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| Mirrors > Home > MPE Home > Th. List > ring0cl | Structured version Visualization version GIF version | ||
| Description: The zero element of a ring belongs to its base set. (Contributed by Mario Carneiro, 12-Jan-2014.) | 
| Ref | Expression | 
|---|---|
| ring0cl.b | ⊢ 𝐵 = (Base‘𝑅) | 
| ring0cl.z | ⊢ 0 = (0g‘𝑅) | 
| Ref | Expression | 
|---|---|
| ring0cl | ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ringgrp 20235 | . 2 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
| 2 | ring0cl.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
| 3 | ring0cl.z | . . 3 ⊢ 0 = (0g‘𝑅) | |
| 4 | 2, 3 | grpidcl 18983 | . 2 ⊢ (𝑅 ∈ Grp → 0 ∈ 𝐵) | 
| 5 | 1, 4 | syl 17 | 1 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 Basecbs 17247 0gc0g 17484 Grpcgrp 18951 Ringcrg 20230 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-riota 7388 df-ov 7434 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-ring 20232 | 
| This theorem is referenced by: dvdsr01 20371 dvdsr02 20372 irredn0 20423 isnzr2 20518 isnzr2hash 20519 ringelnzr 20523 0ring 20526 01eq0ring 20530 01eq0ringOLD 20531 zrrnghm 20536 cntzsubr 20606 domneq0r 20724 ringen1zr 20779 imadrhmcl 20798 abv0 20824 abvtrivd 20833 lmod0cl 20886 lmod0vs 20893 lmodvs0 20894 rhmpreimaidl 21287 lpi0 21336 frlmphllem 21800 frlmphl 21801 uvcvvcl2 21808 uvcff 21811 psr1cl 21981 mvrf 22005 mplmon 22053 mplmonmul 22054 mplcoe1 22055 evlslem3 22104 coe1z 22266 coe1tmfv2 22278 ply1scl0OLD 22294 ply1scln0 22295 ply1chr 22310 gsummoncoe1 22312 rhmmpl 22387 rhmply1vr1 22391 mamumat1cl 22445 dmatsubcl 22504 dmatmulcl 22506 scmatscmiddistr 22514 marrepcl 22570 mdetr0 22611 mdetunilem8 22625 mdetunilem9 22626 maducoeval2 22646 maduf 22647 madutpos 22648 madugsum 22649 marep01ma 22666 smadiadetlem4 22675 smadiadetglem2 22678 1elcpmat 22721 m2cpminv0 22767 decpmataa0 22774 monmatcollpw 22785 pmatcollpw3fi1lem1 22792 pmatcollpw3fi1lem2 22793 chfacfisf 22860 cphsubrglem 25211 mdegaddle 26113 ply1divex 26176 r1pid2 26201 facth1 26206 fta1blem 26210 abvcxp 27659 rloccring 33274 elrspunidl 33456 elrspunsn 33457 rhmimaidl 33460 qsidomlem2 33481 ply1degltel 33615 ply1degleel 33616 ply1degltlss 33617 gsummoncoe1fzo 33618 ply1gsumz 33619 r1p0 33626 r1pid2OLD 33629 r1pquslmic 33631 zrhcntr 33980 lfl0sc 39083 lflsc0N 39084 baerlem3lem1 41709 ricdrng1 42538 rhmpsr 42562 evl0 42567 evlsbagval 42576 selvvvval 42595 frlmpwfi 43110 mnringmulrcld 44247 zlidlring 48150 cznrng 48177 linc0scn0 48340 linc1 48342 | 
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