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Theorem lsmless12 15296
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 14950 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
21ad2antrr 708 . . 3  |-  ( ( ( S  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  /\  ( R  C_  S  /\  T  C_  U ) )  ->  G  e.  Grp )
3 eqid 2437 . . . . 5  |-  ( Base `  G )  =  (
Base `  G )
43subgss 14946 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  ( Base `  G ) )
54ad2antrr 708 . . 3  |-  ( ( ( S  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  /\  ( R  C_  S  /\  T  C_  U ) )  ->  S  C_  ( Base `  G
) )
6 simprr 735 . . . 4  |-  ( ( ( S  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  /\  ( R  C_  S  /\  T  C_  U ) )  ->  T  C_  U )
73subgss 14946 . . . . 5  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  ( Base `  G ) )
87ad2antlr 709 . . . 4  |-  ( ( ( S  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  /\  ( R  C_  S  /\  T  C_  U ) )  ->  U  C_  ( Base `  G
) )
96, 8sstrd 3359 . . 3  |-  ( ( ( S  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  /\  ( R  C_  S  /\  T  C_  U ) )  ->  T  C_  ( Base `  G
) )
10 simprl 734 . . 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 15279 . . 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 1188 . 2  |-  ( ( ( S  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  /\  ( R  C_  S  /\  T  C_  U ) )  -> 
( R  .(+)  T ) 
C_  ( S  .(+)  T ) )
14 simpll 732 . . 3  |-  ( ( ( S  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  /\  ( R  C_  S  /\  T  C_  U ) )  ->  S  e.  (SubGrp `  G
) )
15 simplr 733 . . 3  |-  ( ( ( S  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  /\  ( R  C_  S  /\  T  C_  U ) )  ->  U  e.  (SubGrp `  G
) )
1611lsmless2 15295 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  U )  ->  ( S  .(+)  T )  C_  ( S  .(+) 
U ) )
1714, 15, 6, 16syl3anc 1185 . 2  |-  ( ( ( S  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  /\  ( R  C_  S  /\  T  C_  U ) )  -> 
( S  .(+)  T ) 
C_  ( S  .(+)  U ) )
1813, 17sstrd 3359 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 360    = wceq 1653    e. wcel 1726    C_ wss 3321   ` cfv 5455  (class class class)co 6082   Basecbs 13470   Grpcgrp 14686  SubGrpcsubg 14939   LSSumclsm 15269
This theorem is referenced by:  lsmlub  15298  dochexmidlem2  32260
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-subg 14942  df-lsm 15271
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