Theorem subsubrng 20523

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 20511	. . . . 5 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng)
2	1	adantr 480	. . . 4 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑆)) → 𝑅 ∈ Rng)
3		eqid 2735	. . . . . . . . 9 ⊢ (Base‘𝑆) = (Base‘𝑆)
4	3	subrngss 20508	. . . . . . . 8 ⊢ (𝐵 ∈ (SubRng‘𝑆) → 𝐵 ⊆ (Base‘𝑆))
5	4	adantl 481	. . . . . . 7 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑆)) → 𝐵 ⊆ (Base‘𝑆))
6		subsubrng.s	. . . . . . . . 9 ⊢ 𝑆 = (𝑅 ↾s 𝐴)
7	6	subrngbas 20514	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 = (Base‘𝑆))
8	7	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑆)) → 𝐴 = (Base‘𝑆))
9	5, 8	sseqtrrd 3996	. . . . . 6 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑆)) → 𝐵 ⊆ 𝐴)
10	6	oveq1i 7415	. . . . . . 7 ⊢ (𝑆 ↾s 𝐵) = ((𝑅 ↾s 𝐴) ↾s 𝐵)
11		ressabs 17269	. . . . . . 7 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ⊆ 𝐴) → ((𝑅 ↾s 𝐴) ↾s 𝐵) = (𝑅 ↾s 𝐵))
12	10, 11	eqtrid 2782	. . . . . 6 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ⊆ 𝐴) → (𝑆 ↾s 𝐵) = (𝑅 ↾s 𝐵))
13	9, 12	syldan 591	. . . . 5 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑆)) → (𝑆 ↾s 𝐵) = (𝑅 ↾s 𝐵))
14		eqid 2735	. . . . . . 7 ⊢ (𝑆 ↾s 𝐵) = (𝑆 ↾s 𝐵)
15	14	subrngrng 20510	. . . . . 6 ⊢ (𝐵 ∈ (SubRng‘𝑆) → (𝑆 ↾s 𝐵) ∈ Rng)
16	15	adantl 481	. . . . 5 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑆)) → (𝑆 ↾s 𝐵) ∈ Rng)
17	13, 16	eqeltrrd 2835	. . . 4 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑆)) → (𝑅 ↾s 𝐵) ∈ Rng)
18		eqid 2735	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
19	18	subrngss 20508	. . . . . 6 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
20	19	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑆)) → 𝐴 ⊆ (Base‘𝑅))
21	9, 20	sstrd 3969	. . . 4 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑆)) → 𝐵 ⊆ (Base‘𝑅))
22	18	issubrng 20507	. . . 4 ⊢ (𝐵 ∈ (SubRng‘𝑅) ↔ (𝑅 ∈ Rng ∧ (𝑅 ↾s 𝐵) ∈ Rng ∧ 𝐵 ⊆ (Base‘𝑅)))
23	2, 17, 21, 22	syl3anbrc 1344	. . 3 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑆)) → 𝐵 ∈ (SubRng‘𝑅))
24	23, 9	jca 511	. 2 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑆)) → (𝐵 ∈ (SubRng‘𝑅) ∧ 𝐵 ⊆ 𝐴))
25	6	subrngrng 20510	. . . 4 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝑆 ∈ Rng)
26	25	adantr 480	. . 3 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝐵 ∈ (SubRng‘𝑅) ∧ 𝐵 ⊆ 𝐴)) → 𝑆 ∈ Rng)
27	12	adantrl 716	. . . 4 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝐵 ∈ (SubRng‘𝑅) ∧ 𝐵 ⊆ 𝐴)) → (𝑆 ↾s 𝐵) = (𝑅 ↾s 𝐵))
28		eqid 2735	. . . . . 6 ⊢ (𝑅 ↾s 𝐵) = (𝑅 ↾s 𝐵)
29	28	subrngrng 20510	. . . . 5 ⊢ (𝐵 ∈ (SubRng‘𝑅) → (𝑅 ↾s 𝐵) ∈ Rng)
30	29	ad2antrl 728	. . . 4 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝐵 ∈ (SubRng‘𝑅) ∧ 𝐵 ⊆ 𝐴)) → (𝑅 ↾s 𝐵) ∈ Rng)
31	27, 30	eqeltrd 2834	. . 3 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝐵 ∈ (SubRng‘𝑅) ∧ 𝐵 ⊆ 𝐴)) → (𝑆 ↾s 𝐵) ∈ Rng)
32		simprr 772	. . . 4 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝐵 ∈ (SubRng‘𝑅) ∧ 𝐵 ⊆ 𝐴)) → 𝐵 ⊆ 𝐴)
33	7	adantr 480	. . . 4 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝐵 ∈ (SubRng‘𝑅) ∧ 𝐵 ⊆ 𝐴)) → 𝐴 = (Base‘𝑆))
34	32, 33	sseqtrd 3995	. . 3 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝐵 ∈ (SubRng‘𝑅) ∧ 𝐵 ⊆ 𝐴)) → 𝐵 ⊆ (Base‘𝑆))
35	3	issubrng 20507	. . 3 ⊢ (𝐵 ∈ (SubRng‘𝑆) ↔ (𝑆 ∈ Rng ∧ (𝑆 ↾s 𝐵) ∈ Rng ∧ 𝐵 ⊆ (Base‘𝑆)))
36	26, 31, 34, 35	syl3anbrc 1344	. 2 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝐵 ∈ (SubRng‘𝑅) ∧ 𝐵 ⊆ 𝐴)) → 𝐵 ∈ (SubRng‘𝑆))
37	24, 36	impbida 800	1 ⊢ (𝐴 ∈ (SubRng‘𝑅) → (𝐵 ∈ (SubRng‘𝑆) ↔ (𝐵 ∈ (SubRng‘𝑅) ∧ 𝐵 ⊆ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ⊆ wss 3926  ‘cfv 6531  (class class class)co 7405  Basecbs 17228   ↾_s cress 17251  Rngcrng 20112  SubRngcsubrng 20505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-1cn 11187  ax-addcl 11189

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-nn 12241  df-sets 17183  df-slot 17201  df-ndx 17213  df-base 17229  df-ress 17252  df-subg 19106  df-abl 19764  df-rng 20113  df-subrng 20506

This theorem is referenced by:  subsubrng2  20524