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Theorem lsmless2 16164
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 15691 . . 3  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
213ad2ant1 1009 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  U )  ->  G  e.  Grp )
3 eqid 2443 . . . 4  |-  ( Base `  G )  =  (
Base `  G )
43subgss 15687 . . 3  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  ( Base `  G ) )
543ad2ant1 1009 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  U )  ->  S  C_  ( Base `  G ) )
63subgss 15687 . . 3  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  ( Base `  G ) )
763ad2ant2 1010 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  U )  ->  U  C_  ( Base `  G ) )
8 simp3 990 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  U )  ->  T  C_  U
)
9 lsmub1.p . . 3  |-  .(+)  =  (
LSSum `  G )
103, 9lsmless2x 16149 . 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 1221 1  |-  ( ( S  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  U )  ->  ( S  .(+)  T )  C_  ( S  .(+) 
U ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1369    e. wcel 1756    C_ wss 3333   ` cfv 5423  (class class class)co 6096   Basecbs 14179   Grpcgrp 15415  SubGrpcsubg 15680   LSSumclsm 16138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-1st 6582  df-2nd 6583  df-subg 15683  df-lsm 16140
This theorem is referenced by:  lsmless12  16165  lsmmod  16177  lsmelval2  17171  lsmsat  32658  lsatcvat3  32702  cdlemn5pre  34850  dvh3dim3N  35099
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