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Theorem subsubg 16350
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 16332 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
21adantr 465 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  (SubGrp `  H )
)  ->  G  e.  Grp )
3 eqid 2457 . . . . . . . 8  |-  ( Base `  H )  =  (
Base `  H )
43subgss 16328 . . . . . . 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 16331 . . . . . . 7  |-  ( S  e.  (SubGrp `  G
)  ->  S  =  ( Base `  H )
)
87adantr 465 . . . . . 6  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  (SubGrp `  H )
)  ->  S  =  ( Base `  H )
)
95, 8sseqtr4d 3536 . . . . 5  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  (SubGrp `  H )
)  ->  A  C_  S
)
10 eqid 2457 . . . . . . 7  |-  ( Base `  G )  =  (
Base `  G )
1110subgss 16328 . . . . . 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 3509 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  (SubGrp `  H )
)  ->  A  C_  ( Base `  G ) )
146oveq1i 6306 . . . . . . 7  |-  ( Hs  A )  =  ( ( Gs  S )s  A )
15 ressabs 14709 . . . . . . 7  |-  ( ( S  e.  (SubGrp `  G )  /\  A  C_  S )  ->  (
( Gs  S )s  A )  =  ( Gs  A ) )
1614, 15syl5eq 2510 . . . . . 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 2457 . . . . . . 7  |-  ( Hs  A )  =  ( Hs  A )
1918subggrp 16330 . . . . . 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 2546 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  (SubGrp `  H )
)  ->  ( Gs  A
)  e.  Grp )
2210issubg 16327 . . . 4  |-  ( A  e.  (SubGrp `  G
)  <->  ( G  e. 
Grp  /\  A  C_  ( Base `  G )  /\  ( Gs  A )  e.  Grp ) )
232, 13, 21, 22syl3anbrc 1180 . . 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 16330 . . . 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 757 . . . 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 3535 . . 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 2457 . . . . . 6  |-  ( Gs  A )  =  ( Gs  A )
3231subggrp 16330 . . . . 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 2545 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  ( A  e.  (SubGrp `  G
)  /\  A  C_  S
) )  ->  ( Hs  A )  e.  Grp )
353issubg 16327 . . 3  |-  ( A  e.  (SubGrp `  H
)  <->  ( H  e. 
Grp  /\  A  C_  ( Base `  H )  /\  ( Hs  A )  e.  Grp ) )
3626, 29, 34, 35syl3anbrc 1180 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  ( A  e.  (SubGrp `  G
)  /\  A  C_  S
) )  ->  A  e.  (SubGrp `  H )
)
3724, 36impbida 832 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 1395    e. wcel 1819    C_ wss 3471   ` cfv 5594  (class class class)co 6296   Basecbs 14643   ↾s cress 14644   Grpcgrp 16179  SubGrpcsubg 16321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-i2m1 9577  ax-1ne0 9578  ax-rrecex 9581  ax-cnre 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-recs 7060  df-rdg 7094  df-nn 10557  df-ndx 14646  df-slot 14647  df-base 14648  df-sets 14649  df-ress 14650  df-subg 16324
This theorem is referenced by:  nmznsg  16371  subgslw  16762  subgdmdprd  17207  subgdprd  17208  ablfac1c  17248  pgpfaclem1  17258  pgpfaclem2  17259  ablfaclem3  17264
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