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Theorem lsmval 15272
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 14939 . . 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 14935 . . 3  |-  ( T  e.  (SubGrp `  G
)  ->  T  C_  B
)
54adantr 452 . 2  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  ->  T  C_  B
)
63subgss 14935 . . 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 15263 . 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 1652    e. wcel 1725    C_ wss 3312   ran crn 4871   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   Basecbs 13459   +g cplusg 13519   Grpcgrp 14675  SubGrpcsubg 14928   LSSumclsm 15258
This theorem is referenced by:  lsmidm  15286  lsmass  15292
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-subg 14931  df-lsm 15260
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