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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 8016 | . 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 1480 ℝcr 7612 -cneg 7927 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-setind 4447 ax-resscn 7705 ax-1cn 7706 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-addass 7715 ax-distr 7717 ax-i2m1 7718 ax-0id 7721 ax-rnegex 7722 ax-cnre 7724 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-sub 7928 df-neg 7929 |
This theorem is referenced by: eqord2 8239 possumd 8324 reapmul1 8350 reapneg 8352 apneg 8366 mulext1 8367 recgt0 8601 prodgt0 8603 prodge0 8605 negiso 8706 nnnegz 9050 peano2z 9083 supinfneg 9383 infsupneg 9384 monoord2 10243 recj 10632 reneg 10633 imcj 10640 imneg 10641 cjap 10671 resqrexlemcalc3 10781 resqrexlemgt0 10785 abslt 10853 absle 10854 minmax 10994 mincl 10995 lemininf 10998 ltmininf 10999 bdtri 11004 xrmaxaddlem 11022 xrminrpcl 11036 climge0 11087 cos12dec 11463 absefib 11466 efieq1re 11467 dvdslelemd 11530 infssuzex 11631 ivthdec 12780 coseq0negpitopi 12906 cosq34lt1 12920 |
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