Theorem subsubrng 14020

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 14009	. . . . 5 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng)
2	1	adantr 276	. . . 4 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑆)) → 𝑅 ∈ Rng)
3		eqid 2206	. . . . . . . . 9 ⊢ (Base‘𝑆) = (Base‘𝑆)
4	3	subrngss 14006	. . . . . . . 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 14012	. . . . . . . 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 3233	. . . . . 6 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑆)) → 𝐵 ⊆ 𝐴)
10	6	oveq1i 5961	. . . . . . 7 ⊢ (𝑆 ↾s 𝐵) = ((𝑅 ↾s 𝐴) ↾s 𝐵)
11		ressabsg 12952	. . . . . . . . 9 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ⊆ 𝐴 ∧ 𝑅 ∈ Rng) → ((𝑅 ↾s 𝐴) ↾s 𝐵) = (𝑅 ↾s 𝐵))
12	11	3expa 1206	. . . . . . . 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 2251	. . . . . 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 2206	. . . . . . 7 ⊢ (𝑆 ↾s 𝐵) = (𝑆 ↾s 𝐵)
17	16	subrngrng 14008	. . . . . 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 2284	. . . 4 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑆)) → (𝑅 ↾s 𝐵) ∈ Rng)
20		eqid 2206	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
21	20	subrngss 14006	. . . . . 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 3204	. . . 4 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑆)) → 𝐵 ⊆ (Base‘𝑅))
24	20	issubrng 14005	. . . 4 ⊢ (𝐵 ∈ (SubRng‘𝑅) ↔ (𝑅 ∈ Rng ∧ (𝑅 ↾s 𝐵) ∈ Rng ∧ 𝐵 ⊆ (Base‘𝑅)))
25	2, 19, 23, 24	syl3anbrc 1184	. . 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 14008	. . . 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 2206	. . . . . 6 ⊢ (𝑅 ↾s 𝐵) = (𝑅 ↾s 𝐵)
31	30	subrngrng 14008	. . . . 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 2283	. . 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 3232	. . 3 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝐵 ∈ (SubRng‘𝑅) ∧ 𝐵 ⊆ 𝐴)) → 𝐵 ⊆ (Base‘𝑆))
37	3	issubrng 14005	. . 3 ⊢ (𝐵 ∈ (SubRng‘𝑆) ↔ (𝑆 ∈ Rng ∧ (𝑆 ↾s 𝐵) ∈ Rng ∧ 𝐵 ⊆ (Base‘𝑆)))
38	28, 33, 36, 37	syl3anbrc 1184	. 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 1373   ∈ wcel 2177   ⊆ wss 3167  ‘cfv 5276  (class class class)co 5951  Basecbs 12876   ↾_s cress 12877  Rngcrng 13738  SubRngcsubrng 14003

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4166  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-setind 4589  ax-cnex 8023  ax-resscn 8024  ax-1re 8026  ax-addrcl 8029

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-nul 3462  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-int 3888  df-br 4048  df-opab 4110  df-mpt 4111  df-id 4344  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-fv 5284  df-ov 5954  df-oprab 5955  df-mpo 5956  df-inn 9044  df-2 9102  df-3 9103  df-ndx 12879  df-slot 12880  df-base 12882  df-sets 12883  df-iress 12884  df-plusg 12966  df-mulr 12967  df-subg 13550  df-abl 13667  df-rng 13739  df-subrng 14004

This theorem is referenced by:  subsubrng2  14021