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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 11417 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 + 𝐵) ∈ ℤ) | |
4 | 1, 2, 3 | syl2anc 693 | 1 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 (class class class)co 6650 + caddc 9939 ℤcz 11377 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 |
This theorem is referenced by: zadd2cl 11490 qaddcl 11804 elincfzoext 12525 eluzgtdifelfzo 12529 fladdz 12626 seqshft2 12827 expaddzlem 12903 sqoddm1div8 13028 ccatass 13371 lswccatn0lsw 13373 cshf1 13556 2cshw 13559 2cshwcshw 13571 fsumrev2 14514 isumshft 14571 divcnvshft 14587 dvds2ln 15014 sadadd3 15183 sadaddlem 15188 sadadd 15189 bezoutlem4 15259 lcmgcdlem 15319 divgcdcoprm0 15379 hashdvds 15480 pythagtriplem4 15524 pythagtriplem11 15530 pcaddlem 15592 gzmulcl 15642 4sqlem8 15649 4sqlem10 15651 4sqlem11 15659 4sqlem14 15662 4sqlem16 15664 prmgaplem7 15761 prmgaplem8 15762 gsumccat 17378 mulgdir 17573 mndodconglem 17960 chfacfscmulfsupp 20664 chfacfpmmulfsupp 20668 ulmshftlem 24143 ulmshft 24144 dchrptlem2 24990 lgsqrlem2 25072 lgsquad2lem1 25109 2lgsoddprmlem2 25134 2sqlem4 25146 2sqlem8 25151 crctcshwlkn0lem5 26706 numclwlk2lem2f 27236 ex-ind-dvds 27318 2sqmod 29648 archirngz 29743 archiabllem2c 29749 qqhghm 30032 qqhrhm 30033 fsum2dsub 30685 breprexplemc 30710 divcnvlin 31618 caushft 33557 pell1234qrmulcl 37419 jm2.18 37555 jm2.19lem3 37558 jm2.19lem4 37559 jm2.25 37566 inductionexd 38453 fzisoeu 39514 uzubioo 39794 wallispilem4 40285 etransclem44 40495 gbowgt5 41650 mogoldbb 41673 nnsum4primesevenALTV 41689 2zlidl 41934 |
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