![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 11876 | . 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 2081 (class class class)co 7021 + caddc 10391 ℤcz 11834 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5099 ax-nul 5106 ax-pow 5162 ax-pr 5226 ax-un 7324 ax-resscn 10445 ax-1cn 10446 ax-icn 10447 ax-addcl 10448 ax-addrcl 10449 ax-mulcl 10450 ax-mulrcl 10451 ax-mulcom 10452 ax-addass 10453 ax-mulass 10454 ax-distr 10455 ax-i2m1 10456 ax-1ne0 10457 ax-1rid 10458 ax-rnegex 10459 ax-rrecex 10460 ax-cnre 10461 ax-pre-lttri 10462 ax-pre-lttrn 10463 ax-pre-ltadd 10464 ax-pre-mulgt0 10465 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3710 df-csb 3816 df-dif 3866 df-un 3868 df-in 3870 df-ss 3878 df-pss 3880 df-nul 4216 df-if 4386 df-pw 4459 df-sn 4477 df-pr 4479 df-tp 4481 df-op 4483 df-uni 4750 df-iun 4831 df-br 4967 df-opab 5029 df-mpt 5046 df-tr 5069 df-id 5353 df-eprel 5358 df-po 5367 df-so 5368 df-fr 5407 df-we 5409 df-xp 5454 df-rel 5455 df-cnv 5456 df-co 5457 df-dm 5458 df-rn 5459 df-res 5460 df-ima 5461 df-pred 6028 df-ord 6074 df-on 6075 df-lim 6076 df-suc 6077 df-iota 6194 df-fun 6232 df-fn 6233 df-f 6234 df-f1 6235 df-fo 6236 df-f1o 6237 df-fv 6238 df-riota 6982 df-ov 7024 df-oprab 7025 df-mpo 7026 df-om 7442 df-wrecs 7803 df-recs 7865 df-rdg 7903 df-er 8144 df-en 8363 df-dom 8364 df-sdom 8365 df-pnf 10528 df-mnf 10529 df-xr 10530 df-ltxr 10531 df-le 10532 df-sub 10724 df-neg 10725 df-nn 11492 df-n0 11751 df-z 11835 |
This theorem is referenced by: zadd2cl 11949 qaddcl 12219 elincfzoext 12950 eluzgtdifelfzo 12954 fladdz 13050 seqshft2 13251 expaddzlem 13327 sqoddm1div8 13459 ccatass 13791 lswccatn0lsw 13794 cshf1 14013 2cshw 14016 2cshwcshw 14028 fsumrev2 14975 isumshft 15032 divcnvshft 15048 dvds2ln 15480 sadadd3 15648 sadaddlem 15653 sadadd 15654 bezoutlem4 15724 lcmgcdlem 15784 divgcdcoprm0 15843 hashdvds 15946 pythagtriplem4 15990 pythagtriplem11 15996 pcaddlem 16058 gzmulcl 16108 4sqlem8 16115 4sqlem10 16117 4sqlem11 16125 4sqlem14 16128 4sqlem16 16130 prmgaplem7 16227 prmgaplem8 16228 gsumccat 17822 mulgdir 18018 mndodconglem 18405 chfacfscmulfsupp 21156 chfacfpmmulfsupp 21160 ulmshftlem 24665 ulmshft 24666 dchrptlem2 25528 lgsqrlem2 25610 lgsquad2lem1 25647 2lgsoddprmlem2 25672 2sqlem4 25684 2sqlem8 25689 2sqmod 25699 crctcshwlkn0lem5 27284 numclwlk2lem2f 27853 ex-ind-dvds 27937 cshwrnid 30314 archirngz 30461 archiabllem2c 30467 qqhghm 30851 qqhrhm 30852 fsum2dsub 31500 breprexplemc 31525 divcnvlin 32579 caushft 34594 pell1234qrmulcl 38963 jm2.18 39096 jm2.19lem3 39099 jm2.19lem4 39100 jm2.25 39107 inductionexd 40016 fzisoeu 41134 uzubioo 41411 wallispilem4 41922 etransclem44 42132 gbowgt5 43436 mogoldbb 43459 nnsum4primesevenALTV 43475 2zlidl 43710 |
Copyright terms: Public domain | W3C validator |