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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 8041 | . 2 |
4 | addcl 7870 | . . 3 | |
5 | subadd 8093 | . . 3 | |
6 | 4, 1, 2, 5 | syl3anc 1227 | . 2 |
7 | 3, 6 | mpbird 166 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1342 wcel 2135 (class class class)co 5837 cc 7743 caddc 7748 cmin 8061 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4095 ax-pow 4148 ax-pr 4182 ax-setind 4509 ax-resscn 7837 ax-1cn 7838 ax-icn 7840 ax-addcl 7841 ax-addrcl 7842 ax-mulcl 7843 ax-addcom 7845 ax-addass 7847 ax-distr 7849 ax-i2m1 7850 ax-0id 7853 ax-rnegex 7854 ax-cnre 7856 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2724 df-sbc 2948 df-dif 3114 df-un 3116 df-in 3118 df-ss 3125 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-uni 3785 df-br 3978 df-opab 4039 df-id 4266 df-xp 4605 df-rel 4606 df-cnv 4607 df-co 4608 df-dm 4609 df-iota 5148 df-fun 5185 df-fv 5191 df-riota 5793 df-ov 5840 df-oprab 5841 df-mpo 5842 df-sub 8063 |
This theorem is referenced by: pncan2 8097 addsubass 8100 pncan3oi 8106 subid1 8110 nppcan2 8121 pncand 8202 nn1m1nn 8867 nnsub 8888 elnn0nn 9148 zrevaddcl 9233 nzadd 9235 elz2 9254 qrevaddcl 9574 irradd 9576 fzrev3 10013 elfzp1b 10023 fzrevral3 10033 fzval3 10130 subsq2 10553 bcp1nk 10665 bcp1m1 10668 bcpasc 10669 shftlem 10745 shftval5 10758 fsump1 11348 mptfzshft 11370 telfsumo 11394 fsumparts 11398 bcxmas 11417 isum1p 11420 geolim 11439 mertenslem2 11464 mertensabs 11465 eftlub 11618 effsumlt 11620 eirraplem 11704 dvdsadd 11762 prmind2 12038 fldivp1 12264 prmpwdvds 12271 pockthlem 12272 dvexp 13233 abssinper 13325 |
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