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Theorem lsmless2 15253
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 14908 . . 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 2408 . . . 4  |-  ( Base `  G )  =  (
Base `  G )
43subgss 14904 . . 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 14904 . . 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 15238 . 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 1649    e. wcel 1721    C_ wss 3284   ` cfv 5417  (class class class)co 6044   Basecbs 13428   Grpcgrp 14644  SubGrpcsubg 14897   LSSumclsm 15227
This theorem is referenced by:  lsmless12  15254  lsmmod  15266  lsmelval2  16116  lsmsat  29495  lsatcvat3  29539  cdlemn5pre  31687  dvh3dim3N  31936
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-subg 14900  df-lsm 15229
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