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| Mirrors > Home > MPE Home > Th. List > rnglidl0 | Structured version Visualization version GIF version | ||
| Description: Every non-unital ring contains a zero ideal. (Contributed by AV, 19-Feb-2025.) | 
| Ref | Expression | 
|---|---|
| rnglidl0.u | ⊢ 𝑈 = (LIdeal‘𝑅) | 
| rnglidl0.z | ⊢ 0 = (0g‘𝑅) | 
| Ref | Expression | 
|---|---|
| rnglidl0 | ⊢ (𝑅 ∈ Rng → { 0 } ∈ 𝑈) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqid 2736 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
| 2 | rnglidl0.z | . . . 4 ⊢ 0 = (0g‘𝑅) | |
| 3 | 1, 2 | rng0cl 20161 | . . 3 ⊢ (𝑅 ∈ Rng → 0 ∈ (Base‘𝑅)) | 
| 4 | 3 | snssd 4808 | . 2 ⊢ (𝑅 ∈ Rng → { 0 } ⊆ (Base‘𝑅)) | 
| 5 | 2 | fvexi 6919 | . . . 4 ⊢ 0 ∈ V | 
| 6 | 5 | a1i 11 | . . 3 ⊢ (𝑅 ∈ Rng → 0 ∈ V) | 
| 7 | 6 | snn0d 4774 | . 2 ⊢ (𝑅 ∈ Rng → { 0 } ≠ ∅) | 
| 8 | eqid 2736 | . . . . . . . 8 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
| 9 | 1, 8, 2 | rngrz 20164 | . . . . . . 7 ⊢ ((𝑅 ∈ Rng ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥(.r‘𝑅) 0 ) = 0 ) | 
| 10 | 9 | oveq1d 7447 | . . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑥 ∈ (Base‘𝑅)) → ((𝑥(.r‘𝑅) 0 )(+g‘𝑅) 0 ) = ( 0 (+g‘𝑅) 0 )) | 
| 11 | rnggrp 20156 | . . . . . . . 8 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp) | |
| 12 | 1, 2 | grpidcl 18984 | . . . . . . . 8 ⊢ (𝑅 ∈ Grp → 0 ∈ (Base‘𝑅)) | 
| 13 | eqid 2736 | . . . . . . . . 9 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
| 14 | 1, 13, 2 | grprid 18987 | . . . . . . . 8 ⊢ ((𝑅 ∈ Grp ∧ 0 ∈ (Base‘𝑅)) → ( 0 (+g‘𝑅) 0 ) = 0 ) | 
| 15 | 11, 12, 14 | syl2anc2 585 | . . . . . . 7 ⊢ (𝑅 ∈ Rng → ( 0 (+g‘𝑅) 0 ) = 0 ) | 
| 16 | 15 | adantr 480 | . . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑥 ∈ (Base‘𝑅)) → ( 0 (+g‘𝑅) 0 ) = 0 ) | 
| 17 | 10, 16 | eqtrd 2776 | . . . . 5 ⊢ ((𝑅 ∈ Rng ∧ 𝑥 ∈ (Base‘𝑅)) → ((𝑥(.r‘𝑅) 0 )(+g‘𝑅) 0 ) = 0 ) | 
| 18 | 5 | elsn2 4664 | . . . . 5 ⊢ (((𝑥(.r‘𝑅) 0 )(+g‘𝑅) 0 ) ∈ { 0 } ↔ ((𝑥(.r‘𝑅) 0 )(+g‘𝑅) 0 ) = 0 ) | 
| 19 | 17, 18 | sylibr 234 | . . . 4 ⊢ ((𝑅 ∈ Rng ∧ 𝑥 ∈ (Base‘𝑅)) → ((𝑥(.r‘𝑅) 0 )(+g‘𝑅) 0 ) ∈ { 0 }) | 
| 20 | oveq2 7440 | . . . . . . . . 9 ⊢ (𝑦 = 0 → (𝑥(.r‘𝑅)𝑦) = (𝑥(.r‘𝑅) 0 )) | |
| 21 | 20 | oveq1d 7447 | . . . . . . . 8 ⊢ (𝑦 = 0 → ((𝑥(.r‘𝑅)𝑦)(+g‘𝑅)𝑧) = ((𝑥(.r‘𝑅) 0 )(+g‘𝑅)𝑧)) | 
| 22 | 21 | eleq1d 2825 | . . . . . . 7 ⊢ (𝑦 = 0 → (((𝑥(.r‘𝑅)𝑦)(+g‘𝑅)𝑧) ∈ { 0 } ↔ ((𝑥(.r‘𝑅) 0 )(+g‘𝑅)𝑧) ∈ { 0 })) | 
| 23 | 22 | ralbidv 3177 | . . . . . 6 ⊢ (𝑦 = 0 → (∀𝑧 ∈ { 0 } ((𝑥(.r‘𝑅)𝑦)(+g‘𝑅)𝑧) ∈ { 0 } ↔ ∀𝑧 ∈ { 0 } ((𝑥(.r‘𝑅) 0 )(+g‘𝑅)𝑧) ∈ { 0 })) | 
| 24 | 5, 23 | ralsn 4680 | . . . . 5 ⊢ (∀𝑦 ∈ { 0 }∀𝑧 ∈ { 0 } ((𝑥(.r‘𝑅)𝑦)(+g‘𝑅)𝑧) ∈ { 0 } ↔ ∀𝑧 ∈ { 0 } ((𝑥(.r‘𝑅) 0 )(+g‘𝑅)𝑧) ∈ { 0 }) | 
| 25 | oveq2 7440 | . . . . . . 7 ⊢ (𝑧 = 0 → ((𝑥(.r‘𝑅) 0 )(+g‘𝑅)𝑧) = ((𝑥(.r‘𝑅) 0 )(+g‘𝑅) 0 )) | |
| 26 | 25 | eleq1d 2825 | . . . . . 6 ⊢ (𝑧 = 0 → (((𝑥(.r‘𝑅) 0 )(+g‘𝑅)𝑧) ∈ { 0 } ↔ ((𝑥(.r‘𝑅) 0 )(+g‘𝑅) 0 ) ∈ { 0 })) | 
| 27 | 5, 26 | ralsn 4680 | . . . . 5 ⊢ (∀𝑧 ∈ { 0 } ((𝑥(.r‘𝑅) 0 )(+g‘𝑅)𝑧) ∈ { 0 } ↔ ((𝑥(.r‘𝑅) 0 )(+g‘𝑅) 0 ) ∈ { 0 }) | 
| 28 | 24, 27 | bitri 275 | . . . 4 ⊢ (∀𝑦 ∈ { 0 }∀𝑧 ∈ { 0 } ((𝑥(.r‘𝑅)𝑦)(+g‘𝑅)𝑧) ∈ { 0 } ↔ ((𝑥(.r‘𝑅) 0 )(+g‘𝑅) 0 ) ∈ { 0 }) | 
| 29 | 19, 28 | sylibr 234 | . . 3 ⊢ ((𝑅 ∈ Rng ∧ 𝑥 ∈ (Base‘𝑅)) → ∀𝑦 ∈ { 0 }∀𝑧 ∈ { 0 } ((𝑥(.r‘𝑅)𝑦)(+g‘𝑅)𝑧) ∈ { 0 }) | 
| 30 | 29 | ralrimiva 3145 | . 2 ⊢ (𝑅 ∈ Rng → ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ { 0 }∀𝑧 ∈ { 0 } ((𝑥(.r‘𝑅)𝑦)(+g‘𝑅)𝑧) ∈ { 0 }) | 
| 31 | rnglidl0.u | . . 3 ⊢ 𝑈 = (LIdeal‘𝑅) | |
| 32 | 31, 1, 13, 8 | islidl 21226 | . 2 ⊢ ({ 0 } ∈ 𝑈 ↔ ({ 0 } ⊆ (Base‘𝑅) ∧ { 0 } ≠ ∅ ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ { 0 }∀𝑧 ∈ { 0 } ((𝑥(.r‘𝑅)𝑦)(+g‘𝑅)𝑧) ∈ { 0 })) | 
| 33 | 4, 7, 30, 32 | syl3anbrc 1343 | 1 ⊢ (𝑅 ∈ Rng → { 0 } ∈ 𝑈) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 ∀wral 3060 Vcvv 3479 ⊆ wss 3950 ∅c0 4332 {csn 4625 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 +gcplusg 17298 .rcmulr 17299 0gc0g 17485 Grpcgrp 18952 Rngcrng 20150 LIdealclidl 21217 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-sca 17314 df-vsca 17315 df-ip 17316 df-0g 17487 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-grp 18955 df-abl 19802 df-mgp 20139 df-rng 20151 df-lss 20931 df-sra 21173 df-rgmod 21174 df-lidl 21219 | 
| This theorem is referenced by: lidl0 21241 | 
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