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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 11457 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − 𝐵) ∈ ℂ) | |
2 | simpr 484 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ ℂ) | |
3 | 1, 2 | addcomd 11414 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) + 𝐵) = (𝐵 + (𝐴 − 𝐵))) |
4 | pncan3 11466 | . . 3 ⊢ ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (𝐵 + (𝐴 − 𝐵)) = 𝐴) | |
5 | 4 | ancoms 458 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐵 + (𝐴 − 𝐵)) = 𝐴) |
6 | 3, 5 | eqtrd 2764 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) + 𝐵) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 (class class class)co 7402 ℂcc 11105 + caddc 11110 − cmin 11442 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-po 5579 df-so 5580 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11248 df-mnf 11249 df-ltxr 11251 df-sub 11444 |
This theorem is referenced by: addsubass 11468 npncan 11479 nppcan 11480 nnpcan 11481 subcan2 11483 nnncan 11493 npcand 11573 nn1suc 12232 zlem1lt 12612 zltlem1 12613 peano5uzi 12649 nummac 12720 uzp1 12861 peano2uzr 12885 qbtwnre 13176 fz01en 13527 fzsuc2 13557 fseq1m1p1 13574 predfz 13624 fzoss2 13658 fzoaddel2 13686 fzosplitsnm1 13705 fldiv 13823 modfzo0difsn 13906 seqm1 13983 monoord2 13997 sermono 13998 seqf1olem1 14005 seqf1olem2 14006 seqz 14014 expm1t 14054 expubnd 14140 bcm1k 14273 bcn2 14277 hashfzo 14387 hashbclem 14409 hashf1 14416 seqcoll 14423 swrdfv2 14609 swrdspsleq 14613 swrdlsw 14615 addlenrevpfx 14638 ccatpfx 14649 cshwlen 14747 cshwidxmodr 14752 cshwidxm 14756 swrd2lsw 14901 shftlem 15013 shftfval 15015 seqshft 15030 iserex 15601 serf0 15625 iseralt 15629 sumrblem 15655 fsumm1 15695 mptfzshft 15722 binomlem 15773 binom1dif 15777 isumsplit 15784 climcndslem1 15793 binomrisefac 15984 bpolycl 15994 bpolysum 15995 bpolydiflem 15996 bpoly2 15999 bpoly3 16000 fsumcube 16002 ruclem12 16183 dvdssub2 16243 4sqlem19 16897 vdwapun 16908 vdwapid1 16909 vdwlem5 16919 vdwlem8 16922 vdwnnlem2 16930 ramub1lem2 16961 1259lem4 17068 1259prm 17070 2503prm 17074 4001prm 17079 gsumsgrpccat 18757 sylow1lem1 19510 efgsres 19650 efgredleme 19655 gsummptshft 19848 ablsimpgfindlem1 20021 icccvx 24799 reparphti 24847 reparphtiOLD 24848 ovolunlem1 25350 advlog 26507 cxpaddlelem 26605 ang180lem1 26660 ang180lem3 26662 asinlem2 26720 tanatan 26770 ppiub 27056 perfect1 27080 lgsquad2lem1 27236 rplogsumlem1 27336 selberg2lem 27402 logdivbnd 27408 pntrsumo1 27417 pntrsumbnd2 27419 ax5seglem3 28661 ax5seglem5 28663 axbtwnid 28669 axlowdimlem16 28687 axeuclidlem 28692 axcontlem2 28695 crctcshwlkn0lem6 29541 clwwlknonex2lem2 29833 clwwlknonex2 29834 eucrctshift 29968 cvmliftlem7 34773 nndivsub 35833 ltflcei 36970 itg2addnclem3 37035 mettrifi 37119 irrapxlem1 42074 rmspecsqrtnq 42158 jm2.24nn 42212 jm2.18 42241 jm2.23 42249 jm2.27c 42260 monoord2xrv 44704 itgsinexp 45181 2elfz2melfz 46536 sbgoldbwt 46955 sgoldbeven3prm 46961 evengpop3 46976 evengpoap3 46977 zlmodzxzsub 47250 ackval42 47595 |
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