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Mirrors > Home > MPE Home > Th. List > divnegd | Structured version Visualization version GIF version |
Description: Move negative sign inside of a division. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
div1d.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
divcld.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
divcld.3 | ⊢ (𝜑 → 𝐵 ≠ 0) |
Ref | Expression |
---|---|
divnegd | ⊢ (𝜑 → -(𝐴 / 𝐵) = (-𝐴 / 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | div1d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | divcld.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
3 | divcld.3 | . 2 ⊢ (𝜑 → 𝐵 ≠ 0) | |
4 | divneg 11332 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → -(𝐴 / 𝐵) = (-𝐴 / 𝐵)) | |
5 | 1, 2, 3, 4 | syl3anc 1367 | 1 ⊢ (𝜑 → -(𝐴 / 𝐵) = (-𝐴 / 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 (class class class)co 7156 ℂcc 10535 0cc0 10537 -cneg 10871 / cdiv 11297 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 |
This theorem is referenced by: qnegcl 12366 negmod0 13247 sinhval 15507 tanhbnd 15514 bitsfzo 15784 bitscmp 15787 pcneg 16210 dvrec 24552 dvsincos 24578 logtayl2 25245 logbrec 25360 cosangneg2d 25385 isosctrlem2 25397 angpieqvdlem 25406 dcubic2 25422 mcubic 25425 amgmlem 25567 basellem5 25662 pntpbnd1 26162 numdenneg 30533 divnumden2 30534 dvacos 34994 areacirc 35002 itgsincmulx 42279 dirkertrigeqlem3 42405 fourierdlem24 42436 fourierdlem26 42438 fourierdlem30 42442 fourierdlem39 42451 fourierdlem43 42455 fourierdlem44 42456 fourierdlem89 42500 fourierdlem91 42502 sqwvfourb 42534 etransclem47 42586 sharhght 43142 quad1 43805 requad1 43807 1subrec1sub 44712 eenglngeehlnmlem2 44745 line2 44759 itschlc0xyqsol 44774 |
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