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Theorem issubrng2 20450
Description: Characterize the subrings of a ring by closure properties. (Contributed by AV, 15-Feb-2025.)
Hypotheses
Ref Expression
issubrng2.b 𝐵 = (Base‘𝑅)
issubrng2.t · = (.r𝑅)
Assertion
Ref Expression
issubrng2 (𝑅 ∈ Rng → (𝐴 ∈ (SubRng‘𝑅) ↔ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝑅,𝑦   𝑥, · ,𝑦
Allowed substitution hints:   𝐵(𝑥,𝑦)

Proof of Theorem issubrng2
Dummy variables 𝑣 𝑢 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 subrngsubg 20444 . . 3 (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅))
2 issubrng2.t . . . . . 6 · = (.r𝑅)
32subrngmcl 20449 . . . . 5 ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝑥𝐴𝑦𝐴) → (𝑥 · 𝑦) ∈ 𝐴)
433expb 1119 . . . 4 ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝑥𝐴𝑦𝐴)) → (𝑥 · 𝑦) ∈ 𝐴)
54ralrimivva 3199 . . 3 (𝐴 ∈ (SubRng‘𝑅) → ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)
61, 5jca 511 . 2 (𝐴 ∈ (SubRng‘𝑅) → (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴))
7 simpl 482 . . . 4 ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝑅 ∈ Rng)
8 simprl 768 . . . . . 6 ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝐴 ∈ (SubGrp‘𝑅))
9 eqid 2731 . . . . . . 7 (𝑅s 𝐴) = (𝑅s 𝐴)
109subgbas 19050 . . . . . 6 (𝐴 ∈ (SubGrp‘𝑅) → 𝐴 = (Base‘(𝑅s 𝐴)))
118, 10syl 17 . . . . 5 ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝐴 = (Base‘(𝑅s 𝐴)))
12 eqid 2731 . . . . . . 7 (+g𝑅) = (+g𝑅)
139, 12ressplusg 17242 . . . . . 6 (𝐴 ∈ (SubGrp‘𝑅) → (+g𝑅) = (+g‘(𝑅s 𝐴)))
148, 13syl 17 . . . . 5 ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (+g𝑅) = (+g‘(𝑅s 𝐴)))
159, 2ressmulr 17259 . . . . . 6 (𝐴 ∈ (SubGrp‘𝑅) → · = (.r‘(𝑅s 𝐴)))
168, 15syl 17 . . . . 5 ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → · = (.r‘(𝑅s 𝐴)))
17 rngabl 20053 . . . . . 6 (𝑅 ∈ Rng → 𝑅 ∈ Abel)
189subgabl 19749 . . . . . 6 ((𝑅 ∈ Abel ∧ 𝐴 ∈ (SubGrp‘𝑅)) → (𝑅s 𝐴) ∈ Abel)
1917, 8, 18syl2an2r 682 . . . . 5 ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑅s 𝐴) ∈ Abel)
20 simprr 770 . . . . . . 7 ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)
21 oveq1 7419 . . . . . . . . 9 (𝑥 = 𝑢 → (𝑥 · 𝑦) = (𝑢 · 𝑦))
2221eleq1d 2817 . . . . . . . 8 (𝑥 = 𝑢 → ((𝑥 · 𝑦) ∈ 𝐴 ↔ (𝑢 · 𝑦) ∈ 𝐴))
23 oveq2 7420 . . . . . . . . 9 (𝑦 = 𝑣 → (𝑢 · 𝑦) = (𝑢 · 𝑣))
2423eleq1d 2817 . . . . . . . 8 (𝑦 = 𝑣 → ((𝑢 · 𝑦) ∈ 𝐴 ↔ (𝑢 · 𝑣) ∈ 𝐴))
2522, 24rspc2v 3622 . . . . . . 7 ((𝑢𝐴𝑣𝐴) → (∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴 → (𝑢 · 𝑣) ∈ 𝐴))
2620, 25syl5com 31 . . . . . 6 ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → ((𝑢𝐴𝑣𝐴) → (𝑢 · 𝑣) ∈ 𝐴))
27263impib 1115 . . . . 5 (((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ 𝑢𝐴𝑣𝐴) → (𝑢 · 𝑣) ∈ 𝐴)
28 issubrng2.b . . . . . . . . . . 11 𝐵 = (Base‘𝑅)
2928subgss 19047 . . . . . . . . . 10 (𝐴 ∈ (SubGrp‘𝑅) → 𝐴𝐵)
308, 29syl 17 . . . . . . . . 9 ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝐴𝐵)
3130sseld 3981 . . . . . . . 8 ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑢𝐴𝑢𝐵))
3230sseld 3981 . . . . . . . 8 ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑣𝐴𝑣𝐵))
3330sseld 3981 . . . . . . . 8 ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑤𝐴𝑤𝐵))
3431, 32, 333anim123d 1442 . . . . . . 7 ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → ((𝑢𝐴𝑣𝐴𝑤𝐴) → (𝑢𝐵𝑣𝐵𝑤𝐵)))
3534imp 406 . . . . . 6 (((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢𝐴𝑣𝐴𝑤𝐴)) → (𝑢𝐵𝑣𝐵𝑤𝐵))
3628, 2rngass 20057 . . . . . . 7 ((𝑅 ∈ Rng ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢 · 𝑣) · 𝑤) = (𝑢 · (𝑣 · 𝑤)))
3736adantlr 712 . . . . . 6 (((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢 · 𝑣) · 𝑤) = (𝑢 · (𝑣 · 𝑤)))
3835, 37syldan 590 . . . . 5 (((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢𝐴𝑣𝐴𝑤𝐴)) → ((𝑢 · 𝑣) · 𝑤) = (𝑢 · (𝑣 · 𝑤)))
3928, 12, 2rngdi 20058 . . . . . . 7 ((𝑅 ∈ Rng ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → (𝑢 · (𝑣(+g𝑅)𝑤)) = ((𝑢 · 𝑣)(+g𝑅)(𝑢 · 𝑤)))
4039adantlr 712 . . . . . 6 (((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → (𝑢 · (𝑣(+g𝑅)𝑤)) = ((𝑢 · 𝑣)(+g𝑅)(𝑢 · 𝑤)))
4135, 40syldan 590 . . . . 5 (((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢𝐴𝑣𝐴𝑤𝐴)) → (𝑢 · (𝑣(+g𝑅)𝑤)) = ((𝑢 · 𝑣)(+g𝑅)(𝑢 · 𝑤)))
4228, 12, 2rngdir 20059 . . . . . . 7 ((𝑅 ∈ Rng ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢(+g𝑅)𝑣) · 𝑤) = ((𝑢 · 𝑤)(+g𝑅)(𝑣 · 𝑤)))
4342adantlr 712 . . . . . 6 (((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢(+g𝑅)𝑣) · 𝑤) = ((𝑢 · 𝑤)(+g𝑅)(𝑣 · 𝑤)))
4435, 43syldan 590 . . . . 5 (((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢𝐴𝑣𝐴𝑤𝐴)) → ((𝑢(+g𝑅)𝑣) · 𝑤) = ((𝑢 · 𝑤)(+g𝑅)(𝑣 · 𝑤)))
4511, 14, 16, 19, 27, 38, 41, 44isrngd 20071 . . . 4 ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑅s 𝐴) ∈ Rng)
4628issubrng 20439 . . . 4 (𝐴 ∈ (SubRng‘𝑅) ↔ (𝑅 ∈ Rng ∧ (𝑅s 𝐴) ∈ Rng ∧ 𝐴𝐵))
477, 45, 30, 46syl3anbrc 1342 . . 3 ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝐴 ∈ (SubRng‘𝑅))
4847ex 412 . 2 (𝑅 ∈ Rng → ((𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴) → 𝐴 ∈ (SubRng‘𝑅)))
496, 48impbid2 225 1 (𝑅 ∈ Rng → (𝐴 ∈ (SubRng‘𝑅) ↔ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1086   = wceq 1540  wcel 2105  wral 3060  wss 3948  cfv 6543  (class class class)co 7412  Basecbs 17151  s cress 17180  +gcplusg 17204  .rcmulr 17205  SubGrpcsubg 19040  Abelcabl 19694  Rngcrng 20050  SubRngcsubrng 20437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-er 8709  df-en 8946  df-dom 8947  df-sdom 8948  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-nn 12220  df-2 12282  df-3 12283  df-sets 17104  df-slot 17122  df-ndx 17134  df-base 17152  df-ress 17181  df-plusg 17217  df-mulr 17218  df-mgm 18568  df-sgrp 18647  df-grp 18861  df-subg 19043  df-cmn 19695  df-abl 19696  df-mgp 20033  df-rng 20051  df-subrng 20438
This theorem is referenced by:  opprsubrng  20451  subrngint  20452  rhmimasubrng  20458  cntzsubrng  20459  pzriprnglem5  21258
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