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Theorem subsubg 15716
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 15698 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
21adantr 465 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  (SubGrp `  H )
)  ->  G  e.  Grp )
3 eqid 2443 . . . . . . . 8  |-  ( Base `  H )  =  (
Base `  H )
43subgss 15694 . . . . . . 7  |-  ( A  e.  (SubGrp `  H
)  ->  A  C_  ( Base `  H ) )
54adantl 466 . . . . . 6  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  (SubGrp `  H )
)  ->  A  C_  ( Base `  H ) )
6 subsubg.h . . . . . . . 8  |-  H  =  ( Gs  S )
76subgbas 15697 . . . . . . 7  |-  ( S  e.  (SubGrp `  G
)  ->  S  =  ( Base `  H )
)
87adantr 465 . . . . . 6  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  (SubGrp `  H )
)  ->  S  =  ( Base `  H )
)
95, 8sseqtr4d 3405 . . . . 5  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  (SubGrp `  H )
)  ->  A  C_  S
)
10 eqid 2443 . . . . . . 7  |-  ( Base `  G )  =  (
Base `  G )
1110subgss 15694 . . . . . 6  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  ( Base `  G ) )
1211adantr 465 . . . . 5  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  (SubGrp `  H )
)  ->  S  C_  ( Base `  G ) )
139, 12sstrd 3378 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  (SubGrp `  H )
)  ->  A  C_  ( Base `  G ) )
146oveq1i 6113 . . . . . . 7  |-  ( Hs  A )  =  ( ( Gs  S )s  A )
15 ressabs 14248 . . . . . . 7  |-  ( ( S  e.  (SubGrp `  G )  /\  A  C_  S )  ->  (
( Gs  S )s  A )  =  ( Gs  A ) )
1614, 15syl5eq 2487 . . . . . 6  |-  ( ( S  e.  (SubGrp `  G )  /\  A  C_  S )  ->  ( Hs  A )  =  ( Gs  A ) )
179, 16syldan 470 . . . . 5  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  (SubGrp `  H )
)  ->  ( Hs  A
)  =  ( Gs  A ) )
18 eqid 2443 . . . . . . 7  |-  ( Hs  A )  =  ( Hs  A )
1918subggrp 15696 . . . . . 6  |-  ( A  e.  (SubGrp `  H
)  ->  ( Hs  A
)  e.  Grp )
2019adantl 466 . . . . 5  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  (SubGrp `  H )
)  ->  ( Hs  A
)  e.  Grp )
2117, 20eqeltrrd 2518 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  (SubGrp `  H )
)  ->  ( Gs  A
)  e.  Grp )
2210issubg 15693 . . . 4  |-  ( A  e.  (SubGrp `  G
)  <->  ( G  e. 
Grp  /\  A  C_  ( Base `  G )  /\  ( Gs  A )  e.  Grp ) )
232, 13, 21, 22syl3anbrc 1172 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  (SubGrp `  H )
)  ->  A  e.  (SubGrp `  G ) )
2423, 9jca 532 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  (SubGrp `  H )
)  ->  ( A  e.  (SubGrp `  G )  /\  A  C_  S ) )
256subggrp 15696 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  H  e.  Grp )
2625adantr 465 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  ( A  e.  (SubGrp `  G
)  /\  A  C_  S
) )  ->  H  e.  Grp )
27 simprr 756 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  ( A  e.  (SubGrp `  G
)  /\  A  C_  S
) )  ->  A  C_  S )
287adantr 465 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  ( A  e.  (SubGrp `  G
)  /\  A  C_  S
) )  ->  S  =  ( Base `  H
) )
2927, 28sseqtrd 3404 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  ( A  e.  (SubGrp `  G
)  /\  A  C_  S
) )  ->  A  C_  ( Base `  H
) )
3016adantrl 715 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  ( A  e.  (SubGrp `  G
)  /\  A  C_  S
) )  ->  ( Hs  A )  =  ( Gs  A ) )
31 eqid 2443 . . . . . 6  |-  ( Gs  A )  =  ( Gs  A )
3231subggrp 15696 . . . . 5  |-  ( A  e.  (SubGrp `  G
)  ->  ( Gs  A
)  e.  Grp )
3332ad2antrl 727 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  ( A  e.  (SubGrp `  G
)  /\  A  C_  S
) )  ->  ( Gs  A )  e.  Grp )
3430, 33eqeltrd 2517 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  ( A  e.  (SubGrp `  G
)  /\  A  C_  S
) )  ->  ( Hs  A )  e.  Grp )
353issubg 15693 . . 3  |-  ( A  e.  (SubGrp `  H
)  <->  ( H  e. 
Grp  /\  A  C_  ( Base `  H )  /\  ( Hs  A )  e.  Grp ) )
3626, 29, 34, 35syl3anbrc 1172 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  ( A  e.  (SubGrp `  G
)  /\  A  C_  S
) )  ->  A  e.  (SubGrp `  H )
)
3724, 36impbida 828 1  |-  ( S  e.  (SubGrp `  G
)  ->  ( A  e.  (SubGrp `  H )  <->  ( A  e.  (SubGrp `  G )  /\  A  C_  S ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    C_ wss 3340   ` cfv 5430  (class class class)co 6103   Basecbs 14186   ↾s cress 14187   Grpcgrp 15422  SubGrpcsubg 15687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-i2m1 9362  ax-1ne0 9363  ax-rrecex 9366  ax-cnre 9367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-recs 6844  df-rdg 6878  df-nn 10335  df-ndx 14189  df-slot 14190  df-base 14191  df-sets 14192  df-ress 14193  df-subg 15690
This theorem is referenced by:  nmznsg  15737  subgslw  16127  subgdmdprd  16543  subgdprd  16544  ablfac1c  16584  pgpfaclem1  16594  pgpfaclem2  16595  ablfaclem3  16600
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