Theorem renegcli 8396

Description: Closure law for negative of reals. (Note: this inference proof style and the deduction theorem usage in renegcl 8395 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

Step	Hyp	Ref	Expression
1		renegcl.1	. 2 |- A e. RR
2		renegcl 8395	. 2 |- ( A e. RR -> -u A e. RR )
3	1, 2	ax-mp 5	1 |- -u A e. RR

Syntax hints:   e. wcel 2200   RRcr 7986   -ucneg 8306

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-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-setind 4626  ax-resscn 8079  ax-1cn 8080  ax-icn 8082  ax-addcl 8083  ax-addrcl 8084  ax-mulcl 8085  ax-addcom 8087  ax-addass 8089  ax-distr 8091  ax-i2m1 8092  ax-0id 8095  ax-rnegex 8096  ax-cnre 8098

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-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 3888  df-br 4083  df-opab 4145  df-id 4381  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-iota 5274  df-fun 5316  df-fv 5322  df-riota 5947  df-ov 5997  df-oprab 5998  df-mpo 5999  df-sub 8307  df-neg 8308

This theorem is referenced by:  resubcli  8397  inelr  8719  cju  9096  neg1rr  9204  sincos2sgn  12263  neghalfpire  15452  coseq0negpitopi  15495  negpitopissre  15514  rpabscxpbnd  15599  ex-fl  16019