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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 18577 | . 2 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring) | |
2 | eqid 2610 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | eqid 2610 | . . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
4 | eqid 2610 | . . . 4 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅) | |
5 | 2, 3, 4 | drngnidl 19050 | . . 3 ⊢ (𝑅 ∈ DivRing → (LIdeal‘𝑅) = {{(0g‘𝑅)}, (Base‘𝑅)}) |
6 | eqid 2610 | . . . . . 6 ⊢ (LPIdeal‘𝑅) = (LPIdeal‘𝑅) | |
7 | 6, 3 | lpi0 19068 | . . . . 5 ⊢ (𝑅 ∈ Ring → {(0g‘𝑅)} ∈ (LPIdeal‘𝑅)) |
8 | 6, 2 | lpi1 19069 | . . . . 5 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) ∈ (LPIdeal‘𝑅)) |
9 | snex 4835 | . . . . . . 7 ⊢ {(0g‘𝑅)} ∈ V | |
10 | fvex 6113 | . . . . . . 7 ⊢ (Base‘𝑅) ∈ V | |
11 | 9, 10 | prss 4291 | . . . . . 6 ⊢ (({(0g‘𝑅)} ∈ (LPIdeal‘𝑅) ∧ (Base‘𝑅) ∈ (LPIdeal‘𝑅)) ↔ {{(0g‘𝑅)}, (Base‘𝑅)} ⊆ (LPIdeal‘𝑅)) |
12 | 11 | bicomi 213 | . . . . 5 ⊢ ({{(0g‘𝑅)}, (Base‘𝑅)} ⊆ (LPIdeal‘𝑅) ↔ ({(0g‘𝑅)} ∈ (LPIdeal‘𝑅) ∧ (Base‘𝑅) ∈ (LPIdeal‘𝑅))) |
13 | 7, 8, 12 | sylanbrc 695 | . . . 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 3602 | . 2 ⊢ (𝑅 ∈ DivRing → (LIdeal‘𝑅) ⊆ (LPIdeal‘𝑅)) |
16 | 6, 4 | islpir2 19072 | . 2 ⊢ (𝑅 ∈ LPIR ↔ (𝑅 ∈ Ring ∧ (LIdeal‘𝑅) ⊆ (LPIdeal‘𝑅))) |
17 | 1, 15, 16 | sylanbrc 695 | 1 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ LPIR) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 ⊆ wss 3540 {csn 4125 {cpr 4127 ‘cfv 5804 Basecbs 15695 0gc0g 15923 Ringcrg 18370 DivRingcdr 18570 LIdealclidl 18991 LPIdealclpidl 19062 LPIRclpir 19063 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-ip 15786 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-sbg 17250 df-subg 17414 df-mgp 18313 df-ur 18325 df-ring 18372 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-invr 18495 df-drng 18572 df-subrg 18601 df-lmod 18688 df-lss 18754 df-lsp 18793 df-sra 18993 df-rgmod 18994 df-lidl 18995 df-rsp 18996 df-lpidl 19064 df-lpir 19065 |
This theorem is referenced by: (None) |
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