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Theorem subsubg 19180
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 19162 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
21adantr 480 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → 𝐺 ∈ Grp)
3 eqid 2735 . . . . . . . 8 (Base‘𝐻) = (Base‘𝐻)
43subgss 19158 . . . . . . 7 (𝐴 ∈ (SubGrp‘𝐻) → 𝐴 ⊆ (Base‘𝐻))
54adantl 481 . . . . . 6 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → 𝐴 ⊆ (Base‘𝐻))
6 subsubg.h . . . . . . . 8 𝐻 = (𝐺s 𝑆)
76subgbas 19161 . . . . . . 7 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻))
87adantr 480 . . . . . 6 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → 𝑆 = (Base‘𝐻))
95, 8sseqtrrd 4037 . . . . 5 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → 𝐴𝑆)
10 eqid 2735 . . . . . . 7 (Base‘𝐺) = (Base‘𝐺)
1110subgss 19158 . . . . . 6 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
1211adantr 480 . . . . 5 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → 𝑆 ⊆ (Base‘𝐺))
139, 12sstrd 4006 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → 𝐴 ⊆ (Base‘𝐺))
146oveq1i 7441 . . . . . . 7 (𝐻s 𝐴) = ((𝐺s 𝑆) ↾s 𝐴)
15 ressabs 17295 . . . . . . 7 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑆) → ((𝐺s 𝑆) ↾s 𝐴) = (𝐺s 𝐴))
1614, 15eqtrid 2787 . . . . . 6 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑆) → (𝐻s 𝐴) = (𝐺s 𝐴))
179, 16syldan 591 . . . . 5 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → (𝐻s 𝐴) = (𝐺s 𝐴))
18 eqid 2735 . . . . . . 7 (𝐻s 𝐴) = (𝐻s 𝐴)
1918subggrp 19160 . . . . . 6 (𝐴 ∈ (SubGrp‘𝐻) → (𝐻s 𝐴) ∈ Grp)
2019adantl 481 . . . . 5 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → (𝐻s 𝐴) ∈ Grp)
2117, 20eqeltrrd 2840 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → (𝐺s 𝐴) ∈ Grp)
2210issubg 19157 . . . 4 (𝐴 ∈ (SubGrp‘𝐺) ↔ (𝐺 ∈ Grp ∧ 𝐴 ⊆ (Base‘𝐺) ∧ (𝐺s 𝐴) ∈ Grp))
232, 13, 21, 22syl3anbrc 1342 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → 𝐴 ∈ (SubGrp‘𝐺))
2423, 9jca 511 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑆))
256subggrp 19160 . . . 4 (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
2625adantr 480 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑆)) → 𝐻 ∈ Grp)
27 simprr 773 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑆)) → 𝐴𝑆)
287adantr 480 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑆)) → 𝑆 = (Base‘𝐻))
2927, 28sseqtrd 4036 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑆)) → 𝐴 ⊆ (Base‘𝐻))
3016adantrl 716 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑆)) → (𝐻s 𝐴) = (𝐺s 𝐴))
31 eqid 2735 . . . . . 6 (𝐺s 𝐴) = (𝐺s 𝐴)
3231subggrp 19160 . . . . 5 (𝐴 ∈ (SubGrp‘𝐺) → (𝐺s 𝐴) ∈ Grp)
3332ad2antrl 728 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑆)) → (𝐺s 𝐴) ∈ Grp)
3430, 33eqeltrd 2839 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑆)) → (𝐻s 𝐴) ∈ Grp)
353issubg 19157 . . 3 (𝐴 ∈ (SubGrp‘𝐻) ↔ (𝐻 ∈ Grp ∧ 𝐴 ⊆ (Base‘𝐻) ∧ (𝐻s 𝐴) ∈ Grp))
3626, 29, 34, 35syl3anbrc 1342 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑆)) → 𝐴 ∈ (SubGrp‘𝐻))
3724, 36impbida 801 1 (𝑆 ∈ (SubGrp‘𝐺) → (𝐴 ∈ (SubGrp‘𝐻) ↔ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑆)))
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  Grpcgrp 18964  SubGrpcsubg 19151
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
This theorem is referenced by:  nmznsg  19199  subgslw  19649  subgdmdprd  20069  subgdprd  20070  ablfac1c  20106  pgpfaclem1  20116  pgpfaclem2  20117  ablfaclem3  20122
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