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Mirrors > Home > ILE Home > Th. List > zaddcl | GIF version |
Description: Closure of addition of integers. (Contributed by NM, 9-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.) |
Ref | Expression |
---|---|
zaddcl | ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 + 𝑁) ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elz 9253 | . . . 4 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ))) | |
2 | 1 | simprbi 275 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)) |
3 | 2 | adantl 277 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)) |
4 | zcn 9256 | . . . . . . 7 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ) | |
5 | 4 | adantr 276 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 ∈ ℂ) |
6 | 5 | addid1d 8104 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 + 0) = 𝑀) |
7 | simpl 109 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 ∈ ℤ) | |
8 | 6, 7 | eqeltrd 2254 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 + 0) ∈ ℤ) |
9 | oveq2 5882 | . . . . 5 ⊢ (𝑁 = 0 → (𝑀 + 𝑁) = (𝑀 + 0)) | |
10 | 9 | eleq1d 2246 | . . . 4 ⊢ (𝑁 = 0 → ((𝑀 + 𝑁) ∈ ℤ ↔ (𝑀 + 0) ∈ ℤ)) |
11 | 8, 10 | syl5ibrcom 157 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 = 0 → (𝑀 + 𝑁) ∈ ℤ)) |
12 | zaddcllempos 9288 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℤ) | |
13 | 12 | ex 115 | . . . 4 ⊢ (𝑀 ∈ ℤ → (𝑁 ∈ ℕ → (𝑀 + 𝑁) ∈ ℤ)) |
14 | 13 | adantr 276 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ ℕ → (𝑀 + 𝑁) ∈ ℤ)) |
15 | zre 9255 | . . . 4 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
16 | zaddcllemneg 9290 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℤ) | |
17 | 16 | 3expia 1205 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℝ) → (-𝑁 ∈ ℕ → (𝑀 + 𝑁) ∈ ℤ)) |
18 | 15, 17 | sylan2 286 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (-𝑁 ∈ ℕ → (𝑀 + 𝑁) ∈ ℤ)) |
19 | 11, 14, 18 | 3jaod 1304 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℤ)) |
20 | 3, 19 | mpd 13 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 + 𝑁) ∈ ℤ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∨ w3o 977 = wceq 1353 ∈ wcel 2148 (class class class)co 5874 ℂcc 7808 ℝcr 7809 0cc0 7810 + caddc 7813 -cneg 8127 ℕcn 8917 ℤcz 9251 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-addcom 7910 ax-addass 7912 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-0id 7918 ax-rnegex 7919 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-ltadd 7926 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-br 4004 df-opab 4065 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-iota 5178 df-fun 5218 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-pnf 7992 df-mnf 7993 df-xr 7994 df-ltxr 7995 df-le 7996 df-sub 8128 df-neg 8129 df-inn 8918 df-n0 9175 df-z 9252 |
This theorem is referenced by: zsubcl 9292 zrevaddcl 9301 zdivadd 9340 zaddcld 9377 eluzaddi 9552 eluzsubi 9553 eluzadd 9554 nn0pzuz 9585 fzen 10040 fzaddel 10056 fzrev3 10084 fzrevral3 10104 elfzmlbp 10129 fzoaddel 10189 zpnn0elfzo 10204 elfzomelpfzo 10228 fzoshftral 10235 climshftlemg 11305 fsumzcl 11405 summodnegmod 11824 dvds2ln 11826 dvds2add 11827 dvdsadd 11838 dvdsadd2b 11842 addmodlteqALT 11859 3dvdsdec 11864 3dvds2dec 11865 opoe 11894 opeo 11896 ndvdsadd 11930 pythagtriplem9 12267 difsqpwdvds 12331 gzaddcl 12369 zsubrg 13406 zringmulg 13419 |
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