Theorem apneg 8397

Description: Negation respects apartness. (Contributed by Jim Kingdon, 14-Feb-2020.)

Assertion

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

Proof of Theorem apneg

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

Step	Hyp	Ref	Expression
1		cnre 7786	. . 3 |- ( B e. CC -> E. z e. RR E. w e. RR B = ( z + ( _i x. w ) ) )
2	1	adantl 275	. 2 |- ( ( A e. CC /\ B e. CC ) -> E. z e. RR E. w e. RR B = ( z + ( _i x. w ) ) )
3		cnre 7786	. . . . . 6 |- ( A e. CC -> E. x e. RR E. y e. RR A = ( x + ( _i x. y ) ) )
4	3	ad3antrrr 484	. . . . 5 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) -> E. x e. RR E. y e. RR A = ( x + ( _i x. y ) ) )
5		simpr 109	. . . . . . . . 9 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> A = ( x + ( _i x. y ) ) )
6		simpllr 524	. . . . . . . . 9 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> B = ( z + ( _i x. w ) ) )
7	5, 6	breq12d 3950	. . . . . . . 8 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( A # B <-> ( x + ( _i x. y ) ) # ( z + ( _i x. w ) ) ) )
8		simplrl 525	. . . . . . . . 9 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> x e. RR )
9		simplrr 526	. . . . . . . . 9 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> y e. RR )
10		simprl 521	. . . . . . . . . 10 |- ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) -> z e. RR )
11	10	ad3antrrr 484	. . . . . . . . 9 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> z e. RR )
12		simprr 522	. . . . . . . . . 10 |- ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) -> w e. RR )
13	12	ad3antrrr 484	. . . . . . . . 9 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> w e. RR )
14		apreim 8389	. . . . . . . . 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 ) ) )
15	8, 9, 11, 13, 14	syl22anc 1218	. . . . . . . 8 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( ( x + ( _i x. y ) ) # ( z + ( _i x. w ) ) <-> ( x # z \/ y # w ) ) )
16	8	renegcld 8166	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> -u x e. RR )
17	9	renegcld 8166	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> -u y e. RR )
18	11	renegcld 8166	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> -u z e. RR )
19	13	renegcld 8166	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> -u w e. RR )
20		apreim 8389	. . . . . . . . . 10 |- ( ( ( -u x e. RR /\ -u y e. RR ) /\ ( -u z e. RR /\ -u w e. RR ) ) -> ( ( -u x + ( _i x. -u y ) ) # ( -u z + ( _i x. -u w ) ) <-> ( -u x # -u z \/ -u y # -u w ) ) )
21	16, 17, 18, 19, 20	syl22anc 1218	. . . . . . . . 9 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( ( -u x + ( _i x. -u y ) ) # ( -u z + ( _i x. -u w ) ) <-> ( -u x # -u z \/ -u y # -u w ) ) )
22	8	recnd 7818	. . . . . . . . . . . 12 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> x e. CC )
23		ax-icn 7739	. . . . . . . . . . . . . 14 |- _i e. CC
24	23	a1i 9	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> _i e. CC )
25	9	recnd 7818	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> y e. CC )
26	24, 25	mulcld 7810	. . . . . . . . . . . 12 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( _i x. y ) e. CC )
27	22, 26	negdid 8110	. . . . . . . . . . 11 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> -u ( x + ( _i x. y ) ) = ( -u x + -u ( _i x. y ) ) )
28	5	negeqd 7981	. . . . . . . . . . 11 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> -u A = -u ( x + ( _i x. y ) ) )
29	24, 25	mulneg2d 8198	. . . . . . . . . . . 12 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( _i x. -u y ) = -u ( _i x. y ) )
30	29	oveq2d 5798	. . . . . . . . . . 11 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( -u x + ( _i x. -u y ) ) = ( -u x + -u ( _i x. y ) ) )
31	27, 28, 30	3eqtr4d 2183	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> -u A = ( -u x + ( _i x. -u y ) ) )
32	11	recnd 7818	. . . . . . . . . . . 12 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> z e. CC )
33	13	recnd 7818	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> w e. CC )
34	24, 33	mulcld 7810	. . . . . . . . . . . 12 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( _i x. w ) e. CC )
35	32, 34	negdid 8110	. . . . . . . . . . 11 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> -u ( z + ( _i x. w ) ) = ( -u z + -u ( _i x. w ) ) )
36	6	negeqd 7981	. . . . . . . . . . 11 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> -u B = -u ( z + ( _i x. w ) ) )
37	24, 33	mulneg2d 8198	. . . . . . . . . . . 12 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( _i x. -u w ) = -u ( _i x. w ) )
38	37	oveq2d 5798	. . . . . . . . . . 11 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( -u z + ( _i x. -u w ) ) = ( -u z + -u ( _i x. w ) ) )
39	35, 36, 38	3eqtr4d 2183	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> -u B = ( -u z + ( _i x. -u w ) ) )
40	31, 39	breq12d 3950	. . . . . . . . 9 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( -u A # -u B <-> ( -u x + ( _i x. -u y ) ) # ( -u z + ( _i x. -u w ) ) ) )
41		reapneg 8383	. . . . . . . . . . 11 |- ( ( x e. RR /\ z e. RR ) -> ( x # z <-> -u x # -u z ) )
42	8, 11, 41	syl2anc 409	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( x # z <-> -u x # -u z ) )
43		reapneg 8383	. . . . . . . . . . 11 |- ( ( y e. RR /\ w e. RR ) -> ( y # w <-> -u y # -u w ) )
44	9, 13, 43	syl2anc 409	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( y # w <-> -u y # -u w ) )
45	42, 44	orbi12d 783	. . . . . . . . 9 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( ( x # z \/ y # w ) <-> ( -u x # -u z \/ -u y # -u w ) ) )
46	21, 40, 45	3bitr4rd 220	. . . . . . . 8 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( ( x # z \/ y # w ) <-> -u A # -u B ) )
47	7, 15, 46	3bitrd 213	. . . . . . 7 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( A # B <-> -u A # -u B ) )
48	47	ex 114	. . . . . 6 |- ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) -> ( A = ( x + ( _i x. y ) ) -> ( A # B <-> -u A # -u B ) ) )
49	48	rexlimdvva 2560	. . . . 5 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) -> ( E. x e. RR E. y e. RR A = ( x + ( _i x. y ) ) -> ( A # B <-> -u A # -u B ) ) )
50	4, 49	mpd 13	. . . 4 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) -> ( A # B <-> -u A # -u B ) )
51	50	ex 114	. . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) -> ( B = ( z + ( _i x. w ) ) -> ( A # B <-> -u A # -u B ) ) )
52	51	rexlimdvva 2560	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( E. z e. RR E. w e. RR B = ( z + ( _i x. w ) ) -> ( A # B <-> -u A # -u B ) ) )
53	2, 52	mpd 13	1 |- ( ( A e. CC /\ B e. CC ) -> ( A # B <-> -u A # -u B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698   = wceq 1332   e. wcel 1481   E.wrex 2418   class class class wbr 3937  (class class class)co 5782   CCcc 7642   RRcr 7643   _ici 7646   + caddc 7647   x. cmul 7649   -ucneg 7958   # cap 8367

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-mulrcl 7743  ax-addcom 7744  ax-mulcom 7745  ax-addass 7746  ax-mulass 7747  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-1rid 7751  ax-0id 7752  ax-rnegex 7753  ax-precex 7754  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760  ax-pre-mulgt0 7761

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-iota 5096  df-fun 5133  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-pnf 7826  df-mnf 7827  df-ltxr 7829  df-sub 7959  df-neg 7960  df-reap 8361  df-ap 8368

This theorem is referenced by:  mulext1  8398  negap0  8416  div2subap  8620  cjap  10710  geosergap  11307