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Theorem issubrg 18603
 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 18601 . . 3 SubRing = (𝑟 ∈ Ring ↦ {𝑠 ∈ 𝒫 (Base‘𝑟) ∣ ((𝑟s 𝑠) ∈ Ring ∧ (1r𝑟) ∈ 𝑠)})
21mptrcl 6198 . 2 (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
3 simpll 786 . 2 (((𝑅 ∈ Ring ∧ (𝑅s 𝐴) ∈ Ring) ∧ (𝐴𝐵1𝐴)) → 𝑅 ∈ Ring)
4 fveq2 6103 . . . . . . . 8 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
5 issubrg.b . . . . . . . 8 𝐵 = (Base‘𝑅)
64, 5syl6eqr 2662 . . . . . . 7 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
76pweqd 4113 . . . . . 6 (𝑟 = 𝑅 → 𝒫 (Base‘𝑟) = 𝒫 𝐵)
8 oveq1 6556 . . . . . . . 8 (𝑟 = 𝑅 → (𝑟s 𝑠) = (𝑅s 𝑠))
98eleq1d 2672 . . . . . . 7 (𝑟 = 𝑅 → ((𝑟s 𝑠) ∈ Ring ↔ (𝑅s 𝑠) ∈ Ring))
10 fveq2 6103 . . . . . . . . 9 (𝑟 = 𝑅 → (1r𝑟) = (1r𝑅))
11 issubrg.i . . . . . . . . 9 1 = (1r𝑅)
1210, 11syl6eqr 2662 . . . . . . . 8 (𝑟 = 𝑅 → (1r𝑟) = 1 )
1312eleq1d 2672 . . . . . . 7 (𝑟 = 𝑅 → ((1r𝑟) ∈ 𝑠1𝑠))
149, 13anbi12d 743 . . . . . 6 (𝑟 = 𝑅 → (((𝑟s 𝑠) ∈ Ring ∧ (1r𝑟) ∈ 𝑠) ↔ ((𝑅s 𝑠) ∈ Ring ∧ 1𝑠)))
157, 14rabeqbidv 3168 . . . . 5 (𝑟 = 𝑅 → {𝑠 ∈ 𝒫 (Base‘𝑟) ∣ ((𝑟s 𝑠) ∈ Ring ∧ (1r𝑟) ∈ 𝑠)} = {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅s 𝑠) ∈ Ring ∧ 1𝑠)})
16 fvex 6113 . . . . . . . 8 (Base‘𝑅) ∈ V
175, 16eqeltri 2684 . . . . . . 7 𝐵 ∈ V
1817pwex 4774 . . . . . 6 𝒫 𝐵 ∈ V
1918rabex 4740 . . . . 5 {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅s 𝑠) ∈ Ring ∧ 1𝑠)} ∈ V
2015, 1, 19fvmpt 6191 . . . 4 (𝑅 ∈ Ring → (SubRing‘𝑅) = {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅s 𝑠) ∈ Ring ∧ 1𝑠)})
2120eleq2d 2673 . . 3 (𝑅 ∈ Ring → (𝐴 ∈ (SubRing‘𝑅) ↔ 𝐴 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅s 𝑠) ∈ Ring ∧ 1𝑠)}))
22 oveq2 6557 . . . . . . . 8 (𝑠 = 𝐴 → (𝑅s 𝑠) = (𝑅s 𝐴))
2322eleq1d 2672 . . . . . . 7 (𝑠 = 𝐴 → ((𝑅s 𝑠) ∈ Ring ↔ (𝑅s 𝐴) ∈ Ring))
24 eleq2 2677 . . . . . . 7 (𝑠 = 𝐴 → ( 1𝑠1𝐴))
2523, 24anbi12d 743 . . . . . 6 (𝑠 = 𝐴 → (((𝑅s 𝑠) ∈ Ring ∧ 1𝑠) ↔ ((𝑅s 𝐴) ∈ Ring ∧ 1𝐴)))
2625elrab 3331 . . . . 5 (𝐴 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅s 𝑠) ∈ Ring ∧ 1𝑠)} ↔ (𝐴 ∈ 𝒫 𝐵 ∧ ((𝑅s 𝐴) ∈ Ring ∧ 1𝐴)))
2717elpw2 4755 . . . . . 6 (𝐴 ∈ 𝒫 𝐵𝐴𝐵)
2827anbi1i 727 . . . . 5 ((𝐴 ∈ 𝒫 𝐵 ∧ ((𝑅s 𝐴) ∈ Ring ∧ 1𝐴)) ↔ (𝐴𝐵 ∧ ((𝑅s 𝐴) ∈ Ring ∧ 1𝐴)))
29 an12 834 . . . . 5 ((𝐴𝐵 ∧ ((𝑅s 𝐴) ∈ Ring ∧ 1𝐴)) ↔ ((𝑅s 𝐴) ∈ Ring ∧ (𝐴𝐵1𝐴)))
3026, 28, 293bitri 285 . . . 4 (𝐴 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅s 𝑠) ∈ Ring ∧ 1𝑠)} ↔ ((𝑅s 𝐴) ∈ Ring ∧ (𝐴𝐵1𝐴)))
31 ibar 524 . . . . 5 (𝑅 ∈ Ring → ((𝑅s 𝐴) ∈ Ring ↔ (𝑅 ∈ Ring ∧ (𝑅s 𝐴) ∈ Ring)))
3231anbi1d 737 . . . 4 (𝑅 ∈ Ring → (((𝑅s 𝐴) ∈ Ring ∧ (𝐴𝐵1𝐴)) ↔ ((𝑅 ∈ Ring ∧ (𝑅s 𝐴) ∈ Ring) ∧ (𝐴𝐵1𝐴))))
3330, 32syl5bb 271 . . 3 (𝑅 ∈ Ring → (𝐴 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅s 𝑠) ∈ Ring ∧ 1𝑠)} ↔ ((𝑅 ∈ Ring ∧ (𝑅s 𝐴) ∈ Ring) ∧ (𝐴𝐵1𝐴))))
3421, 33bitrd 267 . 2 (𝑅 ∈ Ring → (𝐴 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅s 𝐴) ∈ Ring) ∧ (𝐴𝐵1𝐴))))
352, 3, 34pm5.21nii 367 1 (𝐴 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅s 𝐴) ∈ Ring) ∧ (𝐴𝐵1𝐴)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173   ⊆ wss 3540  𝒫 cpw 4108  ‘cfv 5804  (class class class)co 6549  Basecbs 15695   ↾s cress 15696  1rcur 18324  Ringcrg 18370  SubRingcsubrg 18599 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-subrg 18601 This theorem is referenced by:  subrgss  18604  subrgid  18605  subrgring  18606  subrgrcl  18608  subrg1cl  18611  issubrg2  18623  subsubrg  18629  subrgpropd  18637  issubassa  19145  subrgpsr  19240  cphsubrglem  22785
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