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Mirrors > Home > MPE Home > Th. List > pncan | 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 |
---|---|
pncan | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) − 𝐵) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 484 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ ℂ) | |
2 | simpl 482 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐴 ∈ ℂ) | |
3 | 1, 2 | addcomd 11414 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐵 + 𝐴) = (𝐴 + 𝐵)) |
4 | addcl 11189 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ) | |
5 | subadd 11461 | . . 3 ⊢ (((𝐴 + 𝐵) ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (((𝐴 + 𝐵) − 𝐵) = 𝐴 ↔ (𝐵 + 𝐴) = (𝐴 + 𝐵))) | |
6 | 4, 1, 2, 5 | syl3anc 1368 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((𝐴 + 𝐵) − 𝐵) = 𝐴 ↔ (𝐵 + 𝐴) = (𝐴 + 𝐵))) |
7 | 3, 6 | mpbird 257 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) − 𝐵) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ 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: pncan2 11465 addsubass 11468 pncan3oi 11474 subid1 11478 nppcan2 11489 pncand 11570 nn1m1nn 12231 nnsub 12254 elnn0nn 12512 elz2 12574 zrevaddcl 12605 nzadd 12608 qrevaddcl 12953 irradd 12955 fzrev3 13565 elfzp1b 13576 fzrevral3 13586 fzval3 13699 seqf1olem1 14005 seqf1olem2 14006 bcp1nk 14275 bcp1m1 14278 bcpasc 14279 hashbclem 14409 ccatalpha 14541 wrdind 14670 wrd2ind 14671 2cshwcshw 14774 shftlem 15013 shftval5 15023 isershft 15608 isercoll2 15613 mptfzshft 15722 telfsumo 15746 fsumparts 15750 bcxmas 15779 isum1p 15785 geolim 15814 mertenslem2 15829 mertens 15830 fsumkthpow 15998 eftlub 16051 effsumlt 16053 eirrlem 16146 dvdsadd 16244 prmind2 16621 iserodd 16769 fldivp1 16831 prmpwdvds 16838 pockthlem 16839 prmreclem4 16853 prmreclem6 16855 4sqlem11 16889 vdwapun 16908 ramub1lem1 16960 ramcl 16963 efgsval2 19645 efgsrel 19646 shft2rab 25361 uniioombllem3 25438 uniioombllem4 25439 dvexp 25809 dvfsumlem1 25884 degltp1le 25933 ply1divex 25996 plyaddlem1 26069 plymullem1 26070 dvply1 26140 dvply2g 26141 vieta1lem2 26167 aaliou3lem7 26205 dvradcnv 26276 pserdvlem2 26284 abssinper 26374 advlogexp 26508 atantayl3 26790 leibpilem2 26792 emcllem2 26848 harmonicbnd4 26862 basellem8 26939 ppiprm 27002 ppinprm 27003 chtprm 27004 chtnprm 27005 chpp1 27006 chtub 27064 perfectlem1 27081 perfectlem2 27082 perfect 27083 bcp1ctr 27131 lgsvalmod 27168 lgseisen 27231 lgsquadlem1 27232 lgsquad2lem1 27236 2sqlem10 27280 rplogsumlem1 27336 selberg2lem 27402 logdivbnd 27408 pntrsumo1 27417 pntpbnd2 27439 clwwlkf1 29774 subfacp1lem5 34666 subfacp1lem6 34667 subfacval2 34669 subfaclim 34670 cvmliftlem7 34773 cvmliftlem10 34776 mblfinlem2 37020 itg2addnclem3 37035 fdc 37107 mettrifi 37119 heiborlem4 37176 heiborlem6 37178 lzenom 42022 2nn0ind 42198 jm2.17a 42213 jm2.17b 42214 jm2.17c 42215 evensumeven 46885 perfectALTVlem2 46900 perfectALTV 46901 |
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