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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 8150 | . 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 2135 ℝcr 7743 -cneg 8061 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-setind 4508 ax-resscn 7836 ax-1cn 7837 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-distr 7848 ax-i2m1 7849 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-sub 8062 df-neg 8063 |
This theorem is referenced by: eqord2 8373 possumd 8458 reapmul1 8484 reapneg 8486 apneg 8500 mulext1 8501 recgt0 8736 prodgt0 8738 prodge0 8740 negiso 8841 nnnegz 9185 peano2z 9218 nn0negleid 9250 difgtsumgt 9251 supinfneg 9524 infsupneg 9525 monoord2 10402 recj 10795 reneg 10796 imcj 10803 imneg 10804 cjap 10834 resqrexlemcalc3 10944 resqrexlemgt0 10948 abslt 11016 absle 11017 minmax 11157 mincl 11158 lemininf 11161 ltmininf 11162 bdtri 11167 xrmaxaddlem 11187 xrminrpcl 11201 climge0 11252 cos12dec 11694 absefib 11697 efieq1re 11698 dvdslelemd 11766 infssuzex 11867 zsupssdc 11872 ivthdec 13163 coseq0negpitopi 13298 cosq34lt1 13312 rpabscxpbnd 13400 |
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