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Mathbox for Alexander van der Vekens |
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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 42663. (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 20182 | . 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 42782 | . 2 ⊢ 𝐸 ∈ 𝑈 |
5 | 2zrng.r | . . 3 ⊢ 𝑅 = (ℤring ↾s 𝐸) | |
6 | 3, 5 | lidlrng 42775 | . 2 ⊢ ((ℤring ∈ Ring ∧ 𝐸 ∈ 𝑈) → 𝑅 ∈ Rng) |
7 | 1, 4, 6 | mp2an 685 | 1 ⊢ 𝑅 ∈ Rng |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1658 ∈ wcel 2166 ∃wrex 3119 {crab 3122 ‘cfv 6124 (class class class)co 6906 · cmul 10258 2c2 11407 ℤcz 11705 ↾s cress 16224 Ringcrg 18902 LIdealclidl 19532 ℤringzring 20179 Rngcrng 42722 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-rep 4995 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 ax-addf 10332 ax-mulf 10333 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rmo 3126 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-int 4699 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-om 7328 df-1st 7429 df-2nd 7430 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-1o 7827 df-oadd 7831 df-er 8010 df-en 8224 df-dom 8225 df-sdom 8226 df-fin 8227 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-nn 11352 df-2 11415 df-3 11416 df-4 11417 df-5 11418 df-6 11419 df-7 11420 df-8 11421 df-9 11422 df-n0 11620 df-z 11706 df-dec 11823 df-uz 11970 df-fz 12621 df-struct 16225 df-ndx 16226 df-slot 16227 df-base 16229 df-sets 16230 df-ress 16231 df-plusg 16319 df-mulr 16320 df-starv 16321 df-sca 16322 df-vsca 16323 df-ip 16324 df-tset 16325 df-ple 16326 df-ds 16328 df-unif 16329 df-0g 16456 df-mgm 17596 df-sgrp 17638 df-mnd 17649 df-grp 17780 df-minusg 17781 df-sbg 17782 df-subg 17943 df-cmn 18549 df-abl 18550 df-mgp 18845 df-ur 18857 df-ring 18904 df-cring 18905 df-subrg 19135 df-lmod 19222 df-lss 19290 df-sra 19534 df-rgmod 19535 df-lidl 19536 df-cnfld 20108 df-zring 20180 df-rng0 42723 |
This theorem is referenced by: (None) |
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