Theorem renegcli 10342

Description: Closure law for negative of reals. (Note: this inference proof style and the deduction theorem usage in renegcl 10344 is deprecated, but is retained for its demonstration value.) (Contributed by NM, 17-Jan-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Hypothesis

Ref	Expression
renegcl.1	|- A e. RR

Assertion

Ref	Expression
renegcli	|- -u A e. RR

Proof of Theorem renegcli

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		renegcl.1	. 2 |- A e. RR
2		ax-rnegex 10007	. 2 |- ( A e. RR -> E. x e. RR ( A + x ) = 0 )
3		recn 10026	. . . . 5 |- ( x e. RR -> x e. CC )
4		df-neg 10269	. . . . . . 7 |- -u A = ( 0 - A )
5	4	eqeq1i 2627	. . . . . 6 |- ( -u A = x <-> ( 0 - A ) = x )
6		0cn 10032	. . . . . . 7 |- 0 e. CC
7	1	recni 10052	. . . . . . 7 |- A e. CC
8		subadd 10284	. . . . . . 7 |- ( ( 0 e. CC /\ A e. CC /\ x e. CC ) -> ( ( 0 - A ) = x <-> ( A + x ) = 0 ) )
9	6, 7, 8	mp3an12 1414	. . . . . 6 |- ( x e. CC -> ( ( 0 - A ) = x <-> ( A + x ) = 0 ) )
10	5, 9	syl5bb 272	. . . . 5 |- ( x e. CC -> ( -u A = x <-> ( A + x ) = 0 ) )
11	3, 10	syl 17	. . . 4 |- ( x e. RR -> ( -u A = x <-> ( A + x ) = 0 ) )
12		eleq1a 2696	. . . 4 |- ( x e. RR -> ( -u A = x -> -u A e. RR ) )
13	11, 12	sylbird 250	. . 3 |- ( x e. RR -> ( ( A + x ) = 0 -> -u A e. RR ) )
14	13	rexlimiv 3027	. 2 |- ( E. x e. RR ( A + x ) = 0 -> -u A e. RR )
15	1, 2, 14	mp2b 10	1 |- -u A e. RR

Syntax hints:   <-> wb 196   = wceq 1483   e. wcel 1990   E.wrex 2913  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   + caddc 9939   - cmin 10266   -ucneg 10267

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-ltxr 10079  df-sub 10268  df-neg 10269

This theorem is referenced by:  resubcli  10343  renegcl  10344  recgt0ii  10929  inelr  11010  cju  11016  neg1rr  11125  sincos2sgn  14924  dvdslelem  15031  divalglem1  15117  divalglem6  15121  modsubi  15776  neghalfpire  24217  coseq0negpitopi  24255  pige3  24269  negpitopissre  24286  eff1o  24295  ellogrn  24306  logimclad  24319  logneg  24334  logcj  24352  argregt0  24356  argrege0  24357  argimgt0  24358  argimlt0  24359  logimul  24360  logneg2  24361  logcnlem3  24390  dvloglem  24394  logf1o2  24396  efopnlem2  24403  cxpsqrtlem  24448  abscxpbnd  24494  logreclem  24500  ang180lem2  24540  asinneg  24613  asinsin  24619  asin1  24621  asinrecl  24629  atanlogaddlem  24640  atanlogsublem  24642  atanlogsub  24643  atantan  24650  atanbndlem  24652  birthday  24681  ppiub  24929  lgsdir2lem1  25050  ex-fl  27304  ex-ceil  27305  normlem2  27968  logdivsqrle  30728  logi  31620  bj-pinftyccb  33108  bj-minftyccb  33112  bj-pinftynminfty  33114  cos2h  33400  tan2h  33401  renegclALT  34249  fourierdlem5  40329  fourierdlem9  40333  fourierdlem18  40342  fourierdlem24  40348  fourierdlem38  40362  fourierdlem40  40364  fourierdlem43  40367  fourierdlem44  40368  fourierdlem46  40369  fourierdlem50  40373  fourierdlem62  40385  fourierdlem66  40389  fourierdlem74  40397  fourierdlem75  40398  fourierdlem76  40399  fourierdlem77  40400  fourierdlem78  40401  fourierdlem83  40406  fourierdlem85  40408  fourierdlem87  40410  fourierdlem88  40411  fourierdlem93  40416  fourierdlem94  40417  fourierdlem95  40418  fourierdlem101  40424  fourierdlem102  40425  fourierdlem103  40426  fourierdlem104  40427  fourierdlem111  40434  fourierdlem112  40435  fourierdlem113  40436  fourierdlem114  40437  sqwvfoura  40445  sqwvfourb  40446  fouriersw  40448  fouriercn  40449