Theorem apreim 8877

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 527	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> A e. RR )
2	1	recnd 8302	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> A e. CC )
3		ax-icn 8222	. . . . . . 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 529	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> B e. RR )
6	5	recnd 8302	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> B e. CC )
7	4, 6	mulcld 8294	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( _i x. B ) e. CC )
8	2, 7	addcld 8293	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( A + ( _i x. B ) ) e. CC )
9		simprl 531	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> C e. RR )
10	9	recnd 8302	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> C e. CC )
11		simprr 533	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> D e. RR )
12	11	recnd 8302	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> D e. CC )
13	4, 12	mulcld 8294	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( _i x. D ) e. CC )
14	10, 13	addcld 8293	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( C + ( _i x. D ) ) e. CC )
15		eqeq1 2239	. . . . . . . . 9 |- ( x = ( A + ( _i x. B ) ) -> ( x = ( r + ( _i x. s ) ) <-> ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) ) )
16	15	anbi1d 465	. . . . . . . 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 465	. . . . . . 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 2567	. . . . . 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 2567	. . . . 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 2239	. . . . . . . . 9 |- ( y = ( C + ( _i x. D ) ) -> ( y = ( t + ( _i x. u ) ) <-> ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) )
21	20	anbi2d 464	. . . . . . . 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 465	. . . . . . 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 2567	. . . . . 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 2567	. . . . 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 8856	. . . . 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 4387	. . . 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 411	. . 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 533	. . . . . . 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 492	. . . . . . . . . 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 492	. . . . . . . . . 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 8861	. . . . . . . . . 10 |- ( ( A e. RR /\ C e. RR ) -> ( A # C <-> A #ℝ C ) )
32	29, 30, 31	syl2anc 411	. . . . . . . . 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 492	. . . . . . . . . 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 492	. . . . . . . . . 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 8861	. . . . . . . . . 10 |- ( ( B e. RR /\ D e. RR ) -> ( B # D <-> B #ℝ D ) )
36	33, 34, 35	syl2anc 411	. . . . . . . . 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 801	. . . . . . . 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 539	. . . . . . . . . . . 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 536	. . . . . . . . . . . . 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 8876	. . . . . . . . . . . . 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 1273	. . . . . . . . . . . 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 147	. . . . . . . . . . 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 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 ) ) ) -> A = r )
44		simprlr 540	. . . . . . . . . . . 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 529	. . . . . . . . . . . . 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 8876	. . . . . . . . . . . . 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 1273	. . . . . . . . . . . 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 147	. . . . . . . . . . 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 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 ) ) ) -> C = t )
50	43, 49	breq12d 4122	. . . . . . . . 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 114	. . . . . . . . . 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 114	. . . . . . . . . 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 4122	. . . . . . . . 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 801	. . . . . . . 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 188	. . . . . . 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 167	. . . . . 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 115	. . . . 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 2668	. . . 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 2668	. . 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 150	. 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 509	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( A # C <-> A #ℝ C ) )
62	35	ad2ant2l 508	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( B # D <-> B #ℝ D ) )
63	61, 62	orbi12d 801	. . . . 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 454	. . . 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 2232	. . . . . . . . . . . 12 |- ( A + ( _i x. B ) ) = ( A + ( _i x. B ) )
66		eqid 2232	. . . . . . . . . . . 12 |- ( C + ( _i x. D ) ) = ( C + ( _i x. D ) )
67	65, 66	pm3.2i 272	. . . . . . . . . . 11 |- ( ( A + ( _i x. B ) ) = ( A + ( _i x. B ) ) /\ ( C + ( _i x. D ) ) = ( C + ( _i x. D ) ) )
68	67	biantrur 303	. . . . . . . . . 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 6057	. . . . . . . . . . . . . 14 |- ( t = C -> ( t + ( _i x. u ) ) = ( C + ( _i x. u ) ) )
70	69	eqeq2d 2244	. . . . . . . . . . . . 13 |- ( t = C -> ( ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) <-> ( C + ( _i x. D ) ) = ( C + ( _i x. u ) ) ) )
71	70	anbi2d 464	. . . . . . . . . . . 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 4113	. . . . . . . . . . . . 13 |- ( t = C -> ( A #ℝ t <-> A #ℝ C ) )
73	72	orbi1d 799	. . . . . . . . . . . 12 |- ( t = C -> ( ( A #ℝ t \/ B #ℝ u ) <-> ( A #ℝ C \/ B #ℝ u ) ) )
74	71, 73	anbi12d 473	. . . . . . . . . . 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 6058	. . . . . . . . . . . . . . 15 |- ( u = D -> ( _i x. u ) = ( _i x. D ) )
76	75	oveq2d 6066	. . . . . . . . . . . . . 14 |- ( u = D -> ( C + ( _i x. u ) ) = ( C + ( _i x. D ) ) )
77	76	eqeq2d 2244	. . . . . . . . . . . . 13 |- ( u = D -> ( ( C + ( _i x. D ) ) = ( C + ( _i x. u ) ) <-> ( C + ( _i x. D ) ) = ( C + ( _i x. D ) ) ) )
78	77	anbi2d 464	. . . . . . . . . . . 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 4113	. . . . . . . . . . . . 13 |- ( u = D -> ( B #ℝ u <-> B #ℝ D ) )
80	79	orbi2d 798	. . . . . . . . . . . 12 |- ( u = D -> ( ( A #ℝ C \/ B #ℝ u ) <-> ( A #ℝ C \/ B #ℝ D ) ) )
81	78, 80	anbi12d 473	. . . . . . . . . . 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 2936	. . . . . . . . . 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 1312	. . . . . . . . 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 1230	. . . . . . . 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 6057	. . . . . . . . . . . . 13 |- ( r = A -> ( r + ( _i x. s ) ) = ( A + ( _i x. s ) ) )
86	85	eqeq2d 2244	. . . . . . . . . . . 12 |- ( r = A -> ( ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) <-> ( A + ( _i x. B ) ) = ( A + ( _i x. s ) ) ) )
87	86	anbi1d 465	. . . . . . . . . . 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 4112	. . . . . . . . . . . 12 |- ( r = A -> ( r #ℝ t <-> A #ℝ t ) )
89	88	orbi1d 799	. . . . . . . . . . 11 |- ( r = A -> ( ( r #ℝ t \/ s #ℝ u ) <-> ( A #ℝ t \/ s #ℝ u ) ) )
90	87, 89	anbi12d 473	. . . . . . . . . 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 2567	. . . . . . . . 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 6058	. . . . . . . . . . . . . 14 |- ( s = B -> ( _i x. s ) = ( _i x. B ) )
93	92	oveq2d 6066	. . . . . . . . . . . . 13 |- ( s = B -> ( A + ( _i x. s ) ) = ( A + ( _i x. B ) ) )
94	93	eqeq2d 2244	. . . . . . . . . . . 12 |- ( s = B -> ( ( A + ( _i x. B ) ) = ( A + ( _i x. s ) ) <-> ( A + ( _i x. B ) ) = ( A + ( _i x. B ) ) ) )
95	94	anbi1d 465	. . . . . . . . . . 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 4112	. . . . . . . . . . . 12 |- ( s = B -> ( s #ℝ u <-> B #ℝ u ) )
97	96	orbi2d 798	. . . . . . . . . . 11 |- ( s = B -> ( ( A #ℝ t \/ s #ℝ u ) <-> ( A #ℝ t \/ B #ℝ u ) ) )
98	95, 97	anbi12d 473	. . . . . . . . . 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 2567	. . . . . . . . 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 2936	. . . . . . . 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 1309	. . . . . . 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 1230	. . . . . 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 400	. . . . 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 276	. . . . 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 167	. . . 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 121	. . 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 115	. 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 129	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 104   <-> wb 105   \/ wo 716   = wceq 1398   e. wcel 2203   E.wrex 2521   class class class wbr 4109  (class class class)co 6050   CCcc 8125   RRcr 8126   _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:  apirr  8879  apsym  8880  apcotr  8881  apadd1  8882  apneg  8885  mulext1  8886  apti  8896  recexaplem2  8926  crap0  9232  iap0  9461  cjap  11591  cnreim  11663  absext  11748