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Theorem subsubg 19167
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 19149 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
21adantr 480 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → 𝐺 ∈ Grp)
3 eqid 2737 . . . . . . . 8 (Base‘𝐻) = (Base‘𝐻)
43subgss 19145 . . . . . . 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 19148 . . . . . . 7 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻))
87adantr 480 . . . . . 6 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → 𝑆 = (Base‘𝐻))
95, 8sseqtrrd 4021 . . . . 5 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → 𝐴𝑆)
10 eqid 2737 . . . . . . 7 (Base‘𝐺) = (Base‘𝐺)
1110subgss 19145 . . . . . 6 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
1211adantr 480 . . . . 5 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → 𝑆 ⊆ (Base‘𝐺))
139, 12sstrd 3994 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → 𝐴 ⊆ (Base‘𝐺))
146oveq1i 7441 . . . . . . 7 (𝐻s 𝐴) = ((𝐺s 𝑆) ↾s 𝐴)
15 ressabs 17294 . . . . . . 7 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑆) → ((𝐺s 𝑆) ↾s 𝐴) = (𝐺s 𝐴))
1614, 15eqtrid 2789 . . . . . 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 2737 . . . . . . 7 (𝐻s 𝐴) = (𝐻s 𝐴)
1918subggrp 19147 . . . . . 6 (𝐴 ∈ (SubGrp‘𝐻) → (𝐻s 𝐴) ∈ Grp)
2019adantl 481 . . . . 5 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → (𝐻s 𝐴) ∈ Grp)
2117, 20eqeltrrd 2842 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → (𝐺s 𝐴) ∈ Grp)
2210issubg 19144 . . . 4 (𝐴 ∈ (SubGrp‘𝐺) ↔ (𝐺 ∈ Grp ∧ 𝐴 ⊆ (Base‘𝐺) ∧ (𝐺s 𝐴) ∈ Grp))
232, 13, 21, 22syl3anbrc 1344 . . 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 19147 . . . 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 4020 . . 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 2737 . . . . . 6 (𝐺s 𝐴) = (𝐺s 𝐴)
3231subggrp 19147 . . . . 5 (𝐴 ∈ (SubGrp‘𝐺) → (𝐺s 𝐴) ∈ Grp)
3332ad2antrl 728 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑆)) → (𝐺s 𝐴) ∈ Grp)
3430, 33eqeltrd 2841 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑆)) → (𝐻s 𝐴) ∈ Grp)
353issubg 19144 . . 3 (𝐴 ∈ (SubGrp‘𝐻) ↔ (𝐻 ∈ Grp ∧ 𝐴 ⊆ (Base‘𝐻) ∧ (𝐻s 𝐴) ∈ Grp))
3626, 29, 34, 35syl3anbrc 1344 . 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 1540  wcel 2108  wss 3951  cfv 6561  (class class class)co 7431  Basecbs 17247  s cress 17274  Grpcgrp 18951  SubGrpcsubg 19138
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
This theorem is referenced by:  nmznsg  19186  subgslw  19634  subgdmdprd  20054  subgdprd  20055  ablfac1c  20091  pgpfaclem1  20101  pgpfaclem2  20102  ablfaclem3  20107
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