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Theorem lsmval 14975
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 14642 . . 3  |-  ( T  e.  (SubGrp `  G
)  ->  G  e.  Grp )
21adantr 451 . 2  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  ->  G  e.  Grp )
3 lsmval.v . . . 4  |-  B  =  ( Base `  G
)
43subgss 14638 . . 3  |-  ( T  e.  (SubGrp `  G
)  ->  T  C_  B
)
54adantr 451 . 2  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  ->  T  C_  B
)
63subgss 14638 . . 3  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  B
)
76adantl 452 . 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 14966 . 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 1182 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 358    = wceq 1632    e. wcel 1696    C_ wss 3165   ran crn 4706   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   Basecbs 13164   +g cplusg 13224   Grpcgrp 14378  SubGrpcsubg 14631   LSSumclsm 14961
This theorem is referenced by:  lsmidm  14989  lsmass  14995
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-subg 14634  df-lsm 14963
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