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Mirrors > Home > MPE Home > Th. List > drnglpir | Structured version Visualization version GIF version |
Description: Division rings are principal ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.) |
Ref | Expression |
---|---|
drnglpir | ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ LPIR) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | drngring 19146 | . 2 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring) | |
2 | eqid 2778 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | eqid 2778 | . . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
4 | eqid 2778 | . . . 4 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅) | |
5 | 2, 3, 4 | drngnidl 19626 | . . 3 ⊢ (𝑅 ∈ DivRing → (LIdeal‘𝑅) = {{(0g‘𝑅)}, (Base‘𝑅)}) |
6 | eqid 2778 | . . . . . 6 ⊢ (LPIdeal‘𝑅) = (LPIdeal‘𝑅) | |
7 | 6, 3 | lpi0 19644 | . . . . 5 ⊢ (𝑅 ∈ Ring → {(0g‘𝑅)} ∈ (LPIdeal‘𝑅)) |
8 | 6, 2 | lpi1 19645 | . . . . 5 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) ∈ (LPIdeal‘𝑅)) |
9 | snex 5140 | . . . . . . 7 ⊢ {(0g‘𝑅)} ∈ V | |
10 | fvex 6459 | . . . . . . 7 ⊢ (Base‘𝑅) ∈ V | |
11 | 9, 10 | prss 4582 | . . . . . 6 ⊢ (({(0g‘𝑅)} ∈ (LPIdeal‘𝑅) ∧ (Base‘𝑅) ∈ (LPIdeal‘𝑅)) ↔ {{(0g‘𝑅)}, (Base‘𝑅)} ⊆ (LPIdeal‘𝑅)) |
12 | 11 | bicomi 216 | . . . . 5 ⊢ ({{(0g‘𝑅)}, (Base‘𝑅)} ⊆ (LPIdeal‘𝑅) ↔ ({(0g‘𝑅)} ∈ (LPIdeal‘𝑅) ∧ (Base‘𝑅) ∈ (LPIdeal‘𝑅))) |
13 | 7, 8, 12 | sylanbrc 578 | . . . 4 ⊢ (𝑅 ∈ Ring → {{(0g‘𝑅)}, (Base‘𝑅)} ⊆ (LPIdeal‘𝑅)) |
14 | 1, 13 | syl 17 | . . 3 ⊢ (𝑅 ∈ DivRing → {{(0g‘𝑅)}, (Base‘𝑅)} ⊆ (LPIdeal‘𝑅)) |
15 | 5, 14 | eqsstrd 3858 | . 2 ⊢ (𝑅 ∈ DivRing → (LIdeal‘𝑅) ⊆ (LPIdeal‘𝑅)) |
16 | 6, 4 | islpir2 19648 | . 2 ⊢ (𝑅 ∈ LPIR ↔ (𝑅 ∈ Ring ∧ (LIdeal‘𝑅) ⊆ (LPIdeal‘𝑅))) |
17 | 1, 15, 16 | sylanbrc 578 | 1 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ LPIR) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∈ wcel 2107 ⊆ wss 3792 {csn 4398 {cpr 4400 ‘cfv 6135 Basecbs 16255 0gc0g 16486 Ringcrg 18934 DivRingcdr 19139 LIdealclidl 19567 LPIdealclpidl 19638 LPIRclpir 19639 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-tpos 7634 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-sca 16354 df-vsca 16355 df-ip 16356 df-0g 16488 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-grp 17812 df-minusg 17813 df-sbg 17814 df-subg 17975 df-mgp 18877 df-ur 18889 df-ring 18936 df-oppr 19010 df-dvdsr 19028 df-unit 19029 df-invr 19059 df-drng 19141 df-subrg 19170 df-lmod 19257 df-lss 19325 df-lsp 19367 df-sra 19569 df-rgmod 19570 df-lidl 19571 df-rsp 19572 df-lpidl 19640 df-lpir 19641 |
This theorem is referenced by: (None) |
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