ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  apreap Unicode version
Theorem apreap 8861
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 2239 . . . . . . . 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 2567 . . . . 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 2567 . . . 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 2239 . . . . . . . 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 2567 . . . . 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 2567 . . . 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 8856 . . . 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 4387 . . 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 535 . . . . . . . . . . . . 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 537 . . . . . . . . . . . . 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 538 . . . . . . . . . . . . 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 539 . . . . . . . . . . . 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 8860 . . . . . . . . . . . 12 |- ( ( ( A e. RR /\ r e. RR ) /\ ( s e. RR /\ A = ( r + ( _i x. s ) ) ) ) -> ( r = A /\ s = 0 ) )
2114, 16, 18, 19, 20syl22anc 1275 . . . . . . . . . . 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 536 . . . . . . . . . . . . 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 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 ) ) ) -> t e. RR )
26 simplrr 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 ) ) ) -> u e. RR )
27 simprlr 540 . . . . . . . . . . . 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 8860 . . . . . . . . . . . 12 |- ( ( ( B e. RR /\ t e. RR ) /\ ( u e. RR /\ B = ( t + ( _i x. u ) ) ) ) -> ( t = B /\ u = 0 ) )
2924, 25, 26, 27, 28syl22anc 1275 . . . . . . . . . . 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 2268 . . . . . . . . 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 8853 . . . . . . . . . 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 533 . . . . . . . 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 1386 . . . . . . 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 4140 . . . . . 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 2668 . . . 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 2668 . . 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 8222 . . . . . . . 8 |- _i e. CC
4544mul01i 8664 . . . . . . 7 |- ( _i x. 0 ) = 0
4645oveq2i 6061 . . . . . 6 |- ( A + ( _i x. 0 ) ) = ( A + 0 )
47 simp1 1024 . . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> A e. RR )
4847recnd 8302 . . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> A e. CC )
4948addridd 8422 . . . . . 6 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> ( A + 0 ) = A )
5046, 49eqtr2id 2278 . . . . 5 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> A = ( A + ( _i x. 0 ) ) )
5145oveq2i 6061 . . . . . 6 |- ( B + ( _i x. 0 ) ) = ( B + 0 )
52 simp2 1025 . . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> B e. RR )
5352recnd 8302 . . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> B e. CC )
5453addridd 8422 . . . . . 6 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> ( B + 0 ) = B )
5551, 54eqtr2id 2278 . . . . 5 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> B = ( B + ( _i x. 0 ) ) )
56 olc 719 . . . . . . 7 |- ( A #ℝ B -> ( 0 #ℝ 0 \/ A #ℝ B ) )
57563ad2ant3 1047 . . . . . 6 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> ( 0 #ℝ 0 \/ A #ℝ B ) )
5857orcomd 737 . . . . 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 8275 . . . . . . . 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 6066 . . . . . . . . . . . 12 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ u = 0 ) -> ( _i x. u ) = ( _i x. 0 ) )
6362oveq2d 6066 . . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ u = 0 ) -> ( B + ( _i x. u ) ) = ( B + ( _i x. 0 ) ) )
6463eqeq2d 2244 . . . . . . . . . 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 4121 . . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ u = 0 ) -> ( 0 #ℝ u <-> 0 #ℝ 0 ) )
6766orbi2d 798 . . . . . . . . 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 2925 . . . . . . 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 6065 . . . . . . . . . . . 12 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ t = B ) -> ( t + ( _i x. u ) ) = ( B + ( _i x. u ) ) )
7271eqeq2d 2244 . . . . . . . . . . 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 4121 . . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ t = B ) -> ( A #ℝ t <-> A #ℝ B ) )
7574orbi1d 799 . . . . . . . . . 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 2543 . . . . . . . 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 2925 . . . . . . 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 6066 . . . . . . . . . . . 12 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ s = 0 ) -> ( _i x. s ) = ( _i x. 0 ) )
8281oveq2d 6066 . . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ s = 0 ) -> ( A + ( _i x. s ) ) = ( A + ( _i x. 0 ) ) )
8382eqeq2d 2244 . . . . . . . . . 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 4119 . . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ s = 0 ) -> ( s #ℝ u <-> 0 #ℝ u ) )
8685orbi2d 798 . . . . . . . . 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 2567 . . . . . . 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 2925 . . . . . 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 6065 . . . . . . . . . . . 12 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ r = A ) -> ( r + ( _i x. s ) ) = ( A + ( _i x. s ) ) )
9291eqeq2d 2244 . . . . . . . . . . 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 4119 . . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ r = A ) -> ( r #ℝ t <-> A #ℝ t ) )
9594orbi1d 799 . . . . . . . . . 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 2543 . . . . . . . 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 2567 . . . . . . 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 2925 . . . . . 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 1044 . . . . 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 1232 . 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 716   /\ w3a 1005   = wceq 1398   e. wcel 2203   E.wrex 2521   class class class wbr 4109  (class class class)co 6050   RRcr 8126   0cc0 8127   _ici 8129   + caddc 8130   x. cmul 8132   #ℝ creap 8848   # cap 8855
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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244
This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-pnf 8310  df-mnf 8311  df-ltxr 8313  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856
This theorem is referenced by:  reaplt  8862  apreim  8877  apirr  8879  apti  8896  recexap  8927  rerecclap  9004
  Copyright terms: Public domain W3C validator