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Mirrors > Home > MPE Home > Th. List > mulm1 | Structured version Visualization version GIF version |
Description: Product with minus one is negative. (Contributed by NM, 16-Nov-1999.) |
Ref | Expression |
---|---|
mulm1 | ⊢ (𝐴 ∈ ℂ → (-1 · 𝐴) = -𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1cn 10311 | . . 3 ⊢ 1 ∈ ℂ | |
2 | mulneg1 10791 | . . 3 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (-1 · 𝐴) = -(1 · 𝐴)) | |
3 | 1, 2 | mpan 683 | . 2 ⊢ (𝐴 ∈ ℂ → (-1 · 𝐴) = -(1 · 𝐴)) |
4 | mulid2 10356 | . . 3 ⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴) | |
5 | 4 | negeqd 10596 | . 2 ⊢ (𝐴 ∈ ℂ → -(1 · 𝐴) = -𝐴) |
6 | 3, 5 | eqtrd 2862 | 1 ⊢ (𝐴 ∈ ℂ → (-1 · 𝐴) = -𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 (class class class)co 6906 ℂcc 10251 1c1 10254 · cmul 10258 -cneg 10587 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-br 4875 df-opab 4937 df-mpt 4954 df-id 5251 df-po 5264 df-so 5265 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-er 8010 df-en 8224 df-dom 8225 df-sdom 8226 df-pnf 10394 df-mnf 10395 df-ltxr 10397 df-sub 10588 df-neg 10589 |
This theorem is referenced by: addneg1mul 10797 mulm1i 10800 mulm1d 10807 div2neg 11075 sqrtneglem 14385 sqreulem 14477 sinhval 15257 coshval 15258 demoivreALT 15304 sinmpi 24640 cosmpi 24641 sinppi 24642 cosppi 24643 cxpsqrt 24849 relogbdiv 24920 angneg 24944 lgsdir2lem4 25467 cnnvm 28093 cncph 28230 hvm1neg 28445 hvsubdistr2 28463 lnfnsubi 29461 dvasin 34040 |
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