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Theorem lsmfval 16261
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 3087 . . 3  |-  ( G  e.  V  ->  G  e.  _V )
3 fveq2 5802 . . . . . . 7  |-  ( w  =  G  ->  ( Base `  w )  =  ( Base `  G
) )
4 lsmfval.v . . . . . . 7  |-  B  =  ( Base `  G
)
53, 4syl6eqr 2513 . . . . . 6  |-  ( w  =  G  ->  ( Base `  w )  =  B )
65pweqd 3976 . . . . 5  |-  ( w  =  G  ->  ~P ( Base `  w )  =  ~P B )
7 fveq2 5802 . . . . . . . . . 10  |-  ( w  =  G  ->  ( +g  `  w )  =  ( +g  `  G
) )
8 lsmfval.a . . . . . . . . . 10  |-  .+  =  ( +g  `  G )
97, 8syl6eqr 2513 . . . . . . . . 9  |-  ( w  =  G  ->  ( +g  `  w )  = 
.+  )
109oveqd 6220 . . . . . . . 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 6262 . . . . . 6  |-  ( w  =  G  ->  (
x  e.  t ,  y  e.  u  |->  ( x ( +g  `  w
) y ) )  =  ( x  e.  t ,  y  e.  u  |->  ( x  .+  y ) ) )
1312rneqd 5178 . . . . 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 6260 . . . 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 16259 . . . 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 5812 . . . . . . 7  |-  ( Base `  G )  e.  _V
174, 16eqeltri 2538 . . . . . 6  |-  B  e. 
_V
1817pwex 4586 . . . . 5  |-  ~P B  e.  _V
1918, 18mpt2ex 6763 . . . 4  |-  ( t  e.  ~P B ,  u  e.  ~P B  |->  ran  ( x  e.  t ,  y  e.  u  |->  ( x  .+  y ) ) )  e.  _V
2014, 15, 19fvmpt 5886 . . 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 2507 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 1370    e. wcel 1758   _Vcvv 3078   ~Pcpw 3971   ran crn 4952   ` cfv 5529  (class class class)co 6203    |-> cmpt2 6205   Basecbs 14295   +g cplusg 14360   LSSumclsm 16257
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-1st 6690  df-2nd 6691  df-lsm 16259
This theorem is referenced by:  lsmvalx  16262  oppglsm  16265  lsmpropd  16298
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