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Theorem apreap 8878
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 2241 . . . . . . . 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 2569 . . . . 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 2569 . . . 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 2241 . . . . . . . 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 2569 . . . . 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 2569 . . . 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 8873 . . . 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 4392 . . 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 8877 . . . . . . . . . . . 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 8877 . . . . . . . . . . . 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 2270 . . . . . . . . 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 8870 . . . . . . . . . 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 4145 . . . . . 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 2670 . . . 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 2670 . . 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 8238 . . . . . . . 8 |- _i e. CC
4544mul01i 8681 . . . . . . 7 |- ( _i x. 0 ) = 0
4645oveq2i 6069 . . . . . 6 |- ( A + ( _i x. 0 ) ) = ( A + 0 )
47 simp1 1024 . . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> A e. RR )
4847recnd 8318 . . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> A e. CC )
4948addridd 8438 . . . . . 6 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> ( A + 0 ) = A )
5046, 49eqtr2id 2280 . . . . 5 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> A = ( A + ( _i x. 0 ) ) )
5145oveq2i 6069 . . . . . 6 |- ( B + ( _i x. 0 ) ) = ( B + 0 )
52 simp2 1025 . . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> B e. RR )
5352recnd 8318 . . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> B e. CC )
5453addridd 8438 . . . . . 6 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> ( B + 0 ) = B )
5551, 54eqtr2id 2280 . . . . 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 8291 . . . . . . . 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 6074 . . . . . . . . . . . 12 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ u = 0 ) -> ( _i x. u ) = ( _i x. 0 ) )
6362oveq2d 6074 . . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ u = 0 ) -> ( B + ( _i x. u ) ) = ( B + ( _i x. 0 ) ) )
6463eqeq2d 2246 . . . . . . . . . 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 4126 . . . . . . . . . 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 2927 . . . . . . 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 6073 . . . . . . . . . . . 12 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ t = B ) -> ( t + ( _i x. u ) ) = ( B + ( _i x. u ) ) )
7271eqeq2d 2246 . . . . . . . . . . 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 4126 . . . . . . . . . . 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 2545 . . . . . . . 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 2927 . . . . . . 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 6074 . . . . . . . . . . . 12 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ s = 0 ) -> ( _i x. s ) = ( _i x. 0 ) )
8281oveq2d 6074 . . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ s = 0 ) -> ( A + ( _i x. s ) ) = ( A + ( _i x. 0 ) ) )
8382eqeq2d 2246 . . . . . . . . . 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 4124 . . . . . . . . . 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 2569 . . . . . . 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 2927 . . . . . 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 6073 . . . . . . . . . . . 12 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ r = A ) -> ( r + ( _i x. s ) ) = ( A + ( _i x. s ) ) )
9291eqeq2d 2246 . . . . . . . . . . 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 4124 . . . . . . . . . . 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 2545 . . . . . . . 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 2569 . . . . . . 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 2927 . . . . . 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 2205   E.wrex 2523   class class class wbr 4114  (class class class)co 6058   RRcr 8142   0cc0 8143   _ici 8145   + caddc 8146   x. cmul 8148   #ℝ creap 8865   # cap 8872
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 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260
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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-iota 5317  df-fun 5359  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873
This theorem is referenced by:  reaplt  8879  apreim  8894  apirr  8896  apti  8913  recexap  8944  rerecclap  9021
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