Theorem subsubrg 20598

Description: A subring of a subring is a subring. (Contributed by Mario Carneiro, 4-Dec-2014.)

Hypothesis

Ref	Expression
subsubrg.s	⊢ 𝑆 = (𝑅 ↾s 𝐴)

Assertion

Ref	Expression
subsubrg	⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝐵 ∈ (SubRing‘𝑆) ↔ (𝐵 ∈ (SubRing‘𝑅) ∧ 𝐵 ⊆ 𝐴)))

Proof of Theorem subsubrg

Step	Hyp	Ref	Expression
1		subrgrcl 20576	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
2	1	adantr 480	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐵 ∈ (SubRing‘𝑆)) → 𝑅 ∈ Ring)
3		eqid 2737	. . . . . . . . 9 ⊢ (Base‘𝑆) = (Base‘𝑆)
4	3	subrgss 20572	. . . . . . . 8 ⊢ (𝐵 ∈ (SubRing‘𝑆) → 𝐵 ⊆ (Base‘𝑆))
5	4	adantl 481	. . . . . . 7 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐵 ∈ (SubRing‘𝑆)) → 𝐵 ⊆ (Base‘𝑆))
6		subsubrg.s	. . . . . . . . 9 ⊢ 𝑆 = (𝑅 ↾s 𝐴)
7	6	subrgbas 20581	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆))
8	7	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐵 ∈ (SubRing‘𝑆)) → 𝐴 = (Base‘𝑆))
9	5, 8	sseqtrrd 4021	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐵 ∈ (SubRing‘𝑆)) → 𝐵 ⊆ 𝐴)
10	6	oveq1i 7441	. . . . . . 7 ⊢ (𝑆 ↾s 𝐵) = ((𝑅 ↾s 𝐴) ↾s 𝐵)
11		ressabs 17294	. . . . . . 7 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐵 ⊆ 𝐴) → ((𝑅 ↾s 𝐴) ↾s 𝐵) = (𝑅 ↾s 𝐵))
12	10, 11	eqtrid 2789	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐵 ⊆ 𝐴) → (𝑆 ↾s 𝐵) = (𝑅 ↾s 𝐵))
13	9, 12	syldan 591	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐵 ∈ (SubRing‘𝑆)) → (𝑆 ↾s 𝐵) = (𝑅 ↾s 𝐵))
14		eqid 2737	. . . . . . 7 ⊢ (𝑆 ↾s 𝐵) = (𝑆 ↾s 𝐵)
15	14	subrgring 20574	. . . . . 6 ⊢ (𝐵 ∈ (SubRing‘𝑆) → (𝑆 ↾s 𝐵) ∈ Ring)
16	15	adantl 481	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐵 ∈ (SubRing‘𝑆)) → (𝑆 ↾s 𝐵) ∈ Ring)
17	13, 16	eqeltrrd 2842	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐵 ∈ (SubRing‘𝑆)) → (𝑅 ↾s 𝐵) ∈ Ring)
18		eqid 2737	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
19	18	subrgss 20572	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
20	19	adantr 480	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐵 ∈ (SubRing‘𝑆)) → 𝐴 ⊆ (Base‘𝑅))
21	9, 20	sstrd 3994	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐵 ∈ (SubRing‘𝑆)) → 𝐵 ⊆ (Base‘𝑅))
22		eqid 2737	. . . . . . . 8 ⊢ (1r‘𝑅) = (1r‘𝑅)
23	6, 22	subrg1 20582	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (1r‘𝑅) = (1r‘𝑆))
24	23	adantr 480	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐵 ∈ (SubRing‘𝑆)) → (1r‘𝑅) = (1r‘𝑆))
25		eqid 2737	. . . . . . . 8 ⊢ (1r‘𝑆) = (1r‘𝑆)
26	25	subrg1cl 20580	. . . . . . 7 ⊢ (𝐵 ∈ (SubRing‘𝑆) → (1r‘𝑆) ∈ 𝐵)
27	26	adantl 481	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐵 ∈ (SubRing‘𝑆)) → (1r‘𝑆) ∈ 𝐵)
28	24, 27	eqeltrd 2841	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐵 ∈ (SubRing‘𝑆)) → (1r‘𝑅) ∈ 𝐵)
29	21, 28	jca 511	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐵 ∈ (SubRing‘𝑆)) → (𝐵 ⊆ (Base‘𝑅) ∧ (1r‘𝑅) ∈ 𝐵))
30	18, 22	issubrg 20571	. . . 4 ⊢ (𝐵 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅 ↾s 𝐵) ∈ Ring) ∧ (𝐵 ⊆ (Base‘𝑅) ∧ (1r‘𝑅) ∈ 𝐵)))
31	2, 17, 29, 30	syl21anbrc 1345	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐵 ∈ (SubRing‘𝑆)) → 𝐵 ∈ (SubRing‘𝑅))
32	31, 9	jca 511	. 2 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐵 ∈ (SubRing‘𝑆)) → (𝐵 ∈ (SubRing‘𝑅) ∧ 𝐵 ⊆ 𝐴))
33	6	subrgring 20574	. . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
34	33	adantr 480	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝐵 ∈ (SubRing‘𝑅) ∧ 𝐵 ⊆ 𝐴)) → 𝑆 ∈ Ring)
35	12	adantrl 716	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝐵 ∈ (SubRing‘𝑅) ∧ 𝐵 ⊆ 𝐴)) → (𝑆 ↾s 𝐵) = (𝑅 ↾s 𝐵))
36		eqid 2737	. . . . . 6 ⊢ (𝑅 ↾s 𝐵) = (𝑅 ↾s 𝐵)
37	36	subrgring 20574	. . . . 5 ⊢ (𝐵 ∈ (SubRing‘𝑅) → (𝑅 ↾s 𝐵) ∈ Ring)
38	37	ad2antrl 728	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝐵 ∈ (SubRing‘𝑅) ∧ 𝐵 ⊆ 𝐴)) → (𝑅 ↾s 𝐵) ∈ Ring)
39	35, 38	eqeltrd 2841	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝐵 ∈ (SubRing‘𝑅) ∧ 𝐵 ⊆ 𝐴)) → (𝑆 ↾s 𝐵) ∈ Ring)
40		simprr 773	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝐵 ∈ (SubRing‘𝑅) ∧ 𝐵 ⊆ 𝐴)) → 𝐵 ⊆ 𝐴)
41	7	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝐵 ∈ (SubRing‘𝑅) ∧ 𝐵 ⊆ 𝐴)) → 𝐴 = (Base‘𝑆))
42	40, 41	sseqtrd 4020	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝐵 ∈ (SubRing‘𝑅) ∧ 𝐵 ⊆ 𝐴)) → 𝐵 ⊆ (Base‘𝑆))
43	23	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝐵 ∈ (SubRing‘𝑅) ∧ 𝐵 ⊆ 𝐴)) → (1r‘𝑅) = (1r‘𝑆))
44	22	subrg1cl 20580	. . . . . 6 ⊢ (𝐵 ∈ (SubRing‘𝑅) → (1r‘𝑅) ∈ 𝐵)
45	44	ad2antrl 728	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝐵 ∈ (SubRing‘𝑅) ∧ 𝐵 ⊆ 𝐴)) → (1r‘𝑅) ∈ 𝐵)
46	43, 45	eqeltrrd 2842	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝐵 ∈ (SubRing‘𝑅) ∧ 𝐵 ⊆ 𝐴)) → (1r‘𝑆) ∈ 𝐵)
47	42, 46	jca 511	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝐵 ∈ (SubRing‘𝑅) ∧ 𝐵 ⊆ 𝐴)) → (𝐵 ⊆ (Base‘𝑆) ∧ (1r‘𝑆) ∈ 𝐵))
48	3, 25	issubrg 20571	. . 3 ⊢ (𝐵 ∈ (SubRing‘𝑆) ↔ ((𝑆 ∈ Ring ∧ (𝑆 ↾s 𝐵) ∈ Ring) ∧ (𝐵 ⊆ (Base‘𝑆) ∧ (1r‘𝑆) ∈ 𝐵)))
49	34, 39, 47, 48	syl21anbrc 1345	. 2 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝐵 ∈ (SubRing‘𝑅) ∧ 𝐵 ⊆ 𝐴)) → 𝐵 ∈ (SubRing‘𝑆))
50	32, 49	impbida 801	1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝐵 ∈ (SubRing‘𝑆) ↔ (𝐵 ∈ (SubRing‘𝑅) ∧ 𝐵 ⊆ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ⊆ wss 3951  ‘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  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  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-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  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-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-subg 19141  df-mgp 20138  df-ur 20179  df-ring 20232  df-subrg 20570

This theorem is referenced by:  subsubrg2  20599  zringunit  21477  rzgrp  21641  subrgmpl  22050  mplbas2  22060  mplind  22094  lsssra  33639  fedgmullem1  33680  fedgmullem2  33681  fedgmul  33682  fldexttr  33709  fldextrspunlem1  33725  fldextrspunfld  33726  algextdeglem2  33759  algextdeglem4  33761