Theorem apti 8569

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 7944	. . 3 |- ( A e. CC -> E. x e. RR E. y e. RR A = ( x + ( _i x. y ) ) )
2	1	adantr 276	. 2 |- ( ( A e. CC /\ B e. CC ) -> E. x e. RR E. y e. RR A = ( x + ( _i x. y ) ) )
3		cnre 7944	. . . . . . 7 |- ( B e. CC -> E. z e. RR E. w e. RR B = ( z + ( _i x. w ) ) )
4	3	adantl 277	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> E. z e. RR E. w e. RR B = ( z + ( _i x. w ) ) )
5	4	ad2antrr 488	. . . . 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 110	. . . . . . . . . 10 |- ( ( ( A e. CC /\ B e. CC ) /\ ( x e. RR /\ y e. RR ) ) -> ( x e. RR /\ y e. RR ) )
7	6	ad3antrrr 492	. . . . . . . . 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 528	. . . . . . . . 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 8549	. . . . . . . . 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 411	. . . . . . . 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 534	. . . . . . . . 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 110	. . . . . . . . 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 2192	. . . . . . . 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 8550	. . . . . . . . . . . 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 667	. . . . . . . . . . 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 752	. . . . . . . . . . 11 |- ( -. ( x # z \/ y # w ) <-> ( -. x # z /\ -. y # w ) )
17	15, 16	bitrdi 196	. . . . . . . . . 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 411	. . . . . . . . 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 4013	. . . . . . . . . 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 667	. . . . . . . . 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 112	. . . . . . . . . . 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 112	. . . . . . . . . . 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 8526	. . . . . . . . . . . 12 |- ( ( x e. RR /\ z e. RR ) -> ( x = z <-> -. x #ℝ z ) )
24		apreap 8534	. . . . . . . . . . . . 13 |- ( ( x e. RR /\ z e. RR ) -> ( x # z <-> x #ℝ z ) )
25	24	notbid 667	. . . . . . . . . . . 12 |- ( ( x e. RR /\ z e. RR ) -> ( -. x # z <-> -. x #ℝ z ) )
26	23, 25	bitr4d 191	. . . . . . . . . . 11 |- ( ( x e. RR /\ z e. RR ) -> ( x = z <-> -. x # z ) )
27	21, 22, 26	syl2anc 411	. . . . . . . . . 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 114	. . . . . . . . . . 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 114	. . . . . . . . . . 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 8526	. . . . . . . . . . . 12 |- ( ( y e. RR /\ w e. RR ) -> ( y = w <-> -. y #ℝ w ) )
31		apreap 8534	. . . . . . . . . . . . 13 |- ( ( y e. RR /\ w e. RR ) -> ( y # w <-> y #ℝ w ) )
32	31	notbid 667	. . . . . . . . . . . 12 |- ( ( y e. RR /\ w e. RR ) -> ( -. y # w <-> -. y #ℝ w ) )
33	30, 32	bitr4d 191	. . . . . . . . . . 11 |- ( ( y e. RR /\ w e. RR ) -> ( y = w <-> -. y # w ) )
34	28, 29, 33	syl2anc 411	. . . . . . . . . 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 473	. . . . . . . . 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 220	. . . . . . . 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 220	. . . . . . 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 115	. . . . . 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 2602	. . . . 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 115	. . 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 2602	. 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 104   <-> wb 105   \/ wo 708   = wceq 1353   e. wcel 2148   E.wrex 2456   class class class wbr 4000  (class class class)co 5869   CCcc 7800   RRcr 7801   _ici 7804   + caddc 7805   x. cmul 7807   #_ℝ creap 8521   # cap 8528

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-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-iota 5174  df-fun 5214  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-pnf 7984  df-mnf 7985  df-ltxr 7987  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529

This theorem is referenced by:  apne  8570  apcon4bid  8571  cnstab  8592  qapne  9628  expeq0  10537  nn0opthd  10686  recvguniq  10988  climuni  11285  dedekindeu  13768  dedekindicclemicc  13777  ivthinc  13788  limcimo  13801  cnplimclemle  13804  coseq0q4123  13922  cos11  13941  refeq  14432  triap  14433