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Mirrors > Home > MPE Home > Th. List > mulneg2d | Structured version Visualization version GIF version |
Description: Product with negative is negative of product. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
mulm1d.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
mulnegd.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
Ref | Expression |
---|---|
mulneg2d | ⊢ (𝜑 → (𝐴 · -𝐵) = -(𝐴 · 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulm1d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | mulnegd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
3 | mulneg2 11703 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · -𝐵) = -(𝐴 · 𝐵)) | |
4 | 1, 2, 3 | syl2anc 582 | 1 ⊢ (𝜑 → (𝐴 · -𝐵) = -(𝐴 · 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 (class class class)co 7426 ℂcc 11158 · cmul 11165 -cneg 11497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5306 ax-nul 5313 ax-pow 5371 ax-pr 5435 ax-un 7748 ax-resscn 11217 ax-1cn 11218 ax-icn 11219 ax-addcl 11220 ax-addrcl 11221 ax-mulcl 11222 ax-mulrcl 11223 ax-mulcom 11224 ax-addass 11225 ax-mulass 11226 ax-distr 11227 ax-i2m1 11228 ax-1ne0 11229 ax-1rid 11230 ax-rnegex 11231 ax-rrecex 11232 ax-cnre 11233 ax-pre-lttri 11234 ax-pre-lttrn 11235 ax-pre-ltadd 11236 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4916 df-br 5156 df-opab 5218 df-mpt 5239 df-id 5582 df-po 5596 df-so 5597 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 df-fv 6564 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-er 8736 df-en 8977 df-dom 8978 df-sdom 8979 df-pnf 11302 df-mnf 11303 df-ltxr 11305 df-sub 11498 df-neg 11499 |
This theorem is referenced by: prodge0rd 13137 expmulz 14130 discr 14259 sincossq 16180 oexpneg 16349 mulgass 19107 mulgmodid 19109 zringlpirlem3 21456 pjthlem1 25459 dvfsum2 26063 vieta1 26343 advlogexp 26685 logccv 26693 cxpmul2z 26721 abscxpbnd 26784 isosctrlem3 26851 affineequiv3 26856 dcubic1lem 26874 mcubic 26878 amgmlem 27021 ftalem5 27108 pntrlog2bndlem2 27610 brbtwn2 28842 colinearalglem4 28846 pjhthlem1 31327 fwddifnp1 35991 areacirclem1 37411 3cubeslem3r 42362 pellexlem6 42509 pell1234qrreccl 42529 pell14qrdich 42544 rmxyneg 42596 rmxm1 42610 ltmulneg 45025 cosknegpi 45508 itgsinexplem1 45593 dirkerper 45735 sqwvfoura 45867 etransclem46 45919 fmtnorec3 47138 oexpnegALTV 47267 oexpnegnz 47268 2zrngagrp 47644 itschlc0xyqsol 48173 amgmwlem 48568 |
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