Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lidldomnnring | Structured version Visualization version GIF version |
Description: A (left) ideal of a domain which is neither the zero ideal nor the unit ideal is not a unital ring. (Contributed by AV, 18-Feb-2020.) |
Ref | Expression |
---|---|
lidlabl.l | ⊢ 𝐿 = (LIdeal‘𝑅) |
lidlabl.i | ⊢ 𝐼 = (𝑅 ↾s 𝑈) |
zlidlring.b | ⊢ 𝐵 = (Base‘𝑅) |
zlidlring.0 | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
lidldomnnring | ⊢ ((𝑅 ∈ Domn ∧ (𝑈 ∈ 𝐿 ∧ 𝑈 ≠ { 0 } ∧ 𝑈 ≠ 𝐵)) → 𝐼 ∉ Ring) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neanior 2886 | . . . . 5 ⊢ ((𝑈 ≠ { 0 } ∧ 𝑈 ≠ 𝐵) ↔ ¬ (𝑈 = { 0 } ∨ 𝑈 = 𝐵)) | |
2 | 1 | biimpi 206 | . . . 4 ⊢ ((𝑈 ≠ { 0 } ∧ 𝑈 ≠ 𝐵) → ¬ (𝑈 = { 0 } ∨ 𝑈 = 𝐵)) |
3 | 2 | 3adant1 1079 | . . 3 ⊢ ((𝑈 ∈ 𝐿 ∧ 𝑈 ≠ { 0 } ∧ 𝑈 ≠ 𝐵) → ¬ (𝑈 = { 0 } ∨ 𝑈 = 𝐵)) |
4 | 3 | adantl 482 | . 2 ⊢ ((𝑅 ∈ Domn ∧ (𝑈 ∈ 𝐿 ∧ 𝑈 ≠ { 0 } ∧ 𝑈 ≠ 𝐵)) → ¬ (𝑈 = { 0 } ∨ 𝑈 = 𝐵)) |
5 | df-nel 2898 | . . 3 ⊢ (𝐼 ∉ Ring ↔ ¬ 𝐼 ∈ Ring) | |
6 | lidlabl.l | . . . . . 6 ⊢ 𝐿 = (LIdeal‘𝑅) | |
7 | lidlabl.i | . . . . . 6 ⊢ 𝐼 = (𝑅 ↾s 𝑈) | |
8 | zlidlring.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
9 | zlidlring.0 | . . . . . 6 ⊢ 0 = (0g‘𝑅) | |
10 | 6, 7, 8, 9 | uzlidlring 41929 | . . . . 5 ⊢ ((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) → (𝐼 ∈ Ring ↔ (𝑈 = { 0 } ∨ 𝑈 = 𝐵))) |
11 | 10 | 3ad2antr1 1226 | . . . 4 ⊢ ((𝑅 ∈ Domn ∧ (𝑈 ∈ 𝐿 ∧ 𝑈 ≠ { 0 } ∧ 𝑈 ≠ 𝐵)) → (𝐼 ∈ Ring ↔ (𝑈 = { 0 } ∨ 𝑈 = 𝐵))) |
12 | 11 | notbid 308 | . . 3 ⊢ ((𝑅 ∈ Domn ∧ (𝑈 ∈ 𝐿 ∧ 𝑈 ≠ { 0 } ∧ 𝑈 ≠ 𝐵)) → (¬ 𝐼 ∈ Ring ↔ ¬ (𝑈 = { 0 } ∨ 𝑈 = 𝐵))) |
13 | 5, 12 | syl5bb 272 | . 2 ⊢ ((𝑅 ∈ Domn ∧ (𝑈 ∈ 𝐿 ∧ 𝑈 ≠ { 0 } ∧ 𝑈 ≠ 𝐵)) → (𝐼 ∉ Ring ↔ ¬ (𝑈 = { 0 } ∨ 𝑈 = 𝐵))) |
14 | 4, 13 | mpbird 247 | 1 ⊢ ((𝑅 ∈ Domn ∧ (𝑈 ∈ 𝐿 ∧ 𝑈 ≠ { 0 } ∧ 𝑈 ≠ 𝐵)) → 𝐼 ∉ Ring) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∨ wo 383 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∉ wnel 2897 {csn 4177 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 ↾s cress 15858 0gc0g 16100 Ringcrg 18547 LIdealclidl 19170 Domncdomn 19280 |
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-rep 4771 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-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-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 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-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-ip 15959 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-sbg 17427 df-subg 17591 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-subrg 18778 df-lmod 18865 df-lss 18933 df-sra 19172 df-rgmod 19173 df-lidl 19174 df-nzr 19258 df-domn 19284 df-rng0 41875 |
This theorem is referenced by: (None) |
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