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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 8019 | . 2 | |
4 | 1, 2, 3 | syl2anc 408 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wcel 1480 (class class class)co 5767 cr 7612 cmin 7926 |
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: ltsubadd 8187 lesubadd 8189 ltaddsub 8191 leaddsub 8193 lesub1 8211 lesub2 8212 ltsub1 8213 ltsub2 8214 lt2sub 8215 le2sub 8216 rereim 8341 ltmul1a 8346 cru 8357 lemul1a 8609 ztri3or 9090 lincmb01cmp 9779 iccf1o 9780 rebtwn2z 10025 qbtwnrelemcalc 10026 qbtwnre 10027 intfracq 10086 modqval 10090 modqlt 10099 modqsubdir 10159 ser3le 10284 expnbnd 10408 crre 10622 remullem 10636 recvguniqlem 10759 resqrexlemover 10775 resqrexlemcalc2 10780 resqrexlemcalc3 10781 resqrexlemnmsq 10782 resqrexlemnm 10783 resqrexlemcvg 10784 resqrexlemglsq 10787 resqrexlemga 10788 fzomaxdiflem 10877 caubnd2 10882 amgm2 10883 icodiamlt 10945 qdenre 10967 maxabslemab 10971 maxabslemlub 10972 maxltsup 10983 bdtrilem 11003 bdtri 11004 mulcn2 11074 reccn2ap 11075 climle 11096 climsqz 11097 climsqz2 11098 climcvg1nlem 11111 fsumle 11225 cvgratnnlembern 11285 cvgratnnlemsumlt 11290 cvgratnnlemfm 11291 cvgratnnlemrate 11292 cvgratnn 11293 efltim 11393 sin01bnd 11453 sin01gt0 11457 cos12dec 11463 blss2ps 12564 blss2 12565 blssps 12585 blss 12586 ivthinclemlopn 12772 ivthinclemuopn 12774 dvcjbr 12830 sin0pilem1 12851 tangtx 12908 cosordlem 12919 cosq34lt1 12920 cvgcmp2nlemabs 13216 trilpolemlt1 13223 |
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