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Theorem subsubrng 20489
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 20477 . . . . 5 (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng)
21adantr 480 . . . 4 ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑆)) → 𝑅 ∈ Rng)
3 eqid 2727 . . . . . . . . 9 (Base‘𝑆) = (Base‘𝑆)
43subrngss 20474 . . . . . . . 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 20480 . . . . . . . 8 (𝐴 ∈ (SubRng‘𝑅) → 𝐴 = (Base‘𝑆))
87adantr 480 . . . . . . 7 ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑆)) → 𝐴 = (Base‘𝑆))
95, 8sseqtrrd 4019 . . . . . 6 ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑆)) → 𝐵𝐴)
106oveq1i 7424 . . . . . . 7 (𝑆s 𝐵) = ((𝑅s 𝐴) ↾s 𝐵)
11 ressabs 17221 . . . . . . 7 ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵𝐴) → ((𝑅s 𝐴) ↾s 𝐵) = (𝑅s 𝐵))
1210, 11eqtrid 2779 . . . . . 6 ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵𝐴) → (𝑆s 𝐵) = (𝑅s 𝐵))
139, 12syldan 590 . . . . 5 ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑆)) → (𝑆s 𝐵) = (𝑅s 𝐵))
14 eqid 2727 . . . . . . 7 (𝑆s 𝐵) = (𝑆s 𝐵)
1514subrngrng 20476 . . . . . 6 (𝐵 ∈ (SubRng‘𝑆) → (𝑆s 𝐵) ∈ Rng)
1615adantl 481 . . . . 5 ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑆)) → (𝑆s 𝐵) ∈ Rng)
1713, 16eqeltrrd 2829 . . . 4 ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑆)) → (𝑅s 𝐵) ∈ Rng)
18 eqid 2727 . . . . . . 7 (Base‘𝑅) = (Base‘𝑅)
1918subrngss 20474 . . . . . 6 (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
2019adantr 480 . . . . 5 ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑆)) → 𝐴 ⊆ (Base‘𝑅))
219, 20sstrd 3988 . . . 4 ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑆)) → 𝐵 ⊆ (Base‘𝑅))
2218issubrng 20473 . . . 4 (𝐵 ∈ (SubRng‘𝑅) ↔ (𝑅 ∈ Rng ∧ (𝑅s 𝐵) ∈ Rng ∧ 𝐵 ⊆ (Base‘𝑅)))
232, 17, 21, 22syl3anbrc 1341 . . 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 20476 . . . 4 (𝐴 ∈ (SubRng‘𝑅) → 𝑆 ∈ Rng)
2625adantr 480 . . 3 ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝐵 ∈ (SubRng‘𝑅) ∧ 𝐵𝐴)) → 𝑆 ∈ Rng)
2712adantrl 715 . . . 4 ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝐵 ∈ (SubRng‘𝑅) ∧ 𝐵𝐴)) → (𝑆s 𝐵) = (𝑅s 𝐵))
28 eqid 2727 . . . . . 6 (𝑅s 𝐵) = (𝑅s 𝐵)
2928subrngrng 20476 . . . . 5 (𝐵 ∈ (SubRng‘𝑅) → (𝑅s 𝐵) ∈ Rng)
3029ad2antrl 727 . . . 4 ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝐵 ∈ (SubRng‘𝑅) ∧ 𝐵𝐴)) → (𝑅s 𝐵) ∈ Rng)
3127, 30eqeltrd 2828 . . 3 ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝐵 ∈ (SubRng‘𝑅) ∧ 𝐵𝐴)) → (𝑆s 𝐵) ∈ Rng)
32 simprr 772 . . . 4 ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝐵 ∈ (SubRng‘𝑅) ∧ 𝐵𝐴)) → 𝐵𝐴)
337adantr 480 . . . 4 ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝐵 ∈ (SubRng‘𝑅) ∧ 𝐵𝐴)) → 𝐴 = (Base‘𝑆))
3432, 33sseqtrd 4018 . . 3 ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝐵 ∈ (SubRng‘𝑅) ∧ 𝐵𝐴)) → 𝐵 ⊆ (Base‘𝑆))
353issubrng 20473 . . 3 (𝐵 ∈ (SubRng‘𝑆) ↔ (𝑆 ∈ Rng ∧ (𝑆s 𝐵) ∈ Rng ∧ 𝐵 ⊆ (Base‘𝑆)))
3626, 31, 34, 35syl3anbrc 1341 . 2 ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝐵 ∈ (SubRng‘𝑅) ∧ 𝐵𝐴)) → 𝐵 ∈ (SubRng‘𝑆))
3724, 36impbida 800 1 (𝐴 ∈ (SubRng‘𝑅) → (𝐵 ∈ (SubRng‘𝑆) ↔ (𝐵 ∈ (SubRng‘𝑅) ∧ 𝐵𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1534  wcel 2099  wss 3944  cfv 6542  (class class class)co 7414  Basecbs 17171  s cress 17200  Rngcrng 20083  SubRngcsubrng 20471
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11186  ax-1cn 11188  ax-addcl 11190
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-nn 12235  df-sets 17124  df-slot 17142  df-ndx 17154  df-base 17172  df-ress 17201  df-subg 19069  df-abl 19729  df-rng 20084  df-subrng 20472
This theorem is referenced by:  subsubrng2  20490
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