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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 7954 | . . 3 | |
2 | simpr 109 | . . 3 | |
3 | 1, 2 | addcomd 7906 | . 2 |
4 | pncan3 7963 | . . 3 | |
5 | 4 | ancoms 266 | . 2 |
6 | 3, 5 | eqtrd 2170 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 (class class class)co 5767 cc 7611 caddc 7616 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 |
This theorem is referenced by: addsubass 7965 npncan 7976 nppcan 7977 nnpcan 7978 subcan2 7980 nnncan 7990 npcand 8070 nn1suc 8732 zlem1lt 9103 zltlem1 9104 peano5uzti 9152 nummac 9219 uzp1 9352 peano2uzr 9373 fz01en 9826 fzsuc2 9852 fseq1m1p1 9868 fzoss2 9942 fzoaddel2 9963 fzosplitsnm1 9979 fzosplitprm1 10004 modfzo0difsn 10161 seq3m1 10234 monoord2 10243 ser3mono 10244 expm1t 10314 expubnd 10343 bcm1k 10499 bcn2 10503 hashfzo 10561 seq3coll 10578 shftlem 10581 shftfvalg 10583 shftfval 10586 iserex 11101 serf0 11114 fsumm1 11178 mptfzshft 11204 binomlem 11245 binom1dif 11249 isumsplit 11253 dvdssub2 11524 |
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