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Mirrors > Home > ILE Home > Th. List > subcld | Unicode version |
Description: Closure law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
negidd.1 | |
pncand.2 |
Ref | Expression |
---|---|
subcld |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negidd.1 | . 2 | |
2 | pncand.2 | . 2 | |
3 | subcl 7961 | . 2 | |
4 | 1, 2, 3 | syl2anc 408 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wcel 1480 (class class class)co 5774 cc 7618 cmin 7933 |
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 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-setind 4452 ax-resscn 7712 ax-1cn 7713 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 |
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 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-sub 7935 |
This theorem is referenced by: pnpncand 8137 kcnktkm1cn 8145 muleqadd 8429 peano2zm 9092 peano5uzti 9159 modqmuladdnn0 10141 modsumfzodifsn 10169 hashfz 10567 hashfzo 10568 shftfvalg 10590 ovshftex 10591 shftfibg 10592 shftfval 10593 shftdm 10594 shftfib 10595 shftval 10597 2shfti 10603 crre 10629 remim 10632 remullem 10643 resqrexlemover 10782 resqrexlemcalc1 10786 abssubne0 10863 abs3lem 10883 caubnd2 10889 maxabslemlub 10979 maxabslemval 10980 maxcl 10982 minabs 11007 bdtrilem 11010 bdtri 11011 climuni 11062 mulcn2 11081 reccn2ap 11082 cn1lem 11083 climcvg1nlem 11118 fsumparts 11239 arisum2 11268 geosergap 11275 geo2sum2 11284 geoisum1c 11289 cvgratnnlemrate 11299 sinval 11409 sinf 11411 tanval2ap 11420 tanval3ap 11421 sinneg 11433 efival 11439 cos12dec 11474 addcncntoplem 12720 mulcncflem 12759 cnopnap 12763 limcimolemlt 12802 limcimo 12803 cnplimclemle 12806 limccnp2lem 12814 dvlemap 12818 dvconst 12830 dvid 12831 dvcnp2cntop 12832 dvaddxxbr 12834 dvmulxxbr 12835 dvcoapbr 12840 dvcjbr 12841 dvrecap 12846 dveflem 12855 dvef 12856 sin0pilem1 12862 ptolemy 12905 tangtx 12919 cosq34lt1 12931 qdencn 13222 |
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