Theorem subsubrg 20508

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 20486	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
2	1	adantr 480	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐵 ∈ (SubRing‘𝑆)) → 𝑅 ∈ Ring)
3		eqid 2731	. . . . . . . . 9 ⊢ (Base‘𝑆) = (Base‘𝑆)
4	3	subrgss 20482	. . . . . . . 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 20491	. . . . . . . 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 3967	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐵 ∈ (SubRing‘𝑆)) → 𝐵 ⊆ 𝐴)
10	6	oveq1i 7351	. . . . . . 7 ⊢ (𝑆 ↾s 𝐵) = ((𝑅 ↾s 𝐴) ↾s 𝐵)
11		ressabs 17154	. . . . . . 7 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐵 ⊆ 𝐴) → ((𝑅 ↾s 𝐴) ↾s 𝐵) = (𝑅 ↾s 𝐵))
12	10, 11	eqtrid 2778	. . . . . 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 2731	. . . . . . 7 ⊢ (𝑆 ↾s 𝐵) = (𝑆 ↾s 𝐵)
15	14	subrgring 20484	. . . . . 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 2832	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐵 ∈ (SubRing‘𝑆)) → (𝑅 ↾s 𝐵) ∈ Ring)
18		eqid 2731	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
19	18	subrgss 20482	. . . . . . 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 3940	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐵 ∈ (SubRing‘𝑆)) → 𝐵 ⊆ (Base‘𝑅))
22		eqid 2731	. . . . . . . 8 ⊢ (1r‘𝑅) = (1r‘𝑅)
23	6, 22	subrg1 20492	. . . . . . 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 2731	. . . . . . . 8 ⊢ (1r‘𝑆) = (1r‘𝑆)
26	25	subrg1cl 20490	. . . . . . 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 2831	. . . . 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 20481	. . . 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 20484	. . . 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 2731	. . . . . 6 ⊢ (𝑅 ↾s 𝐵) = (𝑅 ↾s 𝐵)
37	36	subrgring 20484	. . . . 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 2831	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝐵 ∈ (SubRing‘𝑅) ∧ 𝐵 ⊆ 𝐴)) → (𝑆 ↾s 𝐵) ∈ Ring)
40		simprr 772	. . . . 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 3966	. . . 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 20490	. . . . . 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 2832	. . . 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 20481	. . 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 800	1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝐵 ∈ (SubRing‘𝑆) ↔ (𝐵 ∈ (SubRing‘𝑅) ∧ 𝐵 ⊆ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111   ⊆ wss 3897  ‘cfv 6476  (class class class)co 7341  Basecbs 17115   ↾_s cress 17136  1_rcur 20094  Ringcrg 20146  SubRingcsubrg 20479

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663  ax-cnex 11057  ax-resscn 11058  ax-1cn 11059  ax-icn 11060  ax-addcl 11061  ax-addrcl 11062  ax-mulcl 11063  ax-mulrcl 11064  ax-mulcom 11065  ax-addass 11066  ax-mulass 11067  ax-distr 11068  ax-i2m1 11069  ax-1ne0 11070  ax-1rid 11071  ax-rnegex 11072  ax-rrecex 11073  ax-cnre 11074  ax-pre-lttri 11075  ax-pre-lttrn 11076  ax-pre-ltadd 11077  ax-pre-mulgt0 11078

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5506  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-we 5566  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-pred 6243  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-er 8617  df-en 8865  df-dom 8866  df-sdom 8867  df-pnf 11143  df-mnf 11144  df-xr 11145  df-ltxr 11146  df-le 11147  df-sub 11341  df-neg 11342  df-nn 12121  df-2 12183  df-3 12184  df-sets 17070  df-slot 17088  df-ndx 17100  df-base 17116  df-ress 17137  df-plusg 17169  df-mulr 17170  df-0g 17340  df-mgm 18543  df-sgrp 18622  df-mnd 18638  df-subg 19031  df-mgp 20054  df-ur 20095  df-ring 20148  df-subrg 20480

This theorem is referenced by:  subsubrg2  20509  zringunit  21398  rzgrp  21555  subrgmpl  21962  mplbas2  21972  mplind  22000  subsdrg  33256  ressply1evls1  33520  lsssra  33592  fedgmullem1  33634  fedgmullem2  33635  fedgmul  33636  fldexttr  33663  fldextrspunlem1  33680  fldextrspunfld  33681  algextdeglem2  33723  algextdeglem4  33725