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Theorem issubrg 20588
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 20587 . . 3 SubRing = (𝑟 ∈ Ring ↦ {𝑠 ∈ 𝒫 (Base‘𝑟) ∣ ((𝑟s 𝑠) ∈ Ring ∧ (1r𝑟) ∈ 𝑠)})
21mptrcl 7025 . 2 (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
3 simpll 767 . 2 (((𝑅 ∈ Ring ∧ (𝑅s 𝐴) ∈ Ring) ∧ (𝐴𝐵1𝐴)) → 𝑅 ∈ Ring)
4 fveq2 6907 . . . . . . . 8 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
5 issubrg.b . . . . . . . 8 𝐵 = (Base‘𝑅)
64, 5eqtr4di 2793 . . . . . . 7 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
76pweqd 4622 . . . . . 6 (𝑟 = 𝑅 → 𝒫 (Base‘𝑟) = 𝒫 𝐵)
8 oveq1 7438 . . . . . . . 8 (𝑟 = 𝑅 → (𝑟s 𝑠) = (𝑅s 𝑠))
98eleq1d 2824 . . . . . . 7 (𝑟 = 𝑅 → ((𝑟s 𝑠) ∈ Ring ↔ (𝑅s 𝑠) ∈ Ring))
10 fveq2 6907 . . . . . . . . 9 (𝑟 = 𝑅 → (1r𝑟) = (1r𝑅))
11 issubrg.i . . . . . . . . 9 1 = (1r𝑅)
1210, 11eqtr4di 2793 . . . . . . . 8 (𝑟 = 𝑅 → (1r𝑟) = 1 )
1312eleq1d 2824 . . . . . . 7 (𝑟 = 𝑅 → ((1r𝑟) ∈ 𝑠1𝑠))
149, 13anbi12d 632 . . . . . 6 (𝑟 = 𝑅 → (((𝑟s 𝑠) ∈ Ring ∧ (1r𝑟) ∈ 𝑠) ↔ ((𝑅s 𝑠) ∈ Ring ∧ 1𝑠)))
157, 14rabeqbidv 3452 . . . . 5 (𝑟 = 𝑅 → {𝑠 ∈ 𝒫 (Base‘𝑟) ∣ ((𝑟s 𝑠) ∈ Ring ∧ (1r𝑟) ∈ 𝑠)} = {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅s 𝑠) ∈ Ring ∧ 1𝑠)})
165fvexi 6921 . . . . . . 7 𝐵 ∈ V
1716pwex 5386 . . . . . 6 𝒫 𝐵 ∈ V
1817rabex 5345 . . . . 5 {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅s 𝑠) ∈ Ring ∧ 1𝑠)} ∈ V
1915, 1, 18fvmpt 7016 . . . 4 (𝑅 ∈ Ring → (SubRing‘𝑅) = {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅s 𝑠) ∈ Ring ∧ 1𝑠)})
2019eleq2d 2825 . . 3 (𝑅 ∈ Ring → (𝐴 ∈ (SubRing‘𝑅) ↔ 𝐴 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅s 𝑠) ∈ Ring ∧ 1𝑠)}))
21 oveq2 7439 . . . . . . . 8 (𝑠 = 𝐴 → (𝑅s 𝑠) = (𝑅s 𝐴))
2221eleq1d 2824 . . . . . . 7 (𝑠 = 𝐴 → ((𝑅s 𝑠) ∈ Ring ↔ (𝑅s 𝐴) ∈ Ring))
23 eleq2 2828 . . . . . . 7 (𝑠 = 𝐴 → ( 1𝑠1𝐴))
2422, 23anbi12d 632 . . . . . 6 (𝑠 = 𝐴 → (((𝑅s 𝑠) ∈ Ring ∧ 1𝑠) ↔ ((𝑅s 𝐴) ∈ Ring ∧ 1𝐴)))
2524elrab 3695 . . . . 5 (𝐴 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅s 𝑠) ∈ Ring ∧ 1𝑠)} ↔ (𝐴 ∈ 𝒫 𝐵 ∧ ((𝑅s 𝐴) ∈ Ring ∧ 1𝐴)))
2616elpw2 5340 . . . . . 6 (𝐴 ∈ 𝒫 𝐵𝐴𝐵)
2726anbi1i 624 . . . . 5 ((𝐴 ∈ 𝒫 𝐵 ∧ ((𝑅s 𝐴) ∈ Ring ∧ 1𝐴)) ↔ (𝐴𝐵 ∧ ((𝑅s 𝐴) ∈ Ring ∧ 1𝐴)))
28 an12 645 . . . . 5 ((𝐴𝐵 ∧ ((𝑅s 𝐴) ∈ Ring ∧ 1𝐴)) ↔ ((𝑅s 𝐴) ∈ Ring ∧ (𝐴𝐵1𝐴)))
2925, 27, 283bitri 297 . . . 4 (𝐴 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅s 𝑠) ∈ Ring ∧ 1𝑠)} ↔ ((𝑅s 𝐴) ∈ Ring ∧ (𝐴𝐵1𝐴)))
30 ibar 528 . . . . 5 (𝑅 ∈ Ring → ((𝑅s 𝐴) ∈ Ring ↔ (𝑅 ∈ Ring ∧ (𝑅s 𝐴) ∈ Ring)))
3130anbi1d 631 . . . 4 (𝑅 ∈ Ring → (((𝑅s 𝐴) ∈ Ring ∧ (𝐴𝐵1𝐴)) ↔ ((𝑅 ∈ Ring ∧ (𝑅s 𝐴) ∈ Ring) ∧ (𝐴𝐵1𝐴))))
3229, 31bitrid 283 . . 3 (𝑅 ∈ Ring → (𝐴 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅s 𝑠) ∈ Ring ∧ 1𝑠)} ↔ ((𝑅 ∈ Ring ∧ (𝑅s 𝐴) ∈ Ring) ∧ (𝐴𝐵1𝐴))))
3320, 32bitrd 279 . 2 (𝑅 ∈ Ring → (𝐴 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅s 𝐴) ∈ Ring) ∧ (𝐴𝐵1𝐴))))
342, 3, 33pm5.21nii 378 1 (𝐴 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅s 𝐴) ∈ Ring) ∧ (𝐴𝐵1𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  wcel 2106  {crab 3433  wss 3963  𝒫 cpw 4605  cfv 6563  (class class class)co 7431  Basecbs 17245  s cress 17274  1rcur 20199  Ringcrg 20251  SubRingcsubrg 20586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-subrg 20587
This theorem is referenced by:  subrgss  20589  subrgid  20590  subrgring  20591  subrgrcl  20593  subrgsubrng  20595  subrg1cl  20597  issubrg2  20609  subsubrg  20615  subrgpropd  20625  issubassa  21905  subrgpsr  22016  cphsubrglem  25225  fldgensdrg  33296  fldgenfldext  33693
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