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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 19313 | . . 3 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐵) |
4 | 1 | fvexi 6679 | . . . . 5 ⊢ 𝐵 ∈ V |
5 | hashen1 13725 | . . . . 5 ⊢ (𝐵 ∈ V → ((♯‘𝐵) = 1 ↔ 𝐵 ≈ 1o)) | |
6 | 4, 5 | ax-mp 5 | . . . 4 ⊢ ((♯‘𝐵) = 1 ↔ 𝐵 ≈ 1o) |
7 | en1eqsn 8742 | . . . . 5 ⊢ (( 0 ∈ 𝐵 ∧ 𝐵 ≈ 1o) → 𝐵 = { 0 }) | |
8 | 7 | ex 415 | . . . 4 ⊢ ( 0 ∈ 𝐵 → (𝐵 ≈ 1o → 𝐵 = { 0 })) |
9 | 6, 8 | syl5bi 244 | . . 3 ⊢ ( 0 ∈ 𝐵 → ((♯‘𝐵) = 1 → 𝐵 = { 0 })) |
10 | 3, 9 | syl 17 | . 2 ⊢ (𝑅 ∈ Ring → ((♯‘𝐵) = 1 → 𝐵 = { 0 })) |
11 | 10 | imp 409 | 1 ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → 𝐵 = { 0 }) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3495 {csn 4561 class class class wbr 5059 ‘cfv 6350 1oc1o 8089 ≈ cen 8500 1c1 10532 ♯chash 13684 Basecbs 16477 0gc0g 16707 Ringcrg 19291 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-hash 13685 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-ring 19293 |
This theorem is referenced by: 0ring01eq 20038 01eq0ring 20039 0ringdif 44134 lindsrng01 44516 |
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