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Mirrors > Home > ILE Home > Th. List > negnegd | GIF 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 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
Ref | Expression |
---|---|
negnegd | ⊢ (𝜑 → --𝐴 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negidd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | negneg 8194 | . 2 ⊢ (𝐴 ∈ ℂ → --𝐴 = 𝐴) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → --𝐴 = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 ℂcc 7797 -cneg 8116 |
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 4118 ax-pow 4171 ax-pr 4206 ax-setind 4533 ax-resscn 7891 ax-1cn 7892 ax-icn 7894 ax-addcl 7895 ax-addrcl 7896 ax-mulcl 7897 ax-addcom 7899 ax-addass 7901 ax-distr 7903 ax-i2m1 7904 ax-0id 7907 ax-rnegex 7908 ax-cnre 7910 |
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 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-iota 5174 df-fun 5214 df-fv 5220 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-sub 8117 df-neg 8118 |
This theorem is referenced by: ltnegcon1 8407 ltnegcon2 8408 lenegcon1 8410 lenegcon2 8411 recexre 8522 zaddcllemneg 9278 zeo 9344 zindd 9357 infrenegsupex 9580 supinfneg 9581 infsupneg 9582 supminfex 9583 negm 9601 xnegneg 9817 ceilid 10298 expnegap0 10511 expaddzaplem 10546 expaddzap 10547 cjcj 10873 negfi 11217 minabs 11225 minclpr 11226 mingeb 11231 sincossq 11737 infssuzex 11930 zsupssdc 11935 pcid 12303 4sqlem10 12365 znnen 12379 mulgnegnn 12879 mulgsubcl 12883 mulgneg 12887 mulgz 12896 mulgass 12905 ptolemy 13909 lgsdir2lem4 14096 |
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