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Mirrors > Home > MPE Home > Th. List > mulneg1 | Structured version Visualization version GIF version |
Description: Product with negative is negative of product. Theorem I.12 of [Apostol] p. 18. (Contributed by NM, 14-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
mulneg1 | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 · 𝐵) = -(𝐴 · 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0cn 11252 | . . . 4 ⊢ 0 ∈ ℂ | |
2 | subdir 11694 | . . . 4 ⊢ ((0 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((0 − 𝐴) · 𝐵) = ((0 · 𝐵) − (𝐴 · 𝐵))) | |
3 | 1, 2 | mp3an1 1444 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((0 − 𝐴) · 𝐵) = ((0 · 𝐵) − (𝐴 · 𝐵))) |
4 | simpr 483 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ ℂ) | |
5 | 4 | mul02d 11458 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (0 · 𝐵) = 0) |
6 | 5 | oveq1d 7438 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((0 · 𝐵) − (𝐴 · 𝐵)) = (0 − (𝐴 · 𝐵))) |
7 | 3, 6 | eqtrd 2765 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((0 − 𝐴) · 𝐵) = (0 − (𝐴 · 𝐵))) |
8 | df-neg 11493 | . . 3 ⊢ -𝐴 = (0 − 𝐴) | |
9 | 8 | oveq1i 7433 | . 2 ⊢ (-𝐴 · 𝐵) = ((0 − 𝐴) · 𝐵) |
10 | df-neg 11493 | . 2 ⊢ -(𝐴 · 𝐵) = (0 − (𝐴 · 𝐵)) | |
11 | 7, 9, 10 | 3eqtr4g 2790 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 · 𝐵) = -(𝐴 · 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 (class class class)co 7423 ℂcc 11152 0cc0 11154 · cmul 11159 − cmin 11490 -cneg 11491 |
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 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 ax-resscn 11211 ax-1cn 11212 ax-icn 11213 ax-addcl 11214 ax-addrcl 11215 ax-mulcl 11216 ax-mulrcl 11217 ax-mulcom 11218 ax-addass 11219 ax-mulass 11220 ax-distr 11221 ax-i2m1 11222 ax-1ne0 11223 ax-1rid 11224 ax-rnegex 11225 ax-rrecex 11226 ax-cnre 11227 ax-pre-lttri 11228 ax-pre-lttrn 11229 ax-pre-ltadd 11230 |
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 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5579 df-po 5593 df-so 5594 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7379 df-ov 7426 df-oprab 7427 df-mpo 7428 df-er 8733 df-en 8974 df-dom 8975 df-sdom 8976 df-pnf 11296 df-mnf 11297 df-ltxr 11299 df-sub 11492 df-neg 11493 |
This theorem is referenced by: mulneg2 11697 mulneg12 11698 mulm1 11701 mulneg1i 11706 mulneg1d 11713 divneg 11953 zmulcl 12658 modcyc2 13922 cjreim 15160 tanval3 16131 dvdsnegb 16271 odd2np1 16338 modgcd 16528 pcexp 16856 cnfldmulg 21387 sinperlem 26500 sineq0 26543 efeq1 26547 asinlem3a 26890 atancj 26930 atantayl 26957 atantayl2 26958 zetacvg 27035 basellem3 27103 basellem9 27109 ipval2 30632 ipasslem2 30757 itg2addnclem3 37322 ftc1anclem6 37347 stoweidlem10 45568 |
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