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Mirrors > Home > ILE Home > Th. List > pncan | Unicode version |
Description: Cancellation law for subtraction. (Contributed by NM, 10-May-2004.) (Revised by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
pncan |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 109 | . . 3 | |
2 | simpl 108 | . . 3 | |
3 | 1, 2 | addcomd 7913 | . 2 |
4 | addcl 7745 | . . 3 | |
5 | subadd 7965 | . . 3 | |
6 | 4, 1, 2, 5 | syl3anc 1216 | . 2 |
7 | 3, 6 | mpbird 166 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 (class class class)co 5774 cc 7618 caddc 7623 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: pncan2 7969 addsubass 7972 pncan3oi 7978 subid1 7982 nppcan2 7993 pncand 8074 nn1m1nn 8738 nnsub 8759 elnn0nn 9019 zrevaddcl 9104 nzadd 9106 elz2 9122 qrevaddcl 9436 irradd 9438 fzrev3 9867 elfzp1b 9877 fzrevral3 9887 fzval3 9981 subsq2 10400 bcp1nk 10508 bcp1m1 10511 bcpasc 10512 shftlem 10588 shftval5 10601 fsump1 11189 mptfzshft 11211 telfsumo 11235 fsumparts 11239 bcxmas 11258 isum1p 11261 geolim 11280 mertenslem2 11305 mertensabs 11306 eftlub 11396 effsumlt 11398 eirraplem 11483 dvdsadd 11536 prmind2 11801 dvexp 12844 abssinper 12927 |
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