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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 11699 | . 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 1540 ∈ wcel 2108 (class class class)co 7431 ℂcc 11153 · cmul 11160 -cneg 11493 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 df-sub 11494 df-neg 11495 |
| This theorem is referenced by: divsubdiv 11983 recgt0 12113 xmulneg1 13311 expmulz 14149 discr1 14278 iseraltlem3 15720 incexclem 15872 incexc 15873 mulgass 19129 cphipval 25277 mbfmulc2lem 25682 mbfmulc2 25698 itg2monolem1 25785 itgmulc2 25869 dvrecg 26011 dvmptdiv 26012 dvexp3 26016 dvfsumlem2 26067 dvfsumlem2OLD 26068 aaliou3lem2 26385 advlogexp 26697 logtayl2 26704 dcubic2 26887 dcubic 26889 ftalem5 27120 lgsdilem 27368 2sqlem4 27465 pntrsumo1 27609 pntrlog2bndlem4 27624 brbtwn2 28920 colinearalglem4 28924 axeuclidlem 28977 constrrtcc 33776 logdivsqrle 34665 fwddifnp1 36166 itgmulc2nc 37695 lcmineqlem10 42039 3cubeslem3r 42698 pellexlem6 42845 jm2.19lem1 43001 jm2.19lem4 43004 jm2.19 43005 binomcxplemnotnn0 44375 sineq0ALT 44957 mulltgt0 45027 fperiodmul 45316 cosknegpi 45884 itgsinexplem1 45969 stoweidlem13 46028 stoweidlem42 46057 fourierdlem39 46161 fourierdlem41 46163 fourierdlem48 46169 fourierdlem49 46170 fourierdlem64 46185 etransclem46 46295 eenglngeehlnmlem1 48658 eenglngeehlnmlem2 48659 rrx2linest 48663 rrx2linest2 48665 line2 48673 itscnhlc0yqe 48680 itschlc0yqe 48681 itsclc0yqsol 48685 itsclinecirc0b 48695 itsclquadb 48697 |
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