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Mirrors > Home > MPE Home > Th. List > lidl0cl | Structured version Visualization version GIF version |
Description: An ideal contains 0. (Contributed by Stefan O'Rear, 3-Jan-2015.) |
Ref | Expression |
---|---|
lidlcl.u | ⊢ 𝑈 = (LIdeal‘𝑅) |
lidl0cl.z | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
lidl0cl | ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → 0 ∈ 𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lidl0cl.z | . . 3 ⊢ 0 = (0g‘𝑅) | |
2 | rlm0 20795 | . . 3 ⊢ (0g‘𝑅) = (0g‘(ringLMod‘𝑅)) | |
3 | 1, 2 | eqtri 2761 | . 2 ⊢ 0 = (0g‘(ringLMod‘𝑅)) |
4 | rlmlmod 20803 | . . 3 ⊢ (𝑅 ∈ Ring → (ringLMod‘𝑅) ∈ LMod) | |
5 | simpr 486 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → 𝐼 ∈ 𝑈) | |
6 | lidlcl.u | . . . . 5 ⊢ 𝑈 = (LIdeal‘𝑅) | |
7 | lidlval 20790 | . . . . 5 ⊢ (LIdeal‘𝑅) = (LSubSp‘(ringLMod‘𝑅)) | |
8 | 6, 7 | eqtri 2761 | . . . 4 ⊢ 𝑈 = (LSubSp‘(ringLMod‘𝑅)) |
9 | 5, 8 | eleqtrdi 2844 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → 𝐼 ∈ (LSubSp‘(ringLMod‘𝑅))) |
10 | eqid 2733 | . . . 4 ⊢ (0g‘(ringLMod‘𝑅)) = (0g‘(ringLMod‘𝑅)) | |
11 | eqid 2733 | . . . 4 ⊢ (LSubSp‘(ringLMod‘𝑅)) = (LSubSp‘(ringLMod‘𝑅)) | |
12 | 10, 11 | lss0cl 20534 | . . 3 ⊢ (((ringLMod‘𝑅) ∈ LMod ∧ 𝐼 ∈ (LSubSp‘(ringLMod‘𝑅))) → (0g‘(ringLMod‘𝑅)) ∈ 𝐼) |
13 | 4, 9, 12 | syl2an2r 684 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → (0g‘(ringLMod‘𝑅)) ∈ 𝐼) |
14 | 3, 13 | eqeltrid 2838 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → 0 ∈ 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ‘cfv 6535 0gc0g 17372 Ringcrg 20038 LModclmod 20448 LSubSpclss 20519 ringLModcrglmod 20759 LIdealclidl 20760 |
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 5281 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-cnex 11153 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 ax-pre-mulgt0 11174 |
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 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7843 df-1st 7962 df-2nd 7963 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-er 8691 df-en 8928 df-dom 8929 df-sdom 8930 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 df-sub 11433 df-neg 11434 df-nn 12200 df-2 12262 df-3 12263 df-4 12264 df-5 12265 df-6 12266 df-7 12267 df-8 12268 df-sets 17084 df-slot 17102 df-ndx 17114 df-base 17132 df-ress 17161 df-plusg 17197 df-mulr 17198 df-sca 17200 df-vsca 17201 df-ip 17202 df-0g 17374 df-mgm 18548 df-sgrp 18597 df-mnd 18613 df-grp 18809 df-minusg 18810 df-sbg 18811 df-subg 18988 df-mgp 19971 df-ur 19988 df-ring 20040 df-subrg 20338 df-lmod 20450 df-lss 20520 df-sra 20762 df-rgmod 20763 df-lidl 20764 |
This theorem is referenced by: lidlsubg 20814 lidlnz 20829 ig1peu 25658 ig1pdvds 25663 intlidl 32491 rhmpreimaidl 32492 0ringidl 32494 idlinsubrg 32500 rhmimaidl 32501 mxidlnzr 32534 mxidlprm 32537 ssmxidllem 32538 zarcls0 32779 hbtlem2 41737 hbtlem5 41741 |
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