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Mirrors > Home > MPE Home > Th. List > npcan | Structured version Visualization version GIF version |
Description: Cancellation law for subtraction. (Contributed by NM, 10-May-2004.) (Revised by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
npcan | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) + 𝐵) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subcl 10879 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − 𝐵) ∈ ℂ) | |
2 | simpr 487 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ ℂ) | |
3 | 1, 2 | addcomd 10836 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) + 𝐵) = (𝐵 + (𝐴 − 𝐵))) |
4 | pncan3 10888 | . . 3 ⊢ ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (𝐵 + (𝐴 − 𝐵)) = 𝐴) | |
5 | 4 | ancoms 461 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐵 + (𝐴 − 𝐵)) = 𝐴) |
6 | 3, 5 | eqtrd 2856 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) + 𝐵) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 (class class class)co 7150 ℂcc 10529 + caddc 10534 − cmin 10864 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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-rab 3147 df-v 3497 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 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-po 5469 df-so 5470 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-ltxr 10674 df-sub 10866 |
This theorem is referenced by: addsubass 10890 npncan 10901 nppcan 10902 nnpcan 10903 subcan2 10905 nnncan 10915 npcand 10995 nn1suc 11653 zlem1lt 12028 zltlem1 12029 peano5uzi 12065 nummac 12137 uzp1 12273 peano2uzr 12297 qbtwnre 12586 fz01en 12929 fzsuc2 12959 fseq1m1p1 12976 predfz 13026 fzoss2 13059 fzoaddel2 13087 fzosplitsnm1 13106 fldiv 13222 modfzo0difsn 13305 seqm1 13381 monoord2 13395 sermono 13396 seqf1olem1 13403 seqf1olem2 13404 seqz 13412 expm1t 13451 expubnd 13535 bcm1k 13669 bcn2 13673 hashfzo 13784 hashbclem 13804 hashf1 13809 seqcoll 13816 swrdfv2 14017 swrdspsleq 14021 swrdlsw 14023 addlenrevpfx 14046 ccatpfx 14057 cshwlen 14155 cshwidxmodr 14160 cshwidxm 14164 swrd2lsw 14308 shftlem 14421 shftfval 14423 seqshft 14438 iserex 15007 serf0 15031 iseralt 15035 sumrblem 15062 fsumm1 15100 mptfzshft 15127 binomlem 15178 binom1dif 15182 isumsplit 15189 climcndslem1 15198 binomrisefac 15390 bpolycl 15400 bpolysum 15401 bpolydiflem 15402 bpoly2 15405 bpoly3 15406 fsumcube 15408 ruclem12 15588 dvdssub2 15645 4sqlem19 16293 vdwapun 16304 vdwapid1 16305 vdwlem5 16315 vdwlem8 16318 vdwnnlem2 16326 ramub1lem2 16357 1259lem4 16461 1259prm 16463 2503prm 16467 4001prm 16472 gsumsgrpccat 17998 gsumccatOLD 17999 sylow1lem1 18717 efgsres 18858 efgredleme 18863 gsummptshft 19050 ablsimpgfindlem1 19223 icccvx 23548 reparphti 23595 ovolunlem1 24092 advlog 25231 cxpaddlelem 25326 ang180lem1 25381 ang180lem3 25383 asinlem2 25441 tanatan 25491 ppiub 25774 perfect1 25798 lgsquad2lem1 25954 rplogsumlem1 26054 selberg2lem 26120 logdivbnd 26126 pntrsumo1 26135 pntrsumbnd2 26137 ax5seglem3 26711 ax5seglem5 26713 axbtwnid 26719 axlowdimlem16 26737 axeuclidlem 26742 axcontlem2 26745 crctcshwlkn0lem6 27587 clwwlknonex2lem2 27881 clwwlknonex2 27882 eucrctshift 28016 cvmliftlem7 32533 nndivsub 33800 ltflcei 34874 itg2addnclem3 34939 mettrifi 35026 irrapxlem1 39412 rmspecsqrtnq 39496 jm2.24nn 39549 jm2.18 39578 jm2.23 39586 jm2.27c 39597 monoord2xrv 41752 itgsinexp 42232 2elfz2melfz 43511 sbgoldbwt 43935 sgoldbeven3prm 43941 evengpop3 43956 evengpoap3 43957 zlmodzxzsub 44401 |
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