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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 10682 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → -(𝐴 − 𝐵) = (𝐵 − 𝐴)) | |
4 | 1, 2, 3 | syl2anc 579 | 1 ⊢ (𝜑 → -(𝐴 − 𝐵) = (𝐵 − 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2106 (class class class)co 6922 ℂcc 10270 − cmin 10606 -cneg 10607 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-po 5274 df-so 5275 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-ltxr 10416 df-sub 10608 df-neg 10609 |
This theorem is referenced by: cjneg 14294 icodiamlt 14582 geo2sum2 15009 bpoly3 15191 sinneg 15278 sinhval 15286 vitalilem1 23812 vitalilem2 23813 itgneg 24007 dvrec 24155 dvferm2lem 24186 dvfsumge 24222 dvfsumlem2 24227 dvfsum2 24234 ftc1lem5 24240 ftc2ditg 24246 plyeq0lem 24403 efif1olem2 24727 ang180 24992 isosctrlem3 24998 isosctr 24999 affineequiv3 25003 angpieqvdlem 25006 chordthmlem 25010 mcubic 25025 quart1lem 25033 quartlem1 25035 atanneg 25085 atancj 25088 efiatan 25090 atanlogsub 25094 efiatan2 25095 2efiatan 25096 atantan 25101 atanbndlem 25103 pntrsumo1 25706 pntrlog2bndlem2 25719 pntrlog2bndlem4 25721 pntibndlem2 25732 brbtwn2 26254 colinearalglem4 26258 axsegconlem9 26274 dipcj 28141 bcm1n 30118 signsplypnf 31227 fsum2dsub 31287 dnibndlem11 33061 itg2addnclem3 34072 itg2gt0cn 34074 congsym 38476 cvgdvgrat 39450 negsubdi3d 40398 lptre2pt 40762 liminflimsupclim 40929 stoweidlem13 41139 dirkertrigeqlem2 41225 fourierdlem26 41259 fourierdlem89 41321 fourierdlem90 41322 fourierdlem91 41323 fourierdlem107 41339 etransclem23 41383 sharhght 41963 sigaradd 41964 cevathlem2 41966 fmtnorec3 42463 1subrec1sub 43423 eenglngeehlnmlem1 43455 eenglngeehlnmlem2 43456 rrx2linest 43460 rrx2linest2 43462 line2 43470 itsclinecirc0b 43492 |
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