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Theorem issubrg 18984
Description: The subring predicate. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Proof shortened by AV, 12-Oct-2020.)
Hypotheses
Ref Expression
issubrg.b 𝐵 = (Base‘𝑅)
issubrg.i 1 = (1r𝑅)
Assertion
Ref Expression
issubrg (𝐴 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅s 𝐴) ∈ Ring) ∧ (𝐴𝐵1𝐴)))

Proof of Theorem issubrg
Dummy variables 𝑠 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-subrg 18982 . . 3 SubRing = (𝑟 ∈ Ring ↦ {𝑠 ∈ 𝒫 (Base‘𝑟) ∣ ((𝑟s 𝑠) ∈ Ring ∧ (1r𝑟) ∈ 𝑠)})
21mptrcl 6510 . 2 (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
3 simpll 774 . 2 (((𝑅 ∈ Ring ∧ (𝑅s 𝐴) ∈ Ring) ∧ (𝐴𝐵1𝐴)) → 𝑅 ∈ Ring)
4 fveq2 6408 . . . . . . . 8 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
5 issubrg.b . . . . . . . 8 𝐵 = (Base‘𝑅)
64, 5syl6eqr 2858 . . . . . . 7 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
76pweqd 4356 . . . . . 6 (𝑟 = 𝑅 → 𝒫 (Base‘𝑟) = 𝒫 𝐵)
8 oveq1 6881 . . . . . . . 8 (𝑟 = 𝑅 → (𝑟s 𝑠) = (𝑅s 𝑠))
98eleq1d 2870 . . . . . . 7 (𝑟 = 𝑅 → ((𝑟s 𝑠) ∈ Ring ↔ (𝑅s 𝑠) ∈ Ring))
10 fveq2 6408 . . . . . . . . 9 (𝑟 = 𝑅 → (1r𝑟) = (1r𝑅))
11 issubrg.i . . . . . . . . 9 1 = (1r𝑅)
1210, 11syl6eqr 2858 . . . . . . . 8 (𝑟 = 𝑅 → (1r𝑟) = 1 )
1312eleq1d 2870 . . . . . . 7 (𝑟 = 𝑅 → ((1r𝑟) ∈ 𝑠1𝑠))
149, 13anbi12d 618 . . . . . 6 (𝑟 = 𝑅 → (((𝑟s 𝑠) ∈ Ring ∧ (1r𝑟) ∈ 𝑠) ↔ ((𝑅s 𝑠) ∈ Ring ∧ 1𝑠)))
157, 14rabeqbidv 3385 . . . . 5 (𝑟 = 𝑅 → {𝑠 ∈ 𝒫 (Base‘𝑟) ∣ ((𝑟s 𝑠) ∈ Ring ∧ (1r𝑟) ∈ 𝑠)} = {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅s 𝑠) ∈ Ring ∧ 1𝑠)})
165fvexi 6422 . . . . . . 7 𝐵 ∈ V
1716pwex 5050 . . . . . 6 𝒫 𝐵 ∈ V
1817rabex 5007 . . . . 5 {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅s 𝑠) ∈ Ring ∧ 1𝑠)} ∈ V
1915, 1, 18fvmpt 6503 . . . 4 (𝑅 ∈ Ring → (SubRing‘𝑅) = {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅s 𝑠) ∈ Ring ∧ 1𝑠)})
2019eleq2d 2871 . . 3 (𝑅 ∈ Ring → (𝐴 ∈ (SubRing‘𝑅) ↔ 𝐴 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅s 𝑠) ∈ Ring ∧ 1𝑠)}))
21 oveq2 6882 . . . . . . . 8 (𝑠 = 𝐴 → (𝑅s 𝑠) = (𝑅s 𝐴))
2221eleq1d 2870 . . . . . . 7 (𝑠 = 𝐴 → ((𝑅s 𝑠) ∈ Ring ↔ (𝑅s 𝐴) ∈ Ring))
23 eleq2 2874 . . . . . . 7 (𝑠 = 𝐴 → ( 1𝑠1𝐴))
2422, 23anbi12d 618 . . . . . 6 (𝑠 = 𝐴 → (((𝑅s 𝑠) ∈ Ring ∧ 1𝑠) ↔ ((𝑅s 𝐴) ∈ Ring ∧ 1𝐴)))
2524elrab 3559 . . . . 5 (𝐴 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅s 𝑠) ∈ Ring ∧ 1𝑠)} ↔ (𝐴 ∈ 𝒫 𝐵 ∧ ((𝑅s 𝐴) ∈ Ring ∧ 1𝐴)))
2616elpw2 5020 . . . . . 6 (𝐴 ∈ 𝒫 𝐵𝐴𝐵)
2726anbi1i 612 . . . . 5 ((𝐴 ∈ 𝒫 𝐵 ∧ ((𝑅s 𝐴) ∈ Ring ∧ 1𝐴)) ↔ (𝐴𝐵 ∧ ((𝑅s 𝐴) ∈ Ring ∧ 1𝐴)))
28 an12 627 . . . . 5 ((𝐴𝐵 ∧ ((𝑅s 𝐴) ∈ Ring ∧ 1𝐴)) ↔ ((𝑅s 𝐴) ∈ Ring ∧ (𝐴𝐵1𝐴)))
2925, 27, 283bitri 288 . . . 4 (𝐴 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅s 𝑠) ∈ Ring ∧ 1𝑠)} ↔ ((𝑅s 𝐴) ∈ Ring ∧ (𝐴𝐵1𝐴)))
30 ibar 520 . . . . 5 (𝑅 ∈ Ring → ((𝑅s 𝐴) ∈ Ring ↔ (𝑅 ∈ Ring ∧ (𝑅s 𝐴) ∈ Ring)))
3130anbi1d 617 . . . 4 (𝑅 ∈ Ring → (((𝑅s 𝐴) ∈ Ring ∧ (𝐴𝐵1𝐴)) ↔ ((𝑅 ∈ Ring ∧ (𝑅s 𝐴) ∈ Ring) ∧ (𝐴𝐵1𝐴))))
3229, 31syl5bb 274 . . 3 (𝑅 ∈ Ring → (𝐴 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅s 𝑠) ∈ Ring ∧ 1𝑠)} ↔ ((𝑅 ∈ Ring ∧ (𝑅s 𝐴) ∈ Ring) ∧ (𝐴𝐵1𝐴))))
3320, 32bitrd 270 . 2 (𝑅 ∈ Ring → (𝐴 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅s 𝐴) ∈ Ring) ∧ (𝐴𝐵1𝐴))))
342, 3, 33pm5.21nii 369 1 (𝐴 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅s 𝐴) ∈ Ring) ∧ (𝐴𝐵1𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384   = wceq 1637  wcel 2156  {crab 3100  wss 3769  𝒫 cpw 4351  cfv 6101  (class class class)co 6874  Basecbs 16068  s cress 16069  1rcur 18703  Ringcrg 18749  SubRingcsubrg 18980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6064  df-fun 6103  df-fv 6109  df-ov 6877  df-subrg 18982
This theorem is referenced by:  subrgss  18985  subrgid  18986  subrgring  18987  subrgrcl  18989  subrg1cl  18992  issubrg2  19004  subsubrg  19010  subrgpropd  19018  issubassa  19533  subrgpsr  19628  cphsubrglem  23189
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