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Mirrors > Home > MPE Home > Th. List > sqneg | Structured version Visualization version GIF version |
Description: The square of the negative of a number. (Contributed by NM, 15-Jan-2006.) |
Ref | Expression |
---|---|
sqneg | ⊢ (𝐴 ∈ ℂ → (-𝐴↑2) = (𝐴↑2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mul2neg 10880 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (-𝐴 · -𝐴) = (𝐴 · 𝐴)) | |
2 | 1 | anidms 559 | . 2 ⊢ (𝐴 ∈ ℂ → (-𝐴 · -𝐴) = (𝐴 · 𝐴)) |
3 | negcl 10686 | . . 3 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) | |
4 | sqval 13296 | . . 3 ⊢ (-𝐴 ∈ ℂ → (-𝐴↑2) = (-𝐴 · -𝐴)) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝐴 ∈ ℂ → (-𝐴↑2) = (-𝐴 · -𝐴)) |
6 | sqval 13296 | . 2 ⊢ (𝐴 ∈ ℂ → (𝐴↑2) = (𝐴 · 𝐴)) | |
7 | 2, 5, 6 | 3eqtr4d 2824 | 1 ⊢ (𝐴 ∈ ℂ → (-𝐴↑2) = (𝐴↑2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ∈ wcel 2050 (class class class)co 6976 ℂcc 10333 · cmul 10340 -cneg 10671 2c2 11495 ↑cexp 13244 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-pss 3845 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-2nd 7502 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-er 8089 df-en 8307 df-dom 8308 df-sdom 8309 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-nn 11440 df-2 11503 df-n0 11708 df-z 11794 df-uz 12059 df-seq 13185 df-exp 13245 |
This theorem is referenced by: sqsubswap 13298 neg1sqe1 13374 binom2sub 13396 discr 13416 reusq0 14683 oexpneg 15554 cos2pi 24765 root1id 25036 dcubic1lem 25122 dcubic 25125 mcubic 25126 dquart 25132 asinlem 25147 asinlem2 25148 sinasin 25168 cosasin 25183 atandmneg 25185 cosatan 25200 atantayl2 25217 2sqlem4 25699 2sqnn0 25716 axlowdimlem16 26446 ex-exp 28007 ipidsq 28264 pell1234qrreccl 38853 pell1234qrdich 38860 pell14qrdich 38868 oexpnegALTV 43216 oexpnegnz 43217 itsclc0yqsollem1 44123 itscnhlinecirc02plem1 44143 |
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