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

Theorem subsubg 14656
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 14642 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
21adantr 451 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  (SubGrp `  H )
)  ->  G  e.  Grp )
3 eqid 2296 . . . . . . . 8  |-  ( Base `  H )  =  (
Base `  H )
43subgss 14638 . . . . . . 7  |-  ( A  e.  (SubGrp `  H
)  ->  A  C_  ( Base `  H ) )
54adantl 452 . . . . . 6  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  (SubGrp `  H )
)  ->  A  C_  ( Base `  H ) )
6 subsubg.h . . . . . . . 8  |-  H  =  ( Gs  S )
76subgbas 14641 . . . . . . 7  |-  ( S  e.  (SubGrp `  G
)  ->  S  =  ( Base `  H )
)
87adantr 451 . . . . . 6  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  (SubGrp `  H )
)  ->  S  =  ( Base `  H )
)
95, 8sseqtr4d 3228 . . . . 5  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  (SubGrp `  H )
)  ->  A  C_  S
)
10 eqid 2296 . . . . . . 7  |-  ( Base `  G )  =  (
Base `  G )
1110subgss 14638 . . . . . 6  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  ( Base `  G ) )
1211adantr 451 . . . . 5  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  (SubGrp `  H )
)  ->  S  C_  ( Base `  G ) )
139, 12sstrd 3202 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  (SubGrp `  H )
)  ->  A  C_  ( Base `  G ) )
146oveq1i 5884 . . . . . . 7  |-  ( Hs  A )  =  ( ( Gs  S )s  A )
15 ressabs 13222 . . . . . . 7  |-  ( ( S  e.  (SubGrp `  G )  /\  A  C_  S )  ->  (
( Gs  S )s  A )  =  ( Gs  A ) )
1614, 15syl5eq 2340 . . . . . 6  |-  ( ( S  e.  (SubGrp `  G )  /\  A  C_  S )  ->  ( Hs  A )  =  ( Gs  A ) )
179, 16syldan 456 . . . . 5  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  (SubGrp `  H )
)  ->  ( Hs  A
)  =  ( Gs  A ) )
18 eqid 2296 . . . . . . 7  |-  ( Hs  A )  =  ( Hs  A )
1918subggrp 14640 . . . . . 6  |-  ( A  e.  (SubGrp `  H
)  ->  ( Hs  A
)  e.  Grp )
2019adantl 452 . . . . 5  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  (SubGrp `  H )
)  ->  ( Hs  A
)  e.  Grp )
2117, 20eqeltrrd 2371 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  (SubGrp `  H )
)  ->  ( Gs  A
)  e.  Grp )
2210issubg 14637 . . . 4  |-  ( A  e.  (SubGrp `  G
)  <->  ( G  e. 
Grp  /\  A  C_  ( Base `  G )  /\  ( Gs  A )  e.  Grp ) )
232, 13, 21, 22syl3anbrc 1136 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  (SubGrp `  H )
)  ->  A  e.  (SubGrp `  G ) )
2423, 9jca 518 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  (SubGrp `  H )
)  ->  ( A  e.  (SubGrp `  G )  /\  A  C_  S ) )
256subggrp 14640 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  H  e.  Grp )
2625adantr 451 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  ( A  e.  (SubGrp `  G
)  /\  A  C_  S
) )  ->  H  e.  Grp )
27 simprr 733 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  ( A  e.  (SubGrp `  G
)  /\  A  C_  S
) )  ->  A  C_  S )
287adantr 451 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  ( A  e.  (SubGrp `  G
)  /\  A  C_  S
) )  ->  S  =  ( Base `  H
) )
2927, 28sseqtrd 3227 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  ( A  e.  (SubGrp `  G
)  /\  A  C_  S
) )  ->  A  C_  ( Base `  H
) )
3016adantrl 696 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  ( A  e.  (SubGrp `  G
)  /\  A  C_  S
) )  ->  ( Hs  A )  =  ( Gs  A ) )
31 eqid 2296 . . . . . 6  |-  ( Gs  A )  =  ( Gs  A )
3231subggrp 14640 . . . . 5  |-  ( A  e.  (SubGrp `  G
)  ->  ( Gs  A
)  e.  Grp )
3332ad2antrl 708 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  ( A  e.  (SubGrp `  G
)  /\  A  C_  S
) )  ->  ( Gs  A )  e.  Grp )
3430, 33eqeltrd 2370 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  ( A  e.  (SubGrp `  G
)  /\  A  C_  S
) )  ->  ( Hs  A )  e.  Grp )
353issubg 14637 . . 3  |-  ( A  e.  (SubGrp `  H
)  <->  ( H  e. 
Grp  /\  A  C_  ( Base `  H )  /\  ( Hs  A )  e.  Grp ) )
3626, 29, 34, 35syl3anbrc 1136 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  ( A  e.  (SubGrp `  G
)  /\  A  C_  S
) )  ->  A  e.  (SubGrp `  H )
)
3724, 36impbida 805 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 176    /\ wa 358    = wceq 1632    e. wcel 1696    C_ wss 3165   ` cfv 5271  (class class class)co 5874   Basecbs 13164   ↾s cress 13165   Grpcgrp 14378  SubGrpcsubg 14631
This theorem is referenced by:  nmznsg  14677  subgslw  14943  subgdmdprd  15285  subgdprd  15286  ablfac1c  15322  pgpfaclem1  15332  pgpfaclem2  15333  ablfaclem3  15338
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-i2m1 8821  ax-1ne0 8822  ax-rrecex 8825  ax-cnre 8826
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-recs 6404  df-rdg 6439  df-nn 9763  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-subg 14634
  Copyright terms: Public domain W3C validator