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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 |
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pncand.2 |
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Ref | Expression |
---|---|
subcld |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negidd.1 |
. 2
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2 | pncand.2 |
. 2
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3 | subcl 7878 |
. 2
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4 | 1, 2, 3 | syl2anc 406 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-pow 4056 ax-pr 4089 ax-setind 4410 ax-resscn 7631 ax-1cn 7632 ax-icn 7634 ax-addcl 7635 ax-addrcl 7636 ax-mulcl 7637 ax-addcom 7639 ax-addass 7641 ax-distr 7643 ax-i2m1 7644 ax-0id 7647 ax-rnegex 7648 ax-cnre 7650 |
This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-fal 1318 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ne 2281 df-ral 2393 df-rex 2394 df-reu 2395 df-rab 2397 df-v 2657 df-sbc 2877 df-dif 3037 df-un 3039 df-in 3041 df-ss 3048 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-br 3894 df-opab 3948 df-id 4173 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-iota 5044 df-fun 5081 df-fv 5087 df-riota 5682 df-ov 5729 df-oprab 5730 df-mpo 5731 df-sub 7852 |
This theorem is referenced by: pnpncand 8050 kcnktkm1cn 8058 muleqadd 8336 peano2zm 8990 peano5uzti 9057 modqmuladdnn0 10028 modsumfzodifsn 10056 hashfz 10454 hashfzo 10455 shftfvalg 10477 ovshftex 10478 shftfibg 10479 shftfval 10480 shftdm 10481 shftfib 10482 shftval 10484 2shfti 10490 crre 10516 remim 10519 remullem 10530 resqrexlemover 10668 resqrexlemcalc1 10672 abssubne0 10749 abs3lem 10769 caubnd2 10775 maxabslemlub 10865 maxabslemval 10866 maxcl 10868 minabs 10893 bdtrilem 10896 bdtri 10897 climuni 10948 mulcn2 10967 reccn2ap 10968 cn1lem 10969 climcvg1nlem 11004 fsumparts 11125 arisum2 11154 geosergap 11161 geo2sum2 11170 geoisum1c 11175 cvgratnnlemrate 11185 sinval 11254 sinf 11256 tanval2ap 11265 tanval3ap 11266 sinneg 11278 efival 11284 addcncntoplem 12531 mulcncflem 12570 limcimolemlt 12583 limcimo 12584 cnplimclemle 12587 limccnp2lem 12595 dvlemap 12598 dvconst 12610 dvid 12611 dvcnp2cntop 12612 dvaddxxbr 12614 qdencn 12903 |
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