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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 11269 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · -𝐵) = -(𝐴 · 𝐵)) | |
4 | 1, 2, 3 | syl2anc 587 | 1 ⊢ (𝜑 → (𝐴 · -𝐵) = -(𝐴 · 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 (class class class)co 7213 ℂcc 10727 · cmul 10734 -cneg 11063 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-po 5468 df-so 5469 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-ltxr 10872 df-sub 11064 df-neg 11065 |
This theorem is referenced by: prodge0rd 12693 expmulz 13681 discr 13807 sincossq 15737 oexpneg 15906 mulgass 18528 mulgmodid 18530 zringlpirlem3 20451 pjthlem1 24334 dvfsum2 24931 vieta1 25205 advlogexp 25543 logccv 25551 cxpmul2z 25579 abscxpbnd 25639 isosctrlem3 25703 affineequiv3 25708 dcubic1lem 25726 mcubic 25730 amgmlem 25872 ftalem5 25959 pntrlog2bndlem2 26459 brbtwn2 26996 colinearalglem4 27000 pjhthlem1 29472 fwddifnp1 34204 areacirclem1 35602 3cubeslem3r 40212 pellexlem6 40359 pell1234qrreccl 40379 pell14qrdich 40394 rmxyneg 40445 rmxm1 40459 ltmulneg 42605 cosknegpi 43085 itgsinexplem1 43170 dirkerper 43312 sqwvfoura 43444 etransclem46 43496 fmtnorec3 44673 oexpnegALTV 44802 oexpnegnz 44803 2zrngagrp 45174 itschlc0xyqsol 45786 amgmwlem 46177 |
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