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Mirrors > Home > MPE Home > Th. List > negnegd | Structured version Visualization version 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 10210 | . 2 ⊢ (𝐴 ∈ ℂ → --𝐴 = 𝐴) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → --𝐴 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ℂcc 9813 -cneg 10146 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-ltxr 9958 df-sub 10147 df-neg 10148 |
This theorem is referenced by: negn0 10338 ltnegcon1 10408 ltnegcon2 10409 lenegcon1 10411 lenegcon2 10412 negfi 10850 fiminre 10851 infm3lem 10860 infrenegsup 10883 zeo 11339 zindd 11354 znnn0nn 11365 supminf 11651 zsupss 11653 max0sub 11901 xnegneg 11919 ceilid 12512 expneg 12730 expaddzlem 12765 expaddz 12766 cjcj 13728 cnpart 13828 risefallfac 14594 sincossq 14745 bitsf1 15006 pcid 15415 4sqlem10 15489 mulgnegnn 17374 mulgsubcl 17378 mulgneg 17383 mulgz 17391 mulgass 17402 ghmmulg 17495 cyggeninv 18108 tgpmulg 21707 xrhmeo 22553 cphsqrtcl3 22795 iblneg 23375 itgneg 23376 ditgswap 23429 lhop2 23582 vieta1lem2 23870 ptolemy 24052 tanabsge 24062 tanord 24088 tanregt0 24089 lognegb 24140 logtayl 24206 logtayl2 24208 cxpmul2z 24237 isosctrlem2 24349 dcubic 24373 dquart 24380 atans2 24458 amgmlem 24516 lgamucov 24564 basellem5 24611 basellem9 24615 lgsdir2lem4 24853 dchrisum0flblem1 24997 ostth3 25127 ipasslem3 27072 ftc1anclem6 32660 rexzrexnn0 36386 acongsym 36561 acongneg2 36562 acongtr 36563 binomcxplemnotnn0 37577 ltmulneg 38556 itgsin0pilem1 38841 itgsinexplem1 38845 itgsincmulx 38866 stoweidlem13 38906 fourierdlem39 39039 fourierdlem43 39043 fourierdlem44 39044 etransclem46 39173 hoicvr 39438 sigariz 39701 sigaradd 39704 amgmwlem 42357 |
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