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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 11256 | . 2 ⊢ (𝐴 ∈ ℂ → 1 ∈ ℂ) | |
| 3 | 1, 2 | pncand 11621 | 1 ⊢ (𝐴 ∈ ℂ → ((𝐴 + 1) − 1) = 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 (class class class)co 7431 ℂcc 11153 1c1 11156 + caddc 11158 − cmin 11492 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 df-sub 11494 |
| This theorem is referenced by: nn0split 13683 nn0disj 13684 elfzom1elp1fzo1 13806 sqoddm1div8 14282 wrdlenccats1lenm1 14660 ccats1pfxeq 14752 ltoddhalfle 16398 pwp1fsum 16428 flodddiv4 16452 prmop1 17076 psdpw 22174 cayhamlem1 22872 2lgslem1c 27437 2lgslem3a 27440 wlklenvm1 29640 wwlknp 29863 wwlknlsw 29867 0enwwlksnge1 29884 wlkiswwlks1 29887 wspthsnwspthsnon 29936 wspthsnonn0vne 29937 elwspths2spth 29987 wwlksext2clwwlk 30076 numclwwlk2lem1lem 30361 numclwlk2lem2f 30396 poimirlem4 37631 poimirlem10 37637 poimirlem19 37646 poimirlem28 37655 sumnnodd 45645 iccpartgtprec 47407 fmtnom1nn 47519 fmtnorec1 47524 sfprmdvdsmersenne 47590 proththdlem 47600 41prothprmlem1 47604 dfodd6 47624 evenp1odd 47627 perfectALTVlem1 47708 isubgr3stgrlem2 47934 gpgvtxedg0 48021 altgsumbcALT 48269 fllog2 48489 nnpw2blen 48501 dig2nn1st 48526 nn0sumshdiglemA 48540 nn0sumshdiglemB 48541 aacllem 49320 |
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