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Mirrors > Home > MPE Home > Th. List > zaddcld | Structured version Visualization version GIF version |
Description: Closure of addition of integers. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
zred.1 | ⊢ (𝜑 → 𝐴 ∈ ℤ) |
zaddcld.1 | ⊢ (𝜑 → 𝐵 ∈ ℤ) |
Ref | Expression |
---|---|
zaddcld | ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zred.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℤ) | |
2 | zaddcld.1 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℤ) | |
3 | zaddcl 12010 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 + 𝐵) ∈ ℤ) | |
4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 (class class class)co 7145 + caddc 10528 ℤcz 11969 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-n0 11886 df-z 11970 |
This theorem is referenced by: zadd2cl 12083 qaddcl 12352 elincfzoext 13083 eluzgtdifelfzo 13087 fladdz 13183 seqshft2 13384 expaddzlem 13460 sqoddm1div8 13592 ccatass 13930 cshf1 14160 2cshw 14163 2cshwcshw 14175 fsumrev2 15125 isumshft 15182 divcnvshft 15198 dvds2ln 15630 sadadd3 15798 sadaddlem 15803 sadadd 15804 bezoutlem4 15878 lcmgcdlem 15938 divgcdcoprm0 15997 hashdvds 16100 pythagtriplem4 16144 pythagtriplem11 16150 pcaddlem 16212 gzmulcl 16262 4sqlem8 16269 4sqlem10 16271 4sqlem11 16279 4sqlem14 16282 4sqlem16 16284 prmgaplem7 16381 prmgaplem8 16382 gsumsgrpccat 17992 gsumccatOLD 17993 mulgdir 18197 mndodconglem 18598 chfacfscmulfsupp 21395 chfacfpmmulfsupp 21399 ulmshftlem 24904 ulmshft 24905 dchrptlem2 25768 lgsqrlem2 25850 lgsquad2lem1 25887 2lgsoddprmlem2 25912 2sqlem4 25924 2sqlem8 25929 2sqmod 25939 crctcshwlkn0lem5 27519 numclwlk2lem2f 28083 ex-ind-dvds 28167 cshwrnid 30562 archirngz 30745 archiabllem2c 30751 qqhghm 31128 qqhrhm 31129 fsum2dsub 31777 breprexplemc 31802 divcnvlin 32861 caushft 34917 pell1234qrmulcl 39330 jm2.18 39463 jm2.19lem3 39466 jm2.19lem4 39467 jm2.25 39474 inductionexd 40383 fzisoeu 41443 uzubioo 41719 wallispilem4 42230 etransclem44 42440 gbowgt5 43804 mogoldbb 43827 nnsum4primesevenALTV 43843 2zlidl 44133 |
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