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Theorem renegcli 7734
Description: Closure law for negative of reals. (Note: this inference proof style and the deduction theorem usage in renegcl 7733 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 ∈ ℝ
Assertion
Ref Expression
renegcli -𝐴 ∈ ℝ

Proof of Theorem renegcli
StepHypRef Expression
1 renegcl.1 . 2 𝐴 ∈ ℝ
2 renegcl 7733 . 2 (𝐴 ∈ ℝ → -𝐴 ∈ ℝ)
31, 2ax-mp 7 1 -𝐴 ∈ ℝ
Colors of variables: wff set class
Syntax hints:  wcel 1438  cr 7339  -cneg 7644
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3955  ax-pow 4007  ax-pr 4034  ax-setind 4351  ax-resscn 7427  ax-1cn 7428  ax-icn 7430  ax-addcl 7431  ax-addrcl 7432  ax-mulcl 7433  ax-addcom 7435  ax-addass 7437  ax-distr 7439  ax-i2m1 7440  ax-0id 7443  ax-rnegex 7444  ax-cnre 7446
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-br 3844  df-opab 3898  df-id 4118  df-xp 4442  df-rel 4443  df-cnv 4444  df-co 4445  df-dm 4446  df-iota 4975  df-fun 5012  df-fv 5018  df-riota 5600  df-ov 5647  df-oprab 5648  df-mpt2 5649  df-sub 7645  df-neg 7646
This theorem is referenced by:  resubcli  7735  inelr  8051  cju  8411  neg1rr  8518  sincos2sgn  11043  ex-fl  11535
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