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Mirrors > Home > MPE Home > Th. List > mulneg1d | Structured version Visualization version GIF version |
Description: Product with negative is negative of product. Theorem I.12 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
mulm1d.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
mulnegd.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
Ref | Expression |
---|---|
mulneg1d | ⊢ (𝜑 → (-𝐴 · 𝐵) = -(𝐴 · 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulm1d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | mulnegd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
3 | mulneg1 11064 | . 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 1528 ∈ wcel 2105 (class class class)co 7145 ℂcc 10523 · cmul 10530 -cneg 10859 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-ltxr 10668 df-sub 10860 df-neg 10861 |
This theorem is referenced by: divsubdiv 11344 recgt0 11474 xmulneg1 12650 expmulz 13463 discr1 13588 iseraltlem3 15028 incexclem 15179 incexc 15180 mulgass 18202 cphipval 23773 mbfmulc2lem 24175 mbfmulc2 24191 itg2monolem1 24278 itgmulc2 24361 dvrecg 24497 dvmptdiv 24498 dvexp3 24502 dvfsumlem2 24551 aaliou3lem2 24859 advlogexp 25165 logtayl2 25172 dcubic2 25349 dcubic 25351 ftalem5 25581 lgsdilem 25827 2sqlem4 25924 pntrsumo1 26068 pntrlog2bndlem4 26083 brbtwn2 26618 colinearalglem4 26622 axeuclidlem 26675 logdivsqrle 31820 fwddifnp1 33523 itgmulc2nc 34841 3cubeslem3r 39162 pellexlem6 39309 jm2.19lem1 39464 jm2.19lem4 39467 jm2.19 39468 binomcxplemnotnn0 40565 sineq0ALT 41148 mulltgt0 41156 fperiodmul 41447 cosknegpi 42026 itgsinexplem1 42115 stoweidlem13 42175 stoweidlem42 42204 fourierdlem39 42308 fourierdlem41 42310 fourierdlem48 42316 fourierdlem49 42317 fourierdlem64 42332 etransclem46 42442 eenglngeehlnmlem1 44652 eenglngeehlnmlem2 44653 rrx2linest 44657 rrx2linest2 44659 line2 44667 itscnhlc0yqe 44674 itschlc0yqe 44675 itsclc0yqsol 44679 itsclinecirc0b 44689 itsclquadb 44691 |
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