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Mirrors > Home > ILE Home > Th. List > npcan | Unicode version |
Description: Cancellation law for subtraction. (Contributed by NM, 10-May-2004.) (Revised by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
npcan |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subcl 7929 | . . 3 | |
2 | simpr 109 | . . 3 | |
3 | 1, 2 | addcomd 7881 | . 2 |
4 | pncan3 7938 | . . 3 | |
5 | 4 | ancoms 266 | . 2 |
6 | 3, 5 | eqtrd 2150 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1316 wcel 1465 (class class class)co 5742 cc 7586 caddc 7591 cmin 7901 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-setind 4422 ax-resscn 7680 ax-1cn 7681 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-distr 7692 ax-i2m1 7693 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-sub 7903 |
This theorem is referenced by: addsubass 7940 npncan 7951 nppcan 7952 nnpcan 7953 subcan2 7955 nnncan 7965 npcand 8045 nn1suc 8703 zlem1lt 9068 zltlem1 9069 peano5uzti 9117 nummac 9184 uzp1 9315 peano2uzr 9336 fz01en 9788 fzsuc2 9814 fseq1m1p1 9830 fzoss2 9904 fzoaddel2 9925 fzosplitsnm1 9941 fzosplitprm1 9966 modfzo0difsn 10123 seq3m1 10196 monoord2 10205 ser3mono 10206 expm1t 10276 expubnd 10305 bcm1k 10461 bcn2 10465 hashfzo 10523 seq3coll 10540 shftlem 10543 shftfvalg 10545 shftfval 10548 iserex 11063 serf0 11076 fsumm1 11140 mptfzshft 11166 binomlem 11207 binom1dif 11211 isumsplit 11215 dvdssub2 11447 |
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