Theorem apreim 7703

Description: Complex apartness in terms of real and imaginary parts. (Contributed by Jim Kingdon, 12-Feb-2020.)

Assertion

Ref	Expression
apreim	|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A + ( _i x. B ) ) # ( C + ( _i x. D ) ) <-> ( A # C \/ B # D ) ) )

Proof of Theorem apreim

Dummy variables r s t u x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpll 495	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> A e. RR )
2	1	recnd 7147	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> A e. CC )
3		ax-icn 7071	. . . . . . 7 |- _i e. CC
4	3	a1i 9	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> _i e. CC )
5		simplr 496	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> B e. RR )
6	5	recnd 7147	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> B e. CC )
7	4, 6	mulcld 7139	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( _i x. B ) e. CC )
8	2, 7	addcld 7138	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( A + ( _i x. B ) ) e. CC )
9		simprl 497	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> C e. RR )
10	9	recnd 7147	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> C e. CC )
11		simprr 498	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> D e. RR )
12	11	recnd 7147	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> D e. CC )
13	4, 12	mulcld 7139	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( _i x. D ) e. CC )
14	10, 13	addcld 7138	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( C + ( _i x. D ) ) e. CC )
15		eqeq1 2087	. . . . . . . . 9 |- ( x = ( A + ( _i x. B ) ) -> ( x = ( r + ( _i x. s ) ) <-> ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) ) )
16	15	anbi1d 452	. . . . . . . 8 |- ( x = ( A + ( _i x. B ) ) -> ( ( x = ( r + ( _i x. s ) ) /\ y = ( t + ( _i x. u ) ) ) <-> ( ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) /\ y = ( t + ( _i x. u ) ) ) ) )
17	16	anbi1d 452	. . . . . . 7 |- ( x = ( A + ( _i x. B ) ) -> ( ( ( x = ( r + ( _i x. s ) ) /\ y = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) <-> ( ( ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) /\ y = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) ) )
18	17	2rexbidv 2391	. . . . . 6 |- ( x = ( A + ( _i x. B ) ) -> ( 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 + ( _i x. B ) ) = ( r + ( _i x. s ) ) /\ y = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) ) )
19	18	2rexbidv 2391	. . . . 5 |- ( x = ( A + ( _i x. B ) ) -> ( 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 + ( _i x. B ) ) = ( r + ( _i x. s ) ) /\ y = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) ) )
20		eqeq1 2087	. . . . . . . . 9 |- ( y = ( C + ( _i x. D ) ) -> ( y = ( t + ( _i x. u ) ) <-> ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) )
21	20	anbi2d 451	. . . . . . . 8 |- ( y = ( C + ( _i x. D ) ) -> ( ( ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) /\ y = ( t + ( _i x. u ) ) ) <-> ( ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) ) )
22	21	anbi1d 452	. . . . . . 7 |- ( y = ( C + ( _i x. D ) ) -> ( ( ( ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) /\ y = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) <-> ( ( ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) ) )
23	22	2rexbidv 2391	. . . . . 6 |- ( y = ( C + ( _i x. D ) ) -> ( E. t e. RR E. u e. RR ( ( ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) /\ y = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) <-> E. t e. RR E. u e. RR ( ( ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) ) )
24	23	2rexbidv 2391	. . . . 5 |- ( y = ( C + ( _i x. D ) ) -> ( E. r e. RR E. s e. RR E. t e. RR E. u e. RR ( ( ( A + ( _i x. B ) ) = ( 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 + ( _i x. B ) ) = ( r + ( _i x. s ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) ) )
25		df-ap 7682	. . . . 5 |- # = { <. 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 ) ) }
26	19, 24, 25	brabg 4024	. . . 4 |- ( ( ( A + ( _i x. B ) ) e. CC /\ ( C + ( _i x. D ) ) e. CC ) -> ( ( A + ( _i x. B ) ) # ( C + ( _i x. D ) ) <-> E. r e. RR E. s e. RR E. t e. RR E. u e. RR ( ( ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) ) )
27	8, 14, 26	syl2anc 403	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A + ( _i x. B ) ) # ( C + ( _i x. D ) ) <-> E. r e. RR E. s e. RR E. t e. RR E. u e. RR ( ( ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) ) )
28		simprr 498	. . . . . . 7 |- ( ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( r e. RR /\ s e. RR ) ) /\ ( t e. RR /\ u e. RR ) ) /\ ( ( ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) ) -> ( r #ℝ t \/ s #ℝ u ) )
29	1	ad3antrrr 475	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( r e. RR /\ s e. RR ) ) /\ ( t e. RR /\ u e. RR ) ) /\ ( ( ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) ) -> A e. RR )
30	9	ad3antrrr 475	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( r e. RR /\ s e. RR ) ) /\ ( t e. RR /\ u e. RR ) ) /\ ( ( ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) ) -> C e. RR )
31		apreap 7687	. . . . . . . . . 10 |- ( ( A e. RR /\ C e. RR ) -> ( A # C <-> A #ℝ C ) )
32	29, 30, 31	syl2anc 403	. . . . . . . . 9 |- ( ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( r e. RR /\ s e. RR ) ) /\ ( t e. RR /\ u e. RR ) ) /\ ( ( ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) ) -> ( A # C <-> A #ℝ C ) )
33	5	ad3antrrr 475	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( r e. RR /\ s e. RR ) ) /\ ( t e. RR /\ u e. RR ) ) /\ ( ( ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) ) -> B e. RR )
34	11	ad3antrrr 475	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( r e. RR /\ s e. RR ) ) /\ ( t e. RR /\ u e. RR ) ) /\ ( ( ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) ) -> D e. RR )
35		apreap 7687	. . . . . . . . . 10 |- ( ( B e. RR /\ D e. RR ) -> ( B # D <-> B #ℝ D ) )
36	33, 34, 35	syl2anc 403	. . . . . . . . 9 |- ( ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( r e. RR /\ s e. RR ) ) /\ ( t e. RR /\ u e. RR ) ) /\ ( ( ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) ) -> ( B # D <-> B #ℝ D ) )
37	32, 36	orbi12d 739	. . . . . . . 8 |- ( ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( r e. RR /\ s e. RR ) ) /\ ( t e. RR /\ u e. RR ) ) /\ ( ( ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) ) -> ( ( A # C \/ B # D ) <-> ( A #ℝ C \/ B #ℝ D ) ) )
38		simprll 503	. . . . . . . . . . . 12 |- ( ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( r e. RR /\ s e. RR ) ) /\ ( t e. RR /\ u e. RR ) ) /\ ( ( ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) ) -> ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) )
39		simpllr 500	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( r e. RR /\ s e. RR ) ) /\ ( t e. RR /\ u e. RR ) ) /\ ( ( ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) ) -> ( r e. RR /\ s e. RR ) )
40		cru 7702	. . . . . . . . . . . . 13 |- ( ( ( A e. RR /\ B e. RR ) /\ ( r e. RR /\ s e. RR ) ) -> ( ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) <-> ( A = r /\ B = s ) ) )
41	29, 33, 39, 40	syl21anc 1168	. . . . . . . . . . . 12 |- ( ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( r e. RR /\ s e. RR ) ) /\ ( t e. RR /\ u e. RR ) ) /\ ( ( ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) ) -> ( ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) <-> ( A = r /\ B = s ) ) )
42	38, 41	mpbid 145	. . . . . . . . . . 11 |- ( ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( r e. RR /\ s e. RR ) ) /\ ( t e. RR /\ u e. RR ) ) /\ ( ( ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) ) -> ( A = r /\ B = s ) )
43	42	simpld 110	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( r e. RR /\ s e. RR ) ) /\ ( t e. RR /\ u e. RR ) ) /\ ( ( ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) ) -> A = r )
44		simprlr 504	. . . . . . . . . . . 12 |- ( ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( r e. RR /\ s e. RR ) ) /\ ( t e. RR /\ u e. RR ) ) /\ ( ( ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) ) -> ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) )
45		simplr 496	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( r e. RR /\ s e. RR ) ) /\ ( t e. RR /\ u e. RR ) ) /\ ( ( ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) ) -> ( t e. RR /\ u e. RR ) )
46		cru 7702	. . . . . . . . . . . . 13 |- ( ( ( C e. RR /\ D e. RR ) /\ ( t e. RR /\ u e. RR ) ) -> ( ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) <-> ( C = t /\ D = u ) ) )
47	30, 34, 45, 46	syl21anc 1168	. . . . . . . . . . . 12 |- ( ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( r e. RR /\ s e. RR ) ) /\ ( t e. RR /\ u e. RR ) ) /\ ( ( ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) ) -> ( ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) <-> ( C = t /\ D = u ) ) )
48	44, 47	mpbid 145	. . . . . . . . . . 11 |- ( ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( r e. RR /\ s e. RR ) ) /\ ( t e. RR /\ u e. RR ) ) /\ ( ( ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) ) -> ( C = t /\ D = u ) )
49	48	simpld 110	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( r e. RR /\ s e. RR ) ) /\ ( t e. RR /\ u e. RR ) ) /\ ( ( ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) ) -> C = t )
50	43, 49	breq12d 3798	. . . . . . . . 9 |- ( ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( r e. RR /\ s e. RR ) ) /\ ( t e. RR /\ u e. RR ) ) /\ ( ( ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) ) -> ( A #ℝ C <-> r #ℝ t ) )
51	42	simprd 112	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( r e. RR /\ s e. RR ) ) /\ ( t e. RR /\ u e. RR ) ) /\ ( ( ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) ) -> B = s )
52	48	simprd 112	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( r e. RR /\ s e. RR ) ) /\ ( t e. RR /\ u e. RR ) ) /\ ( ( ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) ) -> D = u )
53	51, 52	breq12d 3798	. . . . . . . . 9 |- ( ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( r e. RR /\ s e. RR ) ) /\ ( t e. RR /\ u e. RR ) ) /\ ( ( ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) ) -> ( B #ℝ D <-> s #ℝ u ) )
54	50, 53	orbi12d 739	. . . . . . . 8 |- ( ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( r e. RR /\ s e. RR ) ) /\ ( t e. RR /\ u e. RR ) ) /\ ( ( ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) ) -> ( ( A #ℝ C \/ B #ℝ D ) <-> ( r #ℝ t \/ s #ℝ u ) ) )
55	37, 54	bitrd 186	. . . . . . 7 |- ( ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( r e. RR /\ s e. RR ) ) /\ ( t e. RR /\ u e. RR ) ) /\ ( ( ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) ) -> ( ( A # C \/ B # D ) <-> ( r #ℝ t \/ s #ℝ u ) ) )
56	28, 55	mpbird 165	. . . . . 6 |- ( ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( r e. RR /\ s e. RR ) ) /\ ( t e. RR /\ u e. RR ) ) /\ ( ( ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) ) -> ( A # C \/ B # D ) )
57	56	ex 113	. . . . 5 |- ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( r e. RR /\ s e. RR ) ) /\ ( t e. RR /\ u e. RR ) ) -> ( ( ( ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) -> ( A # C \/ B # D ) ) )
58	57	rexlimdvva 2484	. . . 4 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( r e. RR /\ s e. RR ) ) -> ( E. t e. RR E. u e. RR ( ( ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) -> ( A # C \/ B # D ) ) )
59	58	rexlimdvva 2484	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( E. r e. RR E. s e. RR E. t e. RR E. u e. RR ( ( ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) -> ( A # C \/ B # D ) ) )
60	27, 59	sylbid 148	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A + ( _i x. B ) ) # ( C + ( _i x. D ) ) -> ( A # C \/ B # D ) ) )
61	31	ad2ant2r 492	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( A # C <-> A #ℝ C ) )
62	35	ad2ant2l 491	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( B # D <-> B #ℝ D ) )
63	61, 62	orbi12d 739	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A # C \/ B # D ) <-> ( A #ℝ C \/ B #ℝ D ) ) )
64	63	pm5.32i 441	. . . 4 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( A # C \/ B # D ) ) <-> ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( A #ℝ C \/ B #ℝ D ) ) )
65		eqid 2081	. . . . . . . . . . . 12 |- ( A + ( _i x. B ) ) = ( A + ( _i x. B ) )
66		eqid 2081	. . . . . . . . . . . 12 |- ( C + ( _i x. D ) ) = ( C + ( _i x. D ) )
67	65, 66	pm3.2i 266	. . . . . . . . . . 11 |- ( ( A + ( _i x. B ) ) = ( A + ( _i x. B ) ) /\ ( C + ( _i x. D ) ) = ( C + ( _i x. D ) ) )
68	67	biantrur 297	. . . . . . . . . 10 |- ( ( A #ℝ C \/ B #ℝ D ) <-> ( ( ( A + ( _i x. B ) ) = ( A + ( _i x. B ) ) /\ ( C + ( _i x. D ) ) = ( C + ( _i x. D ) ) ) /\ ( A #ℝ C \/ B #ℝ D ) ) )
69		oveq1 5539	. . . . . . . . . . . . . 14 |- ( t = C -> ( t + ( _i x. u ) ) = ( C + ( _i x. u ) ) )
70	69	eqeq2d 2092	. . . . . . . . . . . . 13 |- ( t = C -> ( ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) <-> ( C + ( _i x. D ) ) = ( C + ( _i x. u ) ) ) )
71	70	anbi2d 451	. . . . . . . . . . . 12 |- ( t = C -> ( ( ( A + ( _i x. B ) ) = ( A + ( _i x. B ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) <-> ( ( A + ( _i x. B ) ) = ( A + ( _i x. B ) ) /\ ( C + ( _i x. D ) ) = ( C + ( _i x. u ) ) ) ) )
72		breq2 3789	. . . . . . . . . . . . 13 |- ( t = C -> ( A #ℝ t <-> A #ℝ C ) )
73	72	orbi1d 737	. . . . . . . . . . . 12 |- ( t = C -> ( ( A #ℝ t \/ B #ℝ u ) <-> ( A #ℝ C \/ B #ℝ u ) ) )
74	71, 73	anbi12d 456	. . . . . . . . . . 11 |- ( t = C -> ( ( ( ( A + ( _i x. B ) ) = ( A + ( _i x. B ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) /\ ( A #ℝ t \/ B #ℝ u ) ) <-> ( ( ( A + ( _i x. B ) ) = ( A + ( _i x. B ) ) /\ ( C + ( _i x. D ) ) = ( C + ( _i x. u ) ) ) /\ ( A #ℝ C \/ B #ℝ u ) ) ) )
75		oveq2 5540	. . . . . . . . . . . . . . 15 |- ( u = D -> ( _i x. u ) = ( _i x. D ) )
76	75	oveq2d 5548	. . . . . . . . . . . . . 14 |- ( u = D -> ( C + ( _i x. u ) ) = ( C + ( _i x. D ) ) )
77	76	eqeq2d 2092	. . . . . . . . . . . . 13 |- ( u = D -> ( ( C + ( _i x. D ) ) = ( C + ( _i x. u ) ) <-> ( C + ( _i x. D ) ) = ( C + ( _i x. D ) ) ) )
78	77	anbi2d 451	. . . . . . . . . . . 12 |- ( u = D -> ( ( ( A + ( _i x. B ) ) = ( A + ( _i x. B ) ) /\ ( C + ( _i x. D ) ) = ( C + ( _i x. u ) ) ) <-> ( ( A + ( _i x. B ) ) = ( A + ( _i x. B ) ) /\ ( C + ( _i x. D ) ) = ( C + ( _i x. D ) ) ) ) )
79		breq2 3789	. . . . . . . . . . . . 13 |- ( u = D -> ( B #ℝ u <-> B #ℝ D ) )
80	79	orbi2d 736	. . . . . . . . . . . 12 |- ( u = D -> ( ( A #ℝ C \/ B #ℝ u ) <-> ( A #ℝ C \/ B #ℝ D ) ) )
81	78, 80	anbi12d 456	. . . . . . . . . . 11 |- ( u = D -> ( ( ( ( A + ( _i x. B ) ) = ( A + ( _i x. B ) ) /\ ( C + ( _i x. D ) ) = ( C + ( _i x. u ) ) ) /\ ( A #ℝ C \/ B #ℝ u ) ) <-> ( ( ( A + ( _i x. B ) ) = ( A + ( _i x. B ) ) /\ ( C + ( _i x. D ) ) = ( C + ( _i x. D ) ) ) /\ ( A #ℝ C \/ B #ℝ D ) ) ) )
82	74, 81	rspc2ev 2715	. . . . . . . . . 10 |- ( ( C e. RR /\ D e. RR /\ ( ( ( A + ( _i x. B ) ) = ( A + ( _i x. B ) ) /\ ( C + ( _i x. D ) ) = ( C + ( _i x. D ) ) ) /\ ( A #ℝ C \/ B #ℝ D ) ) ) -> E. t e. RR E. u e. RR ( ( ( A + ( _i x. B ) ) = ( A + ( _i x. B ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) /\ ( A #ℝ t \/ B #ℝ u ) ) )
83	68, 82	syl3an3b 1207	. . . . . . . . 9 |- ( ( C e. RR /\ D e. RR /\ ( A #ℝ C \/ B #ℝ D ) ) -> E. t e. RR E. u e. RR ( ( ( A + ( _i x. B ) ) = ( A + ( _i x. B ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) /\ ( A #ℝ t \/ B #ℝ u ) ) )
84	83	3expa 1138	. . . . . . . 8 |- ( ( ( C e. RR /\ D e. RR ) /\ ( A #ℝ C \/ B #ℝ D ) ) -> E. t e. RR E. u e. RR ( ( ( A + ( _i x. B ) ) = ( A + ( _i x. B ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) /\ ( A #ℝ t \/ B #ℝ u ) ) )
85		oveq1 5539	. . . . . . . . . . . . 13 |- ( r = A -> ( r + ( _i x. s ) ) = ( A + ( _i x. s ) ) )
86	85	eqeq2d 2092	. . . . . . . . . . . 12 |- ( r = A -> ( ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) <-> ( A + ( _i x. B ) ) = ( A + ( _i x. s ) ) ) )
87	86	anbi1d 452	. . . . . . . . . . 11 |- ( r = A -> ( ( ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) <-> ( ( A + ( _i x. B ) ) = ( A + ( _i x. s ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) ) )
88		breq1 3788	. . . . . . . . . . . 12 |- ( r = A -> ( r #ℝ t <-> A #ℝ t ) )
89	88	orbi1d 737	. . . . . . . . . . 11 |- ( r = A -> ( ( r #ℝ t \/ s #ℝ u ) <-> ( A #ℝ t \/ s #ℝ u ) ) )
90	87, 89	anbi12d 456	. . . . . . . . . 10 |- ( r = A -> ( ( ( ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) <-> ( ( ( A + ( _i x. B ) ) = ( A + ( _i x. s ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) /\ ( A #ℝ t \/ s #ℝ u ) ) ) )
91	90	2rexbidv 2391	. . . . . . . . 9 |- ( r = A -> ( E. t e. RR E. u e. RR ( ( ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) <-> E. t e. RR E. u e. RR ( ( ( A + ( _i x. B ) ) = ( A + ( _i x. s ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) /\ ( A #ℝ t \/ s #ℝ u ) ) ) )
92		oveq2 5540	. . . . . . . . . . . . . 14 |- ( s = B -> ( _i x. s ) = ( _i x. B ) )
93	92	oveq2d 5548	. . . . . . . . . . . . 13 |- ( s = B -> ( A + ( _i x. s ) ) = ( A + ( _i x. B ) ) )
94	93	eqeq2d 2092	. . . . . . . . . . . 12 |- ( s = B -> ( ( A + ( _i x. B ) ) = ( A + ( _i x. s ) ) <-> ( A + ( _i x. B ) ) = ( A + ( _i x. B ) ) ) )
95	94	anbi1d 452	. . . . . . . . . . 11 |- ( s = B -> ( ( ( A + ( _i x. B ) ) = ( A + ( _i x. s ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) <-> ( ( A + ( _i x. B ) ) = ( A + ( _i x. B ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) ) )
96		breq1 3788	. . . . . . . . . . . 12 |- ( s = B -> ( s #ℝ u <-> B #ℝ u ) )
97	96	orbi2d 736	. . . . . . . . . . 11 |- ( s = B -> ( ( A #ℝ t \/ s #ℝ u ) <-> ( A #ℝ t \/ B #ℝ u ) ) )
98	95, 97	anbi12d 456	. . . . . . . . . 10 |- ( s = B -> ( ( ( ( A + ( _i x. B ) ) = ( A + ( _i x. s ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) /\ ( A #ℝ t \/ s #ℝ u ) ) <-> ( ( ( A + ( _i x. B ) ) = ( A + ( _i x. B ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) /\ ( A #ℝ t \/ B #ℝ u ) ) ) )
99	98	2rexbidv 2391	. . . . . . . . 9 |- ( s = B -> ( E. t e. RR E. u e. RR ( ( ( A + ( _i x. B ) ) = ( A + ( _i x. s ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) /\ ( A #ℝ t \/ s #ℝ u ) ) <-> E. t e. RR E. u e. RR ( ( ( A + ( _i x. B ) ) = ( A + ( _i x. B ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) /\ ( A #ℝ t \/ B #ℝ u ) ) ) )
100	91, 99	rspc2ev 2715	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ E. t e. RR E. u e. RR ( ( ( A + ( _i x. B ) ) = ( A + ( _i x. B ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) /\ ( A #ℝ t \/ B #ℝ u ) ) ) -> E. r e. RR E. s e. RR E. t e. RR E. u e. RR ( ( ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) )
101	84, 100	syl3an3 1204	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ ( ( C e. RR /\ D e. RR ) /\ ( A #ℝ C \/ B #ℝ D ) ) ) -> E. r e. RR E. s e. RR E. t e. RR E. u e. RR ( ( ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) )
102	101	3expa 1138	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( ( C e. RR /\ D e. RR ) /\ ( A #ℝ C \/ B #ℝ D ) ) ) -> E. r e. RR E. s e. RR E. t e. RR E. u e. RR ( ( ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) )
103	102	anassrs 392	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( A #ℝ C \/ B #ℝ D ) ) -> E. r e. RR E. s e. RR E. t e. RR E. u e. RR ( ( ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) )
104	27	adantr 270	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( A #ℝ C \/ B #ℝ D ) ) -> ( ( A + ( _i x. B ) ) # ( C + ( _i x. D ) ) <-> E. r e. RR E. s e. RR E. t e. RR E. u e. RR ( ( ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) /\ ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) ) )
105	103, 104	mpbird 165	. . . 4 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( A #ℝ C \/ B #ℝ D ) ) -> ( A + ( _i x. B ) ) # ( C + ( _i x. D ) ) )
106	64, 105	sylbi 119	. . 3 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) /\ ( A # C \/ B # D ) ) -> ( A + ( _i x. B ) ) # ( C + ( _i x. D ) ) )
107	106	ex 113	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A # C \/ B # D ) -> ( A + ( _i x. B ) ) # ( C + ( _i x. D ) ) ) )
108	60, 107	impbid 127	1 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A + ( _i x. B ) ) # ( C + ( _i x. D ) ) <-> ( A # C \/ B # D ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   \/ wo 661   = wceq 1284   e. wcel 1433   E.wrex 2349   class class class wbr 3785  (class class class)co 5532   CCcc 6979   RRcr 6980   _ici 6983   + caddc 6984   x. cmul 6986   #_ℝ creap 7674   # cap 7681

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-mulrcl 7075  ax-addcom 7076  ax-mulcom 7077  ax-addass 7078  ax-mulass 7079  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-1rid 7083  ax-0id 7084  ax-rnegex 7085  ax-precex 7086  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092  ax-pre-mulgt0 7093

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-iota 4887  df-fun 4924  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-pnf 7155  df-mnf 7156  df-ltxr 7158  df-sub 7281  df-neg 7282  df-reap 7675  df-ap 7682

This theorem is referenced by:  apirr  7705  apsym  7706  apcotr  7707  apadd1  7708  apneg  7711  mulext1  7712  apti  7722  recexaplem2  7742  crap0  8035  iap0  8254  cjap  9793  absext  9949