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Mirrors > Home > ILE Home > Th. List > renegcld | GIF version |
Description: Closure law for negative of reals. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
renegcld.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
Ref | Expression |
---|---|
renegcld | ⊢ (𝜑 → -𝐴 ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | renegcld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | renegcl 7991 | . 2 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → -𝐴 ∈ ℝ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1465 ℝcr 7587 -cneg 7902 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-setind 4422 ax-resscn 7680 ax-1cn 7681 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-distr 7692 ax-i2m1 7693 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-sub 7903 df-neg 7904 |
This theorem is referenced by: eqord2 8214 possumd 8299 reapmul1 8325 reapneg 8327 apneg 8341 mulext1 8342 recgt0 8576 prodgt0 8578 prodge0 8580 negiso 8681 nnnegz 9025 peano2z 9058 supinfneg 9358 infsupneg 9359 monoord2 10218 recj 10607 reneg 10608 imcj 10615 imneg 10616 cjap 10646 resqrexlemcalc3 10756 resqrexlemgt0 10760 abslt 10828 absle 10829 minmax 10969 mincl 10970 lemininf 10973 ltmininf 10974 bdtri 10979 xrmaxaddlem 10997 xrminrpcl 11011 climge0 11062 cos12dec 11401 absefib 11404 efieq1re 11405 dvdslelemd 11468 infssuzex 11569 ivthdec 12718 coseq0negpitopi 12844 cosq34lt1 12858 |
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