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Theorem subsubrng 20563
Description: A subring of a subring is a subring. (Contributed by AV, 15-Feb-2025.)
Hypothesis
Ref Expression
subsubrng.s 𝑆 = (𝑅s 𝐴)
Assertion
Ref Expression
subsubrng (𝐴 ∈ (SubRng‘𝑅) → (𝐵 ∈ (SubRng‘𝑆) ↔ (𝐵 ∈ (SubRng‘𝑅) ∧ 𝐵𝐴)))

Proof of Theorem subsubrng
StepHypRef Expression
1 subrngrcl 20551 . . . . 5 (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng)
21adantr 480 . . . 4 ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑆)) → 𝑅 ∈ Rng)
3 eqid 2737 . . . . . . . . 9 (Base‘𝑆) = (Base‘𝑆)
43subrngss 20548 . . . . . . . 8 (𝐵 ∈ (SubRng‘𝑆) → 𝐵 ⊆ (Base‘𝑆))
54adantl 481 . . . . . . 7 ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑆)) → 𝐵 ⊆ (Base‘𝑆))
6 subsubrng.s . . . . . . . . 9 𝑆 = (𝑅s 𝐴)
76subrngbas 20554 . . . . . . . 8 (𝐴 ∈ (SubRng‘𝑅) → 𝐴 = (Base‘𝑆))
87adantr 480 . . . . . . 7 ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑆)) → 𝐴 = (Base‘𝑆))
95, 8sseqtrrd 4021 . . . . . 6 ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑆)) → 𝐵𝐴)
106oveq1i 7441 . . . . . . 7 (𝑆s 𝐵) = ((𝑅s 𝐴) ↾s 𝐵)
11 ressabs 17294 . . . . . . 7 ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵𝐴) → ((𝑅s 𝐴) ↾s 𝐵) = (𝑅s 𝐵))
1210, 11eqtrid 2789 . . . . . 6 ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵𝐴) → (𝑆s 𝐵) = (𝑅s 𝐵))
139, 12syldan 591 . . . . 5 ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑆)) → (𝑆s 𝐵) = (𝑅s 𝐵))
14 eqid 2737 . . . . . . 7 (𝑆s 𝐵) = (𝑆s 𝐵)
1514subrngrng 20550 . . . . . 6 (𝐵 ∈ (SubRng‘𝑆) → (𝑆s 𝐵) ∈ Rng)
1615adantl 481 . . . . 5 ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑆)) → (𝑆s 𝐵) ∈ Rng)
1713, 16eqeltrrd 2842 . . . 4 ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑆)) → (𝑅s 𝐵) ∈ Rng)
18 eqid 2737 . . . . . . 7 (Base‘𝑅) = (Base‘𝑅)
1918subrngss 20548 . . . . . 6 (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
2019adantr 480 . . . . 5 ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑆)) → 𝐴 ⊆ (Base‘𝑅))
219, 20sstrd 3994 . . . 4 ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑆)) → 𝐵 ⊆ (Base‘𝑅))
2218issubrng 20547 . . . 4 (𝐵 ∈ (SubRng‘𝑅) ↔ (𝑅 ∈ Rng ∧ (𝑅s 𝐵) ∈ Rng ∧ 𝐵 ⊆ (Base‘𝑅)))
232, 17, 21, 22syl3anbrc 1344 . . 3 ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑆)) → 𝐵 ∈ (SubRng‘𝑅))
2423, 9jca 511 . 2 ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑆)) → (𝐵 ∈ (SubRng‘𝑅) ∧ 𝐵𝐴))
256subrngrng 20550 . . . 4 (𝐴 ∈ (SubRng‘𝑅) → 𝑆 ∈ Rng)
2625adantr 480 . . 3 ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝐵 ∈ (SubRng‘𝑅) ∧ 𝐵𝐴)) → 𝑆 ∈ Rng)
2712adantrl 716 . . . 4 ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝐵 ∈ (SubRng‘𝑅) ∧ 𝐵𝐴)) → (𝑆s 𝐵) = (𝑅s 𝐵))
28 eqid 2737 . . . . . 6 (𝑅s 𝐵) = (𝑅s 𝐵)
2928subrngrng 20550 . . . . 5 (𝐵 ∈ (SubRng‘𝑅) → (𝑅s 𝐵) ∈ Rng)
3029ad2antrl 728 . . . 4 ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝐵 ∈ (SubRng‘𝑅) ∧ 𝐵𝐴)) → (𝑅s 𝐵) ∈ Rng)
3127, 30eqeltrd 2841 . . 3 ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝐵 ∈ (SubRng‘𝑅) ∧ 𝐵𝐴)) → (𝑆s 𝐵) ∈ Rng)
32 simprr 773 . . . 4 ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝐵 ∈ (SubRng‘𝑅) ∧ 𝐵𝐴)) → 𝐵𝐴)
337adantr 480 . . . 4 ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝐵 ∈ (SubRng‘𝑅) ∧ 𝐵𝐴)) → 𝐴 = (Base‘𝑆))
3432, 33sseqtrd 4020 . . 3 ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝐵 ∈ (SubRng‘𝑅) ∧ 𝐵𝐴)) → 𝐵 ⊆ (Base‘𝑆))
353issubrng 20547 . . 3 (𝐵 ∈ (SubRng‘𝑆) ↔ (𝑆 ∈ Rng ∧ (𝑆s 𝐵) ∈ Rng ∧ 𝐵 ⊆ (Base‘𝑆)))
3626, 31, 34, 35syl3anbrc 1344 . 2 ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝐵 ∈ (SubRng‘𝑅) ∧ 𝐵𝐴)) → 𝐵 ∈ (SubRng‘𝑆))
3724, 36impbida 801 1 (𝐴 ∈ (SubRng‘𝑅) → (𝐵 ∈ (SubRng‘𝑆) ↔ (𝐵 ∈ (SubRng‘𝑅) ∧ 𝐵𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wss 3951  cfv 6561  (class class class)co 7431  Basecbs 17247  s cress 17274  Rngcrng 20149  SubRngcsubrng 20545
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 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-1cn 11213  ax-addcl 11215
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-nn 12267  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-subg 19141  df-abl 19801  df-rng 20150  df-subrng 20546
This theorem is referenced by:  subsubrng2  20564
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