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Theorem subsubg 17557
Description: A subgroup of a subgroup is a subgroup. (Contributed by Mario Carneiro, 19-Jan-2015.)
Hypothesis
Ref Expression
subsubg.h 𝐻 = (𝐺s 𝑆)
Assertion
Ref Expression
subsubg (𝑆 ∈ (SubGrp‘𝐺) → (𝐴 ∈ (SubGrp‘𝐻) ↔ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑆)))

Proof of Theorem subsubg
StepHypRef Expression
1 subgrcl 17539 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
21adantr 481 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → 𝐺 ∈ Grp)
3 eqid 2621 . . . . . . . 8 (Base‘𝐻) = (Base‘𝐻)
43subgss 17535 . . . . . . 7 (𝐴 ∈ (SubGrp‘𝐻) → 𝐴 ⊆ (Base‘𝐻))
54adantl 482 . . . . . 6 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → 𝐴 ⊆ (Base‘𝐻))
6 subsubg.h . . . . . . . 8 𝐻 = (𝐺s 𝑆)
76subgbas 17538 . . . . . . 7 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻))
87adantr 481 . . . . . 6 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → 𝑆 = (Base‘𝐻))
95, 8sseqtr4d 3627 . . . . 5 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → 𝐴𝑆)
10 eqid 2621 . . . . . . 7 (Base‘𝐺) = (Base‘𝐺)
1110subgss 17535 . . . . . 6 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
1211adantr 481 . . . . 5 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → 𝑆 ⊆ (Base‘𝐺))
139, 12sstrd 3598 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → 𝐴 ⊆ (Base‘𝐺))
146oveq1i 6625 . . . . . . 7 (𝐻s 𝐴) = ((𝐺s 𝑆) ↾s 𝐴)
15 ressabs 15879 . . . . . . 7 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑆) → ((𝐺s 𝑆) ↾s 𝐴) = (𝐺s 𝐴))
1614, 15syl5eq 2667 . . . . . 6 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑆) → (𝐻s 𝐴) = (𝐺s 𝐴))
179, 16syldan 487 . . . . 5 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → (𝐻s 𝐴) = (𝐺s 𝐴))
18 eqid 2621 . . . . . . 7 (𝐻s 𝐴) = (𝐻s 𝐴)
1918subggrp 17537 . . . . . 6 (𝐴 ∈ (SubGrp‘𝐻) → (𝐻s 𝐴) ∈ Grp)
2019adantl 482 . . . . 5 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → (𝐻s 𝐴) ∈ Grp)
2117, 20eqeltrrd 2699 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → (𝐺s 𝐴) ∈ Grp)
2210issubg 17534 . . . 4 (𝐴 ∈ (SubGrp‘𝐺) ↔ (𝐺 ∈ Grp ∧ 𝐴 ⊆ (Base‘𝐺) ∧ (𝐺s 𝐴) ∈ Grp))
232, 13, 21, 22syl3anbrc 1244 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → 𝐴 ∈ (SubGrp‘𝐺))
2423, 9jca 554 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑆))
256subggrp 17537 . . . 4 (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
2625adantr 481 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑆)) → 𝐻 ∈ Grp)
27 simprr 795 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑆)) → 𝐴𝑆)
287adantr 481 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑆)) → 𝑆 = (Base‘𝐻))
2927, 28sseqtrd 3626 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑆)) → 𝐴 ⊆ (Base‘𝐻))
3016adantrl 751 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑆)) → (𝐻s 𝐴) = (𝐺s 𝐴))
31 eqid 2621 . . . . . 6 (𝐺s 𝐴) = (𝐺s 𝐴)
3231subggrp 17537 . . . . 5 (𝐴 ∈ (SubGrp‘𝐺) → (𝐺s 𝐴) ∈ Grp)
3332ad2antrl 763 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑆)) → (𝐺s 𝐴) ∈ Grp)
3430, 33eqeltrd 2698 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑆)) → (𝐻s 𝐴) ∈ Grp)
353issubg 17534 . . 3 (𝐴 ∈ (SubGrp‘𝐻) ↔ (𝐻 ∈ Grp ∧ 𝐴 ⊆ (Base‘𝐻) ∧ (𝐻s 𝐴) ∈ Grp))
3626, 29, 34, 35syl3anbrc 1244 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑆)) → 𝐴 ∈ (SubGrp‘𝐻))
3724, 36impbida 876 1 (𝑆 ∈ (SubGrp‘𝐺) → (𝐴 ∈ (SubGrp‘𝐻) ↔ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wss 3560  cfv 5857  (class class class)co 6615  Basecbs 15800  s cress 15801  Grpcgrp 17362  SubGrpcsubg 17528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-i2m1 9964  ax-1ne0 9965  ax-rrecex 9968  ax-cnre 9969
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-nn 10981  df-ndx 15803  df-slot 15804  df-base 15805  df-sets 15806  df-ress 15807  df-subg 17531
This theorem is referenced by:  nmznsg  17578  subgslw  17971  subgdmdprd  18373  subgdprd  18374  ablfac1c  18410  pgpfaclem1  18420  pgpfaclem2  18421  ablfaclem3  18426
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