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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 11668 | . . . 4 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (1 / 𝐵) ∈ ℂ) | |
2 | mulneg1 11439 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (1 / 𝐵) ∈ ℂ) → (-𝐴 · (1 / 𝐵)) = -(𝐴 · (1 / 𝐵))) | |
3 | 1, 2 | sylan2 592 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (-𝐴 · (1 / 𝐵)) = -(𝐴 · (1 / 𝐵))) |
4 | 3 | 3impb 1113 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (-𝐴 · (1 / 𝐵)) = -(𝐴 · (1 / 𝐵))) |
5 | negcl 11249 | . . 3 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) | |
6 | divrec 11677 | . . 3 ⊢ ((-𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (-𝐴 / 𝐵) = (-𝐴 · (1 / 𝐵))) | |
7 | 5, 6 | syl3an1 1161 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (-𝐴 / 𝐵) = (-𝐴 · (1 / 𝐵))) |
8 | divrec 11677 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝐴 / 𝐵) = (𝐴 · (1 / 𝐵))) | |
9 | 8 | negeqd 11243 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → -(𝐴 / 𝐵) = -(𝐴 · (1 / 𝐵))) |
10 | 4, 7, 9 | 3eqtr4rd 2784 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → -(𝐴 / 𝐵) = (-𝐴 / 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1537 ∈ wcel 2101 ≠ wne 2938 (class class class)co 7295 ℂcc 10897 0cc0 10899 1c1 10900 · cmul 10904 -cneg 11234 / cdiv 11660 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7608 ax-resscn 10956 ax-1cn 10957 ax-icn 10958 ax-addcl 10959 ax-addrcl 10960 ax-mulcl 10961 ax-mulrcl 10962 ax-mulcom 10963 ax-addass 10964 ax-mulass 10965 ax-distr 10966 ax-i2m1 10967 ax-1ne0 10968 ax-1rid 10969 ax-rnegex 10970 ax-rrecex 10971 ax-cnre 10972 ax-pre-lttri 10973 ax-pre-lttrn 10974 ax-pre-ltadd 10975 ax-pre-mulgt0 10976 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3222 df-reu 3223 df-rab 3224 df-v 3436 df-sbc 3719 df-csb 3835 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4260 df-if 4463 df-pw 4538 df-sn 4565 df-pr 4567 df-op 4571 df-uni 4842 df-br 5078 df-opab 5140 df-mpt 5161 df-id 5491 df-po 5505 df-so 5506 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-iota 6399 df-fun 6449 df-fn 6450 df-f 6451 df-f1 6452 df-fo 6453 df-f1o 6454 df-fv 6455 df-riota 7252 df-ov 7298 df-oprab 7299 df-mpo 7300 df-er 8518 df-en 8754 df-dom 8755 df-sdom 8756 df-pnf 11039 df-mnf 11040 df-xr 11041 df-ltxr 11042 df-le 11043 df-sub 11235 df-neg 11236 df-div 11661 |
This theorem is referenced by: divsubdir 11697 divsubdiv 11719 div2neg 11726 divneg2 11727 divnegd 11792 zeo 12434 efi4p 15874 sinneg 15883 tanneg 15885 cos2bnd 15925 cxpsqrtlem 25885 1cubrlem 26019 atancj 26088 efiatan 26090 atantan 26101 atanbndlem 26103 log2cnv 26122 ppiub 26380 quad3 33656 cos2h 35796 tan2h 35797 lhe4.4ex1a 41971 dirkertrigeqlem3 43676 fourierdlem62 43744 fourierdlem103 43785 fourierswlem 43806 enege 45137 onego 45138 0nodd 45404 |
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