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Theorem lsmless12 14988
Description: Subset implies subgroup sum subset. (Contributed by NM, 14-Jan-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
Hypothesis
Ref Expression
lsmub1.p  |-  .(+)  =  (
LSSum `  G )
Assertion
Ref Expression
lsmless12  |-  ( ( ( S  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  /\  ( R  C_  S  /\  T  C_  U ) )  -> 
( R  .(+)  T ) 
C_  ( S  .(+)  U ) )

Proof of Theorem lsmless12
StepHypRef Expression
1 subgrcl 14642 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
21ad2antrr 706 . . 3  |-  ( ( ( S  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  /\  ( R  C_  S  /\  T  C_  U ) )  ->  G  e.  Grp )
3 eqid 2296 . . . . 5  |-  ( Base `  G )  =  (
Base `  G )
43subgss 14638 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  ( Base `  G ) )
54ad2antrr 706 . . 3  |-  ( ( ( S  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  /\  ( R  C_  S  /\  T  C_  U ) )  ->  S  C_  ( Base `  G
) )
6 simprr 733 . . . 4  |-  ( ( ( S  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  /\  ( R  C_  S  /\  T  C_  U ) )  ->  T  C_  U )
73subgss 14638 . . . . 5  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  ( Base `  G ) )
87ad2antlr 707 . . . 4  |-  ( ( ( S  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  /\  ( R  C_  S  /\  T  C_  U ) )  ->  U  C_  ( Base `  G
) )
96, 8sstrd 3202 . . 3  |-  ( ( ( S  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  /\  ( R  C_  S  /\  T  C_  U ) )  ->  T  C_  ( Base `  G
) )
10 simprl 732 . . 3  |-  ( ( ( S  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  /\  ( R  C_  S  /\  T  C_  U ) )  ->  R  C_  S )
11 lsmub1.p . . . 4  |-  .(+)  =  (
LSSum `  G )
123, 11lsmless1x 14971 . . 3  |-  ( ( ( G  e.  Grp  /\  S  C_  ( Base `  G )  /\  T  C_  ( Base `  G
) )  /\  R  C_  S )  ->  ( R  .(+)  T )  C_  ( S  .(+)  T ) )
132, 5, 9, 10, 12syl31anc 1185 . 2  |-  ( ( ( S  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  /\  ( R  C_  S  /\  T  C_  U ) )  -> 
( R  .(+)  T ) 
C_  ( S  .(+)  T ) )
14 simpll 730 . . 3  |-  ( ( ( S  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  /\  ( R  C_  S  /\  T  C_  U ) )  ->  S  e.  (SubGrp `  G
) )
15 simplr 731 . . 3  |-  ( ( ( S  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  /\  ( R  C_  S  /\  T  C_  U ) )  ->  U  e.  (SubGrp `  G
) )
1611lsmless2 14987 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  U )  ->  ( S  .(+)  T )  C_  ( S  .(+) 
U ) )
1714, 15, 6, 16syl3anc 1182 . 2  |-  ( ( ( S  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  /\  ( R  C_  S  /\  T  C_  U ) )  -> 
( S  .(+)  T ) 
C_  ( S  .(+)  U ) )
1813, 17sstrd 3202 1  |-  ( ( ( S  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  /\  ( R  C_  S  /\  T  C_  U ) )  -> 
( R  .(+)  T ) 
C_  ( S  .(+)  U ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    C_ wss 3165   ` cfv 5271  (class class class)co 5874   Basecbs 13164   Grpcgrp 14378  SubGrpcsubg 14631   LSSumclsm 14961
This theorem is referenced by:  lsmlub  14990  dochexmidlem2  32273
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-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  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-1st 6138  df-2nd 6139  df-subg 14634  df-lsm 14963
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