Theorem aprcl 8665

Description: Reverse closure for apartness. (Contributed by Jim Kingdon, 19-Dec-2023.)

Assertion

Ref	Expression
aprcl	|- ( A # B -> ( A e. CC /\ B e. CC ) )

Proof of Theorem aprcl

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

Step	Hyp	Ref	Expression
1		df-br 4030	. . . 4 |- ( A # B <-> <. A , B >. e. # )
2		eqeq1 2200	. . . . . . . . . 10 |- ( x = ( 1st ` <. A , B >. ) -> ( x = ( r + ( _i x. s ) ) <-> ( 1st ` <. A , B >. ) = ( r + ( _i x. s ) ) ) )
3	2	anbi1d 465	. . . . . . . . 9 |- ( x = ( 1st ` <. A , B >. ) -> ( ( x = ( r + ( _i x. s ) ) /\ y = ( t + ( _i x. u ) ) ) <-> ( ( 1st ` <. A , B >. ) = ( r + ( _i x. s ) ) /\ y = ( t + ( _i x. u ) ) ) ) )
4	3	anbi1d 465	. . . . . . . 8 |- ( x = ( 1st ` <. A , B >. ) -> ( ( ( x = ( r + ( _i x. s ) ) /\ y = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) <-> ( ( ( 1st ` <. A , B >. ) = ( r + ( _i x. s ) ) /\ y = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) ) )
5	4	2rexbidv 2519	. . . . . . 7 |- ( x = ( 1st ` <. A , 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 ( ( ( 1st ` <. A , B >. ) = ( r + ( _i x. s ) ) /\ y = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) ) )
6	5	2rexbidv 2519	. . . . . 6 |- ( x = ( 1st ` <. A , 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 ( ( ( 1st ` <. A , B >. ) = ( r + ( _i x. s ) ) /\ y = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) ) )
7		eqeq1 2200	. . . . . . . . . 10 |- ( y = ( 2nd ` <. A , B >. ) -> ( y = ( t + ( _i x. u ) ) <-> ( 2nd ` <. A , B >. ) = ( t + ( _i x. u ) ) ) )
8	7	anbi2d 464	. . . . . . . . 9 |- ( y = ( 2nd ` <. A , B >. ) -> ( ( ( 1st ` <. A , B >. ) = ( r + ( _i x. s ) ) /\ y = ( t + ( _i x. u ) ) ) <-> ( ( 1st ` <. A , B >. ) = ( r + ( _i x. s ) ) /\ ( 2nd ` <. A , B >. ) = ( t + ( _i x. u ) ) ) ) )
9	8	anbi1d 465	. . . . . . . 8 |- ( y = ( 2nd ` <. A , B >. ) -> ( ( ( ( 1st ` <. A , B >. ) = ( r + ( _i x. s ) ) /\ y = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) <-> ( ( ( 1st ` <. A , B >. ) = ( r + ( _i x. s ) ) /\ ( 2nd ` <. A , B >. ) = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) ) )
10	9	2rexbidv 2519	. . . . . . 7 |- ( y = ( 2nd ` <. A , B >. ) -> ( E. t e. RR E. u e. RR ( ( ( 1st ` <. A , B >. ) = ( r + ( _i x. s ) ) /\ y = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) <-> E. t e. RR E. u e. RR ( ( ( 1st ` <. A , B >. ) = ( r + ( _i x. s ) ) /\ ( 2nd ` <. A , B >. ) = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) ) )
11	10	2rexbidv 2519	. . . . . 6 |- ( y = ( 2nd ` <. A , B >. ) -> ( E. r e. RR E. s e. RR E. t e. RR E. u e. RR ( ( ( 1st ` <. A , 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 ( ( ( 1st ` <. A , B >. ) = ( r + ( _i x. s ) ) /\ ( 2nd ` <. A , B >. ) = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) ) )
12	6, 11	elopabi 6248	. . . . 5 |- ( <. A , B >. e. { <. 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 ) ) } -> E. r e. RR E. s e. RR E. t e. RR E. u e. RR ( ( ( 1st ` <. A , B >. ) = ( r + ( _i x. s ) ) /\ ( 2nd ` <. A , B >. ) = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) )
13		df-ap 8601	. . . . 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 ) ) }
14	12, 13	eleq2s 2288	. . . 4 |- ( <. A , B >. e. # -> E. r e. RR E. s e. RR E. t e. RR E. u e. RR ( ( ( 1st ` <. A , B >. ) = ( r + ( _i x. s ) ) /\ ( 2nd ` <. A , B >. ) = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) )
15	1, 14	sylbi 121	. . 3 |- ( A # B -> E. r e. RR E. s e. RR E. t e. RR E. u e. RR ( ( ( 1st ` <. A , B >. ) = ( r + ( _i x. s ) ) /\ ( 2nd ` <. A , B >. ) = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) )
16		simpl 109	. . . . . . 7 |- ( ( ( ( 1st ` <. A , B >. ) = ( r + ( _i x. s ) ) /\ ( 2nd ` <. A , B >. ) = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) -> ( ( 1st ` <. A , B >. ) = ( r + ( _i x. s ) ) /\ ( 2nd ` <. A , B >. ) = ( t + ( _i x. u ) ) ) )
17	16	reximi 2591	. . . . . 6 |- ( E. u e. RR ( ( ( 1st ` <. A , B >. ) = ( r + ( _i x. s ) ) /\ ( 2nd ` <. A , B >. ) = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) -> E. u e. RR ( ( 1st ` <. A , B >. ) = ( r + ( _i x. s ) ) /\ ( 2nd ` <. A , B >. ) = ( t + ( _i x. u ) ) ) )
18	17	reximi 2591	. . . . 5 |- ( E. t e. RR E. u e. RR ( ( ( 1st ` <. A , B >. ) = ( r + ( _i x. s ) ) /\ ( 2nd ` <. A , B >. ) = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) -> E. t e. RR E. u e. RR ( ( 1st ` <. A , B >. ) = ( r + ( _i x. s ) ) /\ ( 2nd ` <. A , B >. ) = ( t + ( _i x. u ) ) ) )
19	18	reximi 2591	. . . 4 |- ( E. s e. RR E. t e. RR E. u e. RR ( ( ( 1st ` <. A , B >. ) = ( r + ( _i x. s ) ) /\ ( 2nd ` <. A , B >. ) = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) -> E. s e. RR E. t e. RR E. u e. RR ( ( 1st ` <. A , B >. ) = ( r + ( _i x. s ) ) /\ ( 2nd ` <. A , B >. ) = ( t + ( _i x. u ) ) ) )
20	19	reximi 2591	. . 3 |- ( E. r e. RR E. s e. RR E. t e. RR E. u e. RR ( ( ( 1st ` <. A , B >. ) = ( r + ( _i x. s ) ) /\ ( 2nd ` <. A , B >. ) = ( 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 ( ( 1st ` <. A , B >. ) = ( r + ( _i x. s ) ) /\ ( 2nd ` <. A , B >. ) = ( t + ( _i x. u ) ) ) )
21	15, 20	syl 14	. 2 |- ( A # B -> E. r e. RR E. s e. RR E. t e. RR E. u e. RR ( ( 1st ` <. A , B >. ) = ( r + ( _i x. s ) ) /\ ( 2nd ` <. A , B >. ) = ( t + ( _i x. u ) ) ) )
22	13	relopabi 4787	. . . . . . . . . 10 |- Rel #
23	22	brrelex1i 4702	. . . . . . . . 9 |- ( A # B -> A e. _V )
24	23	ad3antrrr 492	. . . . . . . 8 |- ( ( ( ( A # B /\ ( r e. RR /\ s e. RR ) ) /\ ( t e. RR /\ u e. RR ) ) /\ ( ( 1st ` <. A , B >. ) = ( r + ( _i x. s ) ) /\ ( 2nd ` <. A , B >. ) = ( t + ( _i x. u ) ) ) ) -> A e. _V )
25	22	brrelex2i 4703	. . . . . . . . 9 |- ( A # B -> B e. _V )
26	25	ad3antrrr 492	. . . . . . . 8 |- ( ( ( ( A # B /\ ( r e. RR /\ s e. RR ) ) /\ ( t e. RR /\ u e. RR ) ) /\ ( ( 1st ` <. A , B >. ) = ( r + ( _i x. s ) ) /\ ( 2nd ` <. A , B >. ) = ( t + ( _i x. u ) ) ) ) -> B e. _V )
27		op1stg 6203	. . . . . . . 8 |- ( ( A e. _V /\ B e. _V ) -> ( 1st ` <. A , B >. ) = A )
28	24, 26, 27	syl2anc 411	. . . . . . 7 |- ( ( ( ( A # B /\ ( r e. RR /\ s e. RR ) ) /\ ( t e. RR /\ u e. RR ) ) /\ ( ( 1st ` <. A , B >. ) = ( r + ( _i x. s ) ) /\ ( 2nd ` <. A , B >. ) = ( t + ( _i x. u ) ) ) ) -> ( 1st ` <. A , B >. ) = A )
29		simprl 529	. . . . . . . 8 |- ( ( ( ( A # B /\ ( r e. RR /\ s e. RR ) ) /\ ( t e. RR /\ u e. RR ) ) /\ ( ( 1st ` <. A , B >. ) = ( r + ( _i x. s ) ) /\ ( 2nd ` <. A , B >. ) = ( t + ( _i x. u ) ) ) ) -> ( 1st ` <. A , B >. ) = ( r + ( _i x. s ) ) )
30		simprl 529	. . . . . . . . . . 11 |- ( ( A # B /\ ( r e. RR /\ s e. RR ) ) -> r e. RR )
31	30	ad2antrr 488	. . . . . . . . . 10 |- ( ( ( ( A # B /\ ( r e. RR /\ s e. RR ) ) /\ ( t e. RR /\ u e. RR ) ) /\ ( ( 1st ` <. A , B >. ) = ( r + ( _i x. s ) ) /\ ( 2nd ` <. A , B >. ) = ( t + ( _i x. u ) ) ) ) -> r e. RR )
32	31	recnd 8048	. . . . . . . . 9 |- ( ( ( ( A # B /\ ( r e. RR /\ s e. RR ) ) /\ ( t e. RR /\ u e. RR ) ) /\ ( ( 1st ` <. A , B >. ) = ( r + ( _i x. s ) ) /\ ( 2nd ` <. A , B >. ) = ( t + ( _i x. u ) ) ) ) -> r e. CC )
33		ax-icn 7967	. . . . . . . . . . 11 |- _i e. CC
34	33	a1i 9	. . . . . . . . . 10 |- ( ( ( ( A # B /\ ( r e. RR /\ s e. RR ) ) /\ ( t e. RR /\ u e. RR ) ) /\ ( ( 1st ` <. A , B >. ) = ( r + ( _i x. s ) ) /\ ( 2nd ` <. A , B >. ) = ( t + ( _i x. u ) ) ) ) -> _i e. CC )
35		simprr 531	. . . . . . . . . . . 12 |- ( ( A # B /\ ( r e. RR /\ s e. RR ) ) -> s e. RR )
36	35	ad2antrr 488	. . . . . . . . . . 11 |- ( ( ( ( A # B /\ ( r e. RR /\ s e. RR ) ) /\ ( t e. RR /\ u e. RR ) ) /\ ( ( 1st ` <. A , B >. ) = ( r + ( _i x. s ) ) /\ ( 2nd ` <. A , B >. ) = ( t + ( _i x. u ) ) ) ) -> s e. RR )
37	36	recnd 8048	. . . . . . . . . 10 |- ( ( ( ( A # B /\ ( r e. RR /\ s e. RR ) ) /\ ( t e. RR /\ u e. RR ) ) /\ ( ( 1st ` <. A , B >. ) = ( r + ( _i x. s ) ) /\ ( 2nd ` <. A , B >. ) = ( t + ( _i x. u ) ) ) ) -> s e. CC )
38	34, 37	mulcld 8040	. . . . . . . . 9 |- ( ( ( ( A # B /\ ( r e. RR /\ s e. RR ) ) /\ ( t e. RR /\ u e. RR ) ) /\ ( ( 1st ` <. A , B >. ) = ( r + ( _i x. s ) ) /\ ( 2nd ` <. A , B >. ) = ( t + ( _i x. u ) ) ) ) -> ( _i x. s ) e. CC )
39	32, 38	addcld 8039	. . . . . . . 8 |- ( ( ( ( A # B /\ ( r e. RR /\ s e. RR ) ) /\ ( t e. RR /\ u e. RR ) ) /\ ( ( 1st ` <. A , B >. ) = ( r + ( _i x. s ) ) /\ ( 2nd ` <. A , B >. ) = ( t + ( _i x. u ) ) ) ) -> ( r + ( _i x. s ) ) e. CC )
40	29, 39	eqeltrd 2270	. . . . . . 7 |- ( ( ( ( A # B /\ ( r e. RR /\ s e. RR ) ) /\ ( t e. RR /\ u e. RR ) ) /\ ( ( 1st ` <. A , B >. ) = ( r + ( _i x. s ) ) /\ ( 2nd ` <. A , B >. ) = ( t + ( _i x. u ) ) ) ) -> ( 1st ` <. A , B >. ) e. CC )
41	28, 40	eqeltrrd 2271	. . . . . 6 |- ( ( ( ( A # B /\ ( r e. RR /\ s e. RR ) ) /\ ( t e. RR /\ u e. RR ) ) /\ ( ( 1st ` <. A , B >. ) = ( r + ( _i x. s ) ) /\ ( 2nd ` <. A , B >. ) = ( t + ( _i x. u ) ) ) ) -> A e. CC )
42		op2ndg 6204	. . . . . . . 8 |- ( ( A e. _V /\ B e. _V ) -> ( 2nd ` <. A , B >. ) = B )
43	24, 26, 42	syl2anc 411	. . . . . . 7 |- ( ( ( ( A # B /\ ( r e. RR /\ s e. RR ) ) /\ ( t e. RR /\ u e. RR ) ) /\ ( ( 1st ` <. A , B >. ) = ( r + ( _i x. s ) ) /\ ( 2nd ` <. A , B >. ) = ( t + ( _i x. u ) ) ) ) -> ( 2nd ` <. A , B >. ) = B )
44		simprr 531	. . . . . . . 8 |- ( ( ( ( A # B /\ ( r e. RR /\ s e. RR ) ) /\ ( t e. RR /\ u e. RR ) ) /\ ( ( 1st ` <. A , B >. ) = ( r + ( _i x. s ) ) /\ ( 2nd ` <. A , B >. ) = ( t + ( _i x. u ) ) ) ) -> ( 2nd ` <. A , B >. ) = ( t + ( _i x. u ) ) )
45		recn 8005	. . . . . . . . . . . 12 |- ( t e. RR -> t e. CC )
46	45	adantr 276	. . . . . . . . . . 11 |- ( ( t e. RR /\ u e. RR ) -> t e. CC )
47	33	a1i 9	. . . . . . . . . . . 12 |- ( ( t e. RR /\ u e. RR ) -> _i e. CC )
48		recn 8005	. . . . . . . . . . . . 13 |- ( u e. RR -> u e. CC )
49	48	adantl 277	. . . . . . . . . . . 12 |- ( ( t e. RR /\ u e. RR ) -> u e. CC )
50	47, 49	mulcld 8040	. . . . . . . . . . 11 |- ( ( t e. RR /\ u e. RR ) -> ( _i x. u ) e. CC )
51	46, 50	addcld 8039	. . . . . . . . . 10 |- ( ( t e. RR /\ u e. RR ) -> ( t + ( _i x. u ) ) e. CC )
52	51	adantl 277	. . . . . . . . 9 |- ( ( ( A # B /\ ( r e. RR /\ s e. RR ) ) /\ ( t e. RR /\ u e. RR ) ) -> ( t + ( _i x. u ) ) e. CC )
53	52	adantr 276	. . . . . . . 8 |- ( ( ( ( A # B /\ ( r e. RR /\ s e. RR ) ) /\ ( t e. RR /\ u e. RR ) ) /\ ( ( 1st ` <. A , B >. ) = ( r + ( _i x. s ) ) /\ ( 2nd ` <. A , B >. ) = ( t + ( _i x. u ) ) ) ) -> ( t + ( _i x. u ) ) e. CC )
54	44, 53	eqeltrd 2270	. . . . . . 7 |- ( ( ( ( A # B /\ ( r e. RR /\ s e. RR ) ) /\ ( t e. RR /\ u e. RR ) ) /\ ( ( 1st ` <. A , B >. ) = ( r + ( _i x. s ) ) /\ ( 2nd ` <. A , B >. ) = ( t + ( _i x. u ) ) ) ) -> ( 2nd ` <. A , B >. ) e. CC )
55	43, 54	eqeltrrd 2271	. . . . . 6 |- ( ( ( ( A # B /\ ( r e. RR /\ s e. RR ) ) /\ ( t e. RR /\ u e. RR ) ) /\ ( ( 1st ` <. A , B >. ) = ( r + ( _i x. s ) ) /\ ( 2nd ` <. A , B >. ) = ( t + ( _i x. u ) ) ) ) -> B e. CC )
56	41, 55	jca 306	. . . . 5 |- ( ( ( ( A # B /\ ( r e. RR /\ s e. RR ) ) /\ ( t e. RR /\ u e. RR ) ) /\ ( ( 1st ` <. A , B >. ) = ( r + ( _i x. s ) ) /\ ( 2nd ` <. A , B >. ) = ( t + ( _i x. u ) ) ) ) -> ( A e. CC /\ B e. CC ) )
57	56	ex 115	. . . 4 |- ( ( ( A # B /\ ( r e. RR /\ s e. RR ) ) /\ ( t e. RR /\ u e. RR ) ) -> ( ( ( 1st ` <. A , B >. ) = ( r + ( _i x. s ) ) /\ ( 2nd ` <. A , B >. ) = ( t + ( _i x. u ) ) ) -> ( A e. CC /\ B e. CC ) ) )
58	57	rexlimdvva 2619	. . 3 |- ( ( A # B /\ ( r e. RR /\ s e. RR ) ) -> ( E. t e. RR E. u e. RR ( ( 1st ` <. A , B >. ) = ( r + ( _i x. s ) ) /\ ( 2nd ` <. A , B >. ) = ( t + ( _i x. u ) ) ) -> ( A e. CC /\ B e. CC ) ) )
59	58	rexlimdvva 2619	. 2 |- ( A # B -> ( E. r e. RR E. s e. RR E. t e. RR E. u e. RR ( ( 1st ` <. A , B >. ) = ( r + ( _i x. s ) ) /\ ( 2nd ` <. A , B >. ) = ( t + ( _i x. u ) ) ) -> ( A e. CC /\ B e. CC ) ) )
60	21, 59	mpd 13	1 |- ( A # B -> ( A e. CC /\ B e. CC ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 709   = wceq 1364   e. wcel 2164   E.wrex 2473   _Vcvv 2760   <.cop 3621   class class class wbr 4029   {copab 4089   ` cfv 5254  (class class class)co 5918   1stc1st 6191   2ndc2nd 6192   CCcc 7870   RRcr 7871   _ici 7874   + caddc 7875   x. cmul 7877   #_ℝ creap 8593   # cap 8600

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-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-resscn 7964  ax-icn 7967  ax-addcl 7968  ax-mulcl 7970

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-fo 5260  df-fv 5262  df-1st 6193  df-2nd 6194  df-ap 8601

This theorem is referenced by:  apsscn  8666