Theorem subsubrng 14234

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 14223	. . . . 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 2231	. . . . . . . . 9 |- ( Base ` S ) = ( Base ` S )
4	3	subrngss 14220	. . . . . . . 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 14226	. . . . . . . 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 3266	. . . . . 6 |- ( ( A e. (SubRng ` R ) /\ B e. (SubRng ` S ) ) -> B C_ A )
10	6	oveq1i 6028	. . . . . . 7 |- ( S ↾s B ) = ( ( R ↾s A ) ↾s B )
11		ressabsg 13164	. . . . . . . . 9 |- ( ( A e. (SubRng ` R ) /\ B C_ A /\ R e. Rng ) -> ( ( R ↾s A ) ↾s B ) = ( R ↾s B ) )
12	11	3expa 1229	. . . . . . . 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 2276	. . . . . 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 2231	. . . . . . 7 |- ( S ↾s B ) = ( S ↾s B )
17	16	subrngrng 14222	. . . . . 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 2309	. . . 4 |- ( ( A e. (SubRng ` R ) /\ B e. (SubRng ` S ) ) -> ( R ↾s B ) e. Rng )
20		eqid 2231	. . . . . . 7 |- ( Base ` R ) = ( Base ` R )
21	20	subrngss 14220	. . . . . 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 3237	. . . 4 |- ( ( A e. (SubRng ` R ) /\ B e. (SubRng ` S ) ) -> B C_ ( Base ` R ) )
24	20	issubrng 14219	. . . 4 |- ( B e. (SubRng ` R ) <-> ( R e. Rng /\ ( R ↾s B ) e. Rng /\ B C_ ( Base ` R ) ) )
25	2, 19, 23, 24	syl3anbrc 1207	. . 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 14222	. . . 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 2231	. . . . . 6 |- ( R ↾s B ) = ( R ↾s B )
31	30	subrngrng 14222	. . . . 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 2308	. . 3 |- ( ( A e. (SubRng ` R ) /\ ( B e. (SubRng ` R ) /\ B C_ A ) ) -> ( S ↾s B ) e. Rng )
34		simprr 533	. . . 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 3265	. . 3 |- ( ( A e. (SubRng ` R ) /\ ( B e. (SubRng ` R ) /\ B C_ A ) ) -> B C_ ( Base ` S ) )
37	3	issubrng 14219	. . 3 |- ( B e. (SubRng ` S ) <-> ( S e. Rng /\ ( S ↾s B ) e. Rng /\ B C_ ( Base ` S ) ) )
38	28, 33, 36, 37	syl3anbrc 1207	. 2 |- ( ( A e. (SubRng ` R ) /\ ( B e. (SubRng ` R ) /\ B C_ A ) ) -> B e. (SubRng ` S ) )
39	26, 38	impbida 600	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 1397   e. wcel 2202   C_ wss 3200   ` cfv 5326  (class class class)co 6018   Basecbs 13087   ↾_s cress 13088  Rngcrng 13951  SubRngcsubrng 14217

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1re 8126  ax-addrcl 8129

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334  df-ov 6021  df-oprab 6022  df-mpo 6023  df-inn 9144  df-2 9202  df-3 9203  df-ndx 13090  df-slot 13091  df-base 13093  df-sets 13094  df-iress 13095  df-plusg 13178  df-mulr 13179  df-subg 13762  df-abl 13879  df-rng 13952  df-subrng 14218

This theorem is referenced by:  subsubrng2  14235