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Mirrors > Home > ILE Home > Th. List > renegcld | Unicode version |
Description: Closure law for negative of reals. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
renegcld.1 |
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Ref | Expression |
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renegcld |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | renegcld.1 |
. 2
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2 | renegcl 8217 |
. 2
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3 | 1, 2 | syl 14 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-setind 4536 ax-resscn 7902 ax-1cn 7903 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-addcom 7910 ax-addass 7912 ax-distr 7914 ax-i2m1 7915 ax-0id 7918 ax-rnegex 7919 ax-cnre 7921 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-iota 5178 df-fun 5218 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-sub 8129 df-neg 8130 |
This theorem is referenced by: eqord2 8440 possumd 8525 reapmul1 8551 reapneg 8553 apneg 8567 mulext1 8568 recgt0 8806 prodgt0 8808 prodge0 8810 negiso 8911 nnnegz 9255 peano2z 9288 nn0negleid 9320 difgtsumgt 9321 supinfneg 9594 infsupneg 9595 monoord2 10476 recj 10875 reneg 10876 imcj 10883 imneg 10884 cjap 10914 resqrexlemcalc3 11024 resqrexlemgt0 11028 abslt 11096 absle 11097 minmax 11237 mincl 11238 lemininf 11241 ltmininf 11242 bdtri 11247 xrmaxaddlem 11267 xrminrpcl 11281 climge0 11332 cos12dec 11774 absefib 11777 efieq1re 11778 dvdslelemd 11848 infssuzex 11949 zsupssdc 11954 mulgnegnn 12992 ivthdec 14058 coseq0negpitopi 14193 cosq34lt1 14207 rpabscxpbnd 14295 lgsneg 14361 lgsdilem 14364 lgseisenlem1 14386 |
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