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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 11629 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 + 𝐵) ∈ ℤ) | |
4 | 1, 2, 3 | syl2anc 696 | 1 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2139 (class class class)co 6814 + caddc 10151 ℤcz 11589 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-mulcom 10212 ax-addass 10213 ax-mulass 10214 ax-distr 10215 ax-i2m1 10216 ax-1ne0 10217 ax-1rid 10218 ax-rnegex 10219 ax-rrecex 10220 ax-cnre 10221 ax-pre-lttri 10222 ax-pre-lttrn 10223 ax-pre-ltadd 10224 ax-pre-mulgt0 10225 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6775 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-om 7232 df-wrecs 7577 df-recs 7638 df-rdg 7676 df-er 7913 df-en 8124 df-dom 8125 df-sdom 8126 df-pnf 10288 df-mnf 10289 df-xr 10290 df-ltxr 10291 df-le 10292 df-sub 10480 df-neg 10481 df-nn 11233 df-n0 11505 df-z 11590 |
This theorem is referenced by: zadd2cl 11702 qaddcl 12017 elincfzoext 12740 eluzgtdifelfzo 12744 fladdz 12840 seqshft2 13041 expaddzlem 13117 sqoddm1div8 13242 ccatass 13580 lswccatn0lsw 13583 cshf1 13776 2cshw 13779 2cshwcshw 13791 fsumrev2 14733 isumshft 14790 divcnvshft 14806 dvds2ln 15236 sadadd3 15405 sadaddlem 15410 sadadd 15411 bezoutlem4 15481 lcmgcdlem 15541 divgcdcoprm0 15601 hashdvds 15702 pythagtriplem4 15746 pythagtriplem11 15752 pcaddlem 15814 gzmulcl 15864 4sqlem8 15871 4sqlem10 15873 4sqlem11 15881 4sqlem14 15884 4sqlem16 15886 prmgaplem7 15983 prmgaplem8 15984 gsumccat 17599 mulgdir 17794 mndodconglem 18180 chfacfscmulfsupp 20886 chfacfpmmulfsupp 20890 ulmshftlem 24362 ulmshft 24363 dchrptlem2 25210 lgsqrlem2 25292 lgsquad2lem1 25329 2lgsoddprmlem2 25354 2sqlem4 25366 2sqlem8 25371 crctcshwlkn0lem5 26938 numclwlk2lem2f 27559 numclwlk2lem2fOLD 27566 ex-ind-dvds 27650 2sqmod 29978 archirngz 30073 archiabllem2c 30079 qqhghm 30362 qqhrhm 30363 fsum2dsub 31015 breprexplemc 31040 divcnvlin 31946 caushft 33888 pell1234qrmulcl 37939 jm2.18 38075 jm2.19lem3 38078 jm2.19lem4 38079 jm2.25 38086 inductionexd 38973 fzisoeu 40031 uzubioo 40315 wallispilem4 40806 etransclem44 41016 gbowgt5 42178 mogoldbb 42201 nnsum4primesevenALTV 42217 2zlidl 42462 |
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