Theorem apreim 8522

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 524	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> A e. RR )
2	1	recnd 7948	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> A e. CC )
3		ax-icn 7869	. . . . . . 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 525	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> B e. RR )
6	5	recnd 7948	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> B e. CC )
7	4, 6	mulcld 7940	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( _i x. B ) e. CC )
8	2, 7	addcld 7939	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( A + ( _i x. B ) ) e. CC )
9		simprl 526	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> C e. RR )
10	9	recnd 7948	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> C e. CC )
11		simprr 527	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> D e. RR )
12	11	recnd 7948	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> D e. CC )
13	4, 12	mulcld 7940	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( _i x. D ) e. CC )
14	10, 13	addcld 7939	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( C + ( _i x. D ) ) e. CC )
15		eqeq1 2177	. . . . . . . . 9 |- ( x = ( A + ( _i x. B ) ) -> ( x = ( r + ( _i x. s ) ) <-> ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) ) )
16	15	anbi1d 462	. . . . . . . 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 462	. . . . . . 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 2495	. . . . . 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 2495	. . . . 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 2177	. . . . . . . . 9 |- ( y = ( C + ( _i x. D ) ) -> ( y = ( t + ( _i x. u ) ) <-> ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) ) )
21	20	anbi2d 461	. . . . . . . 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 462	. . . . . . 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 2495	. . . . . 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 2495	. . . . 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 8501	. . . . 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 4254	. . . 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 409	. . 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 527	. . . . . . 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 489	. . . . . . . . . 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 489	. . . . . . . . . 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 8506	. . . . . . . . . 10 |- ( ( A e. RR /\ C e. RR ) -> ( A # C <-> A #ℝ C ) )
32	29, 30, 31	syl2anc 409	. . . . . . . . 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 489	. . . . . . . . . 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 489	. . . . . . . . . 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 8506	. . . . . . . . . 10 |- ( ( B e. RR /\ D e. RR ) -> ( B # D <-> B #ℝ D ) )
36	33, 34, 35	syl2anc 409	. . . . . . . . 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 788	. . . . . . . 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 532	. . . . . . . . . . . 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 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 ) ) ) -> ( r e. RR /\ s e. RR ) )
40		cru 8521	. . . . . . . . . . . . 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 1232	. . . . . . . . . . . 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 146	. . . . . . . . . . 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 111	. . . . . . . . . 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 533	. . . . . . . . . . . 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 525	. . . . . . . . . . . . 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 8521	. . . . . . . . . . . . 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 1232	. . . . . . . . . . . 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 146	. . . . . . . . . . 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 111	. . . . . . . . . 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 4002	. . . . . . . . 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 113	. . . . . . . . . 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 113	. . . . . . . . . 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 4002	. . . . . . . . 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 788	. . . . . . . 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 187	. . . . . . 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 166	. . . . . 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 114	. . . . 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 2595	. . . 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 2595	. . 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 149	. 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 506	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( A # C <-> A #ℝ C ) )
62	35	ad2ant2l 505	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( B # D <-> B #ℝ D ) )
63	61, 62	orbi12d 788	. . . . 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 451	. . . 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 2170	. . . . . . . . . . . 12 |- ( A + ( _i x. B ) ) = ( A + ( _i x. B ) )
66		eqid 2170	. . . . . . . . . . . 12 |- ( C + ( _i x. D ) ) = ( C + ( _i x. D ) )
67	65, 66	pm3.2i 270	. . . . . . . . . . 11 |- ( ( A + ( _i x. B ) ) = ( A + ( _i x. B ) ) /\ ( C + ( _i x. D ) ) = ( C + ( _i x. D ) ) )
68	67	biantrur 301	. . . . . . . . . 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 5860	. . . . . . . . . . . . . 14 |- ( t = C -> ( t + ( _i x. u ) ) = ( C + ( _i x. u ) ) )
70	69	eqeq2d 2182	. . . . . . . . . . . . 13 |- ( t = C -> ( ( C + ( _i x. D ) ) = ( t + ( _i x. u ) ) <-> ( C + ( _i x. D ) ) = ( C + ( _i x. u ) ) ) )
71	70	anbi2d 461	. . . . . . . . . . . 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 3993	. . . . . . . . . . . . 13 |- ( t = C -> ( A #ℝ t <-> A #ℝ C ) )
73	72	orbi1d 786	. . . . . . . . . . . 12 |- ( t = C -> ( ( A #ℝ t \/ B #ℝ u ) <-> ( A #ℝ C \/ B #ℝ u ) ) )
74	71, 73	anbi12d 470	. . . . . . . . . . 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 5861	. . . . . . . . . . . . . . 15 |- ( u = D -> ( _i x. u ) = ( _i x. D ) )
76	75	oveq2d 5869	. . . . . . . . . . . . . 14 |- ( u = D -> ( C + ( _i x. u ) ) = ( C + ( _i x. D ) ) )
77	76	eqeq2d 2182	. . . . . . . . . . . . 13 |- ( u = D -> ( ( C + ( _i x. D ) ) = ( C + ( _i x. u ) ) <-> ( C + ( _i x. D ) ) = ( C + ( _i x. D ) ) ) )
78	77	anbi2d 461	. . . . . . . . . . . 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 3993	. . . . . . . . . . . . 13 |- ( u = D -> ( B #ℝ u <-> B #ℝ D ) )
80	79	orbi2d 785	. . . . . . . . . . . 12 |- ( u = D -> ( ( A #ℝ C \/ B #ℝ u ) <-> ( A #ℝ C \/ B #ℝ D ) ) )
81	78, 80	anbi12d 470	. . . . . . . . . . 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 2849	. . . . . . . . . 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 1271	. . . . . . . . 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 1198	. . . . . . . 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 5860	. . . . . . . . . . . . 13 |- ( r = A -> ( r + ( _i x. s ) ) = ( A + ( _i x. s ) ) )
86	85	eqeq2d 2182	. . . . . . . . . . . 12 |- ( r = A -> ( ( A + ( _i x. B ) ) = ( r + ( _i x. s ) ) <-> ( A + ( _i x. B ) ) = ( A + ( _i x. s ) ) ) )
87	86	anbi1d 462	. . . . . . . . . . 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 3992	. . . . . . . . . . . 12 |- ( r = A -> ( r #ℝ t <-> A #ℝ t ) )
89	88	orbi1d 786	. . . . . . . . . . 11 |- ( r = A -> ( ( r #ℝ t \/ s #ℝ u ) <-> ( A #ℝ t \/ s #ℝ u ) ) )
90	87, 89	anbi12d 470	. . . . . . . . . 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 2495	. . . . . . . . 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 5861	. . . . . . . . . . . . . 14 |- ( s = B -> ( _i x. s ) = ( _i x. B ) )
93	92	oveq2d 5869	. . . . . . . . . . . . 13 |- ( s = B -> ( A + ( _i x. s ) ) = ( A + ( _i x. B ) ) )
94	93	eqeq2d 2182	. . . . . . . . . . . 12 |- ( s = B -> ( ( A + ( _i x. B ) ) = ( A + ( _i x. s ) ) <-> ( A + ( _i x. B ) ) = ( A + ( _i x. B ) ) ) )
95	94	anbi1d 462	. . . . . . . . . . 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 3992	. . . . . . . . . . . 12 |- ( s = B -> ( s #ℝ u <-> B #ℝ u ) )
97	96	orbi2d 785	. . . . . . . . . . 11 |- ( s = B -> ( ( A #ℝ t \/ s #ℝ u ) <-> ( A #ℝ t \/ B #ℝ u ) ) )
98	95, 97	anbi12d 470	. . . . . . . . . 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 2495	. . . . . . . . 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 2849	. . . . . . . 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 1268	. . . . . . 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 1198	. . . . . 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 398	. . . . 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 274	. . . . 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 166	. . . 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 120	. . 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 114	. 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 128	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 103   <-> wb 104   \/ wo 703   = wceq 1348   e. wcel 2141   E.wrex 2449   class class class wbr 3989  (class class class)co 5853   CCcc 7772   RRcr 7773   _ici 7776   + caddc 7777   x. cmul 7779   #_ℝ creap 8493   # cap 8500

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-ltxr 7959  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501

This theorem is referenced by:  apirr  8524  apsym  8525  apcotr  8526  apadd1  8527  apneg  8530  mulext1  8531  apti  8541  recexaplem2  8570  crap0  8874  iap0  9101  cjap  10870  cnreim  10942  absext  11027