![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 8170 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4 | 1, 2, 3 | syl2anc 411 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-setind 4548 ax-resscn 7917 ax-1cn 7918 ax-icn 7920 ax-addcl 7921 ax-addrcl 7922 ax-mulcl 7923 ax-addcom 7925 ax-addass 7927 ax-distr 7929 ax-i2m1 7930 ax-0id 7933 ax-rnegex 7934 ax-cnre 7936 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-br 4016 df-opab 4077 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-iota 5190 df-fun 5230 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-sub 8144 |
This theorem is referenced by: pnpncand 8346 kcnktkm1cn 8354 muleqadd 8639 peano2zm 9305 peano5uzti 9375 modqmuladdnn0 10382 modsumfzodifsn 10410 hashfz 10815 hashfzo 10816 shftfvalg 10841 ovshftex 10842 shftfibg 10843 shftfval 10844 shftdm 10845 shftfib 10846 shftval 10848 2shfti 10854 crre 10880 remim 10883 remullem 10894 resqrexlemover 11033 resqrexlemcalc1 11037 abssubne0 11114 abs3lem 11134 caubnd2 11140 maxabslemlub 11230 maxabslemval 11231 maxcl 11233 minabs 11258 bdtrilem 11261 bdtri 11262 climuni 11315 mulcn2 11334 reccn2ap 11335 cn1lem 11336 climcvg1nlem 11371 fsumparts 11492 arisum2 11521 geosergap 11528 geo2sum2 11537 geoisum1c 11542 cvgratnnlemrate 11552 sinval 11724 sinf 11726 tanval2ap 11735 tanval3ap 11736 sinneg 11748 efival 11754 cos12dec 11789 pythagtriplem1 12279 pythagtriplem14 12291 pythagtriplem16 12293 pythagtriplem17 12294 dvdsprmpweqle 12350 4sqlem5 12394 mul4sqlem 12405 addcncntoplem 14404 mulcncflem 14443 cnopnap 14447 limcimolemlt 14486 limcimo 14487 cnplimclemle 14490 limccnp2lem 14498 dvlemap 14502 dvconst 14514 dvid 14515 dvcnp2cntop 14516 dvaddxxbr 14518 dvmulxxbr 14519 dvcoapbr 14524 dvcjbr 14525 dvrecap 14530 dveflem 14540 dvef 14541 sin0pilem1 14555 ptolemy 14598 tangtx 14612 cosq34lt1 14624 lgsdirprm 14788 qdencn 15129 trirec0 15146 apdifflemf 15148 apdifflemr 15149 apdiff 15150 |
Copyright terms: Public domain | W3C validator |