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Mirrors > Home > ILE Home > Th. List > resubcld | Unicode version |
Description: Closure law for subtraction of reals. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
renegcld.1 | |
resubcld.2 |
Ref | Expression |
---|---|
resubcld |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | renegcld.1 | . 2 | |
2 | resubcld.2 | . 2 | |
3 | resubcl 8139 | . 2 | |
4 | 1, 2, 3 | syl2anc 409 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wcel 2128 (class class class)co 5824 cr 7731 cmin 8046 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-setind 4496 ax-resscn 7824 ax-1cn 7825 ax-icn 7827 ax-addcl 7828 ax-addrcl 7829 ax-mulcl 7830 ax-addcom 7832 ax-addass 7834 ax-distr 7836 ax-i2m1 7837 ax-0id 7840 ax-rnegex 7841 ax-cnre 7843 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-opab 4026 df-id 4253 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-iota 5135 df-fun 5172 df-fv 5178 df-riota 5780 df-ov 5827 df-oprab 5828 df-mpo 5829 df-sub 8048 df-neg 8049 |
This theorem is referenced by: ltsubadd 8307 lesubadd 8309 ltaddsub 8311 leaddsub 8313 lesub1 8331 lesub2 8332 ltsub1 8333 ltsub2 8334 lt2sub 8335 le2sub 8336 rereim 8461 ltmul1a 8466 cru 8477 lemul1a 8729 ztri3or 9210 lincmb01cmp 9907 iccf1o 9908 rebtwn2z 10154 qbtwnrelemcalc 10155 qbtwnre 10156 intfracq 10219 modqval 10223 modqlt 10232 modqsubdir 10292 ser3le 10417 expnbnd 10541 crre 10757 remullem 10771 recvguniqlem 10894 resqrexlemover 10910 resqrexlemcalc2 10915 resqrexlemcalc3 10916 resqrexlemnmsq 10917 resqrexlemnm 10918 resqrexlemcvg 10919 resqrexlemglsq 10922 resqrexlemga 10923 fzomaxdiflem 11012 caubnd2 11017 amgm2 11018 icodiamlt 11080 qdenre 11102 maxabslemab 11106 maxabslemlub 11107 maxltsup 11118 bdtrilem 11138 bdtri 11139 mulcn2 11209 reccn2ap 11210 climle 11231 climsqz 11232 climsqz2 11233 climcvg1nlem 11246 fsumle 11360 cvgratnnlembern 11420 cvgratnnlemsumlt 11425 cvgratnnlemfm 11426 cvgratnnlemrate 11427 cvgratnn 11428 efltim 11595 sin01bnd 11654 sin01gt0 11658 cos12dec 11664 blss2ps 12817 blss2 12818 blssps 12838 blss 12839 ivthinclemlopn 13025 ivthinclemuopn 13027 dvcjbr 13083 reeff1oleme 13104 efltlemlt 13106 sin0pilem1 13113 tangtx 13170 cosordlem 13181 cosq34lt1 13182 cvgcmp2nlemabs 13614 iooref1o 13616 trilpolemlt1 13623 trirec0 13626 apdifflemf 13628 |
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