Theorem apti 8950

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 8322	. . 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 8322	. . . . . . 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 492	. . . . 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 496	. . . . . . . . 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 533	. . . . . . . . 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 8930	. . . . . . . . 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 415	. . . . . . . 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 540	. . . . . . . . 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 2253	. . . . . . . 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 8931	. . . . . . . . . . . 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 677	. . . . . . . . . . 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 764	. . . . . . . . . . 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 415	. . . . . . . . 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 4143	. . . . . . . . . 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 677	. . . . . . . . 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 8907	. . . . . . . . . . . 12 |- ( ( x e. RR /\ z e. RR ) -> ( x = z <-> -. x #ℝ z ) )
24		apreap 8915	. . . . . . . . . . . . 13 |- ( ( x e. RR /\ z e. RR ) -> ( x # z <-> x #ℝ z ) )
25	24	notbid 677	. . . . . . . . . . . 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 415	. . . . . . . . . 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 8907	. . . . . . . . . . . 12 |- ( ( y e. RR /\ w e. RR ) -> ( y = w <-> -. y #ℝ w ) )
31		apreap 8915	. . . . . . . . . . . . 13 |- ( ( y e. RR /\ w e. RR ) -> ( y # w <-> y #ℝ w ) )
32	31	notbid 677	. . . . . . . . . . . 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 415	. . . . . . . . . 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 477	. . . . . . . . 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 2676	. . . . 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 2676	. 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 ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   = wceq 1402   e. wcel 2209   E.wrex 2529   class class class wbr 4130  (class class class)co 6085   CCcc 8177   RRcr 8178   _ici 8181   + caddc 8182   x. cmul 8184   #_ℝ creap 8902   # cap 8909

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-iota 5337  df-fun 5379  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-ltxr 8365  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910

This theorem is used by:  apne  8951  apcon4bid  8952  cnstab  8973  aptap  8978  qapne  10039  expeq0  11007  nn0opthd  11160  sq01  11660  recvguniq  11761  climuni  12059  dedekindeu  15724  dedekindicclemicc  15733  ivthinc  15744  limcimo  15766  cnplimclemle  15769  coseq0q4123  15935  cos11  15954  refeq  17073  triap  17078