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Mirrors > Home > MPE Home > Th. List > pncan1 | Structured version Visualization version GIF version |
Description: Cancellation law for addition and subtraction with 1. (Contributed by Alexander van der Vekens, 3-Oct-2018.) |
Ref | Expression |
---|---|
pncan1 | ⊢ (𝐴 ∈ ℂ → ((𝐴 + 1) − 1) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . 2 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ ℂ) | |
2 | 1cnd 10371 | . 2 ⊢ (𝐴 ∈ ℂ → 1 ∈ ℂ) | |
3 | 1, 2 | pncand 10735 | 1 ⊢ (𝐴 ∈ ℂ → ((𝐴 + 1) − 1) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 (class class class)co 6922 ℂcc 10270 1c1 10273 + caddc 10275 − cmin 10606 |
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 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 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 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 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 |
This theorem is referenced by: nn0split 12773 nn0disj 12774 elfzom1elp1fzo1 12887 sqoddm1div8 13349 wrdlenccats1lenm1 13712 ccats1pfxeq 13831 ccats1swrdeqOLD 13832 ltoddhalfle 15489 pwp1fsum 15521 flodddiv4 15543 prmop1 16146 cayhamlem1 21078 2lgslem1c 25570 2lgslem3a 25573 2lgslem3c 25575 2lgslem3d 25576 wlklenvm1 26969 wwlknp 27192 wwlknlsw 27196 0enwwlksnge1 27213 wlkiswwlks1 27216 wspthsnwspthsnon 27296 wspthsnonn0vne 27297 elwspths2spth 27347 wwlksext2clwwlk 27454 numclwwlk2lem1lem 27750 numclwlk2lem2f 27805 numclwlk2lem2fOLD 27808 poimirlem4 34041 poimirlem10 34047 poimirlem19 34056 poimirlem28 34065 sumnnodd 40774 iccpartgtprec 42392 fmtnom1nn 42469 fmtnorec1 42474 sfprmdvdsmersenne 42545 proththdlem 42555 41prothprmlem1 42559 dfodd6 42579 evenp1odd 42582 perfectALTVlem1 42659 altgsumbcALT 43150 fllog2 43381 nnpw2blen 43393 dig2nn1st 43418 nn0sumshdiglemA 43432 nn0sumshdiglemB 43433 aacllem 43657 |
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