MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dpjidcl Structured version   Visualization version   Unicode version

Theorem dpjidcl 17739
Description: The key property of projections: the sum of all the projections of  A is  A. (Contributed by Mario Carneiro, 26-Apr-2016.) (Revised by AV, 14-Jul-2019.)
Hypotheses
Ref Expression
dpjfval.1  |-  ( ph  ->  G dom DProd  S )
dpjfval.2  |-  ( ph  ->  dom  S  =  I )
dpjfval.p  |-  P  =  ( GdProj S )
dpjidcl.3  |-  ( ph  ->  A  e.  ( G DProd 
S ) )
dpjidcl.0  |-  .0.  =  ( 0g `  G )
dpjidcl.w  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  h finSupp  .0.  }
Assertion
Ref Expression
dpjidcl  |-  ( ph  ->  ( ( x  e.  I  |->  ( ( P `
 x ) `  A ) )  e.  W  /\  A  =  ( G  gsumg  ( x  e.  I  |->  ( ( P `  x ) `  A
) ) ) ) )
Distinct variable groups:    x, h,  .0.    h, i, G, x    P, h, x    ph, i, x    h, I, i, x   
x, W    A, h, x    S, h, i, x
Allowed substitution hints:    ph( h)    A( i)    P( i)    W( h, i)    .0. ( i)

Proof of Theorem dpjidcl
Dummy variables  k 
f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dpjidcl.3 . . . 4  |-  ( ph  ->  A  e.  ( G DProd 
S ) )
2 dpjfval.2 . . . . 5  |-  ( ph  ->  dom  S  =  I )
3 dpjidcl.0 . . . . . 6  |-  .0.  =  ( 0g `  G )
4 dpjidcl.w . . . . . 6  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  h finSupp  .0.  }
53, 4eldprd 17684 . . . . 5  |-  ( dom 
S  =  I  -> 
( A  e.  ( G DProd  S )  <->  ( G dom DProd  S  /\  E. f  e.  W  A  =  ( G  gsumg  f ) ) ) )
62, 5syl 17 . . . 4  |-  ( ph  ->  ( A  e.  ( G DProd  S )  <->  ( G dom DProd  S  /\  E. f  e.  W  A  =  ( G  gsumg  f ) ) ) )
71, 6mpbid 215 . . 3  |-  ( ph  ->  ( G dom DProd  S  /\  E. f  e.  W  A  =  ( G  gsumg  f ) ) )
87simprd 469 . 2  |-  ( ph  ->  E. f  e.  W  A  =  ( G  gsumg  f ) )
9 dpjfval.1 . . . . 5  |-  ( ph  ->  G dom DProd  S )
109adantr 471 . . . 4  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  G dom DProd  S )
112adantr 471 . . . 4  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  dom  S  =  I )
129ad2antrr 737 . . . . . 6  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  G dom DProd  S )
132ad2antrr 737 . . . . . 6  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  dom  S  =  I )
14 dpjfval.p . . . . . 6  |-  P  =  ( GdProj S )
15 simpr 467 . . . . . 6  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  x  e.  I )
1612, 13, 14, 15dpjf 17738 . . . . 5  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( P `  x
) : ( G DProd 
S ) --> ( S `
 x ) )
171ad2antrr 737 . . . . 5  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  A  e.  ( G DProd 
S ) )
1816, 17ffvelrnd 6045 . . . 4  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( ( P `  x ) `  A
)  e.  ( S `
 x ) )
199, 2dprddomcld 17681 . . . . . . 7  |-  ( ph  ->  I  e.  _V )
20 mptexg 6159 . . . . . . 7  |-  ( I  e.  _V  ->  (
x  e.  I  |->  ( ( P `  x
) `  A )
)  e.  _V )
2119, 20syl 17 . . . . . 6  |-  ( ph  ->  ( x  e.  I  |->  ( ( P `  x ) `  A
) )  e.  _V )
2221adantr 471 . . . . 5  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  ( x  e.  I  |->  ( ( P `
 x ) `  A ) )  e. 
_V )
23 funmpt 5636 . . . . . 6  |-  Fun  (
x  e.  I  |->  ( ( P `  x
) `  A )
)
2423a1i 11 . . . . 5  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  Fun  ( x  e.  I  |->  ( ( P `  x ) `
 A ) ) )
25 simprl 769 . . . . . 6  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  f  e.  W
)
264, 10, 11, 25dprdffsupp 17695 . . . . 5  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  f finSupp  .0.  )
27 eldifi 3566 . . . . . . . 8  |-  ( x  e.  ( I  \ 
( f supp  .0.  )
)  ->  x  e.  I )
28 eqid 2461 . . . . . . . . . 10  |-  ( proj1 `  G )  =  ( proj1 `  G )
2912, 13, 14, 28, 15dpjval 17737 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( P `  x
)  =  ( ( S `  x ) ( proj1 `  G ) ( G DProd 
( S  |`  (
I  \  { x } ) ) ) ) )
3029fveq1d 5889 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( ( P `  x ) `  A
)  =  ( ( ( S `  x
) ( proj1 `  G ) ( G DProd 
( S  |`  (
I  \  { x } ) ) ) ) `  A ) )
3127, 30sylan2 481 . . . . . . 7  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( f supp  .0.  ) ) )  -> 
( ( P `  x ) `  A
)  =  ( ( ( S `  x
) ( proj1 `  G ) ( G DProd 
( S  |`  (
I  \  { x } ) ) ) ) `  A ) )
32 simplrr 776 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( f supp  .0.  ) ) )  ->  A  =  ( G  gsumg  f ) )
33 eqid 2461 . . . . . . . . . . 11  |-  ( Base `  G )  =  (
Base `  G )
34 eqid 2461 . . . . . . . . . . 11  |-  (Cntz `  G )  =  (Cntz `  G )
35 dprdgrp 17685 . . . . . . . . . . . . 13  |-  ( G dom DProd  S  ->  G  e. 
Grp )
36 grpmnd 16726 . . . . . . . . . . . . 13  |-  ( G  e.  Grp  ->  G  e.  Mnd )
3710, 35, 363syl 18 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  G  e.  Mnd )
3837adantr 471 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( f supp  .0.  ) ) )  ->  G  e.  Mnd )
3919ad2antrr 737 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( f supp  .0.  ) ) )  ->  I  e.  _V )
404, 10, 11, 25, 33dprdff 17693 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  f : I --> ( Base `  G
) )
4140adantr 471 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( f supp  .0.  ) ) )  -> 
f : I --> ( Base `  G ) )
4225adantr 471 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  f  e.  W )
434, 12, 13, 42, 34dprdfcntz 17696 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ran  f  C_  (
(Cntz `  G ) `  ran  f ) )
4427, 43sylan2 481 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( f supp  .0.  ) ) )  ->  ran  f  C_  ( (Cntz `  G ) `  ran  f ) )
45 snssi 4128 . . . . . . . . . . . . 13  |-  ( x  e.  ( I  \ 
( f supp  .0.  )
)  ->  { x }  C_  ( I  \ 
( f supp  .0.  )
) )
4645adantl 472 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( f supp  .0.  ) ) )  ->  { x }  C_  ( I  \  (
f supp  .0.  ) )
)
4746difss2d 3574 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( f supp  .0.  ) ) )  ->  { x }  C_  I )
48 suppssdm 6953 . . . . . . . . . . . . . . 15  |-  ( f supp 
.0.  )  C_  dom  f
49 fdm 5755 . . . . . . . . . . . . . . . 16  |-  ( f : I --> ( Base `  G )  ->  dom  f  =  I )
5040, 49syl 17 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  dom  f  =  I )
5148, 50syl5sseq 3491 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  ( f supp  .0.  )  C_  I )
5251adantr 471 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( f supp  .0.  ) ) )  -> 
( f supp  .0.  )  C_  I )
53 ssconb 3577 . . . . . . . . . . . . 13  |-  ( ( { x }  C_  I  /\  ( f supp  .0.  )  C_  I )  -> 
( { x }  C_  ( I  \  (
f supp  .0.  ) )  <->  ( f supp  .0.  )  C_  ( I  \  { x } ) ) )
5447, 52, 53syl2anc 671 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( f supp  .0.  ) ) )  -> 
( { x }  C_  ( I  \  (
f supp  .0.  ) )  <->  ( f supp  .0.  )  C_  ( I  \  { x } ) ) )
5546, 54mpbid 215 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( f supp  .0.  ) ) )  -> 
( f supp  .0.  )  C_  ( I  \  {
x } ) )
5626adantr 471 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( f supp  .0.  ) ) )  -> 
f finSupp  .0.  )
5733, 3, 34, 38, 39, 41, 44, 55, 56gsumzres 17591 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( f supp  .0.  ) ) )  -> 
( G  gsumg  ( f  |`  (
I  \  { x } ) ) )  =  ( G  gsumg  f ) )
5832, 57eqtr4d 2498 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( f supp  .0.  ) ) )  ->  A  =  ( G  gsumg  ( f  |`  ( I  \  { x } ) ) ) )
59 eqid 2461 . . . . . . . . . . 11  |-  { h  e.  X_ i  e.  ( I  \  { x } ) ( ( S  |`  ( I  \  { x } ) ) `  i )  |  h finSupp  .0.  }  =  { h  e.  X_ i  e.  ( I  \  {
x } ) ( ( S  |`  (
I  \  { x } ) ) `  i )  |  h finSupp  .0.  }
60 difss 3571 . . . . . . . . . . . . . 14  |-  ( I 
\  { x }
)  C_  I
6160a1i 11 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( I  \  {
x } )  C_  I )
6212, 13, 61dprdres 17709 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( G dom DProd  ( S  |`  ( I  \  {
x } ) )  /\  ( G DProd  ( S  |`  ( I  \  { x } ) ) )  C_  ( G DProd  S ) ) )
6362simpld 465 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  G dom DProd  ( S  |`  ( I  \  {
x } ) ) )
6412, 13dprdf2 17687 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  S : I --> (SubGrp `  G ) )
65 fssres 5771 . . . . . . . . . . . . 13  |-  ( ( S : I --> (SubGrp `  G )  /\  (
I  \  { x } )  C_  I
)  ->  ( S  |`  ( I  \  {
x } ) ) : ( I  \  { x } ) --> (SubGrp `  G )
)
6664, 60, 65sylancl 673 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( S  |`  (
I  \  { x } ) ) : ( I  \  {
x } ) --> (SubGrp `  G ) )
67 fdm 5755 . . . . . . . . . . . 12  |-  ( ( S  |`  ( I  \  { x } ) ) : ( I 
\  { x }
) --> (SubGrp `  G )  ->  dom  ( S  |`  ( I  \  { x } ) )  =  ( I  \  {
x } ) )
6866, 67syl 17 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  dom  ( S  |`  ( I  \  { x } ) )  =  ( I  \  {
x } ) )
6940adantr 471 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  f : I --> ( Base `  G ) )
7069feqmptd 5940 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  f  =  ( k  e.  I  |->  ( f `
 k ) ) )
7170reseq1d 5122 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( f  |`  (
I  \  { x } ) )  =  ( ( k  e.  I  |->  ( f `  k ) )  |`  ( I  \  { x } ) ) )
72 resmpt 5172 . . . . . . . . . . . . . 14  |-  ( ( I  \  { x } )  C_  I  ->  ( ( k  e.  I  |->  ( f `  k ) )  |`  ( I  \  { x } ) )  =  ( k  e.  ( I  \  { x } )  |->  ( f `
 k ) ) )
7360, 72ax-mp 5 . . . . . . . . . . . . 13  |-  ( ( k  e.  I  |->  ( f `  k ) )  |`  ( I  \  { x } ) )  =  ( k  e.  ( I  \  { x } ) 
|->  ( f `  k
) )
7471, 73syl6eq 2511 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( f  |`  (
I  \  { x } ) )  =  ( k  e.  ( I  \  { x } )  |->  ( f `
 k ) ) )
75 eldifi 3566 . . . . . . . . . . . . . . 15  |-  ( k  e.  ( I  \  { x } )  ->  k  e.  I
)
764, 12, 13, 42dprdfcl 17694 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I
)  /\  k  e.  I )  ->  (
f `  k )  e.  ( S `  k
) )
7775, 76sylan2 481 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I
)  /\  k  e.  ( I  \  { x } ) )  -> 
( f `  k
)  e.  ( S `
 k ) )
78 fvres 5901 . . . . . . . . . . . . . . 15  |-  ( k  e.  ( I  \  { x } )  ->  ( ( S  |`  ( I  \  {
x } ) ) `
 k )  =  ( S `  k
) )
7978adantl 472 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I
)  /\  k  e.  ( I  \  { x } ) )  -> 
( ( S  |`  ( I  \  { x } ) ) `  k )  =  ( S `  k ) )
8077, 79eleqtrrd 2542 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I
)  /\  k  e.  ( I  \  { x } ) )  -> 
( f `  k
)  e.  ( ( S  |`  ( I  \  { x } ) ) `  k ) )
81 difexg 4564 . . . . . . . . . . . . . . . . 17  |-  ( I  e.  _V  ->  (
I  \  { x } )  e.  _V )
8219, 81syl 17 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( I  \  {
x } )  e. 
_V )
83 mptexg 6159 . . . . . . . . . . . . . . . 16  |-  ( ( I  \  { x } )  e.  _V  ->  ( k  e.  ( I  \  { x } )  |->  ( f `
 k ) )  e.  _V )
8482, 83syl 17 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( k  e.  ( I  \  { x } )  |->  ( f `
 k ) )  e.  _V )
8584ad2antrr 737 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( k  e.  ( I  \  { x } )  |->  ( f `
 k ) )  e.  _V )
86 funmpt 5636 . . . . . . . . . . . . . . 15  |-  Fun  (
k  e.  ( I 
\  { x }
)  |->  ( f `  k ) )
8786a1i 11 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  Fun  ( k  e.  ( I  \  {
x } )  |->  ( f `  k ) ) )
8826adantr 471 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  f finSupp  .0.  )
89 ssdif 3579 . . . . . . . . . . . . . . . . . 18  |-  ( ( I  \  { x } )  C_  I  ->  ( ( I  \  { x } ) 
\  ( f supp  .0.  ) )  C_  (
I  \  ( f supp  .0.  ) ) )
9060, 89ax-mp 5 . . . . . . . . . . . . . . . . 17  |-  ( ( I  \  { x } )  \  (
f supp  .0.  ) )  C_  ( I  \  (
f supp  .0.  ) )
9190sseli 3439 . . . . . . . . . . . . . . . 16  |-  ( k  e.  ( ( I 
\  { x }
)  \  ( f supp  .0.  ) )  ->  k  e.  ( I  \  (
f supp  .0.  ) )
)
92 ssid 3462 . . . . . . . . . . . . . . . . . 18  |-  ( f supp 
.0.  )  C_  (
f supp  .0.  )
9392a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( f supp  .0.  )  C_  ( f supp  .0.  )
)
9419ad2antrr 737 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  I  e.  _V )
95 fvex 5897 . . . . . . . . . . . . . . . . . . 19  |-  ( 0g
`  G )  e. 
_V
963, 95eqeltri 2535 . . . . . . . . . . . . . . . . . 18  |-  .0.  e.  _V
9796a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  .0.  e.  _V )
9869, 93, 94, 97suppssr 6972 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I
)  /\  k  e.  ( I  \  (
f supp  .0.  ) )
)  ->  ( f `  k )  =  .0.  )
9991, 98sylan2 481 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I
)  /\  k  e.  ( ( I  \  { x } ) 
\  ( f supp  .0.  ) ) )  -> 
( f `  k
)  =  .0.  )
10082ad2antrr 737 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( I  \  {
x } )  e. 
_V )
10199, 100suppss2 6975 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( ( k  e.  ( I  \  {
x } )  |->  ( f `  k ) ) supp  .0.  )  C_  ( f supp  .0.  )
)
102 fsuppsssupp 7924 . . . . . . . . . . . . . 14  |-  ( ( ( ( k  e.  ( I  \  {
x } )  |->  ( f `  k ) )  e.  _V  /\  Fun  ( k  e.  ( I  \  { x } )  |->  ( f `
 k ) ) )  /\  ( f finSupp  .0.  /\  ( ( k  e.  ( I  \  { x } ) 
|->  ( f `  k
) ) supp  .0.  )  C_  ( f supp  .0.  )
) )  ->  (
k  e.  ( I 
\  { x }
)  |->  ( f `  k ) ) finSupp  .0.  )
10385, 87, 88, 101, 102syl22anc 1277 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( k  e.  ( I  \  { x } )  |->  ( f `
 k ) ) finSupp  .0.  )
10459, 63, 68, 80, 103dprdwd 17691 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( k  e.  ( I  \  { x } )  |->  ( f `
 k ) )  e.  { h  e.  X_ i  e.  (
I  \  { x } ) ( ( S  |`  ( I  \  { x } ) ) `  i )  |  h finSupp  .0.  } )
10574, 104eqeltrd 2539 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( f  |`  (
I  \  { x } ) )  e. 
{ h  e.  X_ i  e.  ( I  \  { x } ) ( ( S  |`  ( I  \  { x } ) ) `  i )  |  h finSupp  .0.  } )
1063, 59, 63, 68, 105eldprdi 17699 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( G  gsumg  ( f  |`  (
I  \  { x } ) ) )  e.  ( G DProd  ( S  |`  ( I  \  { x } ) ) ) )
10727, 106sylan2 481 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( f supp  .0.  ) ) )  -> 
( G  gsumg  ( f  |`  (
I  \  { x } ) ) )  e.  ( G DProd  ( S  |`  ( I  \  { x } ) ) ) )
10858, 107eqeltrd 2539 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( f supp  .0.  ) ) )  ->  A  e.  ( G DProd  ( S  |`  ( I  \  { x } ) ) ) )
109 eqid 2461 . . . . . . . . . 10  |-  ( +g  `  G )  =  ( +g  `  G )
110 eqid 2461 . . . . . . . . . 10  |-  ( LSSum `  G )  =  (
LSSum `  G )
11164, 15ffvelrnd 6045 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( S `  x
)  e.  (SubGrp `  G ) )
112 dprdsubg 17705 . . . . . . . . . . 11  |-  ( G dom DProd  ( S  |`  ( I  \  { x } ) )  -> 
( G DProd  ( S  |`  ( I  \  {
x } ) ) )  e.  (SubGrp `  G ) )
11363, 112syl 17 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( G DProd  ( S  |`  ( I  \  {
x } ) ) )  e.  (SubGrp `  G ) )
11412, 13, 15, 3dpjdisj 17734 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( ( S `  x )  i^i  ( G DProd  ( S  |`  (
I  \  { x } ) ) ) )  =  {  .0.  } )
11512, 13, 15, 34dpjcntz 17733 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( S `  x
)  C_  ( (Cntz `  G ) `  ( G DProd  ( S  |`  (
I  \  { x } ) ) ) ) )
116109, 110, 3, 34, 111, 113, 114, 115, 28pj1rid 17400 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I
)  /\  A  e.  ( G DProd  ( S  |`  ( I  \  {
x } ) ) ) )  ->  (
( ( S `  x ) ( proj1 `  G )
( G DProd  ( S  |`  ( I  \  {
x } ) ) ) ) `  A
)  =  .0.  )
11727, 116sylanl2 661 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( f supp 
.0.  ) ) )  /\  A  e.  ( G DProd  ( S  |`  ( I  \  { x } ) ) ) )  ->  ( (
( S `  x
) ( proj1 `  G ) ( G DProd 
( S  |`  (
I  \  { x } ) ) ) ) `  A )  =  .0.  )
118108, 117mpdan 679 . . . . . . 7  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( f supp  .0.  ) ) )  -> 
( ( ( S `
 x ) (
proj1 `  G
) ( G DProd  ( S  |`  ( I  \  { x } ) ) ) ) `  A )  =  .0.  )
11931, 118eqtrd 2495 . . . . . 6  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( f supp  .0.  ) ) )  -> 
( ( P `  x ) `  A
)  =  .0.  )
12019adantr 471 . . . . . 6  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  I  e.  _V )
121119, 120suppss2 6975 . . . . 5  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  ( ( x  e.  I  |->  ( ( P `  x ) `
 A ) ) supp 
.0.  )  C_  (
f supp  .0.  ) )
122 fsuppsssupp 7924 . . . . 5  |-  ( ( ( ( x  e.  I  |->  ( ( P `
 x ) `  A ) )  e. 
_V  /\  Fun  ( x  e.  I  |->  ( ( P `  x ) `
 A ) ) )  /\  ( f finSupp  .0.  /\  ( ( x  e.  I  |->  ( ( P `  x ) `
 A ) ) supp 
.0.  )  C_  (
f supp  .0.  ) )
)  ->  ( x  e.  I  |->  ( ( P `  x ) `
 A ) ) finSupp  .0.  )
12322, 24, 26, 121, 122syl22anc 1277 . . . 4  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  ( x  e.  I  |->  ( ( P `
 x ) `  A ) ) finSupp  .0.  )
1244, 10, 11, 18, 123dprdwd 17691 . . 3  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  ( x  e.  I  |->  ( ( P `
 x ) `  A ) )  e.  W )
125 simprr 771 . . . 4  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  A  =  ( G  gsumg  f ) )
12640feqmptd 5940 . . . . . 6  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  f  =  ( x  e.  I  |->  ( f `  x ) ) )
127 simplrr 776 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  A  =  ( G 
gsumg  f ) )
12812, 35, 363syl 18 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  G  e.  Mnd )
1294, 12, 13, 42dprdffsupp 17695 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  f finSupp  .0.  )
130 disjdif 3850 . . . . . . . . . . . . 13  |-  ( { x }  i^i  (
I  \  { x } ) )  =  (/)
131130a1i 11 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( { x }  i^i  ( I  \  {
x } ) )  =  (/) )
132 undif2 3854 . . . . . . . . . . . . 13  |-  ( { x }  u.  (
I  \  { x } ) )  =  ( { x }  u.  I )
13315snssd 4129 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  { x }  C_  I )
134 ssequn1 3615 . . . . . . . . . . . . . 14  |-  ( { x }  C_  I  <->  ( { x }  u.  I )  =  I )
135133, 134sylib 201 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( { x }  u.  I )  =  I )
136132, 135syl5req 2508 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  I  =  ( { x }  u.  (
I  \  { x } ) ) )
13733, 3, 109, 34, 128, 94, 69, 43, 129, 131, 136gsumzsplit 17608 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( G  gsumg  f )  =  ( ( G  gsumg  ( f  |`  { x } ) ) ( +g  `  G ) ( G  gsumg  ( f  |`  (
I  \  { x } ) ) ) ) )
13869, 133feqresmpt 5941 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( f  |`  { x } )  =  ( k  e.  { x }  |->  ( f `  k ) ) )
139138oveq2d 6330 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( G  gsumg  ( f  |`  { x } ) )  =  ( G  gsumg  ( k  e.  {
x }  |->  ( f `
 k ) ) ) )
14069, 15ffvelrnd 6045 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( f `  x
)  e.  ( Base `  G ) )
141 fveq2 5887 . . . . . . . . . . . . . . 15  |-  ( k  =  x  ->  (
f `  k )  =  ( f `  x ) )
14233, 141gsumsn 17635 . . . . . . . . . . . . . 14  |-  ( ( G  e.  Mnd  /\  x  e.  I  /\  ( f `  x
)  e.  ( Base `  G ) )  -> 
( G  gsumg  ( k  e.  {
x }  |->  ( f `
 k ) ) )  =  ( f `
 x ) )
143128, 15, 140, 142syl3anc 1276 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( G  gsumg  ( k  e.  {
x }  |->  ( f `
 k ) ) )  =  ( f `
 x ) )
144139, 143eqtrd 2495 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( G  gsumg  ( f  |`  { x } ) )  =  ( f `  x
) )
145144oveq1d 6329 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( ( G  gsumg  ( f  |`  { x } ) ) ( +g  `  G
) ( G  gsumg  ( f  |`  ( I  \  {
x } ) ) ) )  =  ( ( f `  x
) ( +g  `  G
) ( G  gsumg  ( f  |`  ( I  \  {
x } ) ) ) ) )
146127, 137, 1453eqtrd 2499 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  A  =  ( ( f `  x ) ( +g  `  G
) ( G  gsumg  ( f  |`  ( I  \  {
x } ) ) ) ) )
14712, 13, 15, 110dpjlsm 17735 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( G DProd  S )  =  ( ( S `
 x ) (
LSSum `  G ) ( G DProd  ( S  |`  ( I  \  { x } ) ) ) ) )
14817, 147eleqtrd 2541 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  A  e.  ( ( S `  x ) ( LSSum `  G )
( G DProd  ( S  |`  ( I  \  {
x } ) ) ) ) )
1494, 10, 11, 25dprdfcl 17694 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( f `  x
)  e.  ( S `
 x ) )
150109, 110, 3, 34, 111, 113, 114, 115, 28, 148, 149, 106pj1eq 17398 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( A  =  ( ( f `  x
) ( +g  `  G
) ( G  gsumg  ( f  |`  ( I  \  {
x } ) ) ) )  <->  ( (
( ( S `  x ) ( proj1 `  G )
( G DProd  ( S  |`  ( I  \  {
x } ) ) ) ) `  A
)  =  ( f `
 x )  /\  ( ( ( G DProd 
( S  |`  (
I  \  { x } ) ) ) ( proj1 `  G ) ( S `
 x ) ) `
 A )  =  ( G  gsumg  ( f  |`  (
I  \  { x } ) ) ) ) ) )
151146, 150mpbid 215 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( ( ( ( S `  x ) ( proj1 `  G ) ( G DProd 
( S  |`  (
I  \  { x } ) ) ) ) `  A )  =  ( f `  x )  /\  (
( ( G DProd  ( S  |`  ( I  \  { x } ) ) ) ( proj1 `  G )
( S `  x
) ) `  A
)  =  ( G 
gsumg  ( f  |`  (
I  \  { x } ) ) ) ) )
152151simpld 465 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( ( ( S `
 x ) (
proj1 `  G
) ( G DProd  ( S  |`  ( I  \  { x } ) ) ) ) `  A )  =  ( f `  x ) )
15330, 152eqtrd 2495 . . . . . . 7  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( ( P `  x ) `  A
)  =  ( f `
 x ) )
154153mpteq2dva 4502 . . . . . 6  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  ( x  e.  I  |->  ( ( P `
 x ) `  A ) )  =  ( x  e.  I  |->  ( f `  x
) ) )
155126, 154eqtr4d 2498 . . . . 5  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  f  =  ( x  e.  I  |->  ( ( P `  x
) `  A )
) )
156155oveq2d 6330 . . . 4  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  ( G  gsumg  f )  =  ( G  gsumg  ( x  e.  I  |->  ( ( P `  x ) `
 A ) ) ) )
157125, 156eqtrd 2495 . . 3  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  A  =  ( G  gsumg  ( x  e.  I  |->  ( ( P `  x ) `  A
) ) ) )
158124, 157jca 539 . 2  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  ( ( x  e.  I  |->  ( ( P `  x ) `
 A ) )  e.  W  /\  A  =  ( G  gsumg  ( x  e.  I  |->  ( ( P `  x ) `
 A ) ) ) ) )
1598, 158rexlimddv 2894 1  |-  ( ph  ->  ( ( x  e.  I  |->  ( ( P `
 x ) `  A ) )  e.  W  /\  A  =  ( G  gsumg  ( x  e.  I  |->  ( ( P `  x ) `  A
) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    = wceq 1454    e. wcel 1897   E.wrex 2749   {crab 2752   _Vcvv 3056    \ cdif 3412    u. cun 3413    i^i cin 3414    C_ wss 3415   (/)c0 3742   {csn 3979   class class class wbr 4415    |-> cmpt 4474   dom cdm 4852   ran crn 4853    |` cres 4854   Fun wfun 5594   -->wf 5596   ` cfv 5600  (class class class)co 6314   supp csupp 6940   X_cixp 7547   finSupp cfsupp 7908   Basecbs 15169   +g cplusg 15238   0gc0g 15386    gsumg cgsu 15387   Mndcmnd 16583   Grpcgrp 16717  SubGrpcsubg 16859  Cntzccntz 17017   LSSumclsm 17334   proj1cpj1 17335   DProd cdprd 17673  dProjcdpj 17674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-inf2 8171  ax-cnex 9620  ax-resscn 9621  ax-1cn 9622  ax-icn 9623  ax-addcl 9624  ax-addrcl 9625  ax-mulcl 9626  ax-mulrcl 9627  ax-mulcom 9628  ax-addass 9629  ax-mulass 9630  ax-distr 9631  ax-i2m1 9632  ax-1ne0 9633  ax-1rid 9634  ax-rnegex 9635  ax-rrecex 9636  ax-cnre 9637  ax-pre-lttri 9638  ax-pre-lttrn 9639  ax-pre-ltadd 9640  ax-pre-mulgt0 9641
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-nel 2635  df-ral 2753  df-rex 2754  df-reu 2755  df-rmo 2756  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-int 4248  df-iun 4293  df-iin 4294  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-se 4812  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-isom 5609  df-riota 6276  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-of 6557  df-om 6719  df-1st 6819  df-2nd 6820  df-supp 6941  df-tpos 6998  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-1o 7207  df-oadd 7211  df-er 7388  df-map 7499  df-ixp 7548  df-en 7595  df-dom 7596  df-sdom 7597  df-fin 7598  df-fsupp 7909  df-oi 8050  df-card 8398  df-pnf 9702  df-mnf 9703  df-xr 9704  df-ltxr 9705  df-le 9706  df-sub 9887  df-neg 9888  df-nn 10637  df-2 10695  df-n0 10898  df-z 10966  df-uz 11188  df-fz 11813  df-fzo 11946  df-seq 12245  df-hash 12547  df-ndx 15172  df-slot 15173  df-base 15174  df-sets 15175  df-ress 15176  df-plusg 15251  df-0g 15388  df-gsum 15389  df-mre 15540  df-mrc 15541  df-acs 15543  df-mgm 16536  df-sgrp 16575  df-mnd 16585  df-mhm 16630  df-submnd 16631  df-grp 16721  df-minusg 16722  df-sbg 16723  df-mulg 16724  df-subg 16862  df-ghm 16929  df-gim 16971  df-cntz 17019  df-oppg 17045  df-lsm 17336  df-pj1 17337  df-cmn 17480  df-dprd 17675  df-dpj 17676
This theorem is referenced by:  dpjeq  17740  dpjid  17741
  Copyright terms: Public domain W3C validator