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Theorem lsmval 14959
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 14626 . . 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 14622 . . 3  |-  ( T  e.  (SubGrp `  G
)  ->  T  C_  B
)
54adantr 451 . 2  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  ->  T  C_  B
)
63subgss 14622 . . 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 14950 . 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 1623    e. wcel 1684    C_ wss 3152   ran crn 4690   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   Basecbs 13148   +g cplusg 13208   Grpcgrp 14362  SubGrpcsubg 14615   LSSumclsm 14945
This theorem is referenced by:  lsmidm  14973  lsmass  14979
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-subg 14618  df-lsm 14947
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