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Theorem rnglidlrng 21149
Description: A (left) ideal of a non-unital ring is a non-unital ring. (Contributed by AV, 17-Feb-2020.) Generalization for non-unital rings. The assumption 𝑈 ∈ (SubGrp‘𝑅) is required because a left ideal of a non-unital ring does not have to be a subgroup. (Revised by AV, 11-Mar-2025.)
Hypotheses
Ref Expression
rnglidlabl.l 𝐿 = (LIdeal‘𝑅)
rnglidlabl.i 𝐼 = (𝑅s 𝑈)
Assertion
Ref Expression
rnglidlrng ((𝑅 ∈ Rng ∧ 𝑈𝐿𝑈 ∈ (SubGrp‘𝑅)) → 𝐼 ∈ Rng)

Proof of Theorem rnglidlrng
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rngabl 20102 . . . 4 (𝑅 ∈ Rng → 𝑅 ∈ Abel)
213ad2ant1 1130 . . 3 ((𝑅 ∈ Rng ∧ 𝑈𝐿𝑈 ∈ (SubGrp‘𝑅)) → 𝑅 ∈ Abel)
3 simp3 1135 . . 3 ((𝑅 ∈ Rng ∧ 𝑈𝐿𝑈 ∈ (SubGrp‘𝑅)) → 𝑈 ∈ (SubGrp‘𝑅))
4 rnglidlabl.i . . . 4 𝐼 = (𝑅s 𝑈)
54subgabl 19798 . . 3 ((𝑅 ∈ Abel ∧ 𝑈 ∈ (SubGrp‘𝑅)) → 𝐼 ∈ Abel)
62, 3, 5syl2anc 582 . 2 ((𝑅 ∈ Rng ∧ 𝑈𝐿𝑈 ∈ (SubGrp‘𝑅)) → 𝐼 ∈ Abel)
7 eqid 2728 . . . 4 (0g𝑅) = (0g𝑅)
87subg0cl 19096 . . 3 (𝑈 ∈ (SubGrp‘𝑅) → (0g𝑅) ∈ 𝑈)
9 rnglidlabl.l . . . 4 𝐿 = (LIdeal‘𝑅)
109, 4, 7rnglidlmsgrp 21148 . . 3 ((𝑅 ∈ Rng ∧ 𝑈𝐿 ∧ (0g𝑅) ∈ 𝑈) → (mulGrp‘𝐼) ∈ Smgrp)
118, 10syl3an3 1162 . 2 ((𝑅 ∈ Rng ∧ 𝑈𝐿𝑈 ∈ (SubGrp‘𝑅)) → (mulGrp‘𝐼) ∈ Smgrp)
12 simpl1 1188 . . . . 5 (((𝑅 ∈ Rng ∧ 𝑈𝐿𝑈 ∈ (SubGrp‘𝑅)) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼) ∧ 𝑐 ∈ (Base‘𝐼))) → 𝑅 ∈ Rng)
139, 4lidlssbas 21116 . . . . . . . . 9 (𝑈𝐿 → (Base‘𝐼) ⊆ (Base‘𝑅))
1413sseld 3981 . . . . . . . 8 (𝑈𝐿 → (𝑎 ∈ (Base‘𝐼) → 𝑎 ∈ (Base‘𝑅)))
1513sseld 3981 . . . . . . . 8 (𝑈𝐿 → (𝑏 ∈ (Base‘𝐼) → 𝑏 ∈ (Base‘𝑅)))
1613sseld 3981 . . . . . . . 8 (𝑈𝐿 → (𝑐 ∈ (Base‘𝐼) → 𝑐 ∈ (Base‘𝑅)))
1714, 15, 163anim123d 1439 . . . . . . 7 (𝑈𝐿 → ((𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼) ∧ 𝑐 ∈ (Base‘𝐼)) → (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅) ∧ 𝑐 ∈ (Base‘𝑅))))
18173ad2ant2 1131 . . . . . 6 ((𝑅 ∈ Rng ∧ 𝑈𝐿𝑈 ∈ (SubGrp‘𝑅)) → ((𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼) ∧ 𝑐 ∈ (Base‘𝐼)) → (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅) ∧ 𝑐 ∈ (Base‘𝑅))))
1918imp 405 . . . . 5 (((𝑅 ∈ Rng ∧ 𝑈𝐿𝑈 ∈ (SubGrp‘𝑅)) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼) ∧ 𝑐 ∈ (Base‘𝐼))) → (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅) ∧ 𝑐 ∈ (Base‘𝑅)))
20 eqid 2728 . . . . . 6 (Base‘𝑅) = (Base‘𝑅)
21 eqid 2728 . . . . . 6 (+g𝑅) = (+g𝑅)
22 eqid 2728 . . . . . 6 (.r𝑅) = (.r𝑅)
2320, 21, 22rngdi 20107 . . . . 5 ((𝑅 ∈ Rng ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅) ∧ 𝑐 ∈ (Base‘𝑅))) → (𝑎(.r𝑅)(𝑏(+g𝑅)𝑐)) = ((𝑎(.r𝑅)𝑏)(+g𝑅)(𝑎(.r𝑅)𝑐)))
2412, 19, 23syl2anc 582 . . . 4 (((𝑅 ∈ Rng ∧ 𝑈𝐿𝑈 ∈ (SubGrp‘𝑅)) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼) ∧ 𝑐 ∈ (Base‘𝐼))) → (𝑎(.r𝑅)(𝑏(+g𝑅)𝑐)) = ((𝑎(.r𝑅)𝑏)(+g𝑅)(𝑎(.r𝑅)𝑐)))
2520, 21, 22rngdir 20108 . . . . 5 ((𝑅 ∈ Rng ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅) ∧ 𝑐 ∈ (Base‘𝑅))) → ((𝑎(+g𝑅)𝑏)(.r𝑅)𝑐) = ((𝑎(.r𝑅)𝑐)(+g𝑅)(𝑏(.r𝑅)𝑐)))
2612, 19, 25syl2anc 582 . . . 4 (((𝑅 ∈ Rng ∧ 𝑈𝐿𝑈 ∈ (SubGrp‘𝑅)) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼) ∧ 𝑐 ∈ (Base‘𝐼))) → ((𝑎(+g𝑅)𝑏)(.r𝑅)𝑐) = ((𝑎(.r𝑅)𝑐)(+g𝑅)(𝑏(.r𝑅)𝑐)))
274, 22ressmulr 17295 . . . . . . . . . 10 (𝑈𝐿 → (.r𝑅) = (.r𝐼))
2827eqcomd 2734 . . . . . . . . 9 (𝑈𝐿 → (.r𝐼) = (.r𝑅))
29 eqidd 2729 . . . . . . . . 9 (𝑈𝐿𝑎 = 𝑎)
304, 21ressplusg 17278 . . . . . . . . . . 11 (𝑈𝐿 → (+g𝑅) = (+g𝐼))
3130eqcomd 2734 . . . . . . . . . 10 (𝑈𝐿 → (+g𝐼) = (+g𝑅))
3231oveqd 7443 . . . . . . . . 9 (𝑈𝐿 → (𝑏(+g𝐼)𝑐) = (𝑏(+g𝑅)𝑐))
3328, 29, 32oveq123d 7447 . . . . . . . 8 (𝑈𝐿 → (𝑎(.r𝐼)(𝑏(+g𝐼)𝑐)) = (𝑎(.r𝑅)(𝑏(+g𝑅)𝑐)))
3428oveqd 7443 . . . . . . . . 9 (𝑈𝐿 → (𝑎(.r𝐼)𝑏) = (𝑎(.r𝑅)𝑏))
3528oveqd 7443 . . . . . . . . 9 (𝑈𝐿 → (𝑎(.r𝐼)𝑐) = (𝑎(.r𝑅)𝑐))
3631, 34, 35oveq123d 7447 . . . . . . . 8 (𝑈𝐿 → ((𝑎(.r𝐼)𝑏)(+g𝐼)(𝑎(.r𝐼)𝑐)) = ((𝑎(.r𝑅)𝑏)(+g𝑅)(𝑎(.r𝑅)𝑐)))
3733, 36eqeq12d 2744 . . . . . . 7 (𝑈𝐿 → ((𝑎(.r𝐼)(𝑏(+g𝐼)𝑐)) = ((𝑎(.r𝐼)𝑏)(+g𝐼)(𝑎(.r𝐼)𝑐)) ↔ (𝑎(.r𝑅)(𝑏(+g𝑅)𝑐)) = ((𝑎(.r𝑅)𝑏)(+g𝑅)(𝑎(.r𝑅)𝑐))))
3831oveqd 7443 . . . . . . . . 9 (𝑈𝐿 → (𝑎(+g𝐼)𝑏) = (𝑎(+g𝑅)𝑏))
39 eqidd 2729 . . . . . . . . 9 (𝑈𝐿𝑐 = 𝑐)
4028, 38, 39oveq123d 7447 . . . . . . . 8 (𝑈𝐿 → ((𝑎(+g𝐼)𝑏)(.r𝐼)𝑐) = ((𝑎(+g𝑅)𝑏)(.r𝑅)𝑐))
4128oveqd 7443 . . . . . . . . 9 (𝑈𝐿 → (𝑏(.r𝐼)𝑐) = (𝑏(.r𝑅)𝑐))
4231, 35, 41oveq123d 7447 . . . . . . . 8 (𝑈𝐿 → ((𝑎(.r𝐼)𝑐)(+g𝐼)(𝑏(.r𝐼)𝑐)) = ((𝑎(.r𝑅)𝑐)(+g𝑅)(𝑏(.r𝑅)𝑐)))
4340, 42eqeq12d 2744 . . . . . . 7 (𝑈𝐿 → (((𝑎(+g𝐼)𝑏)(.r𝐼)𝑐) = ((𝑎(.r𝐼)𝑐)(+g𝐼)(𝑏(.r𝐼)𝑐)) ↔ ((𝑎(+g𝑅)𝑏)(.r𝑅)𝑐) = ((𝑎(.r𝑅)𝑐)(+g𝑅)(𝑏(.r𝑅)𝑐))))
4437, 43anbi12d 630 . . . . . 6 (𝑈𝐿 → (((𝑎(.r𝐼)(𝑏(+g𝐼)𝑐)) = ((𝑎(.r𝐼)𝑏)(+g𝐼)(𝑎(.r𝐼)𝑐)) ∧ ((𝑎(+g𝐼)𝑏)(.r𝐼)𝑐) = ((𝑎(.r𝐼)𝑐)(+g𝐼)(𝑏(.r𝐼)𝑐))) ↔ ((𝑎(.r𝑅)(𝑏(+g𝑅)𝑐)) = ((𝑎(.r𝑅)𝑏)(+g𝑅)(𝑎(.r𝑅)𝑐)) ∧ ((𝑎(+g𝑅)𝑏)(.r𝑅)𝑐) = ((𝑎(.r𝑅)𝑐)(+g𝑅)(𝑏(.r𝑅)𝑐)))))
45443ad2ant2 1131 . . . . 5 ((𝑅 ∈ Rng ∧ 𝑈𝐿𝑈 ∈ (SubGrp‘𝑅)) → (((𝑎(.r𝐼)(𝑏(+g𝐼)𝑐)) = ((𝑎(.r𝐼)𝑏)(+g𝐼)(𝑎(.r𝐼)𝑐)) ∧ ((𝑎(+g𝐼)𝑏)(.r𝐼)𝑐) = ((𝑎(.r𝐼)𝑐)(+g𝐼)(𝑏(.r𝐼)𝑐))) ↔ ((𝑎(.r𝑅)(𝑏(+g𝑅)𝑐)) = ((𝑎(.r𝑅)𝑏)(+g𝑅)(𝑎(.r𝑅)𝑐)) ∧ ((𝑎(+g𝑅)𝑏)(.r𝑅)𝑐) = ((𝑎(.r𝑅)𝑐)(+g𝑅)(𝑏(.r𝑅)𝑐)))))
4645adantr 479 . . . 4 (((𝑅 ∈ Rng ∧ 𝑈𝐿𝑈 ∈ (SubGrp‘𝑅)) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼) ∧ 𝑐 ∈ (Base‘𝐼))) → (((𝑎(.r𝐼)(𝑏(+g𝐼)𝑐)) = ((𝑎(.r𝐼)𝑏)(+g𝐼)(𝑎(.r𝐼)𝑐)) ∧ ((𝑎(+g𝐼)𝑏)(.r𝐼)𝑐) = ((𝑎(.r𝐼)𝑐)(+g𝐼)(𝑏(.r𝐼)𝑐))) ↔ ((𝑎(.r𝑅)(𝑏(+g𝑅)𝑐)) = ((𝑎(.r𝑅)𝑏)(+g𝑅)(𝑎(.r𝑅)𝑐)) ∧ ((𝑎(+g𝑅)𝑏)(.r𝑅)𝑐) = ((𝑎(.r𝑅)𝑐)(+g𝑅)(𝑏(.r𝑅)𝑐)))))
4724, 26, 46mpbir2and 711 . . 3 (((𝑅 ∈ Rng ∧ 𝑈𝐿𝑈 ∈ (SubGrp‘𝑅)) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼) ∧ 𝑐 ∈ (Base‘𝐼))) → ((𝑎(.r𝐼)(𝑏(+g𝐼)𝑐)) = ((𝑎(.r𝐼)𝑏)(+g𝐼)(𝑎(.r𝐼)𝑐)) ∧ ((𝑎(+g𝐼)𝑏)(.r𝐼)𝑐) = ((𝑎(.r𝐼)𝑐)(+g𝐼)(𝑏(.r𝐼)𝑐))))
4847ralrimivvva 3201 . 2 ((𝑅 ∈ Rng ∧ 𝑈𝐿𝑈 ∈ (SubGrp‘𝑅)) → ∀𝑎 ∈ (Base‘𝐼)∀𝑏 ∈ (Base‘𝐼)∀𝑐 ∈ (Base‘𝐼)((𝑎(.r𝐼)(𝑏(+g𝐼)𝑐)) = ((𝑎(.r𝐼)𝑏)(+g𝐼)(𝑎(.r𝐼)𝑐)) ∧ ((𝑎(+g𝐼)𝑏)(.r𝐼)𝑐) = ((𝑎(.r𝐼)𝑐)(+g𝐼)(𝑏(.r𝐼)𝑐))))
49 eqid 2728 . . 3 (Base‘𝐼) = (Base‘𝐼)
50 eqid 2728 . . 3 (mulGrp‘𝐼) = (mulGrp‘𝐼)
51 eqid 2728 . . 3 (+g𝐼) = (+g𝐼)
52 eqid 2728 . . 3 (.r𝐼) = (.r𝐼)
5349, 50, 51, 52isrng 20101 . 2 (𝐼 ∈ Rng ↔ (𝐼 ∈ Abel ∧ (mulGrp‘𝐼) ∈ Smgrp ∧ ∀𝑎 ∈ (Base‘𝐼)∀𝑏 ∈ (Base‘𝐼)∀𝑐 ∈ (Base‘𝐼)((𝑎(.r𝐼)(𝑏(+g𝐼)𝑐)) = ((𝑎(.r𝐼)𝑏)(+g𝐼)(𝑎(.r𝐼)𝑐)) ∧ ((𝑎(+g𝐼)𝑏)(.r𝐼)𝑐) = ((𝑎(.r𝐼)𝑐)(+g𝐼)(𝑏(.r𝐼)𝑐)))))
546, 11, 48, 53syl3anbrc 1340 1 ((𝑅 ∈ Rng ∧ 𝑈𝐿𝑈 ∈ (SubGrp‘𝑅)) → 𝐼 ∈ Rng)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  wral 3058  cfv 6553  (class class class)co 7426  Basecbs 17187  s cress 17216  +gcplusg 17240  .rcmulr 17241  0gc0g 17428  Smgrpcsgrp 18685  SubGrpcsubg 19082  Abelcabl 19743  mulGrpcmgp 20081  Rngcrng 20099  LIdealclidl 21109
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 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223
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 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-er 8731  df-en 8971  df-dom 8972  df-sdom 8973  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-nn 12251  df-2 12313  df-3 12314  df-4 12315  df-5 12316  df-6 12317  df-7 12318  df-8 12319  df-sets 17140  df-slot 17158  df-ndx 17170  df-base 17188  df-ress 17217  df-plusg 17253  df-mulr 17254  df-sca 17256  df-vsca 17257  df-ip 17258  df-0g 17430  df-mgm 18607  df-sgrp 18686  df-mnd 18702  df-grp 18900  df-subg 19085  df-cmn 19744  df-abl 19745  df-mgp 20082  df-rng 20100  df-lss 20823  df-sra 21065  df-rgmod 21066  df-lidl 21111
This theorem is referenced by:  rng2idlsubgsubrng  21169  lidlrng  47373
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