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Mirrors > Home > ILE Home > Th. List > mulm1 | 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 7892 | . . 3 ⊢ 1 ∈ ℂ | |
2 | mulneg1 8339 | . . 3 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (-1 · 𝐴) = -(1 · 𝐴)) | |
3 | 1, 2 | mpan 424 | . 2 ⊢ (𝐴 ∈ ℂ → (-1 · 𝐴) = -(1 · 𝐴)) |
4 | mulid2 7943 | . . 3 ⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴) | |
5 | 4 | negeqd 8139 | . 2 ⊢ (𝐴 ∈ ℂ → -(1 · 𝐴) = -𝐴) |
6 | 3, 5 | eqtrd 2210 | 1 ⊢ (𝐴 ∈ ℂ → (-1 · 𝐴) = -𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 (class class class)co 5869 ℂcc 7797 1c1 7800 · cmul 7804 -cneg 8116 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 ax-setind 4533 ax-resscn 7891 ax-1cn 7892 ax-icn 7894 ax-addcl 7895 ax-addrcl 7896 ax-mulcl 7897 ax-addcom 7899 ax-mulcom 7900 ax-addass 7901 ax-mulass 7902 ax-distr 7903 ax-i2m1 7904 ax-1rid 7906 ax-0id 7907 ax-rnegex 7908 ax-cnre 7910 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-iota 5174 df-fun 5214 df-fv 5220 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-sub 8117 df-neg 8118 |
This theorem is referenced by: mulm1i 8347 mulm1d 8354 div2negap 8678 demoivreALT 11762 sinmpi 13896 cosmpi 13897 sinppi 13898 cosppi 13899 rprelogbdiv 14035 lgsdir2lem4 14092 |
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