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Theorem renegcli 8267
Description: Closure law for negative of reals. (Note: this inference proof style and the deduction theorem usage in renegcl 8266 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 8266 . 2 (𝐴 ∈ ℝ → -𝐴 ∈ ℝ)
31, 2ax-mp 5 1 -𝐴 ∈ ℝ
Colors of variables: wff set class
Syntax hints:  wcel 2160  cr 7857  -cneg 8177
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4143  ax-pow 4199  ax-pr 4234  ax-setind 4561  ax-resscn 7950  ax-1cn 7951  ax-icn 7953  ax-addcl 7954  ax-addrcl 7955  ax-mulcl 7956  ax-addcom 7958  ax-addass 7960  ax-distr 7962  ax-i2m1 7963  ax-0id 7966  ax-rnegex 7967  ax-cnre 7969
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2758  df-sbc 2982  df-dif 3151  df-un 3153  df-in 3155  df-ss 3162  df-pw 3599  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3832  df-br 4026  df-opab 4087  df-id 4318  df-xp 4657  df-rel 4658  df-cnv 4659  df-co 4660  df-dm 4661  df-iota 5203  df-fun 5244  df-fv 5250  df-riota 5861  df-ov 5909  df-oprab 5910  df-mpo 5911  df-sub 8178  df-neg 8179
This theorem is referenced by:  resubcli  8268  inelr  8589  cju  8966  neg1rr  9074  sincos2sgn  11883  neghalfpire  14856  coseq0negpitopi  14899  negpitopissre  14918  rpabscxpbnd  15001  ex-fl  15141
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