Theorem issubrg 20487

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 20486	. . 3 ⊢ SubRing = (𝑟 ∈ Ring ↦ {𝑠 ∈ 𝒫 (Base‘𝑟) ∣ ((𝑟 ↾s 𝑠) ∈ Ring ∧ (1r‘𝑟) ∈ 𝑠)})
2	1	mptrcl 6980	. 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
3		simpll 766	. 2 ⊢ (((𝑅 ∈ Ring ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ (𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴)) → 𝑅 ∈ Ring)
4		fveq2 6861	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
5		issubrg.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑅)
6	4, 5	eqtr4di 2783	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
7	6	pweqd 4583	. . . . . 6 ⊢ (𝑟 = 𝑅 → 𝒫 (Base‘𝑟) = 𝒫 𝐵)
8		oveq1 7397	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (𝑟 ↾s 𝑠) = (𝑅 ↾s 𝑠))
9	8	eleq1d 2814	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ((𝑟 ↾s 𝑠) ∈ Ring ↔ (𝑅 ↾s 𝑠) ∈ Ring))
10		fveq2 6861	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = (1r‘𝑅))
11		issubrg.i	. . . . . . . . 9 ⊢ 1 = (1r‘𝑅)
12	10, 11	eqtr4di 2783	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = 1 )
13	12	eleq1d 2814	. . . . . . 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 3427	. . . . 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 6875	. . . . . . 7 ⊢ 𝐵 ∈ V
17	16	pwex 5338	. . . . . 6 ⊢ 𝒫 𝐵 ∈ V
18	17	rabex 5297	. . . . 5 ⊢ {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅 ↾s 𝑠) ∈ Ring ∧ 1 ∈ 𝑠)} ∈ V
19	15, 1, 18	fvmpt 6971	. . . 4 ⊢ (𝑅 ∈ Ring → (SubRing‘𝑅) = {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅 ↾s 𝑠) ∈ Ring ∧ 1 ∈ 𝑠)})
20	19	eleq2d 2815	. . 3 ⊢ (𝑅 ∈ Ring → (𝐴 ∈ (SubRing‘𝑅) ↔ 𝐴 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅 ↾s 𝑠) ∈ Ring ∧ 1 ∈ 𝑠)}))
21		oveq2 7398	. . . . . . . 8 ⊢ (𝑠 = 𝐴 → (𝑅 ↾s 𝑠) = (𝑅 ↾s 𝐴))
22	21	eleq1d 2814	. . . . . . 7 ⊢ (𝑠 = 𝐴 → ((𝑅 ↾s 𝑠) ∈ Ring ↔ (𝑅 ↾s 𝐴) ∈ Ring))
23		eleq2 2818	. . . . . . 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 3662	. . . . 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 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 2109  {crab 3408   ⊆ wss 3917  𝒫 cpw 4566  ‘cfv 6514  (class class class)co 7390  Basecbs 17186   ↾_s cress 17207  1_rcur 20097  Ringcrg 20149  SubRingcsubrg 20485

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fv 6522  df-ov 7393  df-subrg 20486

This theorem is referenced by:  subrgss  20488  subrgid  20489  subrgring  20490  subrgrcl  20492  subrgsubrng  20494  subrg1cl  20496  issubrg2  20508  subsubrg  20514  subrgpropd  20524  issubassa  21783  subrgpsr  21894  cphsubrglem  25084  fldgensdrg  33271  fldgenfldext  33670  fldextrspundgdvdslem  33682  fldextrspundgdvds  33683