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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 10902 | . 2 ⊢ ℕ ⊆ ℂ | |
2 | id 22 | . . 3 ⊢ (ℕ ⊆ ℂ → ℕ ⊆ ℂ) | |
3 | df-n0 11170 | . . 3 ⊢ ℕ0 = (ℕ ∪ {0}) | |
4 | nnaddcl 10919 | . . . 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 11203 | . 2 ⊢ ((ℕ ⊆ ℂ ∧ (𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0)) → (𝑀 + 𝑁) ∈ ℕ0) |
7 | 1, 6 | mpan 702 | 1 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 + 𝑁) ∈ ℕ0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 ⊆ wss 3540 (class class class)co 6549 ℂcc 9813 + caddc 9818 ℕcn 10897 ℕ0cn0 11169 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-ltxr 9958 df-nn 10898 df-n0 11170 |
This theorem is referenced by: nn0addcli 11207 peano2nn0 11210 nn0addcld 11232 nn0readdcl 11234 xnn0xaddcl 11940 difelfznle 12322 elfzodifsumelfzo 12401 modsumfzodifsn 12605 expadd 12764 faclbnd4lem3 12944 faclbnd5 12947 faclbnd6 12948 facavg 12950 ccatlen 13213 ccatalpha 13228 swrdswrdlem 13311 swrdswrd 13312 swrdccatin1 13334 swrdccatin12lem3 13341 swrdccatid 13348 splfv2a 13358 repswswrd 13382 repswccat 13383 cshwcsh2id 13425 fsumnn0cl 14314 bcxmas 14406 nn0risefaccl 14592 eftlub 14678 4sqlem1 15490 psgnunilem2 17738 sylow1lem1 17836 psrbagaddcl 19191 nn0subm 19620 expmhm 19634 dvnadd 23498 ply1divex 23700 coemullem 23810 coemulhi 23814 plymul0or 23840 chtublem 24736 2sqlem7 24949 vdgrf 26425 numclwwlk2lem1 26629 fmtnofac2lem 40018 crctcsh1wlkn0lem4 41016 ply1mulgsumlem1 41968 |
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