Theorem issubrg 20502

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 20501	. . 3 ⊢ SubRing = (𝑟 ∈ Ring ↦ {𝑠 ∈ 𝒫 (Base‘𝑟) ∣ ((𝑟 ↾s 𝑠) ∈ Ring ∧ (1r‘𝑟) ∈ 𝑠)})
2	1	mptrcl 6948	. 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
3		simpll 766	. 2 ⊢ (((𝑅 ∈ Ring ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ (𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴)) → 𝑅 ∈ Ring)
4		fveq2 6832	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
5		issubrg.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑅)
6	4, 5	eqtr4di 2787	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
7	6	pweqd 4569	. . . . . 6 ⊢ (𝑟 = 𝑅 → 𝒫 (Base‘𝑟) = 𝒫 𝐵)
8		oveq1 7363	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (𝑟 ↾s 𝑠) = (𝑅 ↾s 𝑠))
9	8	eleq1d 2819	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ((𝑟 ↾s 𝑠) ∈ Ring ↔ (𝑅 ↾s 𝑠) ∈ Ring))
10		fveq2 6832	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = (1r‘𝑅))
11		issubrg.i	. . . . . . . . 9 ⊢ 1 = (1r‘𝑅)
12	10, 11	eqtr4di 2787	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = 1 )
13	12	eleq1d 2819	. . . . . . 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 3415	. . . . 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 6846	. . . . . . 7 ⊢ 𝐵 ∈ V
17	16	pwex 5323	. . . . . 6 ⊢ 𝒫 𝐵 ∈ V
18	17	rabex 5282	. . . . 5 ⊢ {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅 ↾s 𝑠) ∈ Ring ∧ 1 ∈ 𝑠)} ∈ V
19	15, 1, 18	fvmpt 6939	. . . 4 ⊢ (𝑅 ∈ Ring → (SubRing‘𝑅) = {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅 ↾s 𝑠) ∈ Ring ∧ 1 ∈ 𝑠)})
20	19	eleq2d 2820	. . 3 ⊢ (𝑅 ∈ Ring → (𝐴 ∈ (SubRing‘𝑅) ↔ 𝐴 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅 ↾s 𝑠) ∈ Ring ∧ 1 ∈ 𝑠)}))
21		oveq2 7364	. . . . . . . 8 ⊢ (𝑠 = 𝐴 → (𝑅 ↾s 𝑠) = (𝑅 ↾s 𝐴))
22	21	eleq1d 2819	. . . . . . 7 ⊢ (𝑠 = 𝐴 → ((𝑅 ↾s 𝑠) ∈ Ring ↔ (𝑅 ↾s 𝐴) ∈ Ring))
23		eleq2 2823	. . . . . . 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 3644	. . . . 5 ⊢ (𝐴 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅 ↾s 𝑠) ∈ Ring ∧ 1 ∈ 𝑠)} ↔ (𝐴 ∈ 𝒫 𝐵 ∧ ((𝑅 ↾s 𝐴) ∈ Ring ∧ 1 ∈ 𝐴)))
26	16	elpw2 5277	. . . . . 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 1541   ∈ wcel 2113  {crab 3397   ⊆ wss 3899  𝒫 cpw 4552  ‘cfv 6490  (class class class)co 7356  Basecbs 17134   ↾_s cress 17155  1_rcur 20114  Ringcrg 20166  SubRingcsubrg 20500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fv 6498  df-ov 7359  df-subrg 20501

This theorem is referenced by:  subrgss  20503  subrgid  20504  subrgring  20505  subrgrcl  20507  subrgsubrng  20509  subrg1cl  20511  issubrg2  20523  subsubrg  20529  subrgpropd  20539  issubassa  21820  subrgpsr  21931  cphsubrglem  25131  fldgensdrg  33345  fldgenfldext  33774  fldextrspundgdvdslem  33786  fldextrspundgdvds  33787