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Mirrors > Home > MPE Home > Th. List > nn0addcl | Unicode version |
Description: Closure of addition of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Mario Carneiro, 17-Jul-2014.) |
Ref | Expression |
---|---|
nn0addcl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnsscn 10566 | . 2 | |
2 | id 22 | . . 3 | |
3 | df-n0 10821 | . . 3 | |
4 | nnaddcl 10583 | . . . 4 | |
5 | 4 | adantl 466 | . . 3 |
6 | 2, 3, 5 | un0addcl 10854 | . 2 |
7 | 1, 6 | mpan 670 | 1 |
Colors of variables: wff setvar class |
Syntax hints: -> wi 4 /\ wa 369
e. wcel 1818 C_ wss 3475 (class class class)co 6296
cc 9511 caddc 9516 cn 10561 cn0 10820 |
This theorem is referenced by: nn0addcli 10858 peano2nn0 10861 nn0addcld 10881 nn0readdcl 10883 elfz0addOLD 11805 difelfznle 11818 elfzodifsumelfzo 11882 expadd 12208 faclbnd4lem3 12373 faclbnd5 12376 faclbnd6 12377 facavg 12379 ccatlen 12594 swrdswrdlem 12684 swrdswrd 12685 swrdccatin1 12708 swrdccatin12lem3 12715 swrdccatid 12722 splfv2a 12732 repswswrd 12756 repswccat 12757 cshwcsh2id 12796 fsumnn0cl 13558 bcxmas 13647 eftlub 13844 4sqlem1 14466 psgnunilem2 16520 sylow1lem1 16618 psrbagaddcl 18020 psrbagaddclOLD 18021 nn0subm 18473 expmhm 18485 dvnadd 22332 ply1divex 22537 coemullem 22647 coemulhi 22651 plymul0or 22677 chtublem 23486 2sqlem7 23645 vdgrf 24898 numclwwlk2lem1 25102 relexpsucr 29053 relexpadd 29061 nn0risefaccl 29144 ply1mulgsumlem1 32986 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1618 ax-4 1631 ax-5 1704 ax-6 1747 ax-7 1790 ax-8 1820 ax-9 1822 ax-10 1837 ax-11 1842 ax-12 1854 ax-13 1999 ax-ext 2435 ax-sep 4573 ax-nul 4581 ax-pow 4630 ax-pr 4691 ax-un 6592 ax-resscn 9570 ax-1cn 9571 ax-icn 9572 ax-addcl 9573 ax-addrcl 9574 ax-mulcl 9575 ax-mulrcl 9576 ax-mulcom 9577 ax-addass 9578 ax-mulass 9579 ax-distr 9580 ax-i2m1 9581 ax-1ne0 9582 ax-1rid 9583 ax-rnegex 9584 ax-rrecex 9585 ax-cnre 9586 ax-pre-lttri 9587 ax-pre-lttrn 9588 ax-pre-ltadd 9589 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 974 df-3an 975 df-tru 1398 df-ex 1613 df-nf 1617 df-sb 1740 df-eu 2286 df-mo 2287 df-clab 2443 df-cleq 2449 df-clel 2452 df-nfc 2607 df-ne 2654 df-nel 2655 df-ral 2812 df-rex 2813 df-reu 2814 df-rab 2816 df-v 3111 df-sbc 3328 df-csb 3435 df-dif 3478 df-un 3480 df-in 3482 df-ss 3489 df-pss 3491 df-nul 3785 df-if 3942 df-pw 4014 df-sn 4030 df-pr 4032 df-tp 4034 df-op 4036 df-uni 4250 df-iun 4332 df-br 4453 df-opab 4511 df-mpt 4512 df-tr 4546 df-eprel 4796 df-id 4800 df-po 4805 df-so 4806 df-fr 4843 df-we 4845 df-ord 4886 df-on 4887 df-lim 4888 df-suc 4889 df-xp 5010 df-rel 5011 df-cnv 5012 df-co 5013 df-dm 5014 df-rn 5015 df-res 5016 df-ima 5017 df-iota 5556 df-fun 5595 df-fn 5596 df-f 5597 df-f1 5598 df-fo 5599 df-f1o 5600 df-fv 5601 df-ov 6299 df-om 6701 df-recs 7061 df-rdg 7095 df-er 7330 df-en 7537 df-dom 7538 df-sdom 7539 df-pnf 9651 df-mnf 9652 df-ltxr 9654 df-nn 10562 df-n0 10821 |
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