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Theorem issubrg2 18740
Description: Characterize the subrings of a ring by closure properties. (Contributed by Mario Carneiro, 3-Dec-2014.)
Hypotheses
Ref Expression
issubrg2.b 𝐵 = (Base‘𝑅)
issubrg2.o 1 = (1r𝑅)
issubrg2.t · = (.r𝑅)
Assertion
Ref Expression
issubrg2 (𝑅 ∈ Ring → (𝐴 ∈ (SubRing‘𝑅) ↔ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝑅,𝑦   𝑥, · ,𝑦
Allowed substitution hints:   𝐵(𝑥,𝑦)   1 (𝑥,𝑦)

Proof of Theorem issubrg2
Dummy variables 𝑣 𝑢 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 subrgsubg 18726 . . 3 (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅))
2 issubrg2.o . . . 4 1 = (1r𝑅)
32subrg1cl 18728 . . 3 (𝐴 ∈ (SubRing‘𝑅) → 1𝐴)
4 issubrg2.t . . . . . 6 · = (.r𝑅)
54subrgmcl 18732 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝐴𝑦𝐴) → (𝑥 · 𝑦) ∈ 𝐴)
653expb 1263 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑥𝐴𝑦𝐴)) → (𝑥 · 𝑦) ∈ 𝐴)
76ralrimivva 2967 . . 3 (𝐴 ∈ (SubRing‘𝑅) → ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)
81, 3, 73jca 1240 . 2 (𝐴 ∈ (SubRing‘𝑅) → (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴))
9 simpl 473 . . . . 5 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝑅 ∈ Ring)
10 simpr1 1065 . . . . . . 7 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝐴 ∈ (SubGrp‘𝑅))
11 eqid 2621 . . . . . . . 8 (𝑅s 𝐴) = (𝑅s 𝐴)
1211subgbas 17538 . . . . . . 7 (𝐴 ∈ (SubGrp‘𝑅) → 𝐴 = (Base‘(𝑅s 𝐴)))
1310, 12syl 17 . . . . . 6 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝐴 = (Base‘(𝑅s 𝐴)))
14 eqid 2621 . . . . . . . 8 (+g𝑅) = (+g𝑅)
1511, 14ressplusg 15933 . . . . . . 7 (𝐴 ∈ (SubGrp‘𝑅) → (+g𝑅) = (+g‘(𝑅s 𝐴)))
1610, 15syl 17 . . . . . 6 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (+g𝑅) = (+g‘(𝑅s 𝐴)))
1711, 4ressmulr 15946 . . . . . . 7 (𝐴 ∈ (SubGrp‘𝑅) → · = (.r‘(𝑅s 𝐴)))
1810, 17syl 17 . . . . . 6 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → · = (.r‘(𝑅s 𝐴)))
1911subggrp 17537 . . . . . . 7 (𝐴 ∈ (SubGrp‘𝑅) → (𝑅s 𝐴) ∈ Grp)
2010, 19syl 17 . . . . . 6 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑅s 𝐴) ∈ Grp)
21 simpr3 1067 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)
22 oveq1 6622 . . . . . . . . . 10 (𝑥 = 𝑢 → (𝑥 · 𝑦) = (𝑢 · 𝑦))
2322eleq1d 2683 . . . . . . . . 9 (𝑥 = 𝑢 → ((𝑥 · 𝑦) ∈ 𝐴 ↔ (𝑢 · 𝑦) ∈ 𝐴))
24 oveq2 6623 . . . . . . . . . 10 (𝑦 = 𝑣 → (𝑢 · 𝑦) = (𝑢 · 𝑣))
2524eleq1d 2683 . . . . . . . . 9 (𝑦 = 𝑣 → ((𝑢 · 𝑦) ∈ 𝐴 ↔ (𝑢 · 𝑣) ∈ 𝐴))
2623, 25rspc2v 3311 . . . . . . . 8 ((𝑢𝐴𝑣𝐴) → (∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴 → (𝑢 · 𝑣) ∈ 𝐴))
2721, 26syl5com 31 . . . . . . 7 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → ((𝑢𝐴𝑣𝐴) → (𝑢 · 𝑣) ∈ 𝐴))
28273impib 1259 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ 𝑢𝐴𝑣𝐴) → (𝑢 · 𝑣) ∈ 𝐴)
29 issubrg2.b . . . . . . . . . . . 12 𝐵 = (Base‘𝑅)
3029subgss 17535 . . . . . . . . . . 11 (𝐴 ∈ (SubGrp‘𝑅) → 𝐴𝐵)
3110, 30syl 17 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝐴𝐵)
3231sseld 3587 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑢𝐴𝑢𝐵))
3331sseld 3587 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑣𝐴𝑣𝐵))
3431sseld 3587 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑤𝐴𝑤𝐵))
3532, 33, 343anim123d 1403 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → ((𝑢𝐴𝑣𝐴𝑤𝐴) → (𝑢𝐵𝑣𝐵𝑤𝐵)))
3635imp 445 . . . . . . 7 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢𝐴𝑣𝐴𝑤𝐴)) → (𝑢𝐵𝑣𝐵𝑤𝐵))
3729, 4ringass 18504 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢 · 𝑣) · 𝑤) = (𝑢 · (𝑣 · 𝑤)))
3837adantlr 750 . . . . . . 7 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢 · 𝑣) · 𝑤) = (𝑢 · (𝑣 · 𝑤)))
3936, 38syldan 487 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢𝐴𝑣𝐴𝑤𝐴)) → ((𝑢 · 𝑣) · 𝑤) = (𝑢 · (𝑣 · 𝑤)))
4029, 14, 4ringdi 18506 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → (𝑢 · (𝑣(+g𝑅)𝑤)) = ((𝑢 · 𝑣)(+g𝑅)(𝑢 · 𝑤)))
4140adantlr 750 . . . . . . 7 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → (𝑢 · (𝑣(+g𝑅)𝑤)) = ((𝑢 · 𝑣)(+g𝑅)(𝑢 · 𝑤)))
4236, 41syldan 487 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢𝐴𝑣𝐴𝑤𝐴)) → (𝑢 · (𝑣(+g𝑅)𝑤)) = ((𝑢 · 𝑣)(+g𝑅)(𝑢 · 𝑤)))
4329, 14, 4ringdir 18507 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢(+g𝑅)𝑣) · 𝑤) = ((𝑢 · 𝑤)(+g𝑅)(𝑣 · 𝑤)))
4443adantlr 750 . . . . . . 7 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢(+g𝑅)𝑣) · 𝑤) = ((𝑢 · 𝑤)(+g𝑅)(𝑣 · 𝑤)))
4536, 44syldan 487 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢𝐴𝑣𝐴𝑤𝐴)) → ((𝑢(+g𝑅)𝑣) · 𝑤) = ((𝑢 · 𝑤)(+g𝑅)(𝑣 · 𝑤)))
46 simpr2 1066 . . . . . 6 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 1𝐴)
4732imp 445 . . . . . . 7 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ 𝑢𝐴) → 𝑢𝐵)
4829, 4, 2ringlidm 18511 . . . . . . . 8 ((𝑅 ∈ Ring ∧ 𝑢𝐵) → ( 1 · 𝑢) = 𝑢)
4948adantlr 750 . . . . . . 7 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ 𝑢𝐵) → ( 1 · 𝑢) = 𝑢)
5047, 49syldan 487 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ 𝑢𝐴) → ( 1 · 𝑢) = 𝑢)
5129, 4, 2ringridm 18512 . . . . . . . 8 ((𝑅 ∈ Ring ∧ 𝑢𝐵) → (𝑢 · 1 ) = 𝑢)
5251adantlr 750 . . . . . . 7 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ 𝑢𝐵) → (𝑢 · 1 ) = 𝑢)
5347, 52syldan 487 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ 𝑢𝐴) → (𝑢 · 1 ) = 𝑢)
5413, 16, 18, 20, 28, 39, 42, 45, 46, 50, 53isringd 18525 . . . . 5 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑅s 𝐴) ∈ Ring)
559, 54jca 554 . . . 4 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑅 ∈ Ring ∧ (𝑅s 𝐴) ∈ Ring))
5631, 46jca 554 . . . 4 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝐴𝐵1𝐴))
5729, 2issubrg 18720 . . . 4 (𝐴 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅s 𝐴) ∈ Ring) ∧ (𝐴𝐵1𝐴)))
5855, 56, 57sylanbrc 697 . . 3 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝐴 ∈ (SubRing‘𝑅))
5958ex 450 . 2 (𝑅 ∈ Ring → ((𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴) → 𝐴 ∈ (SubRing‘𝑅)))
608, 59impbid2 216 1 (𝑅 ∈ Ring → (𝐴 ∈ (SubRing‘𝑅) ↔ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2908  wss 3560  cfv 5857  (class class class)co 6615  Basecbs 15800  s cress 15801  +gcplusg 15881  .rcmulr 15882  Grpcgrp 17362  SubGrpcsubg 17528  1rcur 18441  Ringcrg 18487  SubRingcsubrg 18716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-er 7702  df-en 7916  df-dom 7917  df-sdom 7918  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-2 11039  df-3 11040  df-ndx 15803  df-slot 15804  df-base 15805  df-sets 15806  df-ress 15807  df-plusg 15894  df-mulr 15895  df-0g 16042  df-mgm 17182  df-sgrp 17224  df-mnd 17235  df-subg 17531  df-mgp 18430  df-ur 18442  df-ring 18489  df-subrg 18718
This theorem is referenced by:  opprsubrg  18741  subrgint  18742  issubrg3  18748  issubrngd2  19129  mplsubrg  19380  mplind  19442  cnsubrglem  19736  dmatsrng  20247  scmatsrng  20266  scmatsrng1  20269  cpmatsrgpmat  20466
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