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Theorem apreap 8342
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 2144 . . . . . . . 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 #ℝ t \/ s #ℝ u ) ) <-> ( ( A = ( r + ( _i x. s ) ) /\ y = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) ) )
432rexbidv 2458 . . . . 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 2458 . . . 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 2144 . . . . . . . 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 #ℝ t \/ s #ℝ u ) ) <-> ( ( A = ( r + ( _i x. s ) ) /\ B = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) ) )
982rexbidv 2458 . . . . 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 2458 . . . 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 8337 . . . 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 4186 . . 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 522 . . . . . . . . . . . . 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 524 . . . . . . . . . . . . 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 525 . . . . . . . . . . . . 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 526 . . . . . . . . . . . 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 8341 . . . . . . . . . . . 12 |- ( ( ( A e. RR /\ r e. RR ) /\ ( s e. RR /\ A = ( r + ( _i x. s ) ) ) ) -> ( r = A /\ s = 0 ) )
2114, 16, 18, 19, 20syl22anc 1217 . . . . . . . . . . 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 523 . . . . . . . . . . . . 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 524 . . . . . . . . . . . 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 525 . . . . . . . . . . . 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 527 . . . . . . . . . . . 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 8341 . . . . . . . . . . . 12 |- ( ( ( B e. RR /\ t e. RR ) /\ ( u e. RR /\ B = ( t + ( _i x. u ) ) ) ) -> ( t = B /\ u = 0 ) )
2924, 25, 26, 27, 28syl22anc 1217 . . . . . . . . . . 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 2173 . . . . . . . . 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 8334 . . . . . . . . . 10 |- ( ( s e. RR /\ u e. RR ) -> ( s = u <-> -. s #ℝ 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 #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 521 . . . . . . . 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 1327 . . . . . . 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 3954 . . . . . 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 2555 . . . 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 2555 . . 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 7708 . . . . . . . 8 |- _i e. CC
4544mul01i 8146 . . . . . . 7 |- ( _i x. 0 ) = 0
4645oveq2i 5778 . . . . . 6 |- ( A + ( _i x. 0 ) ) = ( A + 0 )
47 simp1 981 . . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> A e. RR )
4847recnd 7787 . . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> A e. CC )
4948addid1d 7904 . . . . . 6 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> ( A + 0 ) = A )
5046, 49syl5req 2183 . . . . 5 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> A = ( A + ( _i x. 0 ) ) )
5145oveq2i 5778 . . . . . 6 |- ( B + ( _i x. 0 ) ) = ( B + 0 )
52 simp2 982 . . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> B e. RR )
5352recnd 7787 . . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> B e. CC )
5453addid1d 7904 . . . . . 6 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> ( B + 0 ) = B )
5551, 54syl5req 2183 . . . . 5 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> B = ( B + ( _i x. 0 ) ) )
56 olc 700 . . . . . . 7 |- ( A #ℝ B -> ( 0 #ℝ 0 \/ A #ℝ B ) )
57563ad2ant3 1004 . . . . . 6 |- ( ( A e. RR /\ B e. RR /\ A #ℝ B ) -> ( 0 #ℝ 0 \/ A #ℝ B ) )
5857orcomd 718 . . . . 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 7760 . . . . . . . 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 5783 . . . . . . . . . . . 12 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ u = 0 ) -> ( _i x. u ) = ( _i x. 0 ) )
6362oveq2d 5783 . . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ u = 0 ) -> ( B + ( _i x. u ) ) = ( B + ( _i x. 0 ) ) )
6463eqeq2d 2149 . . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ u = 0 ) -> ( B = ( B + ( _i x. u ) ) <-> B = ( B + ( _i x. 0 ) ) ) )
6564anbi2d 459 . . . . . . . . 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 3936 . . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ u = 0 ) -> ( 0 #ℝ u <-> 0 #ℝ 0 ) )
6766orbi2d 779 . . . . . . . . 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 464 . . . . . . . 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 2788 . . . . . . 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 5782 . . . . . . . . . . . 12 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ t = B ) -> ( t + ( _i x. u ) ) = ( B + ( _i x. u ) ) )
7271eqeq2d 2149 . . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ t = B ) -> ( B = ( t + ( _i x. u ) ) <-> B = ( B + ( _i x. u ) ) ) )
7372anbi2d 459 . . . . . . . . . 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 3936 . . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ t = B ) -> ( A #ℝ t <-> A #ℝ B ) )
7574orbi1d 780 . . . . . . . . . 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 464 . . . . . . . . 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 2436 . . . . . . . 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 2788 . . . . . . 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 5783 . . . . . . . . . . . 12 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ s = 0 ) -> ( _i x. s ) = ( _i x. 0 ) )
8281oveq2d 5783 . . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ s = 0 ) -> ( A + ( _i x. s ) ) = ( A + ( _i x. 0 ) ) )
8382eqeq2d 2149 . . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ s = 0 ) -> ( A = ( A + ( _i x. s ) ) <-> A = ( A + ( _i x. 0 ) ) ) )
8483anbi1d 460 . . . . . . . . 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 3934 . . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ s = 0 ) -> ( s #ℝ u <-> 0 #ℝ u ) )
8685orbi2d 779 . . . . . . . . 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 464 . . . . . . . 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 2458 . . . . . . 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 2788 . . . . . 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 5782 . . . . . . . . . . . 12 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ r = A ) -> ( r + ( _i x. s ) ) = ( A + ( _i x. s ) ) )
9291eqeq2d 2149 . . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ r = A ) -> ( A = ( r + ( _i x. s ) ) <-> A = ( A + ( _i x. s ) ) ) )
9392anbi1d 460 . . . . . . . . . 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 3934 . . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR /\ A #ℝ B ) /\ r = A ) -> ( r #ℝ t <-> A #ℝ t ) )
9594orbi1d 780 . . . . . . . . . 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 464 . . . . . . . . 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 2436 . . . . . . . 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 2458 . . . . . . 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 2788 . . . . . 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 1001 . . . . 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 1183 . 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 697   /\ w3a 962   = wceq 1331   e. wcel 1480   E.wrex 2415   class class class wbr 3924  (class class class)co 5767   RRcr 7612   0cc0 7613   _ici 7615   + caddc 7616   x. cmul 7618   #ℝ creap 8329   # cap 8336
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-mulrcl 7712  ax-addcom 7713  ax-mulcom 7714  ax-addass 7715  ax-mulass 7716  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-1rid 7720  ax-0id 7721  ax-rnegex 7722  ax-precex 7723  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729  ax-pre-mulgt0 7730
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-iota 5083  df-fun 5120  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-pnf 7795  df-mnf 7796  df-ltxr 7798  df-sub 7928  df-neg 7929  df-reap 8330  df-ap 8337
This theorem is referenced by:  reaplt  8343  apreim  8358  apirr  8360  apti  8377  recexap  8407  rerecclap  8483
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