![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 11682 | . . 3 ⊢ ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (-𝐵 · 𝐴) = -(𝐵 · 𝐴)) | |
2 | 1 | ancoms 457 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐵 · 𝐴) = -(𝐵 · 𝐴)) |
3 | negcl 11492 | . . 3 ⊢ (𝐵 ∈ ℂ → -𝐵 ∈ ℂ) | |
4 | mulcom 11226 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ -𝐵 ∈ ℂ) → (𝐴 · -𝐵) = (-𝐵 · 𝐴)) | |
5 | 3, 4 | sylan2 591 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · -𝐵) = (-𝐵 · 𝐴)) |
6 | mulcom 11226 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴)) | |
7 | 6 | negeqd 11486 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → -(𝐴 · 𝐵) = -(𝐵 · 𝐴)) |
8 | 2, 5, 7 | 3eqtr4d 2775 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · -𝐵) = -(𝐴 · 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 (class class class)co 7419 ℂcc 11138 · cmul 11145 -cneg 11477 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-ltxr 11285 df-sub 11478 df-neg 11479 |
This theorem is referenced by: mulneg12 11684 submul2 11686 mulsub 11689 mulneg2i 11693 mulneg2d 11700 mulle0b 12118 zmulcl 12644 binom2sub 14218 cjreb 15106 recj 15107 reneg 15108 imcj 15115 imneg 15116 ipcnval 15126 cjneg 15130 cnpart 15223 efexp 16081 efmival 16133 tanhbnd 16141 sinsub 16148 cossub 16149 odd2np1 16321 itgneg 25777 dvsincos 25957 sinperlem 26460 efimpi 26471 dcubic2 26821 dcubic 26823 dquart 26830 quartlem1 26834 asinlem2 26846 asinneg 26863 sinasin 26866 cosasin 26881 atanneg 26884 atanlogadd 26891 atanlogsub 26893 cosatan 26898 atantan 26900 atans2 26908 rpvmasum2 27490 ipasslem2 30714 dvasin 37308 pell1234qrdich 42423 rmxm1 42497 |
Copyright terms: Public domain | W3C validator |