Theorem apti 8579

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 7953	. . 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 7953	. . . . . . 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 8559	. . . . . . . . 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 8560	. . . . . . . . . . . 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 4017	. . . . . . . . . 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 8536	. . . . . . . . . . . 12 |- ( ( x e. RR /\ z e. RR ) -> ( x = z <-> -. x #ℝ z ) )
24		apreap 8544	. . . . . . . . . . . . 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 8536	. . . . . . . . . . . 12 |- ( ( y e. RR /\ w e. RR ) -> ( y = w <-> -. y #ℝ w ) )
31		apreap 8544	. . . . . . . . . . . . 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 4004  (class class class)co 5875   CCcc 7809   RRcr 7810   _ici 7813   + caddc 7814   x. cmul 7816   #_ℝ creap 8531   # cap 8538

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 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-mulrcl 7910  ax-addcom 7911  ax-mulcom 7912  ax-addass 7913  ax-mulass 7914  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-1rid 7918  ax-0id 7919  ax-rnegex 7920  ax-precex 7921  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-lttrn 7925  ax-pre-apti 7926  ax-pre-ltadd 7927  ax-pre-mulgt0 7928

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 2740  df-sbc 2964  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-iota 5179  df-fun 5219  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-pnf 7994  df-mnf 7995  df-ltxr 7997  df-sub 8130  df-neg 8131  df-reap 8532  df-ap 8539

This theorem is referenced by:  apne  8580  apcon4bid  8581  cnstab  8602  aptap  8607  qapne  9639  expeq0  10551  nn0opthd  10702  recvguniq  11004  climuni  11301  dedekindeu  14104  dedekindicclemicc  14113  ivthinc  14124  limcimo  14137  cnplimclemle  14140  coseq0q4123  14258  cos11  14277  refeq  14779  triap  14780