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

Theorem lsmval 15211
Description: Subgroup sum value (for a left module or left vector space). (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
Hypotheses
Ref Expression
lsmval.v  |-  B  =  ( Base `  G
)
lsmval.a  |-  .+  =  ( +g  `  G )
lsmval.p  |-  .(+)  =  (
LSSum `  G )
Assertion
Ref Expression
lsmval  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  ->  ( T  .(+) 
U )  =  ran  ( x  e.  T ,  y  e.  U  |->  ( x  .+  y
) ) )
Distinct variable groups:    x, y,  .+    x, T, y    x, U, y    x, B, y   
x, G, y
Allowed substitution hints:    .(+) ( x, y)

Proof of Theorem lsmval
StepHypRef Expression
1 subgrcl 14878 . . 3  |-  ( T  e.  (SubGrp `  G
)  ->  G  e.  Grp )
21adantr 452 . 2  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  ->  G  e.  Grp )
3 lsmval.v . . . 4  |-  B  =  ( Base `  G
)
43subgss 14874 . . 3  |-  ( T  e.  (SubGrp `  G
)  ->  T  C_  B
)
54adantr 452 . 2  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  ->  T  C_  B
)
63subgss 14874 . . 3  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  B
)
76adantl 453 . 2  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  ->  U  C_  B
)
8 lsmval.a . . 3  |-  .+  =  ( +g  `  G )
9 lsmval.p . . 3  |-  .(+)  =  (
LSSum `  G )
103, 8, 9lsmvalx 15202 . 2  |-  ( ( G  e.  Grp  /\  T  C_  B  /\  U  C_  B )  ->  ( T  .(+)  U )  =  ran  ( x  e.  T ,  y  e.  U  |->  ( x  .+  y ) ) )
112, 5, 7, 10syl3anc 1184 1  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  ->  ( T  .(+) 
U )  =  ran  ( x  e.  T ,  y  e.  U  |->  ( x  .+  y
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    C_ wss 3265   ran crn 4821   ` cfv 5396  (class class class)co 6022    e. cmpt2 6024   Basecbs 13398   +g cplusg 13458   Grpcgrp 14614  SubGrpcsubg 14867   LSSumclsm 15197
This theorem is referenced by:  lsmidm  15225  lsmass  15231
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-subg 14870  df-lsm 15199
  Copyright terms: Public domain W3C validator