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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 2731 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | rnglidl0.z | . . . 4 ⊢ 0 = (0g‘𝑅) | |
3 | 1, 2 | rng0cl 20061 | . . 3 ⊢ (𝑅 ∈ Rng → 0 ∈ (Base‘𝑅)) |
4 | 3 | snssd 4812 | . 2 ⊢ (𝑅 ∈ Rng → { 0 } ⊆ (Base‘𝑅)) |
5 | 2 | fvexi 6905 | . . . 4 ⊢ 0 ∈ V |
6 | 5 | a1i 11 | . . 3 ⊢ (𝑅 ∈ Rng → 0 ∈ V) |
7 | 6 | snn0d 4779 | . 2 ⊢ (𝑅 ∈ Rng → { 0 } ≠ ∅) |
8 | eqid 2731 | . . . . . . . 8 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
9 | 1, 8, 2 | rngrz 20064 | . . . . . . 7 ⊢ ((𝑅 ∈ Rng ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥(.r‘𝑅) 0 ) = 0 ) |
10 | 9 | oveq1d 7427 | . . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑥 ∈ (Base‘𝑅)) → ((𝑥(.r‘𝑅) 0 )(+g‘𝑅) 0 ) = ( 0 (+g‘𝑅) 0 )) |
11 | rnggrp 20056 | . . . . . . . 8 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp) | |
12 | 1, 2 | grpidcl 18890 | . . . . . . . 8 ⊢ (𝑅 ∈ Grp → 0 ∈ (Base‘𝑅)) |
13 | eqid 2731 | . . . . . . . . 9 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
14 | 1, 13, 2 | grprid 18893 | . . . . . . . 8 ⊢ ((𝑅 ∈ Grp ∧ 0 ∈ (Base‘𝑅)) → ( 0 (+g‘𝑅) 0 ) = 0 ) |
15 | 11, 12, 14 | syl2anc2 584 | . . . . . . 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 2771 | . . . . 5 ⊢ ((𝑅 ∈ Rng ∧ 𝑥 ∈ (Base‘𝑅)) → ((𝑥(.r‘𝑅) 0 )(+g‘𝑅) 0 ) = 0 ) |
18 | 5 | elsn2 4667 | . . . . 5 ⊢ (((𝑥(.r‘𝑅) 0 )(+g‘𝑅) 0 ) ∈ { 0 } ↔ ((𝑥(.r‘𝑅) 0 )(+g‘𝑅) 0 ) = 0 ) |
19 | 17, 18 | sylibr 233 | . . . 4 ⊢ ((𝑅 ∈ Rng ∧ 𝑥 ∈ (Base‘𝑅)) → ((𝑥(.r‘𝑅) 0 )(+g‘𝑅) 0 ) ∈ { 0 }) |
20 | oveq2 7420 | . . . . . . . . 9 ⊢ (𝑦 = 0 → (𝑥(.r‘𝑅)𝑦) = (𝑥(.r‘𝑅) 0 )) | |
21 | 20 | oveq1d 7427 | . . . . . . . 8 ⊢ (𝑦 = 0 → ((𝑥(.r‘𝑅)𝑦)(+g‘𝑅)𝑧) = ((𝑥(.r‘𝑅) 0 )(+g‘𝑅)𝑧)) |
22 | 21 | eleq1d 2817 | . . . . . . 7 ⊢ (𝑦 = 0 → (((𝑥(.r‘𝑅)𝑦)(+g‘𝑅)𝑧) ∈ { 0 } ↔ ((𝑥(.r‘𝑅) 0 )(+g‘𝑅)𝑧) ∈ { 0 })) |
23 | 22 | ralbidv 3176 | . . . . . 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 4685 | . . . . 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 7420 | . . . . . . 7 ⊢ (𝑧 = 0 → ((𝑥(.r‘𝑅) 0 )(+g‘𝑅)𝑧) = ((𝑥(.r‘𝑅) 0 )(+g‘𝑅) 0 )) | |
26 | 25 | eleq1d 2817 | . . . . . 6 ⊢ (𝑧 = 0 → (((𝑥(.r‘𝑅) 0 )(+g‘𝑅)𝑧) ∈ { 0 } ↔ ((𝑥(.r‘𝑅) 0 )(+g‘𝑅) 0 ) ∈ { 0 })) |
27 | 5, 26 | ralsn 4685 | . . . . 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 233 | . . 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 20985 | . 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 1342 | 1 ⊢ (𝑅 ∈ Rng → { 0 } ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 ∀wral 3060 Vcvv 3473 ⊆ wss 3948 ∅c0 4322 {csn 4628 ‘cfv 6543 (class class class)co 7412 Basecbs 17151 +gcplusg 17204 .rcmulr 17205 0gc0g 17392 Grpcgrp 18858 Rngcrng 20050 LIdealclidl 20932 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-sca 17220 df-vsca 17221 df-ip 17222 df-0g 17394 df-mgm 18568 df-sgrp 18647 df-mnd 18663 df-grp 18861 df-abl 19696 df-mgp 20033 df-rng 20051 df-lss 20691 df-sra 20934 df-rgmod 20935 df-lidl 20936 |
This theorem is referenced by: (None) |
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