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Mirrors > Home > MPE Home > Th. List > mulneg1d | Structured version Visualization version GIF version |
Description: Product with negative is negative of product. Theorem I.12 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
mulm1d.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
mulnegd.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
Ref | Expression |
---|---|
mulneg1d | ⊢ (𝜑 → (-𝐴 · 𝐵) = -(𝐴 · 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulm1d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | mulnegd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
3 | mulneg1 11078 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 · 𝐵) = -(𝐴 · 𝐵)) | |
4 | 1, 2, 3 | syl2anc 586 | 1 ⊢ (𝜑 → (-𝐴 · 𝐵) = -(𝐴 · 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 (class class class)co 7158 ℂcc 10537 · cmul 10544 -cneg 10873 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-ltxr 10682 df-sub 10874 df-neg 10875 |
This theorem is referenced by: divsubdiv 11358 recgt0 11488 xmulneg1 12665 expmulz 13478 discr1 13603 iseraltlem3 15042 incexclem 15193 incexc 15194 mulgass 18266 cphipval 23848 mbfmulc2lem 24250 mbfmulc2 24266 itg2monolem1 24353 itgmulc2 24436 dvrecg 24572 dvmptdiv 24573 dvexp3 24577 dvfsumlem2 24626 aaliou3lem2 24934 advlogexp 25240 logtayl2 25247 dcubic2 25424 dcubic 25426 ftalem5 25656 lgsdilem 25902 2sqlem4 25999 pntrsumo1 26143 pntrlog2bndlem4 26158 brbtwn2 26693 colinearalglem4 26697 axeuclidlem 26750 logdivsqrle 31923 fwddifnp1 33628 itgmulc2nc 34962 3cubeslem3r 39291 pellexlem6 39438 jm2.19lem1 39593 jm2.19lem4 39596 jm2.19 39597 binomcxplemnotnn0 40695 sineq0ALT 41278 mulltgt0 41286 fperiodmul 41578 cosknegpi 42157 itgsinexplem1 42246 stoweidlem13 42305 stoweidlem42 42334 fourierdlem39 42438 fourierdlem41 42440 fourierdlem48 42446 fourierdlem49 42447 fourierdlem64 42462 etransclem46 42572 eenglngeehlnmlem1 44731 eenglngeehlnmlem2 44732 rrx2linest 44736 rrx2linest2 44738 line2 44746 itscnhlc0yqe 44753 itschlc0yqe 44754 itsclc0yqsol 44758 itsclinecirc0b 44768 itsclquadb 44770 |
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