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Theorem lsmfval 17283
Description: The subgroup sum function (for a group or vector space). (Contributed by NM, 28-Jan-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
Hypotheses
Ref Expression
lsmfval.v  |-  B  =  ( Base `  G
)
lsmfval.a  |-  .+  =  ( +g  `  G )
lsmfval.s  |-  .(+)  =  (
LSSum `  G )
Assertion
Ref Expression
lsmfval  |-  ( G  e.  V  ->  .(+)  =  ( t  e.  ~P B ,  u  e.  ~P B  |->  ran  ( x  e.  t ,  y  e.  u  |->  ( x  .+  y ) ) ) )
Distinct variable groups:    u, t, x, y,  .+    t, B, u, x, y    t, G, u, x, y
Allowed substitution hints:    .(+) ( x, y, u, t)    V( x, y, u, t)

Proof of Theorem lsmfval
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 lsmfval.s . 2  |-  .(+)  =  (
LSSum `  G )
2 elex 3053 . . 3  |-  ( G  e.  V  ->  G  e.  _V )
3 fveq2 5863 . . . . . . 7  |-  ( w  =  G  ->  ( Base `  w )  =  ( Base `  G
) )
4 lsmfval.v . . . . . . 7  |-  B  =  ( Base `  G
)
53, 4syl6eqr 2502 . . . . . 6  |-  ( w  =  G  ->  ( Base `  w )  =  B )
65pweqd 3955 . . . . 5  |-  ( w  =  G  ->  ~P ( Base `  w )  =  ~P B )
7 fveq2 5863 . . . . . . . . 9  |-  ( w  =  G  ->  ( +g  `  w )  =  ( +g  `  G
) )
8 lsmfval.a . . . . . . . . 9  |-  .+  =  ( +g  `  G )
97, 8syl6eqr 2502 . . . . . . . 8  |-  ( w  =  G  ->  ( +g  `  w )  = 
.+  )
109oveqd 6305 . . . . . . 7  |-  ( w  =  G  ->  (
x ( +g  `  w
) y )  =  ( x  .+  y
) )
1110mpt2eq3dv 6354 . . . . . 6  |-  ( w  =  G  ->  (
x  e.  t ,  y  e.  u  |->  ( x ( +g  `  w
) y ) )  =  ( x  e.  t ,  y  e.  u  |->  ( x  .+  y ) ) )
1211rneqd 5061 . . . . 5  |-  ( w  =  G  ->  ran  ( x  e.  t ,  y  e.  u  |->  ( x ( +g  `  w ) y ) )  =  ran  (
x  e.  t ,  y  e.  u  |->  ( x  .+  y ) ) )
136, 6, 12mpt2eq123dv 6350 . . . 4  |-  ( w  =  G  ->  (
t  e.  ~P ( Base `  w ) ,  u  e.  ~P ( Base `  w )  |->  ran  ( x  e.  t ,  y  e.  u  |->  ( x ( +g  `  w ) y ) ) )  =  ( t  e.  ~P B ,  u  e.  ~P B  |->  ran  ( x  e.  t ,  y  e.  u  |->  ( x  .+  y ) ) ) )
14 df-lsm 17281 . . . 4  |-  LSSum  =  ( w  e.  _V  |->  ( t  e.  ~P ( Base `  w ) ,  u  e.  ~P ( Base `  w )  |->  ran  ( x  e.  t ,  y  e.  u  |->  ( x ( +g  `  w ) y ) ) ) )
15 fvex 5873 . . . . . . 7  |-  ( Base `  G )  e.  _V
164, 15eqeltri 2524 . . . . . 6  |-  B  e. 
_V
1716pwex 4585 . . . . 5  |-  ~P B  e.  _V
1817, 17mpt2ex 6867 . . . 4  |-  ( t  e.  ~P B ,  u  e.  ~P B  |->  ran  ( x  e.  t ,  y  e.  u  |->  ( x  .+  y ) ) )  e.  _V
1913, 14, 18fvmpt 5946 . . 3  |-  ( G  e.  _V  ->  ( LSSum `  G )  =  ( t  e.  ~P B ,  u  e.  ~P B  |->  ran  (
x  e.  t ,  y  e.  u  |->  ( x  .+  y ) ) ) )
202, 19syl 17 . 2  |-  ( G  e.  V  ->  ( LSSum `  G )  =  ( t  e.  ~P B ,  u  e.  ~P B  |->  ran  (
x  e.  t ,  y  e.  u  |->  ( x  .+  y ) ) ) )
211, 20syl5eq 2496 1  |-  ( G  e.  V  ->  .(+)  =  ( t  e.  ~P B ,  u  e.  ~P B  |->  ran  ( x  e.  t ,  y  e.  u  |->  ( x  .+  y ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1443    e. wcel 1886   _Vcvv 3044   ~Pcpw 3950   ran crn 4834   ` cfv 5581  (class class class)co 6288    |-> cmpt2 6290   Basecbs 15114   +g cplusg 15183   LSSumclsm 17279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-1st 6790  df-2nd 6791  df-lsm 17281
This theorem is referenced by:  lsmvalx  17284  oppglsm  17287  lsmpropd  17320
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