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Theorem subsubg 14955
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 14941 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
21adantr 452 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  (SubGrp `  H )
)  ->  G  e.  Grp )
3 eqid 2435 . . . . . . . 8  |-  ( Base `  H )  =  (
Base `  H )
43subgss 14937 . . . . . . 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 14940 . . . . . . 7  |-  ( S  e.  (SubGrp `  G
)  ->  S  =  ( Base `  H )
)
87adantr 452 . . . . . 6  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  (SubGrp `  H )
)  ->  S  =  ( Base `  H )
)
95, 8sseqtr4d 3377 . . . . 5  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  (SubGrp `  H )
)  ->  A  C_  S
)
10 eqid 2435 . . . . . . 7  |-  ( Base `  G )  =  (
Base `  G )
1110subgss 14937 . . . . . 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 3350 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  (SubGrp `  H )
)  ->  A  C_  ( Base `  G ) )
146oveq1i 6083 . . . . . . 7  |-  ( Hs  A )  =  ( ( Gs  S )s  A )
15 ressabs 13519 . . . . . . 7  |-  ( ( S  e.  (SubGrp `  G )  /\  A  C_  S )  ->  (
( Gs  S )s  A )  =  ( Gs  A ) )
1614, 15syl5eq 2479 . . . . . 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 2435 . . . . . . 7  |-  ( Hs  A )  =  ( Hs  A )
1918subggrp 14939 . . . . . 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 2510 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  (SubGrp `  H )
)  ->  ( Gs  A
)  e.  Grp )
2210issubg 14936 . . . 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 14939 . . . 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 3376 . . 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 2435 . . . . . 6  |-  ( Gs  A )  =  ( Gs  A )
3231subggrp 14939 . . . . 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 2509 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  ( A  e.  (SubGrp `  G
)  /\  A  C_  S
) )  ->  ( Hs  A )  e.  Grp )
353issubg 14936 . . 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 1652    e. wcel 1725    C_ wss 3312   ` cfv 5446  (class class class)co 6073   Basecbs 13461   ↾s cress 13462   Grpcgrp 14677  SubGrpcsubg 14930
This theorem is referenced by:  nmznsg  14976  subgslw  15242  subgdmdprd  15584  subgdprd  15585  ablfac1c  15621  pgpfaclem1  15631  pgpfaclem2  15632  ablfaclem3  15637
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-i2m1 9050  ax-1ne0 9051  ax-rrecex 9054  ax-cnre 9055
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-recs 6625  df-rdg 6660  df-nn 9993  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-subg 14933
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