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Mirrors > Home > MPE Home > Th. List > negnegi | 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 NM, 8-Feb-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
Ref | Expression |
---|---|
negidi.1 | ⊢ 𝐴 ∈ ℂ |
Ref | Expression |
---|---|
negnegi | ⊢ --𝐴 = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negidi.1 | . 2 ⊢ 𝐴 ∈ ℂ | |
2 | negneg 11515 | . 2 ⊢ (𝐴 ∈ ℂ → --𝐴 = 𝐴) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ --𝐴 = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1540 ∈ wcel 2105 ℂcc 11112 -cneg 11450 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-er 8707 df-en 8944 df-dom 8945 df-sdom 8946 df-pnf 11255 df-mnf 11256 df-ltxr 11258 df-sub 11451 df-neg 11452 |
This theorem is referenced by: negsubdii 11550 negneg1e1 12335 crreczi 14196 sinhval 16102 iblcnlem1 25538 itgcnlem 25540 dvsincos 25734 asinlem3a 26612 asinsin 26634 atanlogsub 26658 atanbnd 26668 atantayl2 26680 basellem8 26829 ex-fl 29968 ex-ceil 29969 sqwvfourb 45244 |
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