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Theorem subsubg 14890
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  e.  (SubGrp `  G
)  ->  ( A  e.  (SubGrp `  H )  <->  ( A  e.  (SubGrp `  G )  /\  A  C_  S ) ) )

Proof of Theorem subsubg
StepHypRef Expression
1 subgrcl 14876 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
21adantr 452 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  (SubGrp `  H )
)  ->  G  e.  Grp )
3 eqid 2387 . . . . . . . 8  |-  ( Base `  H )  =  (
Base `  H )
43subgss 14872 . . . . . . 7  |-  ( A  e.  (SubGrp `  H
)  ->  A  C_  ( Base `  H ) )
54adantl 453 . . . . . 6  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  (SubGrp `  H )
)  ->  A  C_  ( Base `  H ) )
6 subsubg.h . . . . . . . 8  |-  H  =  ( Gs  S )
76subgbas 14875 . . . . . . 7  |-  ( S  e.  (SubGrp `  G
)  ->  S  =  ( Base `  H )
)
87adantr 452 . . . . . 6  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  (SubGrp `  H )
)  ->  S  =  ( Base `  H )
)
95, 8sseqtr4d 3328 . . . . 5  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  (SubGrp `  H )
)  ->  A  C_  S
)
10 eqid 2387 . . . . . . 7  |-  ( Base `  G )  =  (
Base `  G )
1110subgss 14872 . . . . . 6  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  ( Base `  G ) )
1211adantr 452 . . . . 5  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  (SubGrp `  H )
)  ->  S  C_  ( Base `  G ) )
139, 12sstrd 3301 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  (SubGrp `  H )
)  ->  A  C_  ( Base `  G ) )
146oveq1i 6030 . . . . . . 7  |-  ( Hs  A )  =  ( ( Gs  S )s  A )
15 ressabs 13454 . . . . . . 7  |-  ( ( S  e.  (SubGrp `  G )  /\  A  C_  S )  ->  (
( Gs  S )s  A )  =  ( Gs  A ) )
1614, 15syl5eq 2431 . . . . . 6  |-  ( ( S  e.  (SubGrp `  G )  /\  A  C_  S )  ->  ( Hs  A )  =  ( Gs  A ) )
179, 16syldan 457 . . . . 5  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  (SubGrp `  H )
)  ->  ( Hs  A
)  =  ( Gs  A ) )
18 eqid 2387 . . . . . . 7  |-  ( Hs  A )  =  ( Hs  A )
1918subggrp 14874 . . . . . 6  |-  ( A  e.  (SubGrp `  H
)  ->  ( Hs  A
)  e.  Grp )
2019adantl 453 . . . . 5  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  (SubGrp `  H )
)  ->  ( Hs  A
)  e.  Grp )
2117, 20eqeltrrd 2462 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  (SubGrp `  H )
)  ->  ( Gs  A
)  e.  Grp )
2210issubg 14871 . . . 4  |-  ( A  e.  (SubGrp `  G
)  <->  ( G  e. 
Grp  /\  A  C_  ( Base `  G )  /\  ( Gs  A )  e.  Grp ) )
232, 13, 21, 22syl3anbrc 1138 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  (SubGrp `  H )
)  ->  A  e.  (SubGrp `  G ) )
2423, 9jca 519 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  (SubGrp `  H )
)  ->  ( A  e.  (SubGrp `  G )  /\  A  C_  S ) )
256subggrp 14874 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  H  e.  Grp )
2625adantr 452 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  ( A  e.  (SubGrp `  G
)  /\  A  C_  S
) )  ->  H  e.  Grp )
27 simprr 734 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  ( A  e.  (SubGrp `  G
)  /\  A  C_  S
) )  ->  A  C_  S )
287adantr 452 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  ( A  e.  (SubGrp `  G
)  /\  A  C_  S
) )  ->  S  =  ( Base `  H
) )
2927, 28sseqtrd 3327 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  ( A  e.  (SubGrp `  G
)  /\  A  C_  S
) )  ->  A  C_  ( Base `  H
) )
3016adantrl 697 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  ( A  e.  (SubGrp `  G
)  /\  A  C_  S
) )  ->  ( Hs  A )  =  ( Gs  A ) )
31 eqid 2387 . . . . . 6  |-  ( Gs  A )  =  ( Gs  A )
3231subggrp 14874 . . . . 5  |-  ( A  e.  (SubGrp `  G
)  ->  ( Gs  A
)  e.  Grp )
3332ad2antrl 709 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  ( A  e.  (SubGrp `  G
)  /\  A  C_  S
) )  ->  ( Gs  A )  e.  Grp )
3430, 33eqeltrd 2461 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  ( A  e.  (SubGrp `  G
)  /\  A  C_  S
) )  ->  ( Hs  A )  e.  Grp )
353issubg 14871 . . 3  |-  ( A  e.  (SubGrp `  H
)  <->  ( H  e. 
Grp  /\  A  C_  ( Base `  H )  /\  ( Hs  A )  e.  Grp ) )
3626, 29, 34, 35syl3anbrc 1138 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  ( A  e.  (SubGrp `  G
)  /\  A  C_  S
) )  ->  A  e.  (SubGrp `  H )
)
3724, 36impbida 806 1  |-  ( S  e.  (SubGrp `  G
)  ->  ( A  e.  (SubGrp `  H )  <->  ( A  e.  (SubGrp `  G )  /\  A  C_  S ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    C_ wss 3263   ` cfv 5394  (class class class)co 6020   Basecbs 13396   ↾s cress 13397   Grpcgrp 14612  SubGrpcsubg 14865
This theorem is referenced by:  nmznsg  14911  subgslw  15177  subgdmdprd  15519  subgdprd  15520  ablfac1c  15556  pgpfaclem1  15566  pgpfaclem2  15567  ablfaclem3  15572
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-i2m1 8991  ax-1ne0 8992  ax-rrecex 8995  ax-cnre 8996
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-recs 6569  df-rdg 6604  df-nn 9933  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-ress 13403  df-subg 14868
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