Theorem apti 8958

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 8328	. . 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 8328	. . . . . . 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 8938	. . . . . . . . 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 8939	. . . . . . . . . . . 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 4144	. . . . . . . . . 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 8915	. . . . . . . . . . . 12 |- ( ( x e. RR /\ z e. RR ) -> ( x = z <-> -. x #ℝ z ) )
24		apreap 8923	. . . . . . . . . . . . 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 8915	. . . . . . . . . . . 12 |- ( ( y e. RR /\ w e. RR ) -> ( y = w <-> -. y #ℝ w ) )
31		apreap 8923	. . . . . . . . . . . . 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 4131  (class class class)co 6088   CCcc 8183   RRcr 8184   _ici 8187   + caddc 8188   x. cmul 8190   #_ℝ creap 8910   # cap 8917

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 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302

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 4132  df-opab 4194  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-iota 5340  df-fun 5382  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-pnf 8368  df-mnf 8369  df-ltxr 8371  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918

This theorem is used by:  apne  8959  apcon4bid  8960  cnstab  8981  aptap  8986  qapne  10055  expeq0  11028  nn0opthd  11182  sq01  11682  recvguniq  11783  climuni  12084  dedekindeu  15849  dedekindicclemicc  15858  ivthinc  15869  limcimo  15891  cnplimclemle  15894  coseq0q4123  16061  cos11  16081  refeq  17279  triap  17284