![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > rngridlmcl | Structured version Visualization version GIF version |
Description: A right ideal (which is a left ideal over the opposite ring) containing the zero element is closed under right-multiplication by elements of the full non-unital ring. (Contributed by AV, 19-Feb-2025.) |
Ref | Expression |
---|---|
rnglidlmcl.z | β’ 0 = (0gβπ ) |
rnglidlmcl.b | β’ π΅ = (Baseβπ ) |
rnglidlmcl.t | β’ Β· = (.rβπ ) |
rngridlmcl.u | β’ π = (LIdealβ(opprβπ )) |
Ref | Expression |
---|---|
rngridlmcl | β’ (((π β Rng β§ πΌ β π β§ 0 β πΌ) β§ (π β π΅ β§ π β πΌ)) β (π Β· π) β πΌ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rnglidlmcl.b | . . 3 β’ π΅ = (Baseβπ ) | |
2 | rnglidlmcl.t | . . 3 β’ Β· = (.rβπ ) | |
3 | eqid 2725 | . . 3 β’ (opprβπ ) = (opprβπ ) | |
4 | eqid 2725 | . . 3 β’ (.rβ(opprβπ )) = (.rβ(opprβπ )) | |
5 | 1, 2, 3, 4 | opprmul 20278 | . 2 β’ (π(.rβ(opprβπ ))π) = (π Β· π) |
6 | 3 | opprrng 20286 | . . 3 β’ (π β Rng β (opprβπ ) β Rng) |
7 | id 22 | . . 3 β’ (πΌ β π β πΌ β π) | |
8 | rnglidlmcl.z | . . . . 5 β’ 0 = (0gβπ ) | |
9 | 8 | eleq1i 2816 | . . . 4 β’ ( 0 β πΌ β (0gβπ ) β πΌ) |
10 | 9 | biimpi 215 | . . 3 β’ ( 0 β πΌ β (0gβπ ) β πΌ) |
11 | eqid 2725 | . . . . 5 β’ (0gβπ ) = (0gβπ ) | |
12 | 3, 11 | oppr0 20290 | . . . 4 β’ (0gβπ ) = (0gβ(opprβπ )) |
13 | 3, 1 | opprbas 20282 | . . . 4 β’ π΅ = (Baseβ(opprβπ )) |
14 | rngridlmcl.u | . . . 4 β’ π = (LIdealβ(opprβπ )) | |
15 | 12, 13, 4, 14 | rnglidlmcl 21114 | . . 3 β’ ((((opprβπ ) β Rng β§ πΌ β π β§ (0gβπ ) β πΌ) β§ (π β π΅ β§ π β πΌ)) β (π(.rβ(opprβπ ))π) β πΌ) |
16 | 6, 7, 10, 15 | syl3anl 1412 | . 2 β’ (((π β Rng β§ πΌ β π β§ 0 β πΌ) β§ (π β π΅ β§ π β πΌ)) β (π(.rβ(opprβπ ))π) β πΌ) |
17 | 5, 16 | eqeltrrid 2830 | 1 β’ (((π β Rng β§ πΌ β π β§ 0 β πΌ) β§ (π β π΅ β§ π β πΌ)) β (π Β· π) β πΌ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 βcfv 6542 (class class class)co 7415 Basecbs 17177 .rcmulr 17231 0gc0g 17418 Rngcrng 20094 opprcoppr 20274 LIdealclidl 21104 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 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 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-2nd 7990 df-tpos 8228 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-er 8721 df-en 8961 df-dom 8962 df-sdom 8963 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-ress 17207 df-plusg 17243 df-mulr 17244 df-sca 17246 df-vsca 17247 df-ip 17248 df-0g 17420 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-grp 18895 df-cmn 19739 df-abl 19740 df-mgp 20077 df-rng 20095 df-oppr 20275 df-lss 20818 df-sra 21060 df-rgmod 21061 df-lidl 21106 |
This theorem is referenced by: rngqiprngghmlem1 21179 rngqiprngimf 21189 |
Copyright terms: Public domain | W3C validator |