![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 20819 | . . 3 ⊢ (0g‘𝑅) = (0g‘(ringLMod‘𝑅)) | |
3 | 1, 2 | eqtri 2761 | . 2 ⊢ 0 = (0g‘(ringLMod‘𝑅)) |
4 | rlmlmod 20827 | . . 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 20814 | . . . . 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 20557 | . . 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 6544 0gc0g 17385 Ringcrg 20056 LModclmod 20471 LSubSpclss 20542 ringLModcrglmod 20782 LIdealclidl 20783 |
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 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 |
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 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-sca 17213 df-vsca 17214 df-ip 17215 df-0g 17387 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-grp 18822 df-minusg 18823 df-sbg 18824 df-subg 19003 df-mgp 19988 df-ur 20005 df-ring 20058 df-subrg 20317 df-lmod 20473 df-lss 20543 df-sra 20785 df-rgmod 20786 df-lidl 20787 |
This theorem is referenced by: lidlsubg 20838 lidlnz 20853 ig1peu 25689 ig1pdvds 25694 intlidl 32536 rhmpreimaidl 32537 0ringidl 32539 idlinsubrg 32549 rhmimaidl 32550 mxidlnzr 32583 mxidlprm 32586 ssmxidllem 32589 zarcls0 32848 hbtlem2 41866 hbtlem5 41870 |
Copyright terms: Public domain | W3C validator |