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Mirrors > Home > MPE Home > Th. List > mulneg2 | Structured version Visualization version GIF version |
Description: The product with a negative is the negative of the product. (Contributed by NM, 30-Jul-2004.) |
Ref | Expression |
---|---|
mulneg2 | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · -𝐵) = -(𝐴 · 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulneg1 10758 | . . 3 ⊢ ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (-𝐵 · 𝐴) = -(𝐵 · 𝐴)) | |
2 | 1 | ancoms 451 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐵 · 𝐴) = -(𝐵 · 𝐴)) |
3 | negcl 10572 | . . 3 ⊢ (𝐵 ∈ ℂ → -𝐵 ∈ ℂ) | |
4 | mulcom 10310 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ -𝐵 ∈ ℂ) → (𝐴 · -𝐵) = (-𝐵 · 𝐴)) | |
5 | 3, 4 | sylan2 587 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · -𝐵) = (-𝐵 · 𝐴)) |
6 | mulcom 10310 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴)) | |
7 | 6 | negeqd 10566 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → -(𝐴 · 𝐵) = -(𝐵 · 𝐴)) |
8 | 2, 5, 7 | 3eqtr4d 2843 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · -𝐵) = -(𝐴 · 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 (class class class)co 6878 ℂcc 10222 · cmul 10229 -cneg 10557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-po 5233 df-so 5234 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-pnf 10365 df-mnf 10366 df-ltxr 10368 df-sub 10558 df-neg 10559 |
This theorem is referenced by: mulneg12 10760 submul2 10762 mulsub 10765 mulneg2i 10769 mulneg2d 10776 mulle0b 11186 zmulcl 11716 binom2sub 13235 cjreb 14204 recj 14205 reneg 14206 imcj 14213 imneg 14214 ipcnval 14224 cjneg 14228 cnpart 14321 efexp 15167 efmival 15219 tanhbnd 15227 sinsub 15234 cossub 15235 odd2np1 15401 itgneg 23911 dvsincos 24085 sinperlem 24574 efimpi 24585 dcubic2 24923 dcubic 24925 dquart 24932 quartlem1 24936 asinlem2 24948 asinneg 24965 sinasin 24968 cosasin 24983 atanneg 24986 atanlogadd 24993 atanlogsub 24995 cosatan 25000 atantan 25002 atans2 25010 rpvmasum2 25553 ipasslem2 28212 dvasin 33984 pell1234qrdich 38211 rmxm1 38284 |
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