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Mirrors > Home > ILE Home > Th. List > negnegd | Unicode version |
Description: A number is equal to the negative of its negative. Theorem I.4 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
negidd.1 |
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Ref | Expression |
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negnegd |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negidd.1 |
. 2
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2 | negneg 8210 |
. 2
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3 | 1, 2 | syl 14 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-setind 4538 ax-resscn 7906 ax-1cn 7907 ax-icn 7909 ax-addcl 7910 ax-addrcl 7911 ax-mulcl 7912 ax-addcom 7914 ax-addass 7916 ax-distr 7918 ax-i2m1 7919 ax-0id 7922 ax-rnegex 7923 ax-cnre 7925 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2741 df-sbc 2965 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-opab 4067 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-iota 5180 df-fun 5220 df-fv 5226 df-riota 5834 df-ov 5881 df-oprab 5882 df-mpo 5883 df-sub 8133 df-neg 8134 |
This theorem is referenced by: ltnegcon1 8423 ltnegcon2 8424 lenegcon1 8426 lenegcon2 8427 recexre 8538 zaddcllemneg 9295 zeo 9361 zindd 9374 infrenegsupex 9597 supinfneg 9598 infsupneg 9599 supminfex 9600 negm 9618 xnegneg 9836 ceilid 10318 expnegap0 10531 expaddzaplem 10566 expaddzap 10567 cjcj 10895 negfi 11239 minabs 11247 minclpr 11248 mingeb 11253 sincossq 11759 infssuzex 11953 zsupssdc 11958 pcid 12326 4sqlem10 12388 znnen 12402 mulgnegnn 13007 mulgsubcl 13011 mulgneg 13015 mulgz 13025 mulgass 13034 ptolemy 14433 lgsdir2lem4 14620 |
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