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Theorem idlinsubrg 31322
Description: The intersection between an ideal and a subring is an ideal of the subring. (Contributed by Thierry Arnoux, 6-Jul-2024.)
Hypotheses
Ref Expression
idlinsubrg.s 𝑆 = (𝑅s 𝐴)
idlinsubrg.u 𝑈 = (LIdeal‘𝑅)
idlinsubrg.v 𝑉 = (LIdeal‘𝑆)
Assertion
Ref Expression
idlinsubrg ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) → (𝐼𝐴) ∈ 𝑉)

Proof of Theorem idlinsubrg
Dummy variables 𝑎 𝑏 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inss2 4144 . . . 4 (𝐼𝐴) ⊆ 𝐴
2 idlinsubrg.s . . . . 5 𝑆 = (𝑅s 𝐴)
32subrgbas 19809 . . . 4 (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆))
41, 3sseqtrid 3953 . . 3 (𝐴 ∈ (SubRing‘𝑅) → (𝐼𝐴) ⊆ (Base‘𝑆))
54adantr 484 . 2 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) → (𝐼𝐴) ⊆ (Base‘𝑆))
6 subrgrcl 19805 . . . . 5 (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
7 idlinsubrg.u . . . . . 6 𝑈 = (LIdeal‘𝑅)
8 eqid 2737 . . . . . 6 (0g𝑅) = (0g𝑅)
97, 8lidl0cl 20250 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐼𝑈) → (0g𝑅) ∈ 𝐼)
106, 9sylan 583 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) → (0g𝑅) ∈ 𝐼)
11 subrgsubg 19806 . . . . . 6 (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅))
12 subgsubm 18565 . . . . . 6 (𝐴 ∈ (SubGrp‘𝑅) → 𝐴 ∈ (SubMnd‘𝑅))
138subm0cl 18238 . . . . . 6 (𝐴 ∈ (SubMnd‘𝑅) → (0g𝑅) ∈ 𝐴)
1411, 12, 133syl 18 . . . . 5 (𝐴 ∈ (SubRing‘𝑅) → (0g𝑅) ∈ 𝐴)
1514adantr 484 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) → (0g𝑅) ∈ 𝐴)
1610, 15elind 4108 . . 3 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) → (0g𝑅) ∈ (𝐼𝐴))
1716ne0d 4250 . 2 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) → (𝐼𝐴) ≠ ∅)
18 eqid 2737 . . . . . . . . 9 (+g𝑅) = (+g𝑅)
192, 18ressplusg 16834 . . . . . . . 8 (𝐴 ∈ (SubRing‘𝑅) → (+g𝑅) = (+g𝑆))
20 eqid 2737 . . . . . . . . . 10 (.r𝑅) = (.r𝑅)
212, 20ressmulr 16848 . . . . . . . . 9 (𝐴 ∈ (SubRing‘𝑅) → (.r𝑅) = (.r𝑆))
2221oveqd 7230 . . . . . . . 8 (𝐴 ∈ (SubRing‘𝑅) → (𝑥(.r𝑅)𝑎) = (𝑥(.r𝑆)𝑎))
23 eqidd 2738 . . . . . . . 8 (𝐴 ∈ (SubRing‘𝑅) → 𝑏 = 𝑏)
2419, 22, 23oveq123d 7234 . . . . . . 7 (𝐴 ∈ (SubRing‘𝑅) → ((𝑥(.r𝑅)𝑎)(+g𝑅)𝑏) = ((𝑥(.r𝑆)𝑎)(+g𝑆)𝑏))
2524ad4antr 732 . . . . . 6 (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼𝐴)) ∧ 𝑏 ∈ (𝐼𝐴)) → ((𝑥(.r𝑅)𝑎)(+g𝑅)𝑏) = ((𝑥(.r𝑆)𝑎)(+g𝑆)𝑏))
266ad4antr 732 . . . . . . . 8 (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼𝐴)) ∧ 𝑏 ∈ (𝐼𝐴)) → 𝑅 ∈ Ring)
27 simp-4r 784 . . . . . . . 8 (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼𝐴)) ∧ 𝑏 ∈ (𝐼𝐴)) → 𝐼𝑈)
28 eqid 2737 . . . . . . . . . . . . . 14 (Base‘𝑅) = (Base‘𝑅)
2928subrgss 19801 . . . . . . . . . . . . 13 (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
303, 29eqsstrrd 3940 . . . . . . . . . . . 12 (𝐴 ∈ (SubRing‘𝑅) → (Base‘𝑆) ⊆ (Base‘𝑅))
3130adantr 484 . . . . . . . . . . 11 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) → (Base‘𝑆) ⊆ (Base‘𝑅))
3231sselda 3901 . . . . . . . . . 10 (((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑅))
3332ad2antrr 726 . . . . . . . . 9 (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼𝐴)) ∧ 𝑏 ∈ (𝐼𝐴)) → 𝑥 ∈ (Base‘𝑅))
34 inss1 4143 . . . . . . . . . . . 12 (𝐼𝐴) ⊆ 𝐼
3534a1i 11 . . . . . . . . . . 11 (((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) → (𝐼𝐴) ⊆ 𝐼)
3635sselda 3901 . . . . . . . . . 10 ((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼𝐴)) → 𝑎𝐼)
3736adantr 484 . . . . . . . . 9 (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼𝐴)) ∧ 𝑏 ∈ (𝐼𝐴)) → 𝑎𝐼)
387, 28, 20lidlmcl 20255 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝐼𝑈) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑎𝐼)) → (𝑥(.r𝑅)𝑎) ∈ 𝐼)
3926, 27, 33, 37, 38syl22anc 839 . . . . . . . 8 (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼𝐴)) ∧ 𝑏 ∈ (𝐼𝐴)) → (𝑥(.r𝑅)𝑎) ∈ 𝐼)
4034a1i 11 . . . . . . . . 9 ((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼𝐴)) → (𝐼𝐴) ⊆ 𝐼)
4140sselda 3901 . . . . . . . 8 (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼𝐴)) ∧ 𝑏 ∈ (𝐼𝐴)) → 𝑏𝐼)
427, 18lidlacl 20251 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐼𝑈) ∧ ((𝑥(.r𝑅)𝑎) ∈ 𝐼𝑏𝐼)) → ((𝑥(.r𝑅)𝑎)(+g𝑅)𝑏) ∈ 𝐼)
4326, 27, 39, 41, 42syl22anc 839 . . . . . . 7 (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼𝐴)) ∧ 𝑏 ∈ (𝐼𝐴)) → ((𝑥(.r𝑅)𝑎)(+g𝑅)𝑏) ∈ 𝐼)
44 simp-4l 783 . . . . . . . 8 (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼𝐴)) ∧ 𝑏 ∈ (𝐼𝐴)) → 𝐴 ∈ (SubRing‘𝑅))
45 simpr 488 . . . . . . . . . . 11 (((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑆))
463ad2antrr 726 . . . . . . . . . . 11 (((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) → 𝐴 = (Base‘𝑆))
4745, 46eleqtrrd 2841 . . . . . . . . . 10 (((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) → 𝑥𝐴)
4847ad2antrr 726 . . . . . . . . 9 (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼𝐴)) ∧ 𝑏 ∈ (𝐼𝐴)) → 𝑥𝐴)
491a1i 11 . . . . . . . . . . 11 (((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) → (𝐼𝐴) ⊆ 𝐴)
5049sselda 3901 . . . . . . . . . 10 ((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼𝐴)) → 𝑎𝐴)
5150adantr 484 . . . . . . . . 9 (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼𝐴)) ∧ 𝑏 ∈ (𝐼𝐴)) → 𝑎𝐴)
5220subrgmcl 19812 . . . . . . . . 9 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝐴𝑎𝐴) → (𝑥(.r𝑅)𝑎) ∈ 𝐴)
5344, 48, 51, 52syl3anc 1373 . . . . . . . 8 (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼𝐴)) ∧ 𝑏 ∈ (𝐼𝐴)) → (𝑥(.r𝑅)𝑎) ∈ 𝐴)
541a1i 11 . . . . . . . . 9 ((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼𝐴)) → (𝐼𝐴) ⊆ 𝐴)
5554sselda 3901 . . . . . . . 8 (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼𝐴)) ∧ 𝑏 ∈ (𝐼𝐴)) → 𝑏𝐴)
5618subrgacl 19811 . . . . . . . 8 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑥(.r𝑅)𝑎) ∈ 𝐴𝑏𝐴) → ((𝑥(.r𝑅)𝑎)(+g𝑅)𝑏) ∈ 𝐴)
5744, 53, 55, 56syl3anc 1373 . . . . . . 7 (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼𝐴)) ∧ 𝑏 ∈ (𝐼𝐴)) → ((𝑥(.r𝑅)𝑎)(+g𝑅)𝑏) ∈ 𝐴)
5843, 57elind 4108 . . . . . 6 (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼𝐴)) ∧ 𝑏 ∈ (𝐼𝐴)) → ((𝑥(.r𝑅)𝑎)(+g𝑅)𝑏) ∈ (𝐼𝐴))
5925, 58eqeltrrd 2839 . . . . 5 (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼𝐴)) ∧ 𝑏 ∈ (𝐼𝐴)) → ((𝑥(.r𝑆)𝑎)(+g𝑆)𝑏) ∈ (𝐼𝐴))
6059anasss 470 . . . 4 ((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ (𝑎 ∈ (𝐼𝐴) ∧ 𝑏 ∈ (𝐼𝐴))) → ((𝑥(.r𝑆)𝑎)(+g𝑆)𝑏) ∈ (𝐼𝐴))
6160ralrimivva 3112 . . 3 (((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) → ∀𝑎 ∈ (𝐼𝐴)∀𝑏 ∈ (𝐼𝐴)((𝑥(.r𝑆)𝑎)(+g𝑆)𝑏) ∈ (𝐼𝐴))
6261ralrimiva 3105 . 2 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) → ∀𝑥 ∈ (Base‘𝑆)∀𝑎 ∈ (𝐼𝐴)∀𝑏 ∈ (𝐼𝐴)((𝑥(.r𝑆)𝑎)(+g𝑆)𝑏) ∈ (𝐼𝐴))
63 idlinsubrg.v . . 3 𝑉 = (LIdeal‘𝑆)
64 eqid 2737 . . 3 (Base‘𝑆) = (Base‘𝑆)
65 eqid 2737 . . 3 (+g𝑆) = (+g𝑆)
66 eqid 2737 . . 3 (.r𝑆) = (.r𝑆)
6763, 64, 65, 66islidl 20249 . 2 ((𝐼𝐴) ∈ 𝑉 ↔ ((𝐼𝐴) ⊆ (Base‘𝑆) ∧ (𝐼𝐴) ≠ ∅ ∧ ∀𝑥 ∈ (Base‘𝑆)∀𝑎 ∈ (𝐼𝐴)∀𝑏 ∈ (𝐼𝐴)((𝑥(.r𝑆)𝑎)(+g𝑆)𝑏) ∈ (𝐼𝐴)))
685, 17, 62, 67syl3anbrc 1345 1 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) → (𝐼𝐴) ∈ 𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1543  wcel 2110  wne 2940  wral 3061  cin 3865  wss 3866  c0 4237  cfv 6380  (class class class)co 7213  Basecbs 16760  s cress 16784  +gcplusg 16802  .rcmulr 16803  0gc0g 16944  SubMndcsubmnd 18217  SubGrpcsubg 18537  Ringcrg 19562  SubRingcsubrg 19796  LIdealclidl 20207
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-er 8391  df-en 8627  df-dom 8628  df-sdom 8629  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-nn 11831  df-2 11893  df-3 11894  df-4 11895  df-5 11896  df-6 11897  df-7 11898  df-8 11899  df-sets 16717  df-slot 16735  df-ndx 16745  df-base 16761  df-ress 16785  df-plusg 16815  df-mulr 16816  df-sca 16818  df-vsca 16819  df-ip 16820  df-0g 16946  df-mgm 18114  df-sgrp 18163  df-mnd 18174  df-submnd 18219  df-grp 18368  df-minusg 18369  df-sbg 18370  df-subg 18540  df-mgp 19505  df-ur 19517  df-ring 19564  df-subrg 19798  df-lmod 19901  df-lss 19969  df-sra 20209  df-rgmod 20210  df-lidl 20211
This theorem is referenced by: (None)
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