MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lsmless2 Structured version   Unicode version

Theorem lsmless2 16806
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 16332 . . 3  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
213ad2ant1 1017 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  U )  ->  G  e.  Grp )
3 eqid 2457 . . . 4  |-  ( Base `  G )  =  (
Base `  G )
43subgss 16328 . . 3  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  ( Base `  G ) )
543ad2ant1 1017 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  U )  ->  S  C_  ( Base `  G ) )
63subgss 16328 . . 3  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  ( Base `  G ) )
763ad2ant2 1018 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  U )  ->  U  C_  ( Base `  G ) )
8 simp3 998 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  U )  ->  T  C_  U
)
9 lsmub1.p . . 3  |-  .(+)  =  (
LSSum `  G )
103, 9lsmless2x 16791 . 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 1231 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 973    = wceq 1395    e. wcel 1819    C_ wss 3471   ` cfv 5594  (class class class)co 6296   Basecbs 14643   Grpcgrp 16179  SubGrpcsubg 16321   LSSumclsm 16780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-subg 16324  df-lsm 16782
This theorem is referenced by:  lsmless12  16807  lsmmod  16819  lsmelval2  17857  lsmsat  34834  lsatcvat3  34878  cdlemn5pre  37028  dvh3dim3N  37277
  Copyright terms: Public domain W3C validator