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Mirrors > Home > ILE Home > Th. List > negneg | Unicode version |
Description: A number is equal to the negative of its negative. Theorem I.4 of [Apostol] p. 18. (Contributed by NM, 12-Jan-2002.) (Revised by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
negneg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-neg 8053 | . . 3 | |
2 | 0cn 7872 | . . . 4 | |
3 | subneg 8128 | . . . 4 | |
4 | 2, 3 | mpan 421 | . . 3 |
5 | 1, 4 | syl5eq 2202 | . 2 |
6 | addid2 8018 | . 2 | |
7 | 5, 6 | eqtrd 2190 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1335 wcel 2128 (class class class)co 5826 cc 7732 cc0 7734 caddc 7737 cmin 8050 cneg 8051 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-pow 4137 ax-pr 4171 ax-setind 4498 ax-resscn 7826 ax-1cn 7827 ax-icn 7829 ax-addcl 7830 ax-addrcl 7831 ax-mulcl 7832 ax-addcom 7834 ax-addass 7836 ax-distr 7838 ax-i2m1 7839 ax-0id 7842 ax-rnegex 7843 ax-cnre 7845 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-br 3968 df-opab 4028 df-id 4255 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-iota 5137 df-fun 5174 df-fv 5180 df-riota 5782 df-ov 5829 df-oprab 5830 df-mpo 5831 df-sub 8052 df-neg 8053 |
This theorem is referenced by: neg11 8130 negcon1 8131 negreb 8144 negnegi 8149 negnegd 8181 negf1o 8261 mul2neg 8277 divneg2ap 8613 nnnegz 9175 znegclb 9205 expineg2 10437 shftcan2 10746 negfi 11138 dvdsnegb 11715 |
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