ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  apreap Unicode version
Theorem apreap 8506
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 2177 . . . . . . . 8 |- ( x = A -> ( x = ( r + ( _i x. s ) ) <-> A = ( r + ( _i x. s ) ) ) )
21anbi1d 462 . . . . . . 7 |- ( x = A -> ( ( x = ( r + ( _i x. s ) ) /\ y = ( t + ( _i x. u ) ) ) <-> ( A = ( r + ( _i x. s ) ) /\ y = ( t + ( _i x. u ) ) ) ) )
32anbi1d 462 . . . . . 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 2495 . . . . 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 2495 . . . 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 2177 . . . . . . . 8 |- ( y = B -> ( y = ( t + ( _i x. u ) ) <-> B = ( t + ( _i x. u ) ) ) )
76anbi2d 461 . . . . . . 7 |- ( y = B -> ( ( A = ( r + ( _i x. s ) ) /\ y = ( t + ( _i x. u ) ) ) <-> ( A = ( r + ( _i x. s ) ) /\ B = ( t + ( _i x. u ) ) ) ) )
87anbi1d 462 . . . . . 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 2495 . . . . 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 2495 . . . 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 8501 . . . 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 4254 . . 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 528 . . . . . . . . . . . . 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 #ℝ t \/ s #ℝ u ) ) ) -> A e. RR )
15 simplrl 530 . . . . . . . . . . . . 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 #ℝ t \/ s #ℝ u ) ) ) -> r e. RR )
17 simplrr 531 . . . . . . . . . . . . 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 #ℝ t \/ s #ℝ u ) ) ) -> s e. RR )
19 simprll 532 . . . . . . . . . . . 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 8505 . . . . . . . . . . . 12 |- ( ( ( A e. RR /\ r e. RR ) /\ ( s e. RR /\ A = ( r + ( _i x. s ) ) ) ) -> ( r = A /\ s = 0 ) )
2114, 16, 18, 19, 20syl22anc 1234 . . . . . . . . . . 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 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 #ℝ t \/ s #ℝ u ) ) ) -> s = 0 )
23 simpllr 529 . . . . . . . . . . . . 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 #ℝ t \/ s #ℝ u ) ) ) -> B e. RR )
25 simplrl 530 . . . . . . . . . . . 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 531 . . . . . . . . . . . 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 533 . . . . . . . . . . . 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 8505 . . . . . . . . . . . 12 |- ( ( ( B e. RR /\ t e. RR ) /\ ( u e. RR /\ B = ( t + ( _i x. u ) ) ) ) -> ( t = B /\ u = 0 ) )
2924, 25, 26, 27, 28syl22anc 1234 . . . . . . . . . . 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 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 #ℝ t \/ s #ℝ u ) ) ) -> u = 0 )
3122, 30eqtr4d 2206 . . . . . . . . 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 8498 . . . . . . . . . 10 |- ( ( s e. RR /\ u e. RR ) -> ( s = u <-> -. s #ℝ u ) )
3318, 26, 32syl2anc 409 . . . . . . . . 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 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 #ℝ t \/ s #ℝ u ) ) ) -> -. s #ℝ u )
35 simprr 527 . . . . . . . 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 1344 . . . . . . 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 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 #ℝ t \/ s #ℝ 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 #ℝ t \/ s #ℝ u ) ) ) -> t = B )
3936, 37, 383brtr3d 4020 . . . . . 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 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 #ℝ t \/ s #ℝ u ) ) -> A #ℝ B ) )
4140rexlimdvva 2595 . . . 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 2595 . . 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 149 . 2 |- ( ( A e. RR /\ B e. RR ) -> ( A # B -> A #ℝ B ) )
44 ax-icn 7869 . . . . . . . 8 |- _i e. CC
4544mul01i 8310 . . . . . . 7 |- ( _i x. 0 ) = 0
4645oveq2i 5864 . . . . . 6 |- ( A + ( _i x. 0 ) ) = ( A + 0 )
47 simp1 992 . . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> A e. RR )
4847recnd 7948 . . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> A e. CC )
4948addid1d 8068 . . . . . 6 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> ( A + 0 ) = A )
5046, 49eqtr2id 2216 . . . . 5 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> A = ( A + ( _i x. 0 ) ) )
5145oveq2i 5864 . . . . . 6 |- ( B + ( _i x. 0 ) ) = ( B + 0 )
52 simp2 993 . . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> B e. RR )
5352recnd 7948 . . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> B e. CC )
5453addid1d 8068 . . . . . 6 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> ( B + 0 ) = B )
5551, 54eqtr2id 2216 . . . . 5 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> B = ( B + ( _i x. 0 ) ) )
56 olc 706 . . . . . . 7 |- ( A #ℝ B -> ( 0 #ℝ 0 \/ A #ℝ B ) )
57563ad2ant3 1015 . . . . . 6 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> ( 0 #ℝ 0 \/ A #ℝ B ) )
5857orcomd 724 . . . . 5 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> ( A #ℝ B \/ 0 #ℝ 0 ) )
5950, 55, 58jca31 307 . . . 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 7921 . . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> 0 e. RR )
61 simpr 109 . . . . . . . . . . . . 13 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ u = 0 ) -> u = 0 )
6261oveq2d 5869 . . . . . . . . . . . 12 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ u = 0 ) -> ( _i x. u ) = ( _i x. 0 ) )
6362oveq2d 5869 . . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ u = 0 ) -> ( B + ( _i x. u ) ) = ( B + ( _i x. 0 ) ) )
6463eqeq2d 2182 . . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ u = 0 ) -> ( B = ( B + ( _i x. u ) ) <-> B = ( B + ( _i x. 0 ) ) ) )
6564anbi2d 461 . . . . . . . . 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 4001 . . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ u = 0 ) -> ( 0 #ℝ u <-> 0 #ℝ 0 ) )
6766orbi2d 785 . . . . . . . . 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 470 . . . . . . . 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 2838 . . . . . . 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 109 . . . . . . . . . . . . 13 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ t = B ) -> t = B )
7170oveq1d 5868 . . . . . . . . . . . 12 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ t = B ) -> ( t + ( _i x. u ) ) = ( B + ( _i x. u ) ) )
7271eqeq2d 2182 . . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ t = B ) -> ( B = ( t + ( _i x. u ) ) <-> B = ( B + ( _i x. u ) ) ) )
7372anbi2d 461 . . . . . . . . . 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 4001 . . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ t = B ) -> ( A #ℝ t <-> A #ℝ B ) )
7574orbi1d 786 . . . . . . . . . 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 470 . . . . . . . . 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 2471 . . . . . . . 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 2838 . . . . . . 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 109 . . . . . . . . . . . . 13 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ s = 0 ) -> s = 0 )
8180oveq2d 5869 . . . . . . . . . . . 12 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ s = 0 ) -> ( _i x. s ) = ( _i x. 0 ) )
8281oveq2d 5869 . . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ s = 0 ) -> ( A + ( _i x. s ) ) = ( A + ( _i x. 0 ) ) )
8382eqeq2d 2182 . . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ s = 0 ) -> ( A = ( A + ( _i x. s ) ) <-> A = ( A + ( _i x. 0 ) ) ) )
8483anbi1d 462 . . . . . . . . 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 3999 . . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ s = 0 ) -> ( s #ℝ u <-> 0 #ℝ u ) )
8685orbi2d 785 . . . . . . . . 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 470 . . . . . . . 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 2495 . . . . . . 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 2838 . . . . . 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 109 . . . . . . . . . . . . 13 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ r = A ) -> r = A )
9190oveq1d 5868 . . . . . . . . . . . 12 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ r = A ) -> ( r + ( _i x. s ) ) = ( A + ( _i x. s ) ) )
9291eqeq2d 2182 . . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ r = A ) -> ( A = ( r + ( _i x. s ) ) <-> A = ( A + ( _i x. s ) ) ) )
9392anbi1d 462 . . . . . . . . . 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 3999 . . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ r = A ) -> ( r #ℝ t <-> A #ℝ t ) )
9594orbi1d 786 . . . . . . . . . 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 470 . . . . . . . . 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 2471 . . . . . . . 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 2495 . . . . . . 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 2838 . . . . . 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 1012 . . . . 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 168 . . . 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 1200 . 2 |- ( ( A e. RR /\ B e. RR ) -> ( A #ℝ B -> A # B ) )
10543, 104impbid 128 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 103   <-> wb 104   \/ wo 703   /\ w3a 973   = wceq 1348   e. wcel 2141   E.wrex 2449   class class class wbr 3989  (class class class)co 5853   RRcr 7773   0cc0 7774   _ici 7776   + caddc 7777   x. cmul 7779   #ℝ creap 8493   # cap 8500
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-ltxr 7959  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501
This theorem is referenced by:  reaplt  8507  apreim  8522  apirr  8524  apti  8541  recexap  8571  rerecclap  8647
  Copyright terms: Public domain W3C validator