MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  subsubg Structured version   Visualization version   GIF version

Theorem subsubg 17664
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 17646 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
21adantr 480 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → 𝐺 ∈ Grp)
3 eqid 2651 . . . . . . . 8 (Base‘𝐻) = (Base‘𝐻)
43subgss 17642 . . . . . . 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 17645 . . . . . . 7 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻))
87adantr 480 . . . . . 6 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → 𝑆 = (Base‘𝐻))
95, 8sseqtr4d 3675 . . . . 5 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → 𝐴𝑆)
10 eqid 2651 . . . . . . 7 (Base‘𝐺) = (Base‘𝐺)
1110subgss 17642 . . . . . 6 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
1211adantr 480 . . . . 5 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → 𝑆 ⊆ (Base‘𝐺))
139, 12sstrd 3646 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → 𝐴 ⊆ (Base‘𝐺))
146oveq1i 6700 . . . . . . 7 (𝐻s 𝐴) = ((𝐺s 𝑆) ↾s 𝐴)
15 ressabs 15986 . . . . . . 7 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑆) → ((𝐺s 𝑆) ↾s 𝐴) = (𝐺s 𝐴))
1614, 15syl5eq 2697 . . . . . 6 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑆) → (𝐻s 𝐴) = (𝐺s 𝐴))
179, 16syldan 486 . . . . 5 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → (𝐻s 𝐴) = (𝐺s 𝐴))
18 eqid 2651 . . . . . . 7 (𝐻s 𝐴) = (𝐻s 𝐴)
1918subggrp 17644 . . . . . 6 (𝐴 ∈ (SubGrp‘𝐻) → (𝐻s 𝐴) ∈ Grp)
2019adantl 481 . . . . 5 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → (𝐻s 𝐴) ∈ Grp)
2117, 20eqeltrrd 2731 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → (𝐺s 𝐴) ∈ Grp)
2210issubg 17641 . . . 4 (𝐴 ∈ (SubGrp‘𝐺) ↔ (𝐺 ∈ Grp ∧ 𝐴 ⊆ (Base‘𝐺) ∧ (𝐺s 𝐴) ∈ Grp))
232, 13, 21, 22syl3anbrc 1265 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → 𝐴 ∈ (SubGrp‘𝐺))
2423, 9jca 553 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑆))
256subggrp 17644 . . . 4 (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
2625adantr 480 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑆)) → 𝐻 ∈ Grp)
27 simprr 811 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑆)) → 𝐴𝑆)
287adantr 480 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑆)) → 𝑆 = (Base‘𝐻))
2927, 28sseqtrd 3674 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑆)) → 𝐴 ⊆ (Base‘𝐻))
3016adantrl 752 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑆)) → (𝐻s 𝐴) = (𝐺s 𝐴))
31 eqid 2651 . . . . . 6 (𝐺s 𝐴) = (𝐺s 𝐴)
3231subggrp 17644 . . . . 5 (𝐴 ∈ (SubGrp‘𝐺) → (𝐺s 𝐴) ∈ Grp)
3332ad2antrl 764 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑆)) → (𝐺s 𝐴) ∈ Grp)
3430, 33eqeltrd 2730 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑆)) → (𝐻s 𝐴) ∈ Grp)
353issubg 17641 . . 3 (𝐴 ∈ (SubGrp‘𝐻) ↔ (𝐻 ∈ Grp ∧ 𝐴 ⊆ (Base‘𝐻) ∧ (𝐻s 𝐴) ∈ Grp))
3626, 29, 34, 35syl3anbrc 1265 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑆)) → 𝐴 ∈ (SubGrp‘𝐻))
3724, 36impbida 895 1 (𝑆 ∈ (SubGrp‘𝐺) → (𝐴 ∈ (SubGrp‘𝐻) ↔ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wss 3607  cfv 5926  (class class class)co 6690  Basecbs 15904  s cress 15905  Grpcgrp 17469  SubGrpcsubg 17635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-i2m1 10042  ax-1ne0 10043  ax-rrecex 10046  ax-cnre 10047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-nn 11059  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-subg 17638
This theorem is referenced by:  nmznsg  17685  subgslw  18077  subgdmdprd  18479  subgdprd  18480  ablfac1c  18516  pgpfaclem1  18526  pgpfaclem2  18527  ablfaclem3  18532
  Copyright terms: Public domain W3C validator