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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 10318 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − 𝐵) ∈ ℂ) | |
2 | simpr 476 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ ℂ) | |
3 | 1, 2 | addcomd 10276 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) + 𝐵) = (𝐵 + (𝐴 − 𝐵))) |
4 | pncan3 10327 | . . 3 ⊢ ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (𝐵 + (𝐴 − 𝐵)) = 𝐴) | |
5 | 4 | ancoms 468 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐵 + (𝐴 − 𝐵)) = 𝐴) |
6 | 3, 5 | eqtrd 2685 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) + 𝐵) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 (class class class)co 6690 ℂcc 9972 + caddc 9977 − cmin 10304 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-po 5064 df-so 5065 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-ltxr 10117 df-sub 10306 |
This theorem is referenced by: addsubass 10329 npncan 10340 nppcan 10341 nnpcan 10342 subcan2 10344 nnncan 10354 npcand 10434 nn1suc 11079 zlem1lt 11467 zltlem1 11468 peano5uzi 11504 nummac 11596 uzp1 11759 peano2uzr 11781 qbtwnre 12068 fz01en 12407 fzsuc2 12436 fseq1m1p1 12453 predfz 12503 fzoss2 12535 fzoaddel2 12563 fzosplitsnm1 12582 fldiv 12699 modfzo0difsn 12782 seqm1 12858 monoord2 12872 sermono 12873 seqf1olem1 12880 seqf1olem2 12881 seqz 12889 expm1t 12928 expubnd 12961 bcm1k 13142 bcn2 13146 hashfzo 13254 hashbclem 13274 hashf1 13279 seqcoll 13286 addlenrevswrd 13483 swrdfv2 13492 swrdspsleq 13495 swrdlsw 13498 cshwlen 13591 cshwidxmod 13595 cshwidxmodr 13596 cshwidxm 13600 swrd2lsw 13741 shftlem 13852 shftfval 13854 seqshft 13869 iserex 14431 serf0 14455 iseralt 14459 sumrblem 14486 fsumm1 14524 mptfzshft 14554 binomlem 14605 binom1dif 14609 isumsplit 14616 climcndslem1 14625 binomrisefac 14817 bpolycl 14827 bpolysum 14828 bpolydiflem 14829 bpoly2 14832 bpoly3 14833 fsumcube 14835 ruclem12 15014 dvdssub2 15070 4sqlem19 15714 vdwapun 15725 vdwapid1 15726 vdwlem5 15736 vdwlem8 15739 vdwnnlem2 15747 ramub1lem2 15778 1259lem4 15888 1259prm 15890 2503prm 15894 4001prm 15899 gsumccat 17425 sylow1lem1 18059 efgsres 18197 efgredleme 18202 gsummptshft 18382 icccvx 22796 reparphti 22843 ovolunlem1 23311 advlog 24445 cxpaddlelem 24537 ang180lem1 24584 ang180lem3 24586 asinlem2 24641 tanatan 24691 ppiub 24974 perfect1 24998 lgsquad2lem1 25154 rplogsumlem1 25218 selberg2lem 25284 logdivbnd 25290 pntrsumo1 25299 pntrsumbnd2 25301 ax5seglem3 25856 ax5seglem5 25858 axbtwnid 25864 axlowdimlem16 25882 axeuclidlem 25887 axcontlem2 25890 crctcshwlkn0lem6 26763 clwwlknonex2lem2 27083 clwwlknonex2 27084 eucrctshift 27221 cvmliftlem7 31399 nndivsub 32581 ltflcei 33527 itg2addnclem3 33593 mettrifi 33683 irrapxlem1 37703 rmspecsqrtnq 37787 rmspecsqrtnqOLD 37788 jm2.24nn 37843 jm2.18 37872 jm2.23 37880 jm2.27c 37891 monoord2xrv 40027 itgsinexp 40488 2elfz2melfz 41653 addlenrevpfx 41722 sbgoldbwt 41990 sgoldbeven3prm 41996 evengpop3 42011 evengpoap3 42012 zlmodzxzsub 42463 |
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