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Theorem subsubrng 20580
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 20568 . . . . 5 (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng)
21adantr 480 . . . 4 ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑆)) → 𝑅 ∈ Rng)
3 eqid 2735 . . . . . . . . 9 (Base‘𝑆) = (Base‘𝑆)
43subrngss 20565 . . . . . . . 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 20571 . . . . . . . 8 (𝐴 ∈ (SubRng‘𝑅) → 𝐴 = (Base‘𝑆))
87adantr 480 . . . . . . 7 ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑆)) → 𝐴 = (Base‘𝑆))
95, 8sseqtrrd 4037 . . . . . 6 ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑆)) → 𝐵𝐴)
106oveq1i 7441 . . . . . . 7 (𝑆s 𝐵) = ((𝑅s 𝐴) ↾s 𝐵)
11 ressabs 17295 . . . . . . 7 ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵𝐴) → ((𝑅s 𝐴) ↾s 𝐵) = (𝑅s 𝐵))
1210, 11eqtrid 2787 . . . . . 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 2735 . . . . . . 7 (𝑆s 𝐵) = (𝑆s 𝐵)
1514subrngrng 20567 . . . . . 6 (𝐵 ∈ (SubRng‘𝑆) → (𝑆s 𝐵) ∈ Rng)
1615adantl 481 . . . . 5 ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑆)) → (𝑆s 𝐵) ∈ Rng)
1713, 16eqeltrrd 2840 . . . 4 ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑆)) → (𝑅s 𝐵) ∈ Rng)
18 eqid 2735 . . . . . . 7 (Base‘𝑅) = (Base‘𝑅)
1918subrngss 20565 . . . . . 6 (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
2019adantr 480 . . . . 5 ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑆)) → 𝐴 ⊆ (Base‘𝑅))
219, 20sstrd 4006 . . . 4 ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑆)) → 𝐵 ⊆ (Base‘𝑅))
2218issubrng 20564 . . . 4 (𝐵 ∈ (SubRng‘𝑅) ↔ (𝑅 ∈ Rng ∧ (𝑅s 𝐵) ∈ Rng ∧ 𝐵 ⊆ (Base‘𝑅)))
232, 17, 21, 22syl3anbrc 1342 . . 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 20567 . . . 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 2735 . . . . . 6 (𝑅s 𝐵) = (𝑅s 𝐵)
2928subrngrng 20567 . . . . 5 (𝐵 ∈ (SubRng‘𝑅) → (𝑅s 𝐵) ∈ Rng)
3029ad2antrl 728 . . . 4 ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝐵 ∈ (SubRng‘𝑅) ∧ 𝐵𝐴)) → (𝑅s 𝐵) ∈ Rng)
3127, 30eqeltrd 2839 . . 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 4036 . . 3 ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝐵 ∈ (SubRng‘𝑅) ∧ 𝐵𝐴)) → 𝐵 ⊆ (Base‘𝑆))
353issubrng 20564 . . 3 (𝐵 ∈ (SubRng‘𝑆) ↔ (𝑆 ∈ Rng ∧ (𝑆s 𝐵) ∈ Rng ∧ 𝐵 ⊆ (Base‘𝑆)))
3626, 31, 34, 35syl3anbrc 1342 . 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 1537  wcel 2106  wss 3963  cfv 6563  (class class class)co 7431  Basecbs 17245  s cress 17274  Rngcrng 20170  SubRngcsubrng 20562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-1cn 11211  ax-addcl 11213
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-nn 12265  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-subg 19154  df-abl 19816  df-rng 20171  df-subrng 20563
This theorem is referenced by:  subsubrng2  20581
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