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Mirrors > Home > MPE Home > Th. List > divneg | Structured version Visualization version GIF version |
Description: Move negative sign inside of a division. (Contributed by NM, 17-Sep-2004.) |
Ref | Expression |
---|---|
divneg | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → -(𝐴 / 𝐵) = (-𝐴 / 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reccl 11395 | . . . 4 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (1 / 𝐵) ∈ ℂ) | |
2 | mulneg1 11166 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (1 / 𝐵) ∈ ℂ) → (-𝐴 · (1 / 𝐵)) = -(𝐴 · (1 / 𝐵))) | |
3 | 1, 2 | sylan2 596 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (-𝐴 · (1 / 𝐵)) = -(𝐴 · (1 / 𝐵))) |
4 | 3 | 3impb 1116 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (-𝐴 · (1 / 𝐵)) = -(𝐴 · (1 / 𝐵))) |
5 | negcl 10976 | . . 3 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) | |
6 | divrec 11404 | . . 3 ⊢ ((-𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (-𝐴 / 𝐵) = (-𝐴 · (1 / 𝐵))) | |
7 | 5, 6 | syl3an1 1164 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (-𝐴 / 𝐵) = (-𝐴 · (1 / 𝐵))) |
8 | divrec 11404 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝐴 / 𝐵) = (𝐴 · (1 / 𝐵))) | |
9 | 8 | negeqd 10970 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → -(𝐴 / 𝐵) = -(𝐴 · (1 / 𝐵))) |
10 | 4, 7, 9 | 3eqtr4rd 2785 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → -(𝐴 / 𝐵) = (-𝐴 / 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ≠ wne 2935 (class class class)co 7182 ℂcc 10625 0cc0 10627 1c1 10628 · cmul 10632 -cneg 10961 / cdiv 11387 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-resscn 10684 ax-1cn 10685 ax-icn 10686 ax-addcl 10687 ax-addrcl 10688 ax-mulcl 10689 ax-mulrcl 10690 ax-mulcom 10691 ax-addass 10692 ax-mulass 10693 ax-distr 10694 ax-i2m1 10695 ax-1ne0 10696 ax-1rid 10697 ax-rnegex 10698 ax-rrecex 10699 ax-cnre 10700 ax-pre-lttri 10701 ax-pre-lttrn 10702 ax-pre-ltadd 10703 ax-pre-mulgt0 10704 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-br 5041 df-opab 5103 df-mpt 5121 df-id 5439 df-po 5452 df-so 5453 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7139 df-ov 7185 df-oprab 7186 df-mpo 7187 df-er 8332 df-en 8568 df-dom 8569 df-sdom 8570 df-pnf 10767 df-mnf 10768 df-xr 10769 df-ltxr 10770 df-le 10771 df-sub 10962 df-neg 10963 df-div 11388 |
This theorem is referenced by: divsubdir 11424 divsubdiv 11446 div2neg 11453 divneg2 11454 divnegd 11519 zeo 12161 efi4p 15594 sinneg 15603 tanneg 15605 cos2bnd 15645 cxpsqrtlem 25457 1cubrlem 25591 atancj 25660 efiatan 25662 atantan 25673 atanbndlem 25675 log2cnv 25694 ppiub 25952 quad3 33211 cos2h 35423 tan2h 35424 lhe4.4ex1a 41525 dirkertrigeqlem3 43223 fourierdlem62 43291 fourierdlem103 43332 fourierswlem 43353 enege 44678 onego 44679 0nodd 44945 |
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