MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  issubrng Structured version   Visualization version   GIF version

Theorem issubrng 20564
Description: The subring of non-unital ring predicate. (Contributed by AV, 14-Feb-2025.)
Hypothesis
Ref Expression
issubrng.b 𝐵 = (Base‘𝑅)
Assertion
Ref Expression
issubrng (𝐴 ∈ (SubRng‘𝑅) ↔ (𝑅 ∈ Rng ∧ (𝑅s 𝐴) ∈ Rng ∧ 𝐴𝐵))

Proof of Theorem issubrng
Dummy variables 𝑤 𝑠 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-subrng 20563 . . 3 SubRng = (𝑤 ∈ Rng ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤s 𝑠) ∈ Rng})
21mptrcl 7025 . 2 (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng)
3 simp1 1135 . 2 ((𝑅 ∈ Rng ∧ (𝑅s 𝐴) ∈ Rng ∧ 𝐴𝐵) → 𝑅 ∈ Rng)
4 fveq2 6907 . . . . . . 7 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
54pweqd 4622 . . . . . 6 (𝑟 = 𝑅 → 𝒫 (Base‘𝑟) = 𝒫 (Base‘𝑅))
6 oveq1 7438 . . . . . . 7 (𝑟 = 𝑅 → (𝑟s 𝑠) = (𝑅s 𝑠))
76eleq1d 2824 . . . . . 6 (𝑟 = 𝑅 → ((𝑟s 𝑠) ∈ Rng ↔ (𝑅s 𝑠) ∈ Rng))
85, 7rabeqbidv 3452 . . . . 5 (𝑟 = 𝑅 → {𝑠 ∈ 𝒫 (Base‘𝑟) ∣ (𝑟s 𝑠) ∈ Rng} = {𝑠 ∈ 𝒫 (Base‘𝑅) ∣ (𝑅s 𝑠) ∈ Rng})
9 df-subrng 20563 . . . . 5 SubRng = (𝑟 ∈ Rng ↦ {𝑠 ∈ 𝒫 (Base‘𝑟) ∣ (𝑟s 𝑠) ∈ Rng})
10 fvex 6920 . . . . . . 7 (Base‘𝑅) ∈ V
1110pwex 5386 . . . . . 6 𝒫 (Base‘𝑅) ∈ V
1211rabex 5345 . . . . 5 {𝑠 ∈ 𝒫 (Base‘𝑅) ∣ (𝑅s 𝑠) ∈ Rng} ∈ V
138, 9, 12fvmpt 7016 . . . 4 (𝑅 ∈ Rng → (SubRng‘𝑅) = {𝑠 ∈ 𝒫 (Base‘𝑅) ∣ (𝑅s 𝑠) ∈ Rng})
1413eleq2d 2825 . . 3 (𝑅 ∈ Rng → (𝐴 ∈ (SubRng‘𝑅) ↔ 𝐴 ∈ {𝑠 ∈ 𝒫 (Base‘𝑅) ∣ (𝑅s 𝑠) ∈ Rng}))
15 oveq2 7439 . . . . . 6 (𝑠 = 𝐴 → (𝑅s 𝑠) = (𝑅s 𝐴))
1615eleq1d 2824 . . . . 5 (𝑠 = 𝐴 → ((𝑅s 𝑠) ∈ Rng ↔ (𝑅s 𝐴) ∈ Rng))
1716elrab 3695 . . . 4 (𝐴 ∈ {𝑠 ∈ 𝒫 (Base‘𝑅) ∣ (𝑅s 𝑠) ∈ Rng} ↔ (𝐴 ∈ 𝒫 (Base‘𝑅) ∧ (𝑅s 𝐴) ∈ Rng))
18 issubrng.b . . . . . . . . 9 𝐵 = (Base‘𝑅)
1918eqcomi 2744 . . . . . . . 8 (Base‘𝑅) = 𝐵
2019sseq2i 4025 . . . . . . 7 (𝐴 ⊆ (Base‘𝑅) ↔ 𝐴𝐵)
2120anbi2i 623 . . . . . 6 (((𝑅s 𝐴) ∈ Rng ∧ 𝐴 ⊆ (Base‘𝑅)) ↔ ((𝑅s 𝐴) ∈ Rng ∧ 𝐴𝐵))
22 ibar 528 . . . . . 6 (𝑅 ∈ Rng → (((𝑅s 𝐴) ∈ Rng ∧ 𝐴𝐵) ↔ (𝑅 ∈ Rng ∧ ((𝑅s 𝐴) ∈ Rng ∧ 𝐴𝐵))))
2321, 22bitrid 283 . . . . 5 (𝑅 ∈ Rng → (((𝑅s 𝐴) ∈ Rng ∧ 𝐴 ⊆ (Base‘𝑅)) ↔ (𝑅 ∈ Rng ∧ ((𝑅s 𝐴) ∈ Rng ∧ 𝐴𝐵))))
2410elpw2 5340 . . . . . 6 (𝐴 ∈ 𝒫 (Base‘𝑅) ↔ 𝐴 ⊆ (Base‘𝑅))
2524anbi2ci 625 . . . . 5 ((𝐴 ∈ 𝒫 (Base‘𝑅) ∧ (𝑅s 𝐴) ∈ Rng) ↔ ((𝑅s 𝐴) ∈ Rng ∧ 𝐴 ⊆ (Base‘𝑅)))
26 3anass 1094 . . . . 5 ((𝑅 ∈ Rng ∧ (𝑅s 𝐴) ∈ Rng ∧ 𝐴𝐵) ↔ (𝑅 ∈ Rng ∧ ((𝑅s 𝐴) ∈ Rng ∧ 𝐴𝐵)))
2723, 25, 263bitr4g 314 . . . 4 (𝑅 ∈ Rng → ((𝐴 ∈ 𝒫 (Base‘𝑅) ∧ (𝑅s 𝐴) ∈ Rng) ↔ (𝑅 ∈ Rng ∧ (𝑅s 𝐴) ∈ Rng ∧ 𝐴𝐵)))
2817, 27bitrid 283 . . 3 (𝑅 ∈ Rng → (𝐴 ∈ {𝑠 ∈ 𝒫 (Base‘𝑅) ∣ (𝑅s 𝑠) ∈ Rng} ↔ (𝑅 ∈ Rng ∧ (𝑅s 𝐴) ∈ Rng ∧ 𝐴𝐵)))
2914, 28bitrd 279 . 2 (𝑅 ∈ Rng → (𝐴 ∈ (SubRng‘𝑅) ↔ (𝑅 ∈ Rng ∧ (𝑅s 𝐴) ∈ Rng ∧ 𝐴𝐵)))
302, 3, 29pm5.21nii 378 1 (𝐴 ∈ (SubRng‘𝑅) ↔ (𝑅 ∈ Rng ∧ (𝑅s 𝐴) ∈ Rng ∧ 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  {crab 3433  wss 3963  𝒫 cpw 4605  cfv 6563  (class class class)co 7431  Basecbs 17245  s cress 17274  Rngcrng 20170  SubRngcsubrng 20562
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-subrng 20563
This theorem is referenced by:  subrngss  20565  subrngid  20566  subrngrng  20567  subrngrcl  20568  issubrng2  20575  subsubrng  20580  subrngpropd  20585  subrgsubrng  20595  rng2idlsubrng  21293
  Copyright terms: Public domain W3C validator