Theorem subsubrng 13713

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 ∈ (SubRng‘𝑅) → (𝐵 ∈ (SubRng‘𝑆) ↔ (𝐵 ∈ (SubRng‘𝑅) ∧ 𝐵 ⊆ 𝐴)))

Proof of Theorem subsubrng

Step	Hyp	Ref	Expression
1		subrngrcl 13702	. . . . 5 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng)
2	1	adantr 276	. . . 4 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑆)) → 𝑅 ∈ Rng)
3		eqid 2193	. . . . . . . . 9 ⊢ (Base‘𝑆) = (Base‘𝑆)
4	3	subrngss 13699	. . . . . . . 8 ⊢ (𝐵 ∈ (SubRng‘𝑆) → 𝐵 ⊆ (Base‘𝑆))
5	4	adantl 277	. . . . . . 7 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑆)) → 𝐵 ⊆ (Base‘𝑆))
6		subsubrng.s	. . . . . . . . 9 ⊢ 𝑆 = (𝑅 ↾s 𝐴)
7	6	subrngbas 13705	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 = (Base‘𝑆))
8	7	adantr 276	. . . . . . 7 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑆)) → 𝐴 = (Base‘𝑆))
9	5, 8	sseqtrrd 3219	. . . . . 6 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑆)) → 𝐵 ⊆ 𝐴)
10	6	oveq1i 5929	. . . . . . 7 ⊢ (𝑆 ↾s 𝐵) = ((𝑅 ↾s 𝐴) ↾s 𝐵)
11		ressabsg 12697	. . . . . . . . 9 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ⊆ 𝐴 ∧ 𝑅 ∈ Rng) → ((𝑅 ↾s 𝐴) ↾s 𝐵) = (𝑅 ↾s 𝐵))
12	11	3expa 1205	. . . . . . . 8 ⊢ (((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ⊆ 𝐴) ∧ 𝑅 ∈ Rng) → ((𝑅 ↾s 𝐴) ↾s 𝐵) = (𝑅 ↾s 𝐵))
13	1, 12	mpidan 423	. . . . . . 7 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ⊆ 𝐴) → ((𝑅 ↾s 𝐴) ↾s 𝐵) = (𝑅 ↾s 𝐵))
14	10, 13	eqtrid 2238	. . . . . 6 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ⊆ 𝐴) → (𝑆 ↾s 𝐵) = (𝑅 ↾s 𝐵))
15	9, 14	syldan 282	. . . . 5 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑆)) → (𝑆 ↾s 𝐵) = (𝑅 ↾s 𝐵))
16		eqid 2193	. . . . . . 7 ⊢ (𝑆 ↾s 𝐵) = (𝑆 ↾s 𝐵)
17	16	subrngrng 13701	. . . . . 6 ⊢ (𝐵 ∈ (SubRng‘𝑆) → (𝑆 ↾s 𝐵) ∈ Rng)
18	17	adantl 277	. . . . 5 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑆)) → (𝑆 ↾s 𝐵) ∈ Rng)
19	15, 18	eqeltrrd 2271	. . . 4 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑆)) → (𝑅 ↾s 𝐵) ∈ Rng)
20		eqid 2193	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
21	20	subrngss 13699	. . . . . 6 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
22	21	adantr 276	. . . . 5 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑆)) → 𝐴 ⊆ (Base‘𝑅))
23	9, 22	sstrd 3190	. . . 4 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑆)) → 𝐵 ⊆ (Base‘𝑅))
24	20	issubrng 13698	. . . 4 ⊢ (𝐵 ∈ (SubRng‘𝑅) ↔ (𝑅 ∈ Rng ∧ (𝑅 ↾s 𝐵) ∈ Rng ∧ 𝐵 ⊆ (Base‘𝑅)))
25	2, 19, 23, 24	syl3anbrc 1183	. . 3 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑆)) → 𝐵 ∈ (SubRng‘𝑅))
26	25, 9	jca 306	. 2 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑆)) → (𝐵 ∈ (SubRng‘𝑅) ∧ 𝐵 ⊆ 𝐴))
27	6	subrngrng 13701	. . . 4 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝑆 ∈ Rng)
28	27	adantr 276	. . 3 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝐵 ∈ (SubRng‘𝑅) ∧ 𝐵 ⊆ 𝐴)) → 𝑆 ∈ Rng)
29	14	adantrl 478	. . . 4 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝐵 ∈ (SubRng‘𝑅) ∧ 𝐵 ⊆ 𝐴)) → (𝑆 ↾s 𝐵) = (𝑅 ↾s 𝐵))
30		eqid 2193	. . . . . 6 ⊢ (𝑅 ↾s 𝐵) = (𝑅 ↾s 𝐵)
31	30	subrngrng 13701	. . . . 5 ⊢ (𝐵 ∈ (SubRng‘𝑅) → (𝑅 ↾s 𝐵) ∈ Rng)
32	31	ad2antrl 490	. . . 4 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝐵 ∈ (SubRng‘𝑅) ∧ 𝐵 ⊆ 𝐴)) → (𝑅 ↾s 𝐵) ∈ Rng)
33	29, 32	eqeltrd 2270	. . 3 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝐵 ∈ (SubRng‘𝑅) ∧ 𝐵 ⊆ 𝐴)) → (𝑆 ↾s 𝐵) ∈ Rng)
34		simprr 531	. . . 4 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝐵 ∈ (SubRng‘𝑅) ∧ 𝐵 ⊆ 𝐴)) → 𝐵 ⊆ 𝐴)
35	7	adantr 276	. . . 4 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝐵 ∈ (SubRng‘𝑅) ∧ 𝐵 ⊆ 𝐴)) → 𝐴 = (Base‘𝑆))
36	34, 35	sseqtrd 3218	. . 3 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝐵 ∈ (SubRng‘𝑅) ∧ 𝐵 ⊆ 𝐴)) → 𝐵 ⊆ (Base‘𝑆))
37	3	issubrng 13698	. . 3 ⊢ (𝐵 ∈ (SubRng‘𝑆) ↔ (𝑆 ∈ Rng ∧ (𝑆 ↾s 𝐵) ∈ Rng ∧ 𝐵 ⊆ (Base‘𝑆)))
38	28, 33, 36, 37	syl3anbrc 1183	. 2 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝐵 ∈ (SubRng‘𝑅) ∧ 𝐵 ⊆ 𝐴)) → 𝐵 ∈ (SubRng‘𝑆))
39	26, 38	impbida 596	1 ⊢ (𝐴 ∈ (SubRng‘𝑅) → (𝐵 ∈ (SubRng‘𝑆) ↔ (𝐵 ∈ (SubRng‘𝑅) ∧ 𝐵 ⊆ 𝐴)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2164   ⊆ wss 3154  ‘cfv 5255  (class class class)co 5919  Basecbs 12621   ↾_s cress 12622  Rngcrng 13431  SubRngcsubrng 13696

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 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1re 7968  ax-addrcl 7971

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 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-inn 8985  df-2 9043  df-3 9044  df-ndx 12624  df-slot 12625  df-base 12627  df-sets 12628  df-iress 12629  df-plusg 12711  df-mulr 12712  df-subg 13243  df-abl 13360  df-rng 13432  df-subrng 13697

This theorem is referenced by:  subsubrng2  13714