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Theorem lsmfval 16128
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 2976 . . 3  |-  ( G  e.  V  ->  G  e.  _V )
3 fveq2 5686 . . . . . . 7  |-  ( w  =  G  ->  ( Base `  w )  =  ( Base `  G
) )
4 lsmfval.v . . . . . . 7  |-  B  =  ( Base `  G
)
53, 4syl6eqr 2488 . . . . . 6  |-  ( w  =  G  ->  ( Base `  w )  =  B )
65pweqd 3860 . . . . 5  |-  ( w  =  G  ->  ~P ( Base `  w )  =  ~P B )
7 fveq2 5686 . . . . . . . . . 10  |-  ( w  =  G  ->  ( +g  `  w )  =  ( +g  `  G
) )
8 lsmfval.a . . . . . . . . . 10  |-  .+  =  ( +g  `  G )
97, 8syl6eqr 2488 . . . . . . . . 9  |-  ( w  =  G  ->  ( +g  `  w )  = 
.+  )
109oveqd 6103 . . . . . . . 8  |-  ( w  =  G  ->  (
x ( +g  `  w
) y )  =  ( x  .+  y
) )
11103ad2ant1 1009 . . . . . . 7  |-  ( ( w  =  G  /\  x  e.  t  /\  y  e.  u )  ->  ( x ( +g  `  w ) y )  =  ( x  .+  y ) )
1211mpt2eq3dva 6145 . . . . . 6  |-  ( w  =  G  ->  (
x  e.  t ,  y  e.  u  |->  ( x ( +g  `  w
) y ) )  =  ( x  e.  t ,  y  e.  u  |->  ( x  .+  y ) ) )
1312rneqd 5062 . . . . 5  |-  ( w  =  G  ->  ran  ( x  e.  t ,  y  e.  u  |->  ( x ( +g  `  w ) y ) )  =  ran  (
x  e.  t ,  y  e.  u  |->  ( x  .+  y ) ) )
146, 6, 13mpt2eq123dv 6143 . . . 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 ) ) ) )
15 df-lsm 16126 . . . 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 ) ) ) )
16 fvex 5696 . . . . . . 7  |-  ( Base `  G )  e.  _V
174, 16eqeltri 2508 . . . . . 6  |-  B  e. 
_V
1817pwex 4470 . . . . 5  |-  ~P B  e.  _V
1918, 18mpt2ex 6645 . . . 4  |-  ( t  e.  ~P B ,  u  e.  ~P B  |->  ran  ( x  e.  t ,  y  e.  u  |->  ( x  .+  y ) ) )  e.  _V
2014, 15, 19fvmpt 5769 . . 3  |-  ( G  e.  _V  ->  ( LSSum `  G )  =  ( t  e.  ~P B ,  u  e.  ~P B  |->  ran  (
x  e.  t ,  y  e.  u  |->  ( x  .+  y ) ) ) )
212, 20syl 16 . 2  |-  ( G  e.  V  ->  ( LSSum `  G )  =  ( t  e.  ~P B ,  u  e.  ~P B  |->  ran  (
x  e.  t ,  y  e.  u  |->  ( x  .+  y ) ) ) )
221, 21syl5eq 2482 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 1369    e. wcel 1756   _Vcvv 2967   ~Pcpw 3855   ran crn 4836   ` cfv 5413  (class class class)co 6086    e. cmpt2 6088   Basecbs 14166   +g cplusg 14230   LSSumclsm 16124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-1st 6572  df-2nd 6573  df-lsm 16126
This theorem is referenced by:  lsmvalx  16129  oppglsm  16132  lsmpropd  16165
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