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Theorem rnglidlmmgm 13992
Description: The multiplicative group of a (left) ideal of a non-unital ring is a magma. (Contributed by AV, 17-Feb-2020.) Generalization for non-unital rings. The assumption 0𝑈 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)
rnglidlabl.z 0 = (0g𝑅)
Assertion
Ref Expression
rnglidlmmgm ((𝑅 ∈ Rng ∧ 𝑈𝐿0𝑈) → (mulGrp‘𝐼) ∈ Mgm)

Proof of Theorem rnglidlmmgm
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 999 . . . . . 6 ((𝑅 ∈ Rng ∧ 𝑈𝐿0𝑈) → 𝑅 ∈ Rng)
2 rnglidlabl.l . . . . . . . . 9 𝐿 = (LIdeal‘𝑅)
3 rnglidlabl.i . . . . . . . . 9 𝐼 = (𝑅s 𝑈)
42, 3lidlbas 13974 . . . . . . . 8 (𝑈𝐿 → (Base‘𝐼) = 𝑈)
5 eleq1a 2265 . . . . . . . 8 (𝑈𝐿 → ((Base‘𝐼) = 𝑈 → (Base‘𝐼) ∈ 𝐿))
64, 5mpd 13 . . . . . . 7 (𝑈𝐿 → (Base‘𝐼) ∈ 𝐿)
763ad2ant2 1021 . . . . . 6 ((𝑅 ∈ Rng ∧ 𝑈𝐿0𝑈) → (Base‘𝐼) ∈ 𝐿)
84eqcomd 2199 . . . . . . . . 9 (𝑈𝐿𝑈 = (Base‘𝐼))
98eleq2d 2263 . . . . . . . 8 (𝑈𝐿 → ( 0𝑈0 ∈ (Base‘𝐼)))
109biimpa 296 . . . . . . 7 ((𝑈𝐿0𝑈) → 0 ∈ (Base‘𝐼))
11103adant1 1017 . . . . . 6 ((𝑅 ∈ Rng ∧ 𝑈𝐿0𝑈) → 0 ∈ (Base‘𝐼))
121, 7, 113jca 1179 . . . . 5 ((𝑅 ∈ Rng ∧ 𝑈𝐿0𝑈) → (𝑅 ∈ Rng ∧ (Base‘𝐼) ∈ 𝐿0 ∈ (Base‘𝐼)))
132, 3lidlssbas 13973 . . . . . . . . 9 (𝑈𝐿 → (Base‘𝐼) ⊆ (Base‘𝑅))
1413sseld 3178 . . . . . . . 8 (𝑈𝐿 → (𝑎 ∈ (Base‘𝐼) → 𝑎 ∈ (Base‘𝑅)))
15143ad2ant2 1021 . . . . . . 7 ((𝑅 ∈ Rng ∧ 𝑈𝐿0𝑈) → (𝑎 ∈ (Base‘𝐼) → 𝑎 ∈ (Base‘𝑅)))
1615anim1d 336 . . . . . 6 ((𝑅 ∈ Rng ∧ 𝑈𝐿0𝑈) → ((𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼)) → (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝐼))))
1716imp 124 . . . . 5 (((𝑅 ∈ Rng ∧ 𝑈𝐿0𝑈) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼))) → (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝐼)))
18 rnglidlabl.z . . . . . 6 0 = (0g𝑅)
19 eqid 2193 . . . . . 6 (Base‘𝑅) = (Base‘𝑅)
20 eqid 2193 . . . . . 6 (.r𝑅) = (.r𝑅)
2118, 19, 20, 2rnglidlmcl 13976 . . . . 5 (((𝑅 ∈ Rng ∧ (Base‘𝐼) ∈ 𝐿0 ∈ (Base‘𝐼)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝐼))) → (𝑎(.r𝑅)𝑏) ∈ (Base‘𝐼))
2212, 17, 21syl2an2r 595 . . . 4 (((𝑅 ∈ Rng ∧ 𝑈𝐿0𝑈) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼))) → (𝑎(.r𝑅)𝑏) ∈ (Base‘𝐼))
23 simp2 1000 . . . . . . . . 9 ((𝑅 ∈ Rng ∧ 𝑈𝐿0𝑈) → 𝑈𝐿)
243, 20ressmulrg 12762 . . . . . . . . 9 ((𝑈𝐿𝑅 ∈ Rng) → (.r𝑅) = (.r𝐼))
2523, 1, 24syl2anc 411 . . . . . . . 8 ((𝑅 ∈ Rng ∧ 𝑈𝐿0𝑈) → (.r𝑅) = (.r𝐼))
2625eqcomd 2199 . . . . . . 7 ((𝑅 ∈ Rng ∧ 𝑈𝐿0𝑈) → (.r𝐼) = (.r𝑅))
2726oveqd 5935 . . . . . 6 ((𝑅 ∈ Rng ∧ 𝑈𝐿0𝑈) → (𝑎(.r𝐼)𝑏) = (𝑎(.r𝑅)𝑏))
2827eleq1d 2262 . . . . 5 ((𝑅 ∈ Rng ∧ 𝑈𝐿0𝑈) → ((𝑎(.r𝐼)𝑏) ∈ (Base‘𝐼) ↔ (𝑎(.r𝑅)𝑏) ∈ (Base‘𝐼)))
2928adantr 276 . . . 4 (((𝑅 ∈ Rng ∧ 𝑈𝐿0𝑈) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼))) → ((𝑎(.r𝐼)𝑏) ∈ (Base‘𝐼) ↔ (𝑎(.r𝑅)𝑏) ∈ (Base‘𝐼)))
3022, 29mpbird 167 . . 3 (((𝑅 ∈ Rng ∧ 𝑈𝐿0𝑈) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼))) → (𝑎(.r𝐼)𝑏) ∈ (Base‘𝐼))
3130ralrimivva 2576 . 2 ((𝑅 ∈ Rng ∧ 𝑈𝐿0𝑈) → ∀𝑎 ∈ (Base‘𝐼)∀𝑏 ∈ (Base‘𝐼)(𝑎(.r𝐼)𝑏) ∈ (Base‘𝐼))
32 ressex 12683 . . . . . 6 ((𝑅 ∈ Rng ∧ 𝑈𝐿) → (𝑅s 𝑈) ∈ V)
333, 32eqeltrid 2280 . . . . 5 ((𝑅 ∈ Rng ∧ 𝑈𝐿) → 𝐼 ∈ V)
341, 23, 33syl2anc 411 . . . 4 ((𝑅 ∈ Rng ∧ 𝑈𝐿0𝑈) → 𝐼 ∈ V)
35 eqid 2193 . . . . 5 (mulGrp‘𝐼) = (mulGrp‘𝐼)
3635mgpex 13421 . . . 4 (𝐼 ∈ V → (mulGrp‘𝐼) ∈ V)
37 eqid 2193 . . . . 5 (Base‘(mulGrp‘𝐼)) = (Base‘(mulGrp‘𝐼))
38 eqid 2193 . . . . 5 (+g‘(mulGrp‘𝐼)) = (+g‘(mulGrp‘𝐼))
3937, 38ismgm 12940 . . . 4 ((mulGrp‘𝐼) ∈ V → ((mulGrp‘𝐼) ∈ Mgm ↔ ∀𝑎 ∈ (Base‘(mulGrp‘𝐼))∀𝑏 ∈ (Base‘(mulGrp‘𝐼))(𝑎(+g‘(mulGrp‘𝐼))𝑏) ∈ (Base‘(mulGrp‘𝐼))))
4034, 36, 393syl 17 . . 3 ((𝑅 ∈ Rng ∧ 𝑈𝐿0𝑈) → ((mulGrp‘𝐼) ∈ Mgm ↔ ∀𝑎 ∈ (Base‘(mulGrp‘𝐼))∀𝑏 ∈ (Base‘(mulGrp‘𝐼))(𝑎(+g‘(mulGrp‘𝐼))𝑏) ∈ (Base‘(mulGrp‘𝐼))))
41 eqid 2193 . . . . . 6 (Base‘𝐼) = (Base‘𝐼)
4235, 41mgpbasg 13422 . . . . 5 (𝐼 ∈ V → (Base‘𝐼) = (Base‘(mulGrp‘𝐼)))
4334, 42syl 14 . . . 4 ((𝑅 ∈ Rng ∧ 𝑈𝐿0𝑈) → (Base‘𝐼) = (Base‘(mulGrp‘𝐼)))
44 eqid 2193 . . . . . . . . 9 (.r𝐼) = (.r𝐼)
4535, 44mgpplusgg 13420 . . . . . . . 8 (𝐼 ∈ V → (.r𝐼) = (+g‘(mulGrp‘𝐼)))
4634, 45syl 14 . . . . . . 7 ((𝑅 ∈ Rng ∧ 𝑈𝐿0𝑈) → (.r𝐼) = (+g‘(mulGrp‘𝐼)))
4746oveqd 5935 . . . . . 6 ((𝑅 ∈ Rng ∧ 𝑈𝐿0𝑈) → (𝑎(.r𝐼)𝑏) = (𝑎(+g‘(mulGrp‘𝐼))𝑏))
4847, 43eleq12d 2264 . . . . 5 ((𝑅 ∈ Rng ∧ 𝑈𝐿0𝑈) → ((𝑎(.r𝐼)𝑏) ∈ (Base‘𝐼) ↔ (𝑎(+g‘(mulGrp‘𝐼))𝑏) ∈ (Base‘(mulGrp‘𝐼))))
4943, 48raleqbidv 2706 . . . 4 ((𝑅 ∈ Rng ∧ 𝑈𝐿0𝑈) → (∀𝑏 ∈ (Base‘𝐼)(𝑎(.r𝐼)𝑏) ∈ (Base‘𝐼) ↔ ∀𝑏 ∈ (Base‘(mulGrp‘𝐼))(𝑎(+g‘(mulGrp‘𝐼))𝑏) ∈ (Base‘(mulGrp‘𝐼))))
5043, 49raleqbidv 2706 . . 3 ((𝑅 ∈ Rng ∧ 𝑈𝐿0𝑈) → (∀𝑎 ∈ (Base‘𝐼)∀𝑏 ∈ (Base‘𝐼)(𝑎(.r𝐼)𝑏) ∈ (Base‘𝐼) ↔ ∀𝑎 ∈ (Base‘(mulGrp‘𝐼))∀𝑏 ∈ (Base‘(mulGrp‘𝐼))(𝑎(+g‘(mulGrp‘𝐼))𝑏) ∈ (Base‘(mulGrp‘𝐼))))
5140, 50bitr4d 191 . 2 ((𝑅 ∈ Rng ∧ 𝑈𝐿0𝑈) → ((mulGrp‘𝐼) ∈ Mgm ↔ ∀𝑎 ∈ (Base‘𝐼)∀𝑏 ∈ (Base‘𝐼)(𝑎(.r𝐼)𝑏) ∈ (Base‘𝐼)))
5231, 51mpbird 167 1 ((𝑅 ∈ Rng ∧ 𝑈𝐿0𝑈) → (mulGrp‘𝐼) ∈ Mgm)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  w3a 980   = wceq 1364  wcel 2164  wral 2472  Vcvv 2760  cfv 5254  (class class class)co 5918  Basecbs 12618  s cress 12619  +gcplusg 12695  .rcmulr 12696  0gc0g 12867  Mgmcmgm 12937  mulGrpcmgp 13416  Rngcrng 13428  LIdealclidl 13963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-pre-ltirr 7984  ax-pre-lttrn 7986  ax-pre-ltadd 7988
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-pnf 8056  df-mnf 8057  df-ltxr 8059  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-5 9044  df-6 9045  df-7 9046  df-8 9047  df-ndx 12621  df-slot 12622  df-base 12624  df-sets 12625  df-iress 12626  df-plusg 12708  df-mulr 12709  df-sca 12711  df-vsca 12712  df-ip 12713  df-0g 12869  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-grp 13075  df-abl 13357  df-mgp 13417  df-rng 13429  df-lssm 13849  df-sra 13931  df-rgmod 13932  df-lidl 13965
This theorem is referenced by:  rnglidlmsgrp  13993
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