Theorem subsubrg 20483

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 20461	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
2	1	adantr 480	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐵 ∈ (SubRing‘𝑆)) → 𝑅 ∈ Ring)
3		eqid 2729	. . . . . . . . 9 ⊢ (Base‘𝑆) = (Base‘𝑆)
4	3	subrgss 20457	. . . . . . . 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 20466	. . . . . . . 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 3981	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐵 ∈ (SubRing‘𝑆)) → 𝐵 ⊆ 𝐴)
10	6	oveq1i 7379	. . . . . . 7 ⊢ (𝑆 ↾s 𝐵) = ((𝑅 ↾s 𝐴) ↾s 𝐵)
11		ressabs 17194	. . . . . . 7 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐵 ⊆ 𝐴) → ((𝑅 ↾s 𝐴) ↾s 𝐵) = (𝑅 ↾s 𝐵))
12	10, 11	eqtrid 2776	. . . . . 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 2729	. . . . . . 7 ⊢ (𝑆 ↾s 𝐵) = (𝑆 ↾s 𝐵)
15	14	subrgring 20459	. . . . . 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 2829	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐵 ∈ (SubRing‘𝑆)) → (𝑅 ↾s 𝐵) ∈ Ring)
18		eqid 2729	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
19	18	subrgss 20457	. . . . . . 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 3954	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐵 ∈ (SubRing‘𝑆)) → 𝐵 ⊆ (Base‘𝑅))
22		eqid 2729	. . . . . . . 8 ⊢ (1r‘𝑅) = (1r‘𝑅)
23	6, 22	subrg1 20467	. . . . . . 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 2729	. . . . . . . 8 ⊢ (1r‘𝑆) = (1r‘𝑆)
26	25	subrg1cl 20465	. . . . . . 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 2828	. . . . 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 20456	. . . 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 20459	. . . 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 2729	. . . . . 6 ⊢ (𝑅 ↾s 𝐵) = (𝑅 ↾s 𝐵)
37	36	subrgring 20459	. . . . 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 2828	. . 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 3980	. . . 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 20465	. . . . . 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 2829	. . . 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 20456	. . 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 1540   ∈ wcel 2109   ⊆ wss 3911  ‘cfv 6499  (class class class)co 7369  Basecbs 17155   ↾_s cress 17176  1_rcur 20066  Ringcrg 20118  SubRingcsubrg 20454

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 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-er 8648  df-en 8896  df-dom 8897  df-sdom 8898  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-nn 12163  df-2 12225  df-3 12226  df-sets 17110  df-slot 17128  df-ndx 17140  df-base 17156  df-ress 17177  df-plusg 17209  df-mulr 17210  df-0g 17380  df-mgm 18543  df-sgrp 18622  df-mnd 18638  df-subg 19031  df-mgp 20026  df-ur 20067  df-ring 20120  df-subrg 20455

This theorem is referenced by:  subsubrg2  20484  zringunit  21352  rzgrp  21508  subrgmpl  21915  mplbas2  21925  mplind  21953  subsdrg  33221  ressply1evls1  33507  lsssra  33557  fedgmullem1  33598  fedgmullem2  33599  fedgmul  33600  fldexttr  33627  fldextrspunlem1  33643  fldextrspunfld  33644  algextdeglem2  33681  algextdeglem4  33683