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Mirrors > Home > MPE Home > Th. List > lidlacl | Structured version Visualization version GIF version |
Description: An ideal is closed under addition. (Contributed by Stefan O'Rear, 3-Jan-2015.) |
Ref | Expression |
---|---|
lidlcl.u | ⊢ 𝑈 = (LIdeal‘𝑅) |
lidlacl.p | ⊢ + = (+g‘𝑅) |
Ref | Expression |
---|---|
lidlacl | ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (𝑋 + 𝑌) ∈ 𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lidlacl.p | . . . 4 ⊢ + = (+g‘𝑅) | |
2 | rlmplusg 20187 | . . . 4 ⊢ (+g‘𝑅) = (+g‘(ringLMod‘𝑅)) | |
3 | 1, 2 | eqtri 2759 | . . 3 ⊢ + = (+g‘(ringLMod‘𝑅)) |
4 | 3 | oveqi 7204 | . 2 ⊢ (𝑋 + 𝑌) = (𝑋(+g‘(ringLMod‘𝑅))𝑌) |
5 | rlmlmod 20196 | . . . . 5 ⊢ (𝑅 ∈ Ring → (ringLMod‘𝑅) ∈ LMod) | |
6 | 5 | adantr 484 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → (ringLMod‘𝑅) ∈ LMod) |
7 | simpr 488 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → 𝐼 ∈ 𝑈) | |
8 | lidlcl.u | . . . . . 6 ⊢ 𝑈 = (LIdeal‘𝑅) | |
9 | lidlval 20183 | . . . . . 6 ⊢ (LIdeal‘𝑅) = (LSubSp‘(ringLMod‘𝑅)) | |
10 | 8, 9 | eqtri 2759 | . . . . 5 ⊢ 𝑈 = (LSubSp‘(ringLMod‘𝑅)) |
11 | 7, 10 | eleqtrdi 2841 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → 𝐼 ∈ (LSubSp‘(ringLMod‘𝑅))) |
12 | 6, 11 | jca 515 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → ((ringLMod‘𝑅) ∈ LMod ∧ 𝐼 ∈ (LSubSp‘(ringLMod‘𝑅)))) |
13 | eqid 2736 | . . . 4 ⊢ (+g‘(ringLMod‘𝑅)) = (+g‘(ringLMod‘𝑅)) | |
14 | eqid 2736 | . . . 4 ⊢ (LSubSp‘(ringLMod‘𝑅)) = (LSubSp‘(ringLMod‘𝑅)) | |
15 | 13, 14 | lssvacl 19945 | . . 3 ⊢ ((((ringLMod‘𝑅) ∈ LMod ∧ 𝐼 ∈ (LSubSp‘(ringLMod‘𝑅))) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (𝑋(+g‘(ringLMod‘𝑅))𝑌) ∈ 𝐼) |
16 | 12, 15 | sylan 583 | . 2 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (𝑋(+g‘(ringLMod‘𝑅))𝑌) ∈ 𝐼) |
17 | 4, 16 | eqeltrid 2835 | 1 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (𝑋 + 𝑌) ∈ 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ‘cfv 6358 (class class class)co 7191 +gcplusg 16749 Ringcrg 19516 LModclmod 19853 LSubSpclss 19922 ringLModcrglmod 20160 LIdealclidl 20161 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 df-plusg 16762 df-mulr 16763 df-sca 16765 df-vsca 16766 df-ip 16767 df-0g 16900 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-grp 18322 df-subg 18494 df-mgp 19459 df-ur 19471 df-ring 19518 df-subrg 19752 df-lmod 19855 df-lss 19923 df-sra 20163 df-rgmod 20164 df-lidl 20165 |
This theorem is referenced by: lidlsubg 20207 zringlpirlem3 20405 intlidl 31270 rhmpreimaidl 31271 idlinsubrg 31276 mxidlprm 31308 ssmxidllem 31309 hbtlem2 40593 hbtlem5 40597 |
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