Theorem apti 8541

Description: Complex apartness is tight. (Contributed by Jim Kingdon, 21-Feb-2020.)

Assertion

Ref	Expression
apti	|- ( ( A e. CC /\ B e. CC ) -> ( A = B <-> -. A # B ) )

Proof of Theorem apti

Dummy variables w x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnre 7916	. . 3 |- ( A e. CC -> E. x e. RR E. y e. RR A = ( x + ( _i x. y ) ) )
2	1	adantr 274	. 2 |- ( ( A e. CC /\ B e. CC ) -> E. x e. RR E. y e. RR A = ( x + ( _i x. y ) ) )
3		cnre 7916	. . . . . . 7 |- ( B e. CC -> E. z e. RR E. w e. RR B = ( z + ( _i x. w ) ) )
4	3	adantl 275	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> E. z e. RR E. w e. RR B = ( z + ( _i x. w ) ) )
5	4	ad2antrr 485	. . . . 5 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> E. z e. RR E. w e. RR B = ( z + ( _i x. w ) ) )
6		simpr 109	. . . . . . . . . 10 |- ( ( ( A e. CC /\ B e. CC ) /\ ( x e. RR /\ y e. RR ) ) -> ( x e. RR /\ y e. RR ) )
7	6	ad3antrrr 489	. . . . . . . . 9 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) -> ( x e. RR /\ y e. RR ) )
8		simplr 525	. . . . . . . . 9 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) -> ( z e. RR /\ w e. RR ) )
9		cru 8521	. . . . . . . . 9 |- ( ( ( x e. RR /\ y e. RR ) /\ ( z e. RR /\ w e. RR ) ) -> ( ( x + ( _i x. y ) ) = ( z + ( _i x. w ) ) <-> ( x = z /\ y = w ) ) )
10	7, 8, 9	syl2anc 409	. . . . . . . 8 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) -> ( ( x + ( _i x. y ) ) = ( z + ( _i x. w ) ) <-> ( x = z /\ y = w ) ) )
11		simpllr 529	. . . . . . . . 9 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) -> A = ( x + ( _i x. y ) ) )
12		simpr 109	. . . . . . . . 9 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) -> B = ( z + ( _i x. w ) ) )
13	11, 12	eqeq12d 2185	. . . . . . . 8 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) -> ( A = B <-> ( x + ( _i x. y ) ) = ( z + ( _i x. w ) ) ) )
14		apreim 8522	. . . . . . . . . . . 12 |- ( ( ( x e. RR /\ y e. RR ) /\ ( z e. RR /\ w e. RR ) ) -> ( ( x + ( _i x. y ) ) # ( z + ( _i x. w ) ) <-> ( x # z \/ y # w ) ) )
15	14	notbid 662	. . . . . . . . . . 11 |- ( ( ( x e. RR /\ y e. RR ) /\ ( z e. RR /\ w e. RR ) ) -> ( -. ( x + ( _i x. y ) ) # ( z + ( _i x. w ) ) <-> -. ( x # z \/ y # w ) ) )
16		ioran 747	. . . . . . . . . . 11 |- ( -. ( x # z \/ y # w ) <-> ( -. x # z /\ -. y # w ) )
17	15, 16	bitrdi 195	. . . . . . . . . 10 |- ( ( ( x e. RR /\ y e. RR ) /\ ( z e. RR /\ w e. RR ) ) -> ( -. ( x + ( _i x. y ) ) # ( z + ( _i x. w ) ) <-> ( -. x # z /\ -. y # w ) ) )
18	7, 8, 17	syl2anc 409	. . . . . . . . 9 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) -> ( -. ( x + ( _i x. y ) ) # ( z + ( _i x. w ) ) <-> ( -. x # z /\ -. y # w ) ) )
19	11, 12	breq12d 4002	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) -> ( A # B <-> ( x + ( _i x. y ) ) # ( z + ( _i x. w ) ) ) )
20	19	notbid 662	. . . . . . . . 9 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) -> ( -. A # B <-> -. ( x + ( _i x. y ) ) # ( z + ( _i x. w ) ) ) )
21	7	simpld 111	. . . . . . . . . . 11 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) -> x e. RR )
22	8	simpld 111	. . . . . . . . . . 11 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) -> z e. RR )
23		reapti 8498	. . . . . . . . . . . 12 |- ( ( x e. RR /\ z e. RR ) -> ( x = z <-> -. x #ℝ z ) )
24		apreap 8506	. . . . . . . . . . . . 13 |- ( ( x e. RR /\ z e. RR ) -> ( x # z <-> x #ℝ z ) )
25	24	notbid 662	. . . . . . . . . . . 12 |- ( ( x e. RR /\ z e. RR ) -> ( -. x # z <-> -. x #ℝ z ) )
26	23, 25	bitr4d 190	. . . . . . . . . . 11 |- ( ( x e. RR /\ z e. RR ) -> ( x = z <-> -. x # z ) )
27	21, 22, 26	syl2anc 409	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) -> ( x = z <-> -. x # z ) )
28	7	simprd 113	. . . . . . . . . . 11 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) -> y e. RR )
29	8	simprd 113	. . . . . . . . . . 11 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) -> w e. RR )
30		reapti 8498	. . . . . . . . . . . 12 |- ( ( y e. RR /\ w e. RR ) -> ( y = w <-> -. y #ℝ w ) )
31		apreap 8506	. . . . . . . . . . . . 13 |- ( ( y e. RR /\ w e. RR ) -> ( y # w <-> y #ℝ w ) )
32	31	notbid 662	. . . . . . . . . . . 12 |- ( ( y e. RR /\ w e. RR ) -> ( -. y # w <-> -. y #ℝ w ) )
33	30, 32	bitr4d 190	. . . . . . . . . . 11 |- ( ( y e. RR /\ w e. RR ) -> ( y = w <-> -. y # w ) )
34	28, 29, 33	syl2anc 409	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) -> ( y = w <-> -. y # w ) )
35	27, 34	anbi12d 470	. . . . . . . . 9 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) -> ( ( x = z /\ y = w ) <-> ( -. x # z /\ -. y # w ) ) )
36	18, 20, 35	3bitr4d 219	. . . . . . . 8 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) -> ( -. A # B <-> ( x = z /\ y = w ) ) )
37	10, 13, 36	3bitr4d 219	. . . . . . 7 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) -> ( A = B <-> -. A # B ) )
38	37	ex 114	. . . . . 6 |- ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) /\ ( z e. RR /\ w e. RR ) ) -> ( B = ( z + ( _i x. w ) ) -> ( A = B <-> -. A # B ) ) )
39	38	rexlimdvva 2595	. . . . 5 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( E. z e. RR E. w e. RR B = ( z + ( _i x. w ) ) -> ( A = B <-> -. A # B ) ) )
40	5, 39	mpd 13	. . . 4 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( A = B <-> -. A # B ) )
41	40	ex 114	. . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ ( x e. RR /\ y e. RR ) ) -> ( A = ( x + ( _i x. y ) ) -> ( A = B <-> -. A # B ) ) )
42	41	rexlimdvva 2595	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( E. x e. RR E. y e. RR A = ( x + ( _i x. y ) ) -> ( A = B <-> -. A # B ) ) )
43	2, 42	mpd 13	1 |- ( ( A e. CC /\ B e. CC ) -> ( A = B <-> -. A # B ) )

Syntax hints:   -. wn 3   -> 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:  apne  8542  apcon4bid  8543  cnstab  8564  qapne  9598  expeq0  10507  nn0opthd  10656  recvguniq  10959  climuni  11256  dedekindeu  13395  dedekindicclemicc  13404  ivthinc  13415  limcimo  13428  cnplimclemle  13431  coseq0q4123  13549  cos11  13568  refeq  14060  triap  14061