Theorem subsubrng 13558

Description: A subring of a subring is a subring. (Contributed by AV, 15-Feb-2025.)

Hypothesis

Ref	Expression
subsubrng.s	|- S = ( R ↾s A )

Assertion

Ref	Expression
subsubrng	|- ( A e. (SubRng ` R ) -> ( B e. (SubRng ` S ) <-> ( B e. (SubRng ` R ) /\ B C_ A ) ) )

Proof of Theorem subsubrng

Step	Hyp	Ref	Expression
1		subrngrcl 13547	. . . . 5 |- ( A e. (SubRng ` R ) -> R e. Rng )
2	1	adantr 276	. . . 4 |- ( ( A e. (SubRng ` R ) /\ B e. (SubRng ` S ) ) -> R e. Rng )
3		eqid 2189	. . . . . . . . 9 |- ( Base ` S ) = ( Base ` S )
4	3	subrngss 13544	. . . . . . . 8 |- ( B e. (SubRng ` S ) -> B C_ ( Base ` S ) )
5	4	adantl 277	. . . . . . 7 |- ( ( A e. (SubRng ` R ) /\ B e. (SubRng ` S ) ) -> B C_ ( Base ` S ) )
6		subsubrng.s	. . . . . . . . 9 |- S = ( R ↾s A )
7	6	subrngbas 13550	. . . . . . . 8 |- ( A e. (SubRng ` R ) -> A = ( Base ` S ) )
8	7	adantr 276	. . . . . . 7 |- ( ( A e. (SubRng ` R ) /\ B e. (SubRng ` S ) ) -> A = ( Base ` S ) )
9	5, 8	sseqtrrd 3209	. . . . . 6 |- ( ( A e. (SubRng ` R ) /\ B e. (SubRng ` S ) ) -> B C_ A )
10	6	oveq1i 5905	. . . . . . 7 |- ( S ↾s B ) = ( ( R ↾s A ) ↾s B )
11		ressabsg 12585	. . . . . . . . 9 |- ( ( A e. (SubRng ` R ) /\ B C_ A /\ R e. Rng ) -> ( ( R ↾s A ) ↾s B ) = ( R ↾s B ) )
12	11	3expa 1205	. . . . . . . 8 |- ( ( ( A e. (SubRng ` R ) /\ B C_ A ) /\ R e. Rng ) -> ( ( R ↾s A ) ↾s B ) = ( R ↾s B ) )
13	1, 12	mpidan 423	. . . . . . 7 |- ( ( A e. (SubRng ` R ) /\ B C_ A ) -> ( ( R ↾s A ) ↾s B ) = ( R ↾s B ) )
14	10, 13	eqtrid 2234	. . . . . 6 |- ( ( A e. (SubRng ` R ) /\ B C_ A ) -> ( S ↾s B ) = ( R ↾s B ) )
15	9, 14	syldan 282	. . . . 5 |- ( ( A e. (SubRng ` R ) /\ B e. (SubRng ` S ) ) -> ( S ↾s B ) = ( R ↾s B ) )
16		eqid 2189	. . . . . . 7 |- ( S ↾s B ) = ( S ↾s B )
17	16	subrngrng 13546	. . . . . 6 |- ( B e. (SubRng ` S ) -> ( S ↾s B ) e. Rng )
18	17	adantl 277	. . . . 5 |- ( ( A e. (SubRng ` R ) /\ B e. (SubRng ` S ) ) -> ( S ↾s B ) e. Rng )
19	15, 18	eqeltrrd 2267	. . . 4 |- ( ( A e. (SubRng ` R ) /\ B e. (SubRng ` S ) ) -> ( R ↾s B ) e. Rng )
20		eqid 2189	. . . . . . 7 |- ( Base ` R ) = ( Base ` R )
21	20	subrngss 13544	. . . . . 6 |- ( A e. (SubRng ` R ) -> A C_ ( Base ` R ) )
22	21	adantr 276	. . . . 5 |- ( ( A e. (SubRng ` R ) /\ B e. (SubRng ` S ) ) -> A C_ ( Base ` R ) )
23	9, 22	sstrd 3180	. . . 4 |- ( ( A e. (SubRng ` R ) /\ B e. (SubRng ` S ) ) -> B C_ ( Base ` R ) )
24	20	issubrng 13543	. . . 4 |- ( B e. (SubRng ` R ) <-> ( R e. Rng /\ ( R ↾s B ) e. Rng /\ B C_ ( Base ` R ) ) )
25	2, 19, 23, 24	syl3anbrc 1183	. . 3 |- ( ( A e. (SubRng ` R ) /\ B e. (SubRng ` S ) ) -> B e. (SubRng ` R ) )
26	25, 9	jca 306	. 2 |- ( ( A e. (SubRng ` R ) /\ B e. (SubRng ` S ) ) -> ( B e. (SubRng ` R ) /\ B C_ A ) )
27	6	subrngrng 13546	. . . 4 |- ( A e. (SubRng ` R ) -> S e. Rng )
28	27	adantr 276	. . 3 |- ( ( A e. (SubRng ` R ) /\ ( B e. (SubRng ` R ) /\ B C_ A ) ) -> S e. Rng )
29	14	adantrl 478	. . . 4 |- ( ( A e. (SubRng ` R ) /\ ( B e. (SubRng ` R ) /\ B C_ A ) ) -> ( S ↾s B ) = ( R ↾s B ) )
30		eqid 2189	. . . . . 6 |- ( R ↾s B ) = ( R ↾s B )
31	30	subrngrng 13546	. . . . 5 |- ( B e. (SubRng ` R ) -> ( R ↾s B ) e. Rng )
32	31	ad2antrl 490	. . . 4 |- ( ( A e. (SubRng ` R ) /\ ( B e. (SubRng ` R ) /\ B C_ A ) ) -> ( R ↾s B ) e. Rng )
33	29, 32	eqeltrd 2266	. . 3 |- ( ( A e. (SubRng ` R ) /\ ( B e. (SubRng ` R ) /\ B C_ A ) ) -> ( S ↾s B ) e. Rng )
34		simprr 531	. . . 4 |- ( ( A e. (SubRng ` R ) /\ ( B e. (SubRng ` R ) /\ B C_ A ) ) -> B C_ A )
35	7	adantr 276	. . . 4 |- ( ( A e. (SubRng ` R ) /\ ( B e. (SubRng ` R ) /\ B C_ A ) ) -> A = ( Base ` S ) )
36	34, 35	sseqtrd 3208	. . 3 |- ( ( A e. (SubRng ` R ) /\ ( B e. (SubRng ` R ) /\ B C_ A ) ) -> B C_ ( Base ` S ) )
37	3	issubrng 13543	. . 3 |- ( B e. (SubRng ` S ) <-> ( S e. Rng /\ ( S ↾s B ) e. Rng /\ B C_ ( Base ` S ) ) )
38	28, 33, 36, 37	syl3anbrc 1183	. 2 |- ( ( A e. (SubRng ` R ) /\ ( B e. (SubRng ` R ) /\ B C_ A ) ) -> B e. (SubRng ` S ) )
39	26, 38	impbida 596	1 |- ( A e. (SubRng ` R ) -> ( B e. (SubRng ` S ) <-> ( B e. (SubRng ` R ) /\ B C_ A ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2160   C_ wss 3144   ` cfv 5235  (class class class)co 5895   Basecbs 12511   ↾_s cress 12512  Rngcrng 13283  SubRngcsubrng 13541

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-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7931  ax-resscn 7932  ax-1re 7934  ax-addrcl 7937

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-fv 5243  df-ov 5898  df-oprab 5899  df-mpo 5900  df-inn 8949  df-2 9007  df-3 9008  df-ndx 12514  df-slot 12515  df-base 12517  df-sets 12518  df-iress 12519  df-plusg 12599  df-mulr 12600  df-subg 13106  df-abl 13223  df-rng 13284  df-subrng 13542

This theorem is referenced by:  subsubrng2  13559