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Theorem lsmfval 16785
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 3118 . . 3  |-  ( G  e.  V  ->  G  e.  _V )
3 fveq2 5872 . . . . . . 7  |-  ( w  =  G  ->  ( Base `  w )  =  ( Base `  G
) )
4 lsmfval.v . . . . . . 7  |-  B  =  ( Base `  G
)
53, 4syl6eqr 2516 . . . . . 6  |-  ( w  =  G  ->  ( Base `  w )  =  B )
65pweqd 4020 . . . . 5  |-  ( w  =  G  ->  ~P ( Base `  w )  =  ~P B )
7 fveq2 5872 . . . . . . . . 9  |-  ( w  =  G  ->  ( +g  `  w )  =  ( +g  `  G
) )
8 lsmfval.a . . . . . . . . 9  |-  .+  =  ( +g  `  G )
97, 8syl6eqr 2516 . . . . . . . 8  |-  ( w  =  G  ->  ( +g  `  w )  = 
.+  )
109oveqd 6313 . . . . . . 7  |-  ( w  =  G  ->  (
x ( +g  `  w
) y )  =  ( x  .+  y
) )
1110mpt2eq3dv 6362 . . . . . 6  |-  ( w  =  G  ->  (
x  e.  t ,  y  e.  u  |->  ( x ( +g  `  w
) y ) )  =  ( x  e.  t ,  y  e.  u  |->  ( x  .+  y ) ) )
1211rneqd 5240 . . . . 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 6358 . . . 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 16783 . . . 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 5882 . . . . . . 7  |-  ( Base `  G )  e.  _V
164, 15eqeltri 2541 . . . . . 6  |-  B  e. 
_V
1716pwex 4639 . . . . 5  |-  ~P B  e.  _V
1817, 17mpt2ex 6876 . . . 4  |-  ( t  e.  ~P B ,  u  e.  ~P B  |->  ran  ( x  e.  t ,  y  e.  u  |->  ( x  .+  y ) ) )  e.  _V
1913, 14, 18fvmpt 5956 . . 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 16 . 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 2510 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 1395    e. wcel 1819   _Vcvv 3109   ~Pcpw 4015   ran crn 5009   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   Basecbs 14644   +g cplusg 14712   LSSumclsm 16781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-lsm 16783
This theorem is referenced by:  lsmvalx  16786  oppglsm  16789  lsmpropd  16822
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