Theorem issubrg 13348

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 13346	. . 3 ⊢ SubRing = (𝑟 ∈ Ring ↦ {𝑠 ∈ 𝒫 (Base‘𝑟) ∣ ((𝑟 ↾s 𝑠) ∈ Ring ∧ (1r‘𝑟) ∈ 𝑠)})
2	1	mptrcl 5601	. 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
3		simpll 527	. 2 ⊢ (((𝑅 ∈ Ring ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ (𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴)) → 𝑅 ∈ Ring)
4		fveq2 5517	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
5		issubrg.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑅)
6	4, 5	eqtr4di 2228	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
7	6	pweqd 3582	. . . . . 6 ⊢ (𝑟 = 𝑅 → 𝒫 (Base‘𝑟) = 𝒫 𝐵)
8		oveq1 5885	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (𝑟 ↾s 𝑠) = (𝑅 ↾s 𝑠))
9	8	eleq1d 2246	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ((𝑟 ↾s 𝑠) ∈ Ring ↔ (𝑅 ↾s 𝑠) ∈ Ring))
10		fveq2 5517	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = (1r‘𝑅))
11		issubrg.i	. . . . . . . . 9 ⊢ 1 = (1r‘𝑅)
12	10, 11	eqtr4di 2228	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = 1 )
13	12	eleq1d 2246	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ((1r‘𝑟) ∈ 𝑠 ↔ 1 ∈ 𝑠))
14	9, 13	anbi12d 473	. . . . . 6 ⊢ (𝑟 = 𝑅 → (((𝑟 ↾s 𝑠) ∈ Ring ∧ (1r‘𝑟) ∈ 𝑠) ↔ ((𝑅 ↾s 𝑠) ∈ Ring ∧ 1 ∈ 𝑠)))
15	7, 14	rabeqbidv 2734	. . . . 5 ⊢ (𝑟 = 𝑅 → {𝑠 ∈ 𝒫 (Base‘𝑟) ∣ ((𝑟 ↾s 𝑠) ∈ Ring ∧ (1r‘𝑟) ∈ 𝑠)} = {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅 ↾s 𝑠) ∈ Ring ∧ 1 ∈ 𝑠)})
16		id 19	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Ring)
17		basfn 12523	. . . . . . . . 9 ⊢ Base Fn V
18		elex 2750	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ V)
19		funfvex 5534	. . . . . . . . . 10 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
20	19	funfni 5318	. . . . . . . . 9 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
21	17, 18, 20	sylancr 414	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) ∈ V)
22	5, 21	eqeltrid 2264	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝐵 ∈ V)
23	22	pwexd 4183	. . . . . 6 ⊢ (𝑅 ∈ Ring → 𝒫 𝐵 ∈ V)
24		rabexg 4148	. . . . . 6 ⊢ (𝒫 𝐵 ∈ V → {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅 ↾s 𝑠) ∈ Ring ∧ 1 ∈ 𝑠)} ∈ V)
25	23, 24	syl 14	. . . . 5 ⊢ (𝑅 ∈ Ring → {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅 ↾s 𝑠) ∈ Ring ∧ 1 ∈ 𝑠)} ∈ V)
26	1, 15, 16, 25	fvmptd3 5612	. . . 4 ⊢ (𝑅 ∈ Ring → (SubRing‘𝑅) = {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅 ↾s 𝑠) ∈ Ring ∧ 1 ∈ 𝑠)})
27	26	eleq2d 2247	. . 3 ⊢ (𝑅 ∈ Ring → (𝐴 ∈ (SubRing‘𝑅) ↔ 𝐴 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅 ↾s 𝑠) ∈ Ring ∧ 1 ∈ 𝑠)}))
28		oveq2 5886	. . . . . . . 8 ⊢ (𝑠 = 𝐴 → (𝑅 ↾s 𝑠) = (𝑅 ↾s 𝐴))
29	28	eleq1d 2246	. . . . . . 7 ⊢ (𝑠 = 𝐴 → ((𝑅 ↾s 𝑠) ∈ Ring ↔ (𝑅 ↾s 𝐴) ∈ Ring))
30		eleq2 2241	. . . . . . 7 ⊢ (𝑠 = 𝐴 → ( 1 ∈ 𝑠 ↔ 1 ∈ 𝐴))
31	29, 30	anbi12d 473	. . . . . 6 ⊢ (𝑠 = 𝐴 → (((𝑅 ↾s 𝑠) ∈ Ring ∧ 1 ∈ 𝑠) ↔ ((𝑅 ↾s 𝐴) ∈ Ring ∧ 1 ∈ 𝐴)))
32	31	elrab 2895	. . . . 5 ⊢ (𝐴 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅 ↾s 𝑠) ∈ Ring ∧ 1 ∈ 𝑠)} ↔ (𝐴 ∈ 𝒫 𝐵 ∧ ((𝑅 ↾s 𝐴) ∈ Ring ∧ 1 ∈ 𝐴)))
33	32	a1i 9	. . . 4 ⊢ (𝑅 ∈ Ring → (𝐴 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅 ↾s 𝑠) ∈ Ring ∧ 1 ∈ 𝑠)} ↔ (𝐴 ∈ 𝒫 𝐵 ∧ ((𝑅 ↾s 𝐴) ∈ Ring ∧ 1 ∈ 𝐴))))
34		elpw2g 4158	. . . . . 6 ⊢ (𝐵 ∈ V → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))
35	22, 34	syl 14	. . . . 5 ⊢ (𝑅 ∈ Ring → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))
36	35	anbi1d 465	. . . 4 ⊢ (𝑅 ∈ Ring → ((𝐴 ∈ 𝒫 𝐵 ∧ ((𝑅 ↾s 𝐴) ∈ Ring ∧ 1 ∈ 𝐴)) ↔ (𝐴 ⊆ 𝐵 ∧ ((𝑅 ↾s 𝐴) ∈ Ring ∧ 1 ∈ 𝐴))))
37		an12 561	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ ((𝑅 ↾s 𝐴) ∈ Ring ∧ 1 ∈ 𝐴)) ↔ ((𝑅 ↾s 𝐴) ∈ Ring ∧ (𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴)))
38	37	a1i 9	. . . 4 ⊢ (𝑅 ∈ Ring → ((𝐴 ⊆ 𝐵 ∧ ((𝑅 ↾s 𝐴) ∈ Ring ∧ 1 ∈ 𝐴)) ↔ ((𝑅 ↾s 𝐴) ∈ Ring ∧ (𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴))))
39	33, 36, 38	3bitrd 214	. . 3 ⊢ (𝑅 ∈ Ring → (𝐴 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ ((𝑅 ↾s 𝑠) ∈ Ring ∧ 1 ∈ 𝑠)} ↔ ((𝑅 ↾s 𝐴) ∈ Ring ∧ (𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴))))
40		ibar 301	. . . 4 ⊢ (𝑅 ∈ Ring → ((𝑅 ↾s 𝐴) ∈ Ring ↔ (𝑅 ∈ Ring ∧ (𝑅 ↾s 𝐴) ∈ Ring)))
41	40	anbi1d 465	. . 3 ⊢ (𝑅 ∈ Ring → (((𝑅 ↾s 𝐴) ∈ Ring ∧ (𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴)) ↔ ((𝑅 ∈ Ring ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ (𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴))))
42	27, 39, 41	3bitrd 214	. 2 ⊢ (𝑅 ∈ Ring → (𝐴 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ (𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴))))
43	2, 3, 42	pm5.21nii 704	1 ⊢ (𝐴 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ (𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴)))

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2148  {crab 2459  Vcvv 2739   ⊆ wss 3131  𝒫 cpw 3577   Fn wfn 5213  ‘cfv 5218  (class class class)co 5878  Basecbs 12465   ↾_s cress 12466  1_rcur 13148  Ringcrg 13185  SubRingcsubrg 13344

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-cnex 7905  ax-resscn 7906  ax-1re 7908  ax-addrcl 7911

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226  df-ov 5881  df-inn 8923  df-ndx 12468  df-slot 12469  df-base 12471  df-subrg 13346

This theorem is referenced by:  subrgss  13349  subrgid  13350  subrgring  13351  subrgrcl  13353  subrg1cl  13356  issubrg2  13368  subsubrg  13372  subrgpropd  13375