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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 19584 | . . 3 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐵) |
4 | 1 | fvexi 6728 | . . . . 5 ⊢ 𝐵 ∈ V |
5 | hashen1 13934 | . . . . 5 ⊢ (𝐵 ∈ V → ((♯‘𝐵) = 1 ↔ 𝐵 ≈ 1o)) | |
6 | 4, 5 | ax-mp 5 | . . . 4 ⊢ ((♯‘𝐵) = 1 ↔ 𝐵 ≈ 1o) |
7 | en1eqsn 8901 | . . . . 5 ⊢ (( 0 ∈ 𝐵 ∧ 𝐵 ≈ 1o) → 𝐵 = { 0 }) | |
8 | 7 | ex 416 | . . . 4 ⊢ ( 0 ∈ 𝐵 → (𝐵 ≈ 1o → 𝐵 = { 0 })) |
9 | 6, 8 | syl5bi 245 | . . 3 ⊢ ( 0 ∈ 𝐵 → ((♯‘𝐵) = 1 → 𝐵 = { 0 })) |
10 | 3, 9 | syl 17 | . 2 ⊢ (𝑅 ∈ Ring → ((♯‘𝐵) = 1 → 𝐵 = { 0 })) |
11 | 10 | imp 410 | 1 ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → 𝐵 = { 0 }) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 Vcvv 3405 {csn 4538 class class class wbr 5050 ‘cfv 6377 1oc1o 8192 ≈ cen 8620 1c1 10727 ♯chash 13893 Basecbs 16757 0gc0g 16941 Ringcrg 19559 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5189 ax-nul 5196 ax-pow 5255 ax-pr 5319 ax-un 7520 ax-cnex 10782 ax-resscn 10783 ax-1cn 10784 ax-icn 10785 ax-addcl 10786 ax-addrcl 10787 ax-mulcl 10788 ax-mulrcl 10789 ax-mulcom 10790 ax-addass 10791 ax-mulass 10792 ax-distr 10793 ax-i2m1 10794 ax-1ne0 10795 ax-1rid 10796 ax-rnegex 10797 ax-rrecex 10798 ax-cnre 10799 ax-pre-lttri 10800 ax-pre-lttrn 10801 ax-pre-ltadd 10802 ax-pre-mulgt0 10803 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2940 df-nel 3044 df-ral 3063 df-rex 3064 df-reu 3065 df-rmo 3066 df-rab 3067 df-v 3407 df-sbc 3692 df-csb 3809 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-pss 3882 df-nul 4235 df-if 4437 df-pw 4512 df-sn 4539 df-pr 4541 df-tp 4543 df-op 4545 df-uni 4817 df-int 4857 df-iun 4903 df-br 5051 df-opab 5113 df-mpt 5133 df-tr 5159 df-id 5452 df-eprel 5457 df-po 5465 df-so 5466 df-fr 5506 df-we 5508 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6157 df-ord 6213 df-on 6214 df-lim 6215 df-suc 6216 df-iota 6335 df-fun 6379 df-fn 6380 df-f 6381 df-f1 6382 df-fo 6383 df-f1o 6384 df-fv 6385 df-riota 7167 df-ov 7213 df-oprab 7214 df-mpo 7215 df-om 7642 df-1st 7758 df-2nd 7759 df-wrecs 8044 df-recs 8105 df-rdg 8143 df-1o 8199 df-er 8388 df-en 8624 df-dom 8625 df-sdom 8626 df-fin 8627 df-card 9552 df-pnf 10866 df-mnf 10867 df-xr 10868 df-ltxr 10869 df-le 10870 df-sub 11061 df-neg 11062 df-nn 11828 df-n0 12088 df-z 12174 df-uz 12436 df-fz 13093 df-hash 13894 df-0g 16943 df-mgm 18111 df-sgrp 18160 df-mnd 18171 df-grp 18365 df-ring 19561 |
This theorem is referenced by: 0ring01eq 20306 01eq0ring 20307 0ringidl 31316 0ringprmidl 31336 prmidl0 31337 0ringdif 45099 lindsrng01 45480 |
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