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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 11203 | . . 3 ⊢ 1 ∈ ℂ | |
2 | mulneg1 11687 | . . 3 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (-1 · 𝐴) = -(1 · 𝐴)) | |
3 | 1, 2 | mpan 688 | . 2 ⊢ (𝐴 ∈ ℂ → (-1 · 𝐴) = -(1 · 𝐴)) |
4 | mullid 11250 | . . 3 ⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴) | |
5 | 4 | negeqd 11491 | . 2 ⊢ (𝐴 ∈ ℂ → -(1 · 𝐴) = -𝐴) |
6 | 3, 5 | eqtrd 2765 | 1 ⊢ (𝐴 ∈ ℂ → (-1 · 𝐴) = -𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 (class class class)co 7419 ℂcc 11143 1c1 11146 · cmul 11150 -cneg 11482 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11287 df-mnf 11288 df-ltxr 11290 df-sub 11483 df-neg 11484 |
This theorem is referenced by: addneg1mul 11693 mulm1i 11696 mulm1d 11703 div2neg 11975 sqrtneglem 15254 sqreulem 15347 sinhval 16139 coshval 16140 demoivreALT 16186 sinmpi 26472 cosmpi 26473 sinppi 26474 cosppi 26475 cxpsqrt 26687 relogbdiv 26761 angneg 26785 lgsdir2lem4 27311 cnnvm 30569 cncph 30706 hvm1neg 30919 hvsubdistr2 30937 lnfnsubi 31933 dvasin 37310 lcmineqlem1 41634 |
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