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Mirrors > Home > MPE Home > Th. List > nn0nnaddcl | Structured version Visualization version GIF version |
Description: A nonnegative integer plus a positive integer is a positive integer. (Contributed by NM, 22-Dec-2005.) |
Ref | Expression |
---|---|
nn0nnaddcl | ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nncn 11321 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ) | |
2 | nn0cn 11591 | . . . 4 ⊢ (𝑀 ∈ ℕ0 → 𝑀 ∈ ℂ) | |
3 | addcom 10512 | . . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ) → (𝑁 + 𝑀) = (𝑀 + 𝑁)) | |
4 | 1, 2, 3 | syl2an 590 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ0) → (𝑁 + 𝑀) = (𝑀 + 𝑁)) |
5 | nnnn0addcl 11612 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ0) → (𝑁 + 𝑀) ∈ ℕ) | |
6 | 4, 5 | eqeltrrd 2879 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ0) → (𝑀 + 𝑁) ∈ ℕ) |
7 | 6 | ancoms 451 | 1 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 (class class class)co 6878 ℂcc 10222 + caddc 10227 ℕcn 11312 ℕ0cn0 11580 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-ov 6881 df-om 7300 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-pnf 10365 df-mnf 10366 df-ltxr 10368 df-nn 11313 df-n0 11581 |
This theorem is referenced by: nn0p1nn 11621 nnaddm1cl 11724 numnncl 11793 modfzo0difsn 12997 faclbnd4lem1 13333 vdwlem3 16020 vdwlem5 16022 vdwlem6 16023 vdwlem9 16026 mod2xnegi 16108 tdeglem4 24161 basellem5 25163 wwlksext2clwwlk 27373 wwlksext2clwwlkOLD 27374 fmtnodvds 42238 |
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