Theorem apneg 8885

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 8270	. . 3 |- ( B e. CC -> E. z e. RR E. w e. RR B = ( z + ( _i x. w ) ) )
2	1	adantl 277	. 2 |- ( ( A e. CC /\ B e. CC ) -> E. z e. RR E. w e. RR B = ( z + ( _i x. w ) ) )
3		cnre 8270	. . . . . 6 |- ( A e. CC -> E. x e. RR E. y e. RR A = ( x + ( _i x. y ) ) )
4	3	ad3antrrr 492	. . . . 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 110	. . . . . . . . 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 536	. . . . . . . . 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 4122	. . . . . . . 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 537	. . . . . . . . 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 538	. . . . . . . . 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 531	. . . . . . . . . 10 |- ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) -> z e. RR )
11	10	ad3antrrr 492	. . . . . . . . 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 533	. . . . . . . . . 10 |- ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) -> w e. RR )
13	12	ad3antrrr 492	. . . . . . . . 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 8877	. . . . . . . . 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 1275	. . . . . . . 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 8653	. . . . . . . . . 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 8653	. . . . . . . . . 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 8653	. . . . . . . . . 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 8653	. . . . . . . . . 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 8877	. . . . . . . . . 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 1275	. . . . . . . . 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 8302	. . . . . . . . . . . 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 8222	. . . . . . . . . . . . . 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 8302	. . . . . . . . . . . . 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 8294	. . . . . . . . . . . 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 8597	. . . . . . . . . . 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 8468	. . . . . . . . . . 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 8685	. . . . . . . . . . . 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 6066	. . . . . . . . . . 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 2275	. . . . . . . . . 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 8302	. . . . . . . . . . . 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 8302	. . . . . . . . . . . . 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 8294	. . . . . . . . . . . 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 8597	. . . . . . . . . . 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 8468	. . . . . . . . . . 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 8685	. . . . . . . . . . . 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 6066	. . . . . . . . . . 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 2275	. . . . . . . . . 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 4122	. . . . . . . . 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 8871	. . . . . . . . . . 11 |- ( ( x e. RR /\ z e. RR ) -> ( x # z <-> -u x # -u z ) )
42	8, 11, 41	syl2anc 411	. . . . . . . . . 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 8871	. . . . . . . . . . 11 |- ( ( y e. RR /\ w e. RR ) -> ( y # w <-> -u y # -u w ) )
44	9, 13, 43	syl2anc 411	. . . . . . . . . 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 801	. . . . . . . . 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 221	. . . . . . . 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 214	. . . . . . 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 115	. . . . . 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 2668	. . . . 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 115	. . 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 2668	. 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 104   <-> wb 105   \/ wo 716   = wceq 1398   e. wcel 2203   E.wrex 2521   class class class wbr 4109  (class class class)co 6050   CCcc 8125   RRcr 8126   _ici 8129   + caddc 8130   x. cmul 8132   -ucneg 8445   # cap 8855

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-pnf 8310  df-mnf 8311  df-ltxr 8313  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856

This theorem is referenced by:  mulext1  8886  negap0  8904  div2subap  9111  cjap  11591  geosergap  12192