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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 10352 | . 2 ⊢ (𝐴 ∈ ℂ → 1 ∈ ℂ) | |
3 | 1, 2 | pncand 10715 | 1 ⊢ (𝐴 ∈ ℂ → ((𝐴 + 1) − 1) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 (class class class)co 6906 ℂcc 10251 1c1 10254 + caddc 10256 − cmin 10586 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-br 4875 df-opab 4937 df-mpt 4954 df-id 5251 df-po 5264 df-so 5265 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-er 8010 df-en 8224 df-dom 8225 df-sdom 8226 df-pnf 10394 df-mnf 10395 df-ltxr 10397 df-sub 10588 |
This theorem is referenced by: nn0split 12750 nn0disj 12751 elfzom1elp1fzo1 12864 sqoddm1div8 13325 wrdlenccats1lenm1 13683 ccats1pfxeq 13802 ccats1swrdeqOLD 13803 ltoddhalfle 15460 pwp1fsum 15489 flodddiv4 15511 prmop1 16114 cayhamlem1 21042 2lgslem1c 25532 2lgslem3a 25535 2lgslem3c 25537 2lgslem3d 25538 wlklenvm1 26920 wwlknp 27143 wwlknlsw 27147 0enwwlksnge1 27164 wlkiswwlks1 27167 wspthsnwspthsnon 27246 wspthsnonn0vne 27247 elwspths2spth 27297 wwlksext2clwwlk 27410 numclwwlk2lem1lem 27724 numclwlk2lem2f 27781 numclwlk2lem2fOLD 27784 numclwlk2lem2fOLDOLD 27792 poimirlem4 33958 poimirlem10 33964 poimirlem19 33973 poimirlem28 33982 sumnnodd 40658 iccpartgtprec 42245 fmtnom1nn 42275 fmtnorec1 42280 sfprmdvdsmersenne 42351 proththdlem 42361 41prothprmlem1 42365 dfodd6 42381 evenp1odd 42384 perfectALTVlem1 42461 altgsumbcALT 42979 fllog2 43210 nnpw2blen 43222 dig2nn1st 43247 nn0sumshdiglemA 43261 nn0sumshdiglemB 43262 aacllem 43444 |
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