = ( t + ( _i x. u ) ) ) /\ ( A #ℝ t \/ 0 #ℝ u ) ) ) )
7969, 78syld 45 . . . . . 6 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> ( ( ( A = ( A + ( _i x. 0 ) ) /\ B = ( B + ( _i x. 0 ) ) ) /\ ( A #ℝ B \/ 0 #ℝ 0 ) ) -> E. t e. RR E. u e. RR ( ( A = ( A + ( _i x. 0 ) ) /\ B = ( t + ( _i x. u ) ) ) /\ ( A #ℝ t \/ 0 #ℝ u ) ) ) )
80 simpr 110 . . . . . . . . . . . . 13 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ s = 0 ) -> s = 0 )
8180oveq2d 5893 . . . . . . . . . . . 12 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ s = 0 ) -> ( _i x. s ) = ( _i x. 0 ) )
8281oveq2d 5893 . . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ s = 0 ) -> ( A + ( _i x. s ) ) = ( A + ( _i x. 0 ) ) )
8382eqeq2d 2189 . . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ s = 0 ) -> ( A = ( A + ( _i x. s ) ) <-> A = ( A + ( _i x. 0 ) ) ) )
8483anbi1d 465 . . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ s = 0 ) -> ( ( A = ( A + ( _i x. s ) ) /\ B = ( t + ( _i x. u ) ) ) <-> ( A = ( A + ( _i x. 0 ) ) /\ B = ( t + ( _i x. u ) ) ) ) )
8580breq1d 4015 . . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ s = 0 ) -> ( s #ℝ u <-> 0 #ℝ u ) )
8685orbi2d 790 . . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ s = 0 ) -> ( ( A #ℝ t \/ s #ℝ u ) <-> ( A #ℝ t \/ 0 #ℝ u ) ) )
8784, 86anbi12d 473 . . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ s = 0 ) -> ( ( ( A = ( A + ( _i x. s ) ) /\ B = ( t + ( _i x. u ) ) ) /\ ( A #ℝ t \/ s #ℝ u ) ) <-> ( ( A = ( A + ( _i x. 0 ) ) /\ B = ( t + ( _i x. u ) ) ) /\ ( A #ℝ t \/ 0 #ℝ u ) ) ) )
88872rexbidv 2502 . . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ s = 0 ) -> ( E. t e. RR E. u e. RR ( ( A = ( A + ( _i x. s ) ) /\ B = ( t + ( _i x. u ) ) ) /\ ( A #ℝ t \/ s #ℝ u ) ) <-> E. t e. RR E. u e. RR ( ( A = ( A + ( _i x. 0 ) ) /\ B = ( t + ( _i x. u ) ) ) /\ ( A #ℝ t \/ 0 #ℝ u ) ) ) )
8960, 88rspcedv 2847 . . . . . 6 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> ( E. t e. RR E. u e. RR ( ( A = ( A + ( _i x. 0 ) ) /\ B = ( t + ( _i x. u ) ) ) /\ ( A #ℝ t \/ 0 #ℝ u ) ) -> E. s e. RR E. t e. RR E. u e. RR ( ( A = ( A + ( _i x. s ) ) /\ B = ( t + ( _i x. u ) ) ) /\ ( A #ℝ t \/ s #ℝ u ) ) ) )
90 simpr 110 . . . . . . . . . . . . 13 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ r = A ) -> r = A )
9190oveq1d 5892 . . . . . . . . . . . 12 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ r = A ) -> ( r + ( _i x. s ) ) = ( A + ( _i x. s ) ) )
9291eqeq2d 2189 . . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ r = A ) -> ( A = ( r + ( _i x. s ) ) <-> A = ( A + ( _i x. s ) ) ) )
9392anbi1d 465 . . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ r = A ) -> ( ( A = ( r + ( _i x. s ) ) /\ B = ( t + ( _i x. u ) ) ) <-> ( A = ( A + ( _i x. s ) ) /\ B = ( t + ( _i x. u ) ) ) ) )
9490breq1d 4015 . . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ r = A ) -> ( r #ℝ t <-> A #ℝ t ) )
9594orbi1d 791 . . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ r = A ) -> ( ( r #ℝ t \/ s #ℝ u ) <-> ( A #ℝ t \/ s #ℝ u ) ) )
9693, 95anbi12d 473 . . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ r = A ) -> ( ( ( A = ( r + ( _i x. s ) ) /\ B = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) <-> ( ( A = ( A + ( _i x. s ) ) /\ B = ( t + ( _i x. u ) ) ) /\ ( A #ℝ t \/ s #ℝ u ) ) ) )
9796rexbidv 2478 . . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ r = A ) -> ( E. u e. RR ( ( A = ( r + ( _i x. s ) ) /\ B = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) <-> E. u e. RR ( ( A = ( A + ( _i x. s ) ) /\ B = ( t + ( _i x. u ) ) ) /\ ( A #ℝ t \/ s #ℝ u ) ) ) )
98972rexbidv 2502 . . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ 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 #ℝ t \/ s #ℝ u ) ) <-> E. s e. RR E. t e. RR E. u e. RR ( ( A = ( A + ( _i x. s ) ) /\ B = ( t + ( _i x. u ) ) ) /\ ( A #ℝ t \/ s #ℝ u ) ) ) )
9947, 98rspcedv 2847 . . . . . 6 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> ( E. s e. RR E. t e. RR E. u e. RR ( ( A = ( A + ( _i x. s ) ) /\ B = ( t + ( _i x. u ) ) ) /\ ( A #ℝ t \/ s #ℝ 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 #ℝ t \/ s #ℝ u ) ) ) )
10079, 89, 993syld 57 . . . . 5 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> ( ( ( A = ( A + ( _i x. 0 ) ) /\ B = ( B + ( _i x. 0 ) ) ) /\ ( A #ℝ B \/ 0 #ℝ 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 #ℝ t \/ s #ℝ u ) ) ) )
101123adant3 1017 . . . . 5 |- ( ( A e. RR /\ B e. RR /\ A #ℝ 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 #ℝ t \/ s #ℝ u ) ) ) )
102100, 101sylibrd 169 . . . 4 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> ( ( ( A = ( A + ( _i x. 0 ) ) /\ B = ( B + ( _i x. 0 ) ) ) /\ ( A #ℝ B \/ 0 #ℝ 0 ) ) -> A # B ) )
10359, 102mpd 13 . . 3 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> A # B )
1041033expia 1205 . 2 |- ( ( A e. RR /\ B e. RR ) -> ( A #ℝ B -> A # B ) )
10543, 104impbid 129 1 |- ( ( A e. RR /\ B e. RR ) -> ( A # B <-> A #ℝ B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 708   /\ w3a 978   = wceq 1353   e. wcel 2148   E.wrex 2456   class class class wbr 4005  (class class class)co 5877   RRcr 7812   0cc0 7813   _ici 7815   + caddc 7816   x. cmul 7818   #ℝ creap 8533   # cap 8540
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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-pnf 7996  df-mnf 7997  df-ltxr 7999  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541
This theorem is referenced by:  reaplt  8547  apreim  8562  apirr  8564  apti  8581  recexap  8612  rerecclap  8689
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