Theorem eqneg 8879

Description: A number equal to its negative is zero. (Contributed by NM, 12-Jul-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Assertion

Ref	Expression
eqneg	|- ( A e. CC -> ( A = -u A <-> A = 0 ) )

Proof of Theorem eqneg

Step	Hyp	Ref	Expression
1		1p1times 8280	. . 3 |- ( A e. CC -> ( ( 1 + 1 ) x. A ) = ( A + A ) )
2		ax-1cn 8092	. . . . . 6 |- 1 e. CC
3	2, 2	addcli 8150	. . . . 5 |- ( 1 + 1 ) e. CC
4	3	mul01i 8537	. . . 4 |- ( ( 1 + 1 ) x. 0 ) = 0
5		negid 8393	. . . 4 |- ( A e. CC -> ( A + -u A ) = 0 )
6	4, 5	eqtr4id 2281	. . 3 |- ( A e. CC -> ( ( 1 + 1 ) x. 0 ) = ( A + -u A ) )
7	1, 6	eqeq12d 2244	. 2 |- ( A e. CC -> ( ( ( 1 + 1 ) x. A ) = ( ( 1 + 1 ) x. 0 ) <-> ( A + A ) = ( A + -u A ) ) )
8		id 19	. . 3 |- ( A e. CC -> A e. CC )
9		0cnd 8139	. . 3 |- ( A e. CC -> 0 e. CC )
10	3	a1i 9	. . 3 |- ( A e. CC -> ( 1 + 1 ) e. CC )
11		1re 8145	. . . . . 6 |- 1 e. RR
12	11, 11	readdcli 8159	. . . . 5 |- ( 1 + 1 ) e. RR
13		0lt1 8273	. . . . . 6 |- 0 < 1
14	11, 11, 13, 13	addgt0ii 8638	. . . . 5 |- 0 < ( 1 + 1 )
15	12, 14	gt0ap0ii 8775	. . . 4 |- ( 1 + 1 ) # 0
16	15	a1i 9	. . 3 |- ( A e. CC -> ( 1 + 1 ) # 0 )
17	8, 9, 10, 16	mulcanapd 8808	. 2 |- ( A e. CC -> ( ( ( 1 + 1 ) x. A ) = ( ( 1 + 1 ) x. 0 ) <-> A = 0 ) )
18		negcl 8346	. . 3 |- ( A e. CC -> -u A e. CC )
19	8, 8, 18	addcand 8330	. 2 |- ( A e. CC -> ( ( A + A ) = ( A + -u A ) <-> A = -u A ) )
20	7, 17, 19	3bitr3rd 219	1 |- ( A e. CC -> ( A = -u A <-> A = 0 ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1395   e. wcel 2200   class class class wbr 4083  (class class class)co 6001   CCcc 7997   0cc0 7999   1c1 8000   + caddc 8002   x. cmul 8004   -ucneg 8318   # cap 8728

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-id 4384  df-po 4387  df-iso 4388  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729

This theorem is referenced by:  eqnegd  8880  eqnegi  8888