Theorem subsubrng 13770

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 13759	. . . . 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 2196	. . . . . . . . 9 |- ( Base ` S ) = ( Base ` S )
4	3	subrngss 13756	. . . . . . . 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 13762	. . . . . . . 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 3222	. . . . . 6 |- ( ( A e. (SubRng ` R ) /\ B e. (SubRng ` S ) ) -> B C_ A )
10	6	oveq1i 5932	. . . . . . 7 |- ( S ↾s B ) = ( ( R ↾s A ) ↾s B )
11		ressabsg 12754	. . . . . . . . 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 2241	. . . . . 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 2196	. . . . . . 7 |- ( S ↾s B ) = ( S ↾s B )
17	16	subrngrng 13758	. . . . . 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 2274	. . . 4 |- ( ( A e. (SubRng ` R ) /\ B e. (SubRng ` S ) ) -> ( R ↾s B ) e. Rng )
20		eqid 2196	. . . . . . 7 |- ( Base ` R ) = ( Base ` R )
21	20	subrngss 13756	. . . . . 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 3193	. . . 4 |- ( ( A e. (SubRng ` R ) /\ B e. (SubRng ` S ) ) -> B C_ ( Base ` R ) )
24	20	issubrng 13755	. . . 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 13758	. . . 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 2196	. . . . . 6 |- ( R ↾s B ) = ( R ↾s B )
31	30	subrngrng 13758	. . . . 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 2273	. . 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 3221	. . 3 |- ( ( A e. (SubRng ` R ) /\ ( B e. (SubRng ` R ) /\ B C_ A ) ) -> B C_ ( Base ` S ) )
37	3	issubrng 13755	. . 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 2167   C_ wss 3157   ` cfv 5258  (class class class)co 5922   Basecbs 12678   ↾_s cress 12679  Rngcrng 13488  SubRngcsubrng 13753

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1re 7973  ax-addrcl 7976

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-inn 8991  df-2 9049  df-3 9050  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-iress 12686  df-plusg 12768  df-mulr 12769  df-subg 13300  df-abl 13417  df-rng 13489  df-subrng 13754

This theorem is referenced by:  subsubrng2  13771