Theorem subsubrng 14351

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 14340	. . . . 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 2232	. . . . . . . . 9 |- ( Base ` S ) = ( Base ` S )
4	3	subrngss 14337	. . . . . . . 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 14343	. . . . . . . 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 3276	. . . . . 6 |- ( ( A e. (SubRng ` R ) /\ B e. (SubRng ` S ) ) -> B C_ A )
10	6	oveq1i 6059	. . . . . . 7 |- ( S ↾s B ) = ( ( R ↾s A ) ↾s B )
11		ressabsg 13281	. . . . . . . . 9 |- ( ( A e. (SubRng ` R ) /\ B C_ A /\ R e. Rng ) -> ( ( R ↾s A ) ↾s B ) = ( R ↾s B ) )
12	11	3expa 1230	. . . . . . . 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 2277	. . . . . 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 2232	. . . . . . 7 |- ( S ↾s B ) = ( S ↾s B )
17	16	subrngrng 14339	. . . . . 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 2310	. . . 4 |- ( ( A e. (SubRng ` R ) /\ B e. (SubRng ` S ) ) -> ( R ↾s B ) e. Rng )
20		eqid 2232	. . . . . . 7 |- ( Base ` R ) = ( Base ` R )
21	20	subrngss 14337	. . . . . 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 3247	. . . 4 |- ( ( A e. (SubRng ` R ) /\ B e. (SubRng ` S ) ) -> B C_ ( Base ` R ) )
24	20	issubrng 14336	. . . 4 |- ( B e. (SubRng ` R ) <-> ( R e. Rng /\ ( R ↾s B ) e. Rng /\ B C_ ( Base ` R ) ) )
25	2, 19, 23, 24	syl3anbrc 1208	. . 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 14339	. . . 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 2232	. . . . . 6 |- ( R ↾s B ) = ( R ↾s B )
31	30	subrngrng 14339	. . . . 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 2309	. . 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 3275	. . 3 |- ( ( A e. (SubRng ` R ) /\ ( B e. (SubRng ` R ) /\ B C_ A ) ) -> B C_ ( Base ` S ) )
37	3	issubrng 14336	. . 3 |- ( B e. (SubRng ` S ) <-> ( S e. Rng /\ ( S ↾s B ) e. Rng /\ B C_ ( Base ` S ) ) )
38	28, 33, 36, 37	syl3anbrc 1208	. 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 1398   e. wcel 2203   C_ wss 3210   ` cfv 5351  (class class class)co 6049   Basecbs 13204   ↾_s cress 13205  Rngcrng 14068  SubRngcsubrng 14334

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8217  ax-resscn 8218  ax-1re 8220  ax-addrcl 8223

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-fv 5359  df-ov 6052  df-oprab 6053  df-mpo 6054  df-inn 9237  df-2 9295  df-3 9296  df-ndx 13207  df-slot 13208  df-base 13210  df-sets 13211  df-iress 13212  df-plusg 13295  df-mulr 13296  df-subg 13879  df-abl 13996  df-rng 14069  df-subrng 14335

This theorem is referenced by:  subsubrng2  14352