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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 11253 | . 2 ⊢ (𝐴 ∈ ℂ → 1 ∈ ℂ) | |
3 | 1, 2 | pncand 11618 | 1 ⊢ (𝐴 ∈ ℂ → ((𝐴 + 1) − 1) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2105 (class class class)co 7430 ℂcc 11150 1c1 11153 + caddc 11155 − cmin 11489 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-po 5596 df-so 5597 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-ltxr 11297 df-sub 11491 |
This theorem is referenced by: nn0split 13679 nn0disj 13680 elfzom1elp1fzo1 13802 sqoddm1div8 14278 wrdlenccats1lenm1 14656 ccats1pfxeq 14748 ltoddhalfle 16394 pwp1fsum 16424 flodddiv4 16448 prmop1 17071 cayhamlem1 22887 2lgslem1c 27451 2lgslem3a 27454 wlklenvm1 29654 wwlknp 29872 wwlknlsw 29876 0enwwlksnge1 29893 wlkiswwlks1 29896 wspthsnwspthsnon 29945 wspthsnonn0vne 29946 elwspths2spth 29996 wwlksext2clwwlk 30085 numclwwlk2lem1lem 30370 numclwlk2lem2f 30405 poimirlem4 37610 poimirlem10 37616 poimirlem19 37625 poimirlem28 37634 sumnnodd 45585 iccpartgtprec 47344 fmtnom1nn 47456 fmtnorec1 47461 sfprmdvdsmersenne 47527 proththdlem 47537 41prothprmlem1 47541 dfodd6 47561 evenp1odd 47564 perfectALTVlem1 47645 isubgr3stgrlem2 47869 gpgvtxedg0 47955 altgsumbcALT 48197 fllog2 48417 nnpw2blen 48429 dig2nn1st 48454 nn0sumshdiglemA 48468 nn0sumshdiglemB 48469 aacllem 49031 |
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