Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  idlinsubrg Structured version   Visualization version   GIF version

Theorem idlinsubrg 31074
 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 4158 . . . 4 (𝐼𝐴) ⊆ 𝐴
2 idlinsubrg.s . . . . 5 𝑆 = (𝑅s 𝐴)
32subrgbas 19555 . . . 4 (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆))
41, 3sseqtrid 3968 . . 3 (𝐴 ∈ (SubRing‘𝑅) → (𝐼𝐴) ⊆ (Base‘𝑆))
54adantr 484 . 2 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) → (𝐼𝐴) ⊆ (Base‘𝑆))
6 subrgrcl 19551 . . . . 5 (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
7 idlinsubrg.u . . . . . 6 𝑈 = (LIdeal‘𝑅)
8 eqid 2798 . . . . . 6 (0g𝑅) = (0g𝑅)
97, 8lidl0cl 19996 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐼𝑈) → (0g𝑅) ∈ 𝐼)
106, 9sylan 583 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) → (0g𝑅) ∈ 𝐼)
11 subrgsubg 19552 . . . . . 6 (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅))
12 subgsubm 18311 . . . . . 6 (𝐴 ∈ (SubGrp‘𝑅) → 𝐴 ∈ (SubMnd‘𝑅))
138subm0cl 17985 . . . . . 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 4123 . . 3 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) → (0g𝑅) ∈ (𝐼𝐴))
1716ne0d 4253 . 2 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) → (𝐼𝐴) ≠ ∅)
18 eqid 2798 . . . . . . . . 9 (+g𝑅) = (+g𝑅)
192, 18ressplusg 16621 . . . . . . . 8 (𝐴 ∈ (SubRing‘𝑅) → (+g𝑅) = (+g𝑆))
20 eqid 2798 . . . . . . . . . 10 (.r𝑅) = (.r𝑅)
212, 20ressmulr 16634 . . . . . . . . 9 (𝐴 ∈ (SubRing‘𝑅) → (.r𝑅) = (.r𝑆))
2221oveqd 7159 . . . . . . . 8 (𝐴 ∈ (SubRing‘𝑅) → (𝑥(.r𝑅)𝑎) = (𝑥(.r𝑆)𝑎))
23 eqidd 2799 . . . . . . . 8 (𝐴 ∈ (SubRing‘𝑅) → 𝑏 = 𝑏)
2419, 22, 23oveq123d 7163 . . . . . . 7 (𝐴 ∈ (SubRing‘𝑅) → ((𝑥(.r𝑅)𝑎)(+g𝑅)𝑏) = ((𝑥(.r𝑆)𝑎)(+g𝑆)𝑏))
2524ad4antr 731 . . . . . 6 (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼𝐴)) ∧ 𝑏 ∈ (𝐼𝐴)) → ((𝑥(.r𝑅)𝑎)(+g𝑅)𝑏) = ((𝑥(.r𝑆)𝑎)(+g𝑆)𝑏))
266ad4antr 731 . . . . . . . 8 (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼𝐴)) ∧ 𝑏 ∈ (𝐼𝐴)) → 𝑅 ∈ Ring)
27 simp-4r 783 . . . . . . . 8 (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼𝐴)) ∧ 𝑏 ∈ (𝐼𝐴)) → 𝐼𝑈)
28 eqid 2798 . . . . . . . . . . . . . 14 (Base‘𝑅) = (Base‘𝑅)
2928subrgss 19547 . . . . . . . . . . . . 13 (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
303, 29eqsstrrd 3955 . . . . . . . . . . . 12 (𝐴 ∈ (SubRing‘𝑅) → (Base‘𝑆) ⊆ (Base‘𝑅))
3130adantr 484 . . . . . . . . . . 11 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) → (Base‘𝑆) ⊆ (Base‘𝑅))
3231sselda 3916 . . . . . . . . . 10 (((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑅))
3332ad2antrr 725 . . . . . . . . 9 (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼𝐴)) ∧ 𝑏 ∈ (𝐼𝐴)) → 𝑥 ∈ (Base‘𝑅))
34 inss1 4157 . . . . . . . . . . . 12 (𝐼𝐴) ⊆ 𝐼
3534a1i 11 . . . . . . . . . . 11 (((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) → (𝐼𝐴) ⊆ 𝐼)
3635sselda 3916 . . . . . . . . . 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 20001 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝐼𝑈) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑎𝐼)) → (𝑥(.r𝑅)𝑎) ∈ 𝐼)
3926, 27, 33, 37, 38syl22anc 837 . . . . . . . 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 3916 . . . . . . . 8 (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼𝐴)) ∧ 𝑏 ∈ (𝐼𝐴)) → 𝑏𝐼)
427, 18lidlacl 19997 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐼𝑈) ∧ ((𝑥(.r𝑅)𝑎) ∈ 𝐼𝑏𝐼)) → ((𝑥(.r𝑅)𝑎)(+g𝑅)𝑏) ∈ 𝐼)
4326, 27, 39, 41, 42syl22anc 837 . . . . . . 7 (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼𝐴)) ∧ 𝑏 ∈ (𝐼𝐴)) → ((𝑥(.r𝑅)𝑎)(+g𝑅)𝑏) ∈ 𝐼)
44 simp-4l 782 . . . . . . . 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 725 . . . . . . . . . . 11 (((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) → 𝐴 = (Base‘𝑆))
4745, 46eleqtrrd 2893 . . . . . . . . . 10 (((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) → 𝑥𝐴)
4847ad2antrr 725 . . . . . . . . 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 3916 . . . . . . . . . 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 19558 . . . . . . . . 9 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝐴𝑎𝐴) → (𝑥(.r𝑅)𝑎) ∈ 𝐴)
5344, 48, 51, 52syl3anc 1368 . . . . . . . 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 3916 . . . . . . . 8 (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼𝐴)) ∧ 𝑏 ∈ (𝐼𝐴)) → 𝑏𝐴)
5618subrgacl 19557 . . . . . . . 8 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑥(.r𝑅)𝑎) ∈ 𝐴𝑏𝐴) → ((𝑥(.r𝑅)𝑎)(+g𝑅)𝑏) ∈ 𝐴)
5744, 53, 55, 56syl3anc 1368 . . . . . . 7 (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼𝐴)) ∧ 𝑏 ∈ (𝐼𝐴)) → ((𝑥(.r𝑅)𝑎)(+g𝑅)𝑏) ∈ 𝐴)
5843, 57elind 4123 . . . . . 6 (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼𝐴)) ∧ 𝑏 ∈ (𝐼𝐴)) → ((𝑥(.r𝑅)𝑎)(+g𝑅)𝑏) ∈ (𝐼𝐴))
5925, 58eqeltrrd 2891 . . . . 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 3156 . . 3 (((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) → ∀𝑎 ∈ (𝐼𝐴)∀𝑏 ∈ (𝐼𝐴)((𝑥(.r𝑆)𝑎)(+g𝑆)𝑏) ∈ (𝐼𝐴))
6261ralrimiva 3149 . 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 2798 . . 3 (Base‘𝑆) = (Base‘𝑆)
65 eqid 2798 . . 3 (+g𝑆) = (+g𝑆)
66 eqid 2798 . . 3 (.r𝑆) = (.r𝑆)
6763, 64, 65, 66islidl 19995 . 2 ((𝐼𝐴) ∈ 𝑉 ↔ ((𝐼𝐴) ⊆ (Base‘𝑆) ∧ (𝐼𝐴) ≠ ∅ ∧ ∀𝑥 ∈ (Base‘𝑆)∀𝑎 ∈ (𝐼𝐴)∀𝑏 ∈ (𝐼𝐴)((𝑥(.r𝑆)𝑎)(+g𝑆)𝑏) ∈ (𝐼𝐴)))
685, 17, 62, 67syl3anbrc 1340 1 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) → (𝐼𝐴) ∈ 𝑉)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106   ∩ cin 3881   ⊆ wss 3882  ∅c0 4245  ‘cfv 6329  (class class class)co 7142  Basecbs 16492   ↾s cress 16493  +gcplusg 16574  .rcmulr 16575  0gc0g 16722  SubMndcsubmnd 17964  SubGrpcsubg 18283  Ringcrg 19308  SubRingcsubrg 19542  LIdealclidl 19953 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7451  ax-cnex 10597  ax-resscn 10598  ax-1cn 10599  ax-icn 10600  ax-addcl 10601  ax-addrcl 10602  ax-mulcl 10603  ax-mulrcl 10604  ax-mulcom 10605  ax-addass 10606  ax-mulass 10607  ax-distr 10608  ax-i2m1 10609  ax-1ne0 10610  ax-1rid 10611  ax-rnegex 10612  ax-rrecex 10613  ax-cnre 10614  ax-pre-lttri 10615  ax-pre-lttrn 10616  ax-pre-ltadd 10617  ax-pre-mulgt0 10618 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3722  df-csb 3830  df-dif 3885  df-un 3887  df-in 3889  df-ss 3899  df-pss 3901  df-nul 4246  df-if 4428  df-pw 4501  df-sn 4528  df-pr 4530  df-tp 4532  df-op 4534  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5441  df-so 5442  df-fr 5481  df-we 5483  df-xp 5528  df-rel 5529  df-cnv 5530  df-co 5531  df-dm 5532  df-rn 5533  df-res 5534  df-ima 5535  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6288  df-fun 6331  df-fn 6332  df-f 6333  df-f1 6334  df-fo 6335  df-f1o 6336  df-fv 6337  df-riota 7100  df-ov 7145  df-oprab 7146  df-mpo 7147  df-om 7571  df-1st 7681  df-2nd 7682  df-wrecs 7945  df-recs 8006  df-rdg 8044  df-er 8287  df-en 8508  df-dom 8509  df-sdom 8510  df-pnf 10681  df-mnf 10682  df-xr 10683  df-ltxr 10684  df-le 10685  df-sub 10876  df-neg 10877  df-nn 11641  df-2 11703  df-3 11704  df-4 11705  df-5 11706  df-6 11707  df-7 11708  df-8 11709  df-ndx 16495  df-slot 16496  df-base 16498  df-sets 16499  df-ress 16500  df-plusg 16587  df-mulr 16588  df-sca 16590  df-vsca 16591  df-ip 16592  df-0g 16724  df-mgm 17861  df-sgrp 17910  df-mnd 17921  df-submnd 17966  df-grp 18115  df-minusg 18116  df-sbg 18117  df-subg 18286  df-mgp 19251  df-ur 19263  df-ring 19310  df-subrg 19544  df-lmod 19647  df-lss 19715  df-sra 19955  df-rgmod 19956  df-lidl 19957 This theorem is referenced by: (None)
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