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Mirrors > Home > MPE Home > Th. List > 0ring | Structured version Visualization version GIF version |
Description: If a ring has only one element, it is the zero ring. According to Wikipedia ("Zero ring", 14-Apr-2019, https://en.wikipedia.org/wiki/Zero_ring): "The zero ring, denoted {0} or simply 0, consists of the one-element set {0} with the operations + and * defined so that 0 + 0 = 0 and 0 * 0 = 0.". (Contributed by AV, 14-Apr-2019.) |
Ref | Expression |
---|---|
0ring.b | ⊢ 𝐵 = (Base‘𝑅) |
0ring.0 | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
0ring | ⊢ ((𝑅 ∈ Ring ∧ (#‘𝐵) = 1) → 𝐵 = { 0 }) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ring.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
2 | 0ring.0 | . . . 4 ⊢ 0 = (0g‘𝑅) | |
3 | 1, 2 | ring0cl 18569 | . . 3 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐵) |
4 | fvex 6201 | . . . . . 6 ⊢ (Base‘𝑅) ∈ V | |
5 | 1, 4 | eqeltri 2697 | . . . . 5 ⊢ 𝐵 ∈ V |
6 | hashen1 13160 | . . . . 5 ⊢ (𝐵 ∈ V → ((#‘𝐵) = 1 ↔ 𝐵 ≈ 1𝑜)) | |
7 | 5, 6 | ax-mp 5 | . . . 4 ⊢ ((#‘𝐵) = 1 ↔ 𝐵 ≈ 1𝑜) |
8 | en1eqsn 8190 | . . . . 5 ⊢ (( 0 ∈ 𝐵 ∧ 𝐵 ≈ 1𝑜) → 𝐵 = { 0 }) | |
9 | 8 | ex 450 | . . . 4 ⊢ ( 0 ∈ 𝐵 → (𝐵 ≈ 1𝑜 → 𝐵 = { 0 })) |
10 | 7, 9 | syl5bi 232 | . . 3 ⊢ ( 0 ∈ 𝐵 → ((#‘𝐵) = 1 → 𝐵 = { 0 })) |
11 | 3, 10 | syl 17 | . 2 ⊢ (𝑅 ∈ Ring → ((#‘𝐵) = 1 → 𝐵 = { 0 })) |
12 | 11 | imp 445 | 1 ⊢ ((𝑅 ∈ Ring ∧ (#‘𝐵) = 1) → 𝐵 = { 0 }) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 {csn 4177 class class class wbr 4653 ‘cfv 5888 1𝑜c1o 7553 ≈ cen 7952 1c1 9937 #chash 13117 Basecbs 15857 0gc0g 16100 Ringcrg 18547 |
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-cnex 9992 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-rmo 2920 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-int 4476 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-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 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 df-uz 11688 df-fz 12327 df-hash 13118 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-ring 18549 |
This theorem is referenced by: 0ring01eq 19271 01eq0ring 19272 0ringdif 41870 lindsrng01 42257 |
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