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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 10107 | . . 3 ⊢ 1 ∈ ℂ | |
2 | mulneg1 10579 | . . 3 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (-1 · 𝐴) = -(1 · 𝐴)) | |
3 | 1, 2 | mpan 708 | . 2 ⊢ (𝐴 ∈ ℂ → (-1 · 𝐴) = -(1 · 𝐴)) |
4 | mulid2 10151 | . . 3 ⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴) | |
5 | 4 | negeqd 10388 | . 2 ⊢ (𝐴 ∈ ℂ → -(1 · 𝐴) = -𝐴) |
6 | 3, 5 | eqtrd 2758 | 1 ⊢ (𝐴 ∈ ℂ → (-1 · 𝐴) = -𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1596 ∈ wcel 2103 (class class class)co 6765 ℂcc 10047 1c1 10050 · cmul 10054 -cneg 10380 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1835 ax-4 1850 ax-5 1952 ax-6 2018 ax-7 2054 ax-8 2105 ax-9 2112 ax-10 2132 ax-11 2147 ax-12 2160 ax-13 2355 ax-ext 2704 ax-sep 4889 ax-nul 4897 ax-pow 4948 ax-pr 5011 ax-un 7066 ax-resscn 10106 ax-1cn 10107 ax-icn 10108 ax-addcl 10109 ax-addrcl 10110 ax-mulcl 10111 ax-mulrcl 10112 ax-mulcom 10113 ax-addass 10114 ax-mulass 10115 ax-distr 10116 ax-i2m1 10117 ax-1ne0 10118 ax-1rid 10119 ax-rnegex 10120 ax-rrecex 10121 ax-cnre 10122 ax-pre-lttri 10123 ax-pre-lttrn 10124 ax-pre-ltadd 10125 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1599 df-ex 1818 df-nf 1823 df-sb 2011 df-eu 2575 df-mo 2576 df-clab 2711 df-cleq 2717 df-clel 2720 df-nfc 2855 df-ne 2897 df-nel 3000 df-ral 3019 df-rex 3020 df-reu 3021 df-rab 3023 df-v 3306 df-sbc 3542 df-csb 3640 df-dif 3683 df-un 3685 df-in 3687 df-ss 3694 df-nul 4024 df-if 4195 df-pw 4268 df-sn 4286 df-pr 4288 df-op 4292 df-uni 4545 df-br 4761 df-opab 4821 df-mpt 4838 df-id 5128 df-po 5139 df-so 5140 df-xp 5224 df-rel 5225 df-cnv 5226 df-co 5227 df-dm 5228 df-rn 5229 df-res 5230 df-ima 5231 df-iota 5964 df-fun 6003 df-fn 6004 df-f 6005 df-f1 6006 df-fo 6007 df-f1o 6008 df-fv 6009 df-riota 6726 df-ov 6768 df-oprab 6769 df-mpt2 6770 df-er 7862 df-en 8073 df-dom 8074 df-sdom 8075 df-pnf 10189 df-mnf 10190 df-ltxr 10192 df-sub 10381 df-neg 10382 |
This theorem is referenced by: addneg1mul 10585 mulm1i 10588 mulm1d 10595 div2neg 10861 sqrtneglem 14127 sqreulem 14219 sinhval 15004 coshval 15005 demoivreALT 15051 sinmpi 24359 cosmpi 24360 sinppi 24361 cosppi 24362 cxpsqrt 24569 relogbdiv 24637 angneg 24653 lgsdir2lem4 25173 cnnvm 27767 cncph 27904 hvm1neg 28119 hvsubdistr2 28137 lnfnsubi 29135 dvasin 33728 |
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