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Mirrors > Home > MPE Home > Th. List > negsubdi2d | Structured version Visualization version GIF version |
Description: Distribution of negative over subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
negidd.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
pncand.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
Ref | Expression |
---|---|
negsubdi2d | ⊢ (𝜑 → -(𝐴 − 𝐵) = (𝐵 − 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negidd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | pncand.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
3 | negsubdi2 10934 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → -(𝐴 − 𝐵) = (𝐵 − 𝐴)) | |
4 | 1, 2, 3 | syl2anc 587 | 1 ⊢ (𝜑 → -(𝐴 − 𝐵) = (𝐵 − 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 (class class class)co 7135 ℂcc 10524 − cmin 10859 -cneg 10860 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-ltxr 10669 df-sub 10861 df-neg 10862 |
This theorem is referenced by: cjneg 14498 icodiamlt 14787 geo2sum2 15222 bpoly3 15404 sinneg 15491 sinhval 15499 vitalilem1 24212 vitalilem2 24213 itgneg 24407 dvrec 24558 dvferm2lem 24589 dvfsumge 24625 dvfsumlem2 24630 dvfsum2 24637 ftc1lem5 24643 ftc2ditg 24649 plyeq0lem 24807 efif1olem2 25135 ang180 25400 isosctrlem3 25406 isosctr 25407 affineequiv3 25411 angpieqvdlem 25414 chordthmlem 25418 mcubic 25433 quart1lem 25441 quartlem1 25443 atanneg 25493 atancj 25496 efiatan 25498 atanlogsub 25502 efiatan2 25503 2efiatan 25504 atantan 25509 atanbndlem 25511 pntrsumo1 26149 pntrlog2bndlem2 26162 pntrlog2bndlem4 26164 pntibndlem2 26175 brbtwn2 26699 colinearalglem4 26703 axsegconlem9 26719 dipcj 28497 bcm1n 30544 signsplypnf 31930 fsum2dsub 31988 dnibndlem11 33940 irrdifflemf 34739 itg2addnclem3 35110 itg2gt0cn 35112 congsym 39909 cvgdvgrat 41017 negsubdi3d 41925 lptre2pt 42282 liminflimsupclim 42449 stoweidlem13 42655 dirkertrigeqlem2 42741 fourierdlem26 42775 fourierdlem89 42837 fourierdlem90 42838 fourierdlem91 42839 fourierdlem107 42855 etransclem23 42899 sharhght 43479 sigaradd 43480 cevathlem2 43482 fmtnorec3 44065 1subrec1sub 45119 eenglngeehlnmlem1 45151 eenglngeehlnmlem2 45152 rrx2linest 45156 rrx2linest2 45158 line2 45166 itsclinecirc0b 45188 |
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