Theorem aprcl 8411

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 3930	. . . 4 |- ( A # B <-> <. A , B >. e. # )
2		eqeq1 2146	. . . . . . . . . 10 |- ( x = ( 1st ` <. A , B >. ) -> ( x = ( r + ( _i x. s ) ) <-> ( 1st ` <. A , B >. ) = ( r + ( _i x. s ) ) ) )
3	2	anbi1d 460	. . . . . . . . 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 460	. . . . . . . 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 2460	. . . . . . 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 2460	. . . . . 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 2146	. . . . . . . . . 10 |- ( y = ( 2nd ` <. A , B >. ) -> ( y = ( t + ( _i x. u ) ) <-> ( 2nd ` <. A , B >. ) = ( t + ( _i x. u ) ) ) )
8	7	anbi2d 459	. . . . . . . . 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 460	. . . . . . . 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 2460	. . . . . . 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 2460	. . . . . 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 6093	. . . . 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 8347	. . . . 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 2234	. . . 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 2529	. . . . . 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 2529	. . . . 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 2529	. . . 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 2529	. . 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 4665	. . . . . . . . . 10 |- Rel #
23	22	brrelex1i 4582	. . . . . . . . 9 |- ( A # B -> A e. _V )
24	23	ad3antrrr 483	. . . . . . . 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 4583	. . . . . . . . 9 |- ( A # B -> B e. _V )
26	25	ad3antrrr 483	. . . . . . . 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 6048	. . . . . . . 8 |- ( ( A e. _V /\ B e. _V ) -> ( 1st ` <. A , B >. ) = A )
28	24, 26, 27	syl2anc 408	. . . . . . 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 520	. . . . . . . 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 520	. . . . . . . . . . 11 |- ( ( A # B /\ ( r e. RR /\ s e. RR ) ) -> r e. RR )
31	30	ad2antrr 479	. . . . . . . . . 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 7797	. . . . . . . . 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 7718	. . . . . . . . . . 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 521	. . . . . . . . . . . 12 |- ( ( A # B /\ ( r e. RR /\ s e. RR ) ) -> s e. RR )
36	35	ad2antrr 479	. . . . . . . . . . 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 7797	. . . . . . . . . 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 7789	. . . . . . . . 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 7788	. . . . . . . 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 2216	. . . . . . 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 2217	. . . . . 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 6049	. . . . . . . 8 |- ( ( A e. _V /\ B e. _V ) -> ( 2nd ` <. A , B >. ) = B )
43	24, 26, 42	syl2anc 408	. . . . . . 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 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 ) ) ) ) -> ( 2nd ` <. A , B >. ) = ( t + ( _i x. u ) ) )
45		recn 7756	. . . . . . . . . . . 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 7756	. . . . . . . . . . . . 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 7789	. . . . . . . . . . 11 |- ( ( t e. RR /\ u e. RR ) -> ( _i x. u ) e. CC )
51	46, 50	addcld 7788	. . . . . . . . . 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 2216	. . . . . . 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 2217	. . . . . 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 2557	. . 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 2557	. 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 697   = wceq 1331   e. wcel 1480   E.wrex 2417   _Vcvv 2686   <.cop 3530   class class class wbr 3929   {copab 3988   ` cfv 5123  (class class class)co 5774   1stc1st 6036   2ndc2nd 6037   CCcc 7621   RRcr 7622   _ici 7625   + caddc 7626   x. cmul 7628   #_ℝ creap 8339   # cap 8346

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-resscn 7715  ax-icn 7718  ax-addcl 7719  ax-mulcl 7721

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fo 5129  df-fv 5131  df-1st 6038  df-2nd 6039  df-ap 8347

This theorem is referenced by:  apsscn  8412