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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 3036 | . . . . 5 ⊢ ((𝑈 ≠ { 0 } ∧ 𝑈 ≠ 𝐵) ↔ ¬ (𝑈 = { 0 } ∨ 𝑈 = 𝐵)) | |
2 | 1 | biimpi 215 | . . . 4 ⊢ ((𝑈 ≠ { 0 } ∧ 𝑈 ≠ 𝐵) → ¬ (𝑈 = { 0 } ∨ 𝑈 = 𝐵)) |
3 | 2 | 3adant1 1131 | . . 3 ⊢ ((𝑈 ∈ 𝐿 ∧ 𝑈 ≠ { 0 } ∧ 𝑈 ≠ 𝐵) → ¬ (𝑈 = { 0 } ∨ 𝑈 = 𝐵)) |
4 | 3 | adantl 483 | . 2 ⊢ ((𝑅 ∈ Domn ∧ (𝑈 ∈ 𝐿 ∧ 𝑈 ≠ { 0 } ∧ 𝑈 ≠ 𝐵)) → ¬ (𝑈 = { 0 } ∨ 𝑈 = 𝐵)) |
5 | df-nel 3048 | . . 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 46728 | . . . . 5 ⊢ ((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) → (𝐼 ∈ Ring ↔ (𝑈 = { 0 } ∨ 𝑈 = 𝐵))) |
11 | 10 | 3ad2antr1 1189 | . . . 4 ⊢ ((𝑅 ∈ Domn ∧ (𝑈 ∈ 𝐿 ∧ 𝑈 ≠ { 0 } ∧ 𝑈 ≠ 𝐵)) → (𝐼 ∈ Ring ↔ (𝑈 = { 0 } ∨ 𝑈 = 𝐵))) |
12 | 11 | notbid 318 | . . 3 ⊢ ((𝑅 ∈ Domn ∧ (𝑈 ∈ 𝐿 ∧ 𝑈 ≠ { 0 } ∧ 𝑈 ≠ 𝐵)) → (¬ 𝐼 ∈ Ring ↔ ¬ (𝑈 = { 0 } ∨ 𝑈 = 𝐵))) |
13 | 5, 12 | bitrid 283 | . 2 ⊢ ((𝑅 ∈ Domn ∧ (𝑈 ∈ 𝐿 ∧ 𝑈 ≠ { 0 } ∧ 𝑈 ≠ 𝐵)) → (𝐼 ∉ Ring ↔ ¬ (𝑈 = { 0 } ∨ 𝑈 = 𝐵))) |
14 | 4, 13 | mpbird 257 | 1 ⊢ ((𝑅 ∈ Domn ∧ (𝑈 ∈ 𝐿 ∧ 𝑈 ≠ { 0 } ∧ 𝑈 ≠ 𝐵)) → 𝐼 ∉ Ring) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∉ wnel 3047 {csn 4626 ‘cfv 6539 (class class class)co 7403 Basecbs 17139 ↾s cress 17168 0gc0g 17380 Ringcrg 20046 LIdealclidl 20770 Domncdomn 20882 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5283 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-iun 4997 df-br 5147 df-opab 5209 df-mpt 5230 df-tr 5264 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6296 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7359 df-ov 7406 df-oprab 7407 df-mpo 7408 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8260 df-wrecs 8291 df-recs 8365 df-rdg 8404 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11441 df-neg 11442 df-nn 12208 df-2 12270 df-3 12271 df-4 12272 df-5 12273 df-6 12274 df-7 12275 df-8 12276 df-sets 17092 df-slot 17110 df-ndx 17122 df-base 17140 df-ress 17169 df-plusg 17205 df-mulr 17206 df-sca 17208 df-vsca 17209 df-ip 17210 df-0g 17382 df-mgm 18556 df-sgrp 18605 df-mnd 18621 df-grp 18817 df-minusg 18818 df-sbg 18819 df-subg 18996 df-cmn 19642 df-abl 19643 df-mgp 19979 df-ur 19996 df-ring 20048 df-nzr 20280 df-subrg 20348 df-lmod 20460 df-lss 20530 df-sra 20772 df-rgmod 20773 df-lidl 20774 df-domn 20886 df-rng 46583 |
This theorem is referenced by: (None) |
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