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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 10927 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · -𝐵) = -(𝐴 · 𝐵)) | |
4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → (𝐴 · -𝐵) = -(𝐴 · 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1522 ∈ wcel 2080 (class class class)co 7019 ℂcc 10384 · cmul 10391 -cneg 10720 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1778 ax-4 1792 ax-5 1889 ax-6 1948 ax-7 1993 ax-8 2082 ax-9 2090 ax-10 2111 ax-11 2125 ax-12 2140 ax-13 2343 ax-ext 2768 ax-sep 5097 ax-nul 5104 ax-pow 5160 ax-pr 5224 ax-un 7322 ax-resscn 10443 ax-1cn 10444 ax-icn 10445 ax-addcl 10446 ax-addrcl 10447 ax-mulcl 10448 ax-mulrcl 10449 ax-mulcom 10450 ax-addass 10451 ax-mulass 10452 ax-distr 10453 ax-i2m1 10454 ax-1ne0 10455 ax-1rid 10456 ax-rnegex 10457 ax-rrecex 10458 ax-cnre 10459 ax-pre-lttri 10460 ax-pre-lttrn 10461 ax-pre-ltadd 10462 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1763 df-nf 1767 df-sb 2042 df-mo 2575 df-eu 2611 df-clab 2775 df-cleq 2787 df-clel 2862 df-nfc 2934 df-ne 2984 df-nel 3090 df-ral 3109 df-rex 3110 df-reu 3111 df-rab 3113 df-v 3438 df-sbc 3708 df-csb 3814 df-dif 3864 df-un 3866 df-in 3868 df-ss 3876 df-nul 4214 df-if 4384 df-pw 4457 df-sn 4475 df-pr 4477 df-op 4481 df-uni 4748 df-br 4965 df-opab 5027 df-mpt 5044 df-id 5351 df-po 5365 df-so 5366 df-xp 5452 df-rel 5453 df-cnv 5454 df-co 5455 df-dm 5456 df-rn 5457 df-res 5458 df-ima 5459 df-iota 6192 df-fun 6230 df-fn 6231 df-f 6232 df-f1 6233 df-fo 6234 df-f1o 6235 df-fv 6236 df-riota 6980 df-ov 7022 df-oprab 7023 df-mpo 7024 df-er 8142 df-en 8361 df-dom 8362 df-sdom 8363 df-pnf 10526 df-mnf 10527 df-ltxr 10529 df-sub 10721 df-neg 10722 |
This theorem is referenced by: prodge0rd 12346 expmulz 13325 discr 13451 sincossq 15362 oexpneg 15527 mulgass 18018 mulgmodid 18020 zringlpirlem3 20315 pjthlem1 23723 dvfsum2 24314 vieta1 24584 advlogexp 24919 logccv 24927 cxpmul2z 24955 abscxpbnd 25015 isosctrlem3 25079 affineequiv3 25084 dcubic1lem 25102 mcubic 25106 amgmlem 25249 ftalem5 25336 pntrlog2bndlem2 25836 brbtwn2 26374 colinearalglem4 26378 pjhthlem1 28851 fwddifnp1 33229 areacirclem1 34526 pellexlem6 38929 pell1234qrreccl 38949 pell14qrdich 38964 rmxyneg 39015 rmxm1 39029 ltmulneg 41218 cosknegpi 41705 itgsinexplem1 41794 dirkerper 41937 sqwvfoura 42069 etransclem46 42121 fmtnorec3 43206 oexpnegALTV 43338 oexpnegnz 43339 2zrngagrp 43706 itschlc0xyqsol 44249 amgmwlem 44397 |
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