Theorem aprcl 8432

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 3938	. . . 4 |- ( A # B <-> <. A , B >. e. # )
2		eqeq1 2147	. . . . . . . . . 10 |- ( x = ( 1st ` <. A , B >. ) -> ( x = ( r + ( _i x. s ) ) <-> ( 1st ` <. A , B >. ) = ( r + ( _i x. s ) ) ) )
3	2	anbi1d 461	. . . . . . . . 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 461	. . . . . . . 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 2463	. . . . . . 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 2463	. . . . . 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 2147	. . . . . . . . . 10 |- ( y = ( 2nd ` <. A , B >. ) -> ( y = ( t + ( _i x. u ) ) <-> ( 2nd ` <. A , B >. ) = ( t + ( _i x. u ) ) ) )
8	7	anbi2d 460	. . . . . . . . 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 461	. . . . . . . 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 2463	. . . . . . 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 2463	. . . . . 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 6101	. . . . 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 8368	. . . . 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 2235	. . . 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 120	. . 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 108	. . . . . . 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 2532	. . . . . 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 2532	. . . . 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 2532	. . . 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 2532	. . 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 4673	. . . . . . . . . 10 |- Rel #
23	22	brrelex1i 4590	. . . . . . . . 9 |- ( A # B -> A e. _V )
24	23	ad3antrrr 484	. . . . . . . 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 4591	. . . . . . . . 9 |- ( A # B -> B e. _V )
26	25	ad3antrrr 484	. . . . . . . 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 6056	. . . . . . . 8 |- ( ( A e. _V /\ B e. _V ) -> ( 1st ` <. A , B >. ) = A )
28	24, 26, 27	syl2anc 409	. . . . . . 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 521	. . . . . . . 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 521	. . . . . . . . . . 11 |- ( ( A # B /\ ( r e. RR /\ s e. RR ) ) -> r e. RR )
31	30	ad2antrr 480	. . . . . . . . . 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 7818	. . . . . . . . 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 7739	. . . . . . . . . . 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 522	. . . . . . . . . . . 12 |- ( ( A # B /\ ( r e. RR /\ s e. RR ) ) -> s e. RR )
36	35	ad2antrr 480	. . . . . . . . . . 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 7818	. . . . . . . . . 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 7810	. . . . . . . . 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 7809	. . . . . . . 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 2217	. . . . . . 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 2218	. . . . . 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 6057	. . . . . . . 8 |- ( ( A e. _V /\ B e. _V ) -> ( 2nd ` <. A , B >. ) = B )
43	24, 26, 42	syl2anc 409	. . . . . . 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 522	. . . . . . . 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 7777	. . . . . . . . . . . 12 |- ( t e. RR -> t e. CC )
46	45	adantr 274	. . . . . . . . . . 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 7777	. . . . . . . . . . . . 13 |- ( u e. RR -> u e. CC )
49	48	adantl 275	. . . . . . . . . . . 12 |- ( ( t e. RR /\ u e. RR ) -> u e. CC )
50	47, 49	mulcld 7810	. . . . . . . . . . 11 |- ( ( t e. RR /\ u e. RR ) -> ( _i x. u ) e. CC )
51	46, 50	addcld 7809	. . . . . . . . . 10 |- ( ( t e. RR /\ u e. RR ) -> ( t + ( _i x. u ) ) e. CC )
52	51	adantl 275	. . . . . . . . 9 |- ( ( ( A # B /\ ( r e. RR /\ s e. RR ) ) /\ ( t e. RR /\ u e. RR ) ) -> ( t + ( _i x. u ) ) e. CC )
53	52	adantr 274	. . . . . . . 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 2217	. . . . . . 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 2218	. . . . . 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 304	. . . . 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 114	. . . 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 2560	. . 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 2560	. 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 103   \/ wo 698   = wceq 1332   e. wcel 1481   E.wrex 2418   _Vcvv 2689   <.cop 3535   class class class wbr 3937   {copab 3996   ` cfv 5131  (class class class)co 5782   1stc1st 6044   2ndc2nd 6045   CCcc 7642   RRcr 7643   _ici 7646   + caddc 7647   x. cmul 7649   #_ℝ creap 8360   # cap 8367

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-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-resscn 7736  ax-icn 7739  ax-addcl 7740  ax-mulcl 7742

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-fo 5137  df-fv 5139  df-1st 6046  df-2nd 6047  df-ap 8368

This theorem is referenced by:  apsscn  8433