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Mirrors > Home > MPE Home > Th. List > Mathboxes > 2zrng | Structured version Visualization version GIF version |
Description: The ring of integers restricted to the even integers is a non-unital ring, the "ring of even integers". Remark: the structure of the complementary subset of the set of integers, the odd integers, is not even a magma, see oddinmgm 47900. (Contributed by AV, 20-Feb-2020.) |
Ref | Expression |
---|---|
2zrng.e | ⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} |
2zlidl.u | ⊢ 𝑈 = (LIdeal‘ℤring) |
2zrng.r | ⊢ 𝑅 = (ℤring ↾s 𝐸) |
Ref | Expression |
---|---|
2zrng | ⊢ 𝑅 ∈ Rng |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zringring 21485 | . 2 ⊢ ℤring ∈ Ring | |
2 | 2zrng.e | . . 3 ⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} | |
3 | 2zlidl.u | . . 3 ⊢ 𝑈 = (LIdeal‘ℤring) | |
4 | 2, 3 | 2zlidl 47965 | . 2 ⊢ 𝐸 ∈ 𝑈 |
5 | 2zrng.r | . . 3 ⊢ 𝑅 = (ℤring ↾s 𝐸) | |
6 | 3, 5 | lidlrng 47958 | . 2 ⊢ ((ℤring ∈ Ring ∧ 𝐸 ∈ 𝑈) → 𝑅 ∈ Rng) |
7 | 1, 4, 6 | mp2an 691 | 1 ⊢ 𝑅 ∈ Rng |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2108 ∃wrex 3076 {crab 3443 ‘cfv 6575 (class class class)co 7450 · cmul 11191 2c2 12350 ℤcz 12641 ↾s cress 17289 Rngcrng 20181 Ringcrg 20262 LIdealclidl 21241 ℤringczring 21482 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 ax-cnex 11242 ax-resscn 11243 ax-1cn 11244 ax-icn 11245 ax-addcl 11246 ax-addrcl 11247 ax-mulcl 11248 ax-mulrcl 11249 ax-mulcom 11250 ax-addass 11251 ax-mulass 11252 ax-distr 11253 ax-i2m1 11254 ax-1ne0 11255 ax-1rid 11256 ax-rnegex 11257 ax-rrecex 11258 ax-cnre 11259 ax-pre-lttri 11260 ax-pre-lttrn 11261 ax-pre-ltadd 11262 ax-pre-mulgt0 11263 ax-addf 11265 ax-mulf 11266 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6334 df-ord 6400 df-on 6401 df-lim 6402 df-suc 6403 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-riota 7406 df-ov 7453 df-oprab 7454 df-mpo 7455 df-om 7906 df-1st 8032 df-2nd 8033 df-frecs 8324 df-wrecs 8355 df-recs 8429 df-rdg 8468 df-1o 8524 df-er 8765 df-en 9006 df-dom 9007 df-sdom 9008 df-fin 9009 df-pnf 11328 df-mnf 11329 df-xr 11330 df-ltxr 11331 df-le 11332 df-sub 11524 df-neg 11525 df-nn 12296 df-2 12358 df-3 12359 df-4 12360 df-5 12361 df-6 12362 df-7 12363 df-8 12364 df-9 12365 df-n0 12556 df-z 12642 df-dec 12761 df-uz 12906 df-fz 13570 df-struct 17196 df-sets 17213 df-slot 17231 df-ndx 17243 df-base 17261 df-ress 17290 df-plusg 17326 df-mulr 17327 df-starv 17328 df-sca 17329 df-vsca 17330 df-ip 17331 df-tset 17332 df-ple 17333 df-ds 17335 df-unif 17336 df-0g 17503 df-mgm 18680 df-sgrp 18759 df-mnd 18775 df-grp 18978 df-minusg 18979 df-sbg 18980 df-subg 19165 df-cmn 19826 df-abl 19827 df-mgp 20164 df-rng 20182 df-ur 20211 df-ring 20264 df-cring 20265 df-subrng 20574 df-subrg 20599 df-lmod 20884 df-lss 20955 df-sra 21197 df-rgmod 21198 df-lidl 21243 df-cnfld 21390 df-zring 21483 |
This theorem is referenced by: (None) |
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