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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 20077 | . . 3 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐵) |
4 | 1 | fvexi 6902 | . . . . 5 ⊢ 𝐵 ∈ V |
5 | hashen1 14326 | . . . . 5 ⊢ (𝐵 ∈ V → ((♯‘𝐵) = 1 ↔ 𝐵 ≈ 1o)) | |
6 | 4, 5 | ax-mp 5 | . . . 4 ⊢ ((♯‘𝐵) = 1 ↔ 𝐵 ≈ 1o) |
7 | en1eqsn 9270 | . . . . 5 ⊢ (( 0 ∈ 𝐵 ∧ 𝐵 ≈ 1o) → 𝐵 = { 0 }) | |
8 | 7 | ex 413 | . . . 4 ⊢ ( 0 ∈ 𝐵 → (𝐵 ≈ 1o → 𝐵 = { 0 })) |
9 | 6, 8 | biimtrid 241 | . . 3 ⊢ ( 0 ∈ 𝐵 → ((♯‘𝐵) = 1 → 𝐵 = { 0 })) |
10 | 3, 9 | syl 17 | . 2 ⊢ (𝑅 ∈ Ring → ((♯‘𝐵) = 1 → 𝐵 = { 0 })) |
11 | 10 | imp 407 | 1 ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → 𝐵 = { 0 }) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3474 {csn 4627 class class class wbr 5147 ‘cfv 6540 1oc1o 8455 ≈ cen 8932 1c1 11107 ♯chash 14286 Basecbs 17140 0gc0g 17381 Ringcrg 20049 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-hash 14287 df-0g 17383 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-grp 18818 df-ring 20051 |
This theorem is referenced by: 0ring01eq 20296 01eq0ringOLD 20298 0ringsubrg 32366 0ringidl 32527 0ringprmidl 32556 prmidl0 32557 0ringdif 46630 lindsrng01 47102 |
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