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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 7867 | . . 3 ⊢ 1 ∈ ℂ | |
2 | mulneg1 8314 | . . 3 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (-1 · 𝐴) = -(1 · 𝐴)) | |
3 | 1, 2 | mpan 422 | . 2 ⊢ (𝐴 ∈ ℂ → (-1 · 𝐴) = -(1 · 𝐴)) |
4 | mulid2 7918 | . . 3 ⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴) | |
5 | 4 | negeqd 8114 | . 2 ⊢ (𝐴 ∈ ℂ → -(1 · 𝐴) = -𝐴) |
6 | 3, 5 | eqtrd 2203 | 1 ⊢ (𝐴 ∈ ℂ → (-1 · 𝐴) = -𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1348 ∈ wcel 2141 (class class class)co 5853 ℂcc 7772 1c1 7775 · cmul 7779 -cneg 8091 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-setind 4521 ax-resscn 7866 ax-1cn 7867 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-cnre 7885 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-sub 8092 df-neg 8093 |
This theorem is referenced by: mulm1i 8322 mulm1d 8329 div2negap 8652 demoivreALT 11736 sinmpi 13530 cosmpi 13531 sinppi 13532 cosppi 13533 rprelogbdiv 13669 lgsdir2lem4 13726 |
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