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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 12271 | . 2 ⊢ ℕ ⊆ ℂ | |
| 2 | id 22 | . . 3 ⊢ (ℕ ⊆ ℂ → ℕ ⊆ ℂ) | |
| 3 | df-n0 12527 | . . 3 ⊢ ℕ0 = (ℕ ∪ {0}) | |
| 4 | nnaddcl 12289 | . . . 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 12559 | . 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 2108 ⊆ wss 3951 (class class class)co 7431 ℂcc 11153 + caddc 11158 ℕcn 12266 ℕ0cn0 12526 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 df-nn 12267 df-n0 12527 |
| This theorem is referenced by: nn0addcli 12563 peano2nn0 12566 nn0addcld 12591 nn0readdcl 12593 xnn0xaddcl 13277 difelfznle 13682 elfzodifsumelfzo 13770 modsumfzodifsn 13985 expadd 14145 faclbnd4lem3 14334 faclbnd5 14337 faclbnd6 14338 facavg 14340 ccatlen 14613 ccatrn 14627 ccatalpha 14631 swrdccat2 14707 swrdswrdlem 14742 swrdswrd 14743 swrdccatin1 14763 pfxccatin12lem3 14770 splfv2a 14794 repswswrd 14822 repswccat 14824 cshwcsh2id 14867 fsumnn0cl 15772 bcxmas 15871 nn0risefaccl 16058 eftlub 16145 4sqlem1 16986 psgnunilem2 19513 sylow1lem1 19616 nn0subm 21440 expmhm 21454 psrbagaddcl 21944 dvnadd 25965 ply1divex 26176 coemullem 26289 coemulhi 26293 plymul0or 26322 chtublem 27255 2sqlem7 27468 crctcshwlkn0lem4 29833 clwwlkccatlem 30008 fac2xp3 42240 factwoffsmonot 42243 mhphflem 42606 fmtnofac2lem 47555 nn0mnd 48095 ply1mulgsumlem1 48303 |
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