Theorem issubrg 20623

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 20622	. . 3 ⊢ SubRing = (𝑟 ∈ Ring ↦ {𝑠 ∈ 𝒫 (Base‘𝑟) ∣ ((𝑟 ↾s 𝑠) ∈ Ring ∧ (1r‘𝑟) ∈ 𝑠)})
2	1	mptrcl 6987	. 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
3		simpll 776	. 2 ⊢ (((𝑅 ∈ Ring ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ (𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴)) → 𝑅 ∈ Ring)
4		fveq2 6869	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
5		issubrg.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑅)
6	4, 5	eqtr4di 2817	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
7	6	pweqd 4574	. . . . . 6 ⊢ (𝑟 = 𝑅 → 𝒫 (Base‘𝑟) = 𝒫 𝐵)
8		oveq1 7405	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (𝑟 ↾s 𝑠) = (𝑅 ↾s 𝑠))
9	8	eleq1d 2849	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ((𝑟 ↾s 𝑠) ∈ Ring ↔ (𝑅 ↾s 𝑠) ∈ Ring))
10		fveq2 6869	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = (1r‘𝑅))
11		issubrg.i	. . . . . . . . 9 ⊢ 1 = (1r‘𝑅)
12	10, 11	eqtr4di 2817	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = 1 )
13	12	eleq1d 2849	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ((1r‘𝑟) ∈ 𝑠 ↔ 1 ∈ 𝑠))
14	9, 13	anbi12d 641	. . . . . 6 ⊢ (𝑟 = 𝑅 → (((𝑟 ↾s 𝑠) ∈ Ring ∧ (1r‘𝑟) ∈ 𝑠) ↔ ((𝑅 ↾s 𝑠) ∈ Ring ∧ 1 ∈ 𝑠)))
15	7, 14	rabeqbidv 3434	. . . . 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 6883	. . . . . . 7 ⊢ 𝐵 ∈ V
17	16	pwex 5339	. . . . . 6 ⊢ 𝒫 𝐵 ∈ V
18	17	rabex 5297	. . . . 5 ⊢ {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅 ↾s 𝑠) ∈ Ring ∧ 1 ∈ 𝑠)} ∈ V
19	15, 1, 18	fvmpt 6977	. . . 4 ⊢ (𝑅 ∈ Ring → (SubRing‘𝑅) = {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅 ↾s 𝑠) ∈ Ring ∧ 1 ∈ 𝑠)})
20	19	eleq2d 2850	. . 3 ⊢ (𝑅 ∈ Ring → (𝐴 ∈ (SubRing‘𝑅) ↔ 𝐴 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅 ↾s 𝑠) ∈ Ring ∧ 1 ∈ 𝑠)}))
21		oveq2 7406	. . . . . . . 8 ⊢ (𝑠 = 𝐴 → (𝑅 ↾s 𝑠) = (𝑅 ↾s 𝐴))
22	21	eleq1d 2849	. . . . . . 7 ⊢ (𝑠 = 𝐴 → ((𝑅 ↾s 𝑠) ∈ Ring ↔ (𝑅 ↾s 𝐴) ∈ Ring))
23		eleq2 2853	. . . . . . 7 ⊢ (𝑠 = 𝐴 → ( 1 ∈ 𝑠 ↔ 1 ∈ 𝐴))
24	22, 23	anbi12d 641	. . . . . 6 ⊢ (𝑠 = 𝐴 → (((𝑅 ↾s 𝑠) ∈ Ring ∧ 1 ∈ 𝑠) ↔ ((𝑅 ↾s 𝐴) ∈ Ring ∧ 1 ∈ 𝐴)))
25	24	elrab 3652	. . . . 5 ⊢ (𝐴 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅 ↾s 𝑠) ∈ Ring ∧ 1 ∈ 𝑠)} ↔ (𝐴 ∈ 𝒫 𝐵 ∧ ((𝑅 ↾s 𝐴) ∈ Ring ∧ 1 ∈ 𝐴)))
26	16	elpw2 5292	. . . . . 6 ⊢ (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)
27	26	anbi1i 633	. . . . 5 ⊢ ((𝐴 ∈ 𝒫 𝐵 ∧ ((𝑅 ↾s 𝐴) ∈ Ring ∧ 1 ∈ 𝐴)) ↔ (𝐴 ⊆ 𝐵 ∧ ((𝑅 ↾s 𝐴) ∈ Ring ∧ 1 ∈ 𝐴)))
28		an12 655	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ ((𝑅 ↾s 𝐴) ∈ Ring ∧ 1 ∈ 𝐴)) ↔ ((𝑅 ↾s 𝐴) ∈ Ring ∧ (𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴)))
29	25, 27, 28	3bitri 299	. . . 4 ⊢ (𝐴 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅 ↾s 𝑠) ∈ Ring ∧ 1 ∈ 𝑠)} ↔ ((𝑅 ↾s 𝐴) ∈ Ring ∧ (𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴)))
30		ibar 536	. . . . 5 ⊢ (𝑅 ∈ Ring → ((𝑅 ↾s 𝐴) ∈ Ring ↔ (𝑅 ∈ Ring ∧ (𝑅 ↾s 𝐴) ∈ Ring)))
31	30	anbi1d 640	. . . 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 285	. . 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 281	. 2 ⊢ (𝑅 ∈ Ring → (𝐴 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ (𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴))))
34	2, 3, 33	pm5.21nii 380	1 ⊢ (𝐴 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ (𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴)))

Syntax hints:   ↔ wb 208   ∧ wa 399   = wceq 1562   ∈ wcel 2144  {crab 3416   ⊆ wss 3906  𝒫 cpw 4557  ‘cfv 6523  (class class class)co 7398  Basecbs 17247   ↾_s cress 17268  1_rcur 20233  Ringcrg 20285  SubRingcsubrg 20621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fv 6531  df-ov 7401  df-subrg 20622

This theorem is referenced by:  subrgss  20624  subrgid  20625  subrgring  20626  subrgrcl  20628  subrgsubrng  20630  subrg1cl  20632  issubrg2  20644  subsubrg  20650  subrgpropd  20660  issubassa  21921  subrgpsr  22031  cphsubrglem  25241  fldgensdrg  33503  fldgenfldext  33967  fldextrspundgdvdslem  33979  fldextrspundgdvds  33980