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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 10625 | . 2 ⊢ (𝐴 ∈ ℂ → 1 ∈ ℂ) | |
3 | 1, 2 | pncand 10987 | 1 ⊢ (𝐴 ∈ ℂ → ((𝐴 + 1) − 1) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 (class class class)co 7135 ℂcc 10524 1c1 10527 + caddc 10529 − cmin 10859 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-ltxr 10669 df-sub 10861 |
This theorem is referenced by: nn0split 13017 nn0disj 13018 elfzom1elp1fzo1 13132 sqoddm1div8 13600 wrdlenccats1lenm1 13967 ccats1pfxeq 14067 ltoddhalfle 15702 pwp1fsum 15732 flodddiv4 15754 prmop1 16364 cayhamlem1 21471 2lgslem1c 25977 2lgslem3a 25980 wlklenvm1 27411 wwlknp 27629 wwlknlsw 27633 0enwwlksnge1 27650 wlkiswwlks1 27653 wspthsnwspthsnon 27702 wspthsnonn0vne 27703 elwspths2spth 27753 wwlksext2clwwlk 27842 numclwwlk2lem1lem 28127 numclwlk2lem2f 28162 poimirlem4 35061 poimirlem10 35067 poimirlem19 35076 poimirlem28 35085 sumnnodd 42272 iccpartgtprec 43937 fmtnom1nn 44049 fmtnorec1 44054 sfprmdvdsmersenne 44121 proththdlem 44131 41prothprmlem1 44135 dfodd6 44155 evenp1odd 44158 perfectALTVlem1 44239 altgsumbcALT 44755 fllog2 44982 nnpw2blen 44994 dig2nn1st 45019 nn0sumshdiglemA 45033 nn0sumshdiglemB 45034 aacllem 45329 |
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