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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 10636 | . . . 4 ⊢ 0 ∈ ℂ | |
2 | subdir 11077 | . . . 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 487 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ ℂ) | |
5 | 4 | mul02d 10841 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (0 · 𝐵) = 0) |
6 | 5 | oveq1d 7174 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((0 · 𝐵) − (𝐴 · 𝐵)) = (0 − (𝐴 · 𝐵))) |
7 | 3, 6 | eqtrd 2859 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((0 − 𝐴) · 𝐵) = (0 − (𝐴 · 𝐵))) |
8 | df-neg 10876 | . . 3 ⊢ -𝐴 = (0 − 𝐴) | |
9 | 8 | oveq1i 7169 | . 2 ⊢ (-𝐴 · 𝐵) = ((0 − 𝐴) · 𝐵) |
10 | df-neg 10876 | . 2 ⊢ -(𝐴 · 𝐵) = (0 − (𝐴 · 𝐵)) | |
11 | 7, 9, 10 | 3eqtr4g 2884 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 · 𝐵) = -(𝐴 · 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 (class class class)co 7159 ℂcc 10538 0cc0 10540 · cmul 10545 − cmin 10873 -cneg 10874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-po 5477 df-so 5478 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-ltxr 10683 df-sub 10875 df-neg 10876 |
This theorem is referenced by: mulneg2 11080 mulneg12 11081 mulm1 11084 mulneg1i 11089 mulneg1d 11096 divneg 11335 zmulcl 12034 modcyc2 13278 cjreim 14522 tanval3 15490 dvdsnegb 15630 odd2np1 15693 modgcd 15883 pcexp 16199 cnfldmulg 20580 sinperlem 25069 sineq0 25112 efeq1 25116 asinlem3a 25451 atancj 25491 atantayl 25518 atantayl2 25519 zetacvg 25595 basellem3 25663 basellem9 25669 ipval2 28487 ipasslem2 28612 itg2addnclem3 34949 ftc1anclem6 34976 stoweidlem10 42302 |
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