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Mirrors > Home > MPE Home > Th. List > nn0addcl | Structured version Visualization version GIF 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 | ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 + 𝑁) ∈ ℕ0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnsscn 12268 | . 2 ⊢ ℕ ⊆ ℂ | |
2 | id 22 | . . 3 ⊢ (ℕ ⊆ ℂ → ℕ ⊆ ℂ) | |
3 | df-n0 12524 | . . 3 ⊢ ℕ0 = (ℕ ∪ {0}) | |
4 | nnaddcl 12286 | . . . 4 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ) | |
5 | 4 | adantl 481 | . . 3 ⊢ ((ℕ ⊆ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ)) → (𝑀 + 𝑁) ∈ ℕ) |
6 | 2, 3, 5 | un0addcl 12556 | . 2 ⊢ ((ℕ ⊆ ℂ ∧ (𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0)) → (𝑀 + 𝑁) ∈ ℕ0) |
7 | 1, 6 | mpan 690 | 1 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 + 𝑁) ∈ ℕ0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2105 ⊆ wss 3962 (class class class)co 7430 ℂcc 11150 + caddc 11155 ℕcn 12263 ℕ0cn0 12523 |
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-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 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-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 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-ov 7433 df-om 7887 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-ltxr 11297 df-nn 12264 df-n0 12524 |
This theorem is referenced by: nn0addcli 12560 peano2nn0 12563 nn0addcld 12588 nn0readdcl 12590 xnn0xaddcl 13273 difelfznle 13678 elfzodifsumelfzo 13766 modsumfzodifsn 13981 expadd 14141 faclbnd4lem3 14330 faclbnd5 14333 faclbnd6 14334 facavg 14336 ccatlen 14609 ccatrn 14623 ccatalpha 14627 swrdccat2 14703 swrdswrdlem 14738 swrdswrd 14739 swrdccatin1 14759 pfxccatin12lem3 14766 splfv2a 14790 repswswrd 14818 repswccat 14820 cshwcsh2id 14863 fsumnn0cl 15768 bcxmas 15867 nn0risefaccl 16054 eftlub 16141 4sqlem1 16981 psgnunilem2 19527 sylow1lem1 19630 nn0subm 21457 expmhm 21471 psrbagaddcl 21961 dvnadd 25979 ply1divex 26190 coemullem 26303 coemulhi 26307 plymul0or 26336 chtublem 27269 2sqlem7 27482 crctcshwlkn0lem4 29842 clwwlkccatlem 30017 fac2xp3 42220 factwoffsmonot 42223 mhphflem 42582 fmtnofac2lem 47492 nn0mnd 48022 ply1mulgsumlem1 48231 |
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