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Mirrors > Home > MPE Home > Th. List > mulneg1i | Structured version Visualization version GIF version |
Description: Product with negative is negative of product. Theorem I.12 of [Apostol] p. 18. (Contributed by NM, 10-Feb-1995.) (Revised by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
mulm1.1 | ⊢ 𝐴 ∈ ℂ |
mulneg.2 | ⊢ 𝐵 ∈ ℂ |
Ref | Expression |
---|---|
mulneg1i | ⊢ (-𝐴 · 𝐵) = -(𝐴 · 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulm1.1 | . 2 ⊢ 𝐴 ∈ ℂ | |
2 | mulneg.2 | . 2 ⊢ 𝐵 ∈ ℂ | |
3 | mulneg1 10667 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 · 𝐵) = -(𝐴 · 𝐵)) | |
4 | 1, 2, 3 | mp2an 664 | 1 ⊢ (-𝐴 · 𝐵) = -(𝐴 · 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1631 ∈ wcel 2145 (class class class)co 6792 ℂcc 10135 · cmul 10142 -cneg 10468 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7095 ax-resscn 10194 ax-1cn 10195 ax-icn 10196 ax-addcl 10197 ax-addrcl 10198 ax-mulcl 10199 ax-mulrcl 10200 ax-mulcom 10201 ax-addass 10202 ax-mulass 10203 ax-distr 10204 ax-i2m1 10205 ax-1ne0 10206 ax-1rid 10207 ax-rnegex 10208 ax-rrecex 10209 ax-cnre 10210 ax-pre-lttri 10211 ax-pre-lttrn 10212 ax-pre-ltadd 10213 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-mpt 4864 df-id 5157 df-po 5170 df-so 5171 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6753 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-er 7895 df-en 8109 df-dom 8110 df-sdom 8111 df-pnf 10277 df-mnf 10278 df-ltxr 10280 df-sub 10469 df-neg 10470 |
This theorem is referenced by: recgt0ii 11130 crreczi 13195 sinhval 15089 coshval 15090 dvdslelem 15239 divalglem2 15325 divalglem6 15328 gcdaddmlem 15452 ncvspi 23174 ang180lem2 24760 ang180lem3 24761 1cubrlem 24788 asinsinlem 24838 asinsin 24839 asin1 24841 lgsdir2lem5 25274 nvpi 27859 ipasslem10 28031 normlem3 28306 dvasin 33824 zlmodzxzequap 42812 |
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