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Theorem lsmless2 15294
Description: Subset implies subgroup sum subset. (Contributed by NM, 25-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
Hypothesis
Ref Expression
lsmub1.p  |-  .(+)  =  (
LSSum `  G )
Assertion
Ref Expression
lsmless2  |-  ( ( S  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  U )  ->  ( S  .(+)  T )  C_  ( S  .(+) 
U ) )

Proof of Theorem lsmless2
StepHypRef Expression
1 subgrcl 14949 . . 3  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
213ad2ant1 978 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  U )  ->  G  e.  Grp )
3 eqid 2436 . . . 4  |-  ( Base `  G )  =  (
Base `  G )
43subgss 14945 . . 3  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  ( Base `  G ) )
543ad2ant1 978 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  U )  ->  S  C_  ( Base `  G ) )
63subgss 14945 . . 3  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  ( Base `  G ) )
763ad2ant2 979 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  U )  ->  U  C_  ( Base `  G ) )
8 simp3 959 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  U )  ->  T  C_  U
)
9 lsmub1.p . . 3  |-  .(+)  =  (
LSSum `  G )
103, 9lsmless2x 15279 . 2  |-  ( ( ( G  e.  Grp  /\  S  C_  ( Base `  G )  /\  U  C_  ( Base `  G
) )  /\  T  C_  U )  ->  ( S  .(+)  T )  C_  ( S  .(+)  U ) )
112, 5, 7, 8, 10syl31anc 1187 1  |-  ( ( S  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  U )  ->  ( S  .(+)  T )  C_  ( S  .(+) 
U ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1652    e. wcel 1725    C_ wss 3320   ` cfv 5454  (class class class)co 6081   Basecbs 13469   Grpcgrp 14685  SubGrpcsubg 14938   LSSumclsm 15268
This theorem is referenced by:  lsmless12  15295  lsmmod  15307  lsmelval2  16157  lsmsat  29806  lsatcvat3  29850  cdlemn5pre  31998  dvh3dim3N  32247
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-subg 14941  df-lsm 15270
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