Theorem issubrg 20571

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 = (1r‘𝑅)

Assertion

Ref	Expression
issubrg	⊢ (𝐴 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ (𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴)))

Proof of Theorem issubrg

Dummy variables 𝑠 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-subrg 20570	. . 3 ⊢ SubRing = (𝑟 ∈ Ring ↦ {𝑠 ∈ 𝒫 (Base‘𝑟) ∣ ((𝑟 ↾s 𝑠) ∈ Ring ∧ (1r‘𝑟) ∈ 𝑠)})
2	1	mptrcl 7025	. 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
3		simpll 767	. 2 ⊢ (((𝑅 ∈ Ring ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ (𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴)) → 𝑅 ∈ Ring)
4		fveq2 6906	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
5		issubrg.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑅)
6	4, 5	eqtr4di 2795	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
7	6	pweqd 4617	. . . . . 6 ⊢ (𝑟 = 𝑅 → 𝒫 (Base‘𝑟) = 𝒫 𝐵)
8		oveq1 7438	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (𝑟 ↾s 𝑠) = (𝑅 ↾s 𝑠))
9	8	eleq1d 2826	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ((𝑟 ↾s 𝑠) ∈ Ring ↔ (𝑅 ↾s 𝑠) ∈ Ring))
10		fveq2 6906	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = (1r‘𝑅))
11		issubrg.i	. . . . . . . . 9 ⊢ 1 = (1r‘𝑅)
12	10, 11	eqtr4di 2795	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = 1 )
13	12	eleq1d 2826	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ((1r‘𝑟) ∈ 𝑠 ↔ 1 ∈ 𝑠))
14	9, 13	anbi12d 632	. . . . . 6 ⊢ (𝑟 = 𝑅 → (((𝑟 ↾s 𝑠) ∈ Ring ∧ (1r‘𝑟) ∈ 𝑠) ↔ ((𝑅 ↾s 𝑠) ∈ Ring ∧ 1 ∈ 𝑠)))
15	7, 14	rabeqbidv 3455	. . . . 5 ⊢ (𝑟 = 𝑅 → {𝑠 ∈ 𝒫 (Base‘𝑟) ∣ ((𝑟 ↾s 𝑠) ∈ Ring ∧ (1r‘𝑟) ∈ 𝑠)} = {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅 ↾s 𝑠) ∈ Ring ∧ 1 ∈ 𝑠)})
16	5	fvexi 6920	. . . . . . 7 ⊢ 𝐵 ∈ V
17	16	pwex 5380	. . . . . 6 ⊢ 𝒫 𝐵 ∈ V
18	17	rabex 5339	. . . . 5 ⊢ {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅 ↾s 𝑠) ∈ Ring ∧ 1 ∈ 𝑠)} ∈ V
19	15, 1, 18	fvmpt 7016	. . . 4 ⊢ (𝑅 ∈ Ring → (SubRing‘𝑅) = {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅 ↾s 𝑠) ∈ Ring ∧ 1 ∈ 𝑠)})
20	19	eleq2d 2827	. . 3 ⊢ (𝑅 ∈ Ring → (𝐴 ∈ (SubRing‘𝑅) ↔ 𝐴 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅 ↾s 𝑠) ∈ Ring ∧ 1 ∈ 𝑠)}))
21		oveq2 7439	. . . . . . . 8 ⊢ (𝑠 = 𝐴 → (𝑅 ↾s 𝑠) = (𝑅 ↾s 𝐴))
22	21	eleq1d 2826	. . . . . . 7 ⊢ (𝑠 = 𝐴 → ((𝑅 ↾s 𝑠) ∈ Ring ↔ (𝑅 ↾s 𝐴) ∈ Ring))
23		eleq2 2830	. . . . . . 7 ⊢ (𝑠 = 𝐴 → ( 1 ∈ 𝑠 ↔ 1 ∈ 𝐴))
24	22, 23	anbi12d 632	. . . . . 6 ⊢ (𝑠 = 𝐴 → (((𝑅 ↾s 𝑠) ∈ Ring ∧ 1 ∈ 𝑠) ↔ ((𝑅 ↾s 𝐴) ∈ Ring ∧ 1 ∈ 𝐴)))
25	24	elrab 3692	. . . . 5 ⊢ (𝐴 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅 ↾s 𝑠) ∈ Ring ∧ 1 ∈ 𝑠)} ↔ (𝐴 ∈ 𝒫 𝐵 ∧ ((𝑅 ↾s 𝐴) ∈ Ring ∧ 1 ∈ 𝐴)))
26	16	elpw2 5334	. . . . . 6 ⊢ (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)
27	26	anbi1i 624	. . . . 5 ⊢ ((𝐴 ∈ 𝒫 𝐵 ∧ ((𝑅 ↾s 𝐴) ∈ Ring ∧ 1 ∈ 𝐴)) ↔ (𝐴 ⊆ 𝐵 ∧ ((𝑅 ↾s 𝐴) ∈ Ring ∧ 1 ∈ 𝐴)))
28		an12 645	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ ((𝑅 ↾s 𝐴) ∈ Ring ∧ 1 ∈ 𝐴)) ↔ ((𝑅 ↾s 𝐴) ∈ Ring ∧ (𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴)))
29	25, 27, 28	3bitri 297	. . . 4 ⊢ (𝐴 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅 ↾s 𝑠) ∈ Ring ∧ 1 ∈ 𝑠)} ↔ ((𝑅 ↾s 𝐴) ∈ Ring ∧ (𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴)))
30		ibar 528	. . . . 5 ⊢ (𝑅 ∈ Ring → ((𝑅 ↾s 𝐴) ∈ Ring ↔ (𝑅 ∈ Ring ∧ (𝑅 ↾s 𝐴) ∈ Ring)))
31	30	anbi1d 631	. . . 4 ⊢ (𝑅 ∈ Ring → (((𝑅 ↾s 𝐴) ∈ Ring ∧ (𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴)) ↔ ((𝑅 ∈ Ring ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ (𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴))))
32	29, 31	bitrid 283	. . 3 ⊢ (𝑅 ∈ Ring → (𝐴 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅 ↾s 𝑠) ∈ Ring ∧ 1 ∈ 𝑠)} ↔ ((𝑅 ∈ Ring ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ (𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴))))
33	20, 32	bitrd 279	. 2 ⊢ (𝑅 ∈ Ring → (𝐴 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ (𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴))))
34	2, 3, 33	pm5.21nii 378	1 ⊢ (𝐴 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ (𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴)))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  {crab 3436   ⊆ wss 3951  𝒫 cpw 4600  ‘cfv 6561  (class class class)co 7431  Basecbs 17247   ↾_s cress 17274  1_rcur 20178  Ringcrg 20230  SubRingcsubrg 20569

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-subrg 20570

This theorem is referenced by:  subrgss  20572  subrgid  20573  subrgring  20574  subrgrcl  20576  subrgsubrng  20578  subrg1cl  20580  issubrg2  20592  subsubrg  20598  subrgpropd  20608  issubassa  21887  subrgpsr  21998  cphsubrglem  25211  fldgensdrg  33316  fldgenfldext  33718  fldextrspundgdvdslem  33730  fldextrspundgdvds  33731