Theorem subsubrng 14163

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 14152	. . . . 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 2229	. . . . . . . . 9 |- ( Base ` S ) = ( Base ` S )
4	3	subrngss 14149	. . . . . . . 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 14155	. . . . . . . 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 3263	. . . . . 6 |- ( ( A e. (SubRng ` R ) /\ B e. (SubRng ` S ) ) -> B C_ A )
10	6	oveq1i 6004	. . . . . . 7 |- ( S ↾s B ) = ( ( R ↾s A ) ↾s B )
11		ressabsg 13095	. . . . . . . . 9 |- ( ( A e. (SubRng ` R ) /\ B C_ A /\ R e. Rng ) -> ( ( R ↾s A ) ↾s B ) = ( R ↾s B ) )
12	11	3expa 1227	. . . . . . . 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 2274	. . . . . 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 2229	. . . . . . 7 |- ( S ↾s B ) = ( S ↾s B )
17	16	subrngrng 14151	. . . . . 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 2307	. . . 4 |- ( ( A e. (SubRng ` R ) /\ B e. (SubRng ` S ) ) -> ( R ↾s B ) e. Rng )
20		eqid 2229	. . . . . . 7 |- ( Base ` R ) = ( Base ` R )
21	20	subrngss 14149	. . . . . 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 3234	. . . 4 |- ( ( A e. (SubRng ` R ) /\ B e. (SubRng ` S ) ) -> B C_ ( Base ` R ) )
24	20	issubrng 14148	. . . 4 |- ( B e. (SubRng ` R ) <-> ( R e. Rng /\ ( R ↾s B ) e. Rng /\ B C_ ( Base ` R ) ) )
25	2, 19, 23, 24	syl3anbrc 1205	. . 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 14151	. . . 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 2229	. . . . . 6 |- ( R ↾s B ) = ( R ↾s B )
31	30	subrngrng 14151	. . . . 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 2306	. . 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 3262	. . 3 |- ( ( A e. (SubRng ` R ) /\ ( B e. (SubRng ` R ) /\ B C_ A ) ) -> B C_ ( Base ` S ) )
37	3	issubrng 14148	. . 3 |- ( B e. (SubRng ` S ) <-> ( S e. Rng /\ ( S ↾s B ) e. Rng /\ B C_ ( Base ` S ) ) )
38	28, 33, 36, 37	syl3anbrc 1205	. 2 |- ( ( A e. (SubRng ` R ) /\ ( B e. (SubRng ` R ) /\ B C_ A ) ) -> B e. (SubRng ` S ) )
39	26, 38	impbida 598	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 1395   e. wcel 2200   C_ wss 3197   ` cfv 5314  (class class class)co 5994   Basecbs 13018   ↾_s cress 13019  Rngcrng 13881  SubRngcsubrng 14146

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-cnex 8078  ax-resscn 8079  ax-1re 8081  ax-addrcl 8084

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4381  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-fv 5322  df-ov 5997  df-oprab 5998  df-mpo 5999  df-inn 9099  df-2 9157  df-3 9158  df-ndx 13021  df-slot 13022  df-base 13024  df-sets 13025  df-iress 13026  df-plusg 13109  df-mulr 13110  df-subg 13693  df-abl 13810  df-rng 13882  df-subrng 14147

This theorem is referenced by:  subsubrng2  14164