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Theorem dprdfeq0 17257
Description: The zero function is the only function that sums to zero in a direct product. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 14-Jul-2019.)
Hypotheses
Ref Expression
eldprdi.0  |-  .0.  =  ( 0g `  G )
eldprdi.w  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  h finSupp  .0.  }
eldprdi.1  |-  ( ph  ->  G dom DProd  S )
eldprdi.2  |-  ( ph  ->  dom  S  =  I )
eldprdi.3  |-  ( ph  ->  F  e.  W )
Assertion
Ref Expression
dprdfeq0  |-  ( ph  ->  ( ( G  gsumg  F )  =  .0.  <->  F  =  ( x  e.  I  |->  .0.  ) ) )
Distinct variable groups:    x, h, F    h, i, G, x   
h, I, i, x    ph, x    .0. , h, x    S, h, i, x
Allowed substitution hints:    ph( h, i)    F( i)    W( x, h, i)    .0. ( i)

Proof of Theorem dprdfeq0
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eldprdi.w . . . . . . 7  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  h finSupp  .0.  }
2 eldprdi.1 . . . . . . 7  |-  ( ph  ->  G dom DProd  S )
3 eldprdi.2 . . . . . . 7  |-  ( ph  ->  dom  S  =  I )
4 eldprdi.3 . . . . . . 7  |-  ( ph  ->  F  e.  W )
5 eqid 2454 . . . . . . 7  |-  ( Base `  G )  =  (
Base `  G )
61, 2, 3, 4, 5dprdff 17241 . . . . . 6  |-  ( ph  ->  F : I --> ( Base `  G ) )
76feqmptd 5901 . . . . 5  |-  ( ph  ->  F  =  ( x  e.  I  |->  ( F `
 x ) ) )
87adantr 463 . . . 4  |-  ( (
ph  /\  ( G  gsumg  F )  =  .0.  )  ->  F  =  ( x  e.  I  |->  ( F `
 x ) ) )
91, 2, 3, 4dprdfcl 17242 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  I )  ->  ( F `  x )  e.  ( S `  x
) )
109adantlr 712 . . . . . . . 8  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  ( F `  x )  e.  ( S `  x
) )
11 eldprdi.0 . . . . . . . . . . . 12  |-  .0.  =  ( 0g `  G )
122ad2antrr 723 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  G dom DProd  S )
133ad2antrr 723 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  dom  S  =  I )
14 simpr 459 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  x  e.  I )
15 eqid 2454 . . . . . . . . . . . . . 14  |-  ( y  e.  I  |->  if ( y  =  x ,  ( F `  x
) ,  .0.  )
)  =  ( y  e.  I  |->  if ( y  =  x ,  ( F `  x
) ,  .0.  )
)
1611, 1, 12, 13, 14, 10, 15dprdfid 17252 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  (
( y  e.  I  |->  if ( y  =  x ,  ( F `
 x ) ,  .0.  ) )  e.  W  /\  ( G 
gsumg  ( y  e.  I  |->  if ( y  =  x ,  ( F `
 x ) ,  .0.  ) ) )  =  ( F `  x ) ) )
1716simpld 457 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  (
y  e.  I  |->  if ( y  =  x ,  ( F `  x ) ,  .0.  ) )  e.  W
)
184ad2antrr 723 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  F  e.  W )
19 eqid 2454 . . . . . . . . . . . 12  |-  ( -g `  G )  =  (
-g `  G )
2011, 1, 12, 13, 17, 18, 19dprdfsub 17256 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  (
( ( y  e.  I  |->  if ( y  =  x ,  ( F `  x ) ,  .0.  ) )  oF ( -g `  G ) F )  e.  W  /\  ( G  gsumg  ( ( y  e.  I  |->  if ( y  =  x ,  ( F `  x ) ,  .0.  ) )  oF ( -g `  G ) F ) )  =  ( ( G  gsumg  ( y  e.  I  |->  if ( y  =  x ,  ( F `
 x ) ,  .0.  ) ) ) ( -g `  G
) ( G  gsumg  F ) ) ) )
2120simprd 461 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  ( G  gsumg  ( ( y  e.  I  |->  if ( y  =  x ,  ( F `  x ) ,  .0.  ) )  oF ( -g `  G ) F ) )  =  ( ( G  gsumg  ( y  e.  I  |->  if ( y  =  x ,  ( F `
 x ) ,  .0.  ) ) ) ( -g `  G
) ( G  gsumg  F ) ) )
222, 3dprddomcld 17227 . . . . . . . . . . . . 13  |-  ( ph  ->  I  e.  _V )
2322ad2antrr 723 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  I  e.  _V )
24 fvex 5858 . . . . . . . . . . . . . 14  |-  ( F `
 x )  e. 
_V
25 fvex 5858 . . . . . . . . . . . . . . 15  |-  ( 0g
`  G )  e. 
_V
2611, 25eqeltri 2538 . . . . . . . . . . . . . 14  |-  .0.  e.  _V
2724, 26ifex 3997 . . . . . . . . . . . . 13  |-  if ( y  =  x ,  ( F `  x
) ,  .0.  )  e.  _V
2827a1i 11 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  ->  if ( y  =  x ,  ( F `  x ) ,  .0.  )  e.  _V )
29 fvex 5858 . . . . . . . . . . . . 13  |-  ( F `
 y )  e. 
_V
3029a1i 11 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  ->  ( F `  y )  e.  _V )
31 eqidd 2455 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  (
y  e.  I  |->  if ( y  =  x ,  ( F `  x ) ,  .0.  ) )  =  ( y  e.  I  |->  if ( y  =  x ,  ( F `  x ) ,  .0.  ) ) )
326ad2antrr 723 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  F : I --> ( Base `  G ) )
3332feqmptd 5901 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  F  =  ( y  e.  I  |->  ( F `  y ) ) )
3423, 28, 30, 31, 33offval2 6529 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  (
( y  e.  I  |->  if ( y  =  x ,  ( F `
 x ) ,  .0.  ) )  oF ( -g `  G
) F )  =  ( y  e.  I  |->  ( if ( y  =  x ,  ( F `  x ) ,  .0.  ) (
-g `  G )
( F `  y
) ) ) )
3534oveq2d 6286 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  ( G  gsumg  ( ( y  e.  I  |->  if ( y  =  x ,  ( F `  x ) ,  .0.  ) )  oF ( -g `  G ) F ) )  =  ( G 
gsumg  ( y  e.  I  |->  ( if ( y  =  x ,  ( F `  x ) ,  .0.  ) (
-g `  G )
( F `  y
) ) ) ) )
3616simprd 461 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  ( G  gsumg  ( y  e.  I  |->  if ( y  =  x ,  ( F `
 x ) ,  .0.  ) ) )  =  ( F `  x ) )
37 simplr 753 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  ( G  gsumg  F )  =  .0.  )
3836, 37oveq12d 6288 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  (
( G  gsumg  ( y  e.  I  |->  if ( y  =  x ,  ( F `
 x ) ,  .0.  ) ) ) ( -g `  G
) ( G  gsumg  F ) )  =  ( ( F `  x ) ( -g `  G
)  .0.  ) )
39 dprdgrp 17233 . . . . . . . . . . . . 13  |-  ( G dom DProd  S  ->  G  e. 
Grp )
4012, 39syl 16 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  G  e.  Grp )
4132, 14ffvelrnd 6008 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  ( F `  x )  e.  ( Base `  G
) )
425, 11, 19grpsubid1 16322 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  ( F `  x )  e.  ( Base `  G
) )  ->  (
( F `  x
) ( -g `  G
)  .0.  )  =  ( F `  x
) )
4340, 41, 42syl2anc 659 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  (
( F `  x
) ( -g `  G
)  .0.  )  =  ( F `  x
) )
4438, 43eqtrd 2495 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  (
( G  gsumg  ( y  e.  I  |->  if ( y  =  x ,  ( F `
 x ) ,  .0.  ) ) ) ( -g `  G
) ( G  gsumg  F ) )  =  ( F `
 x ) )
4521, 35, 443eqtr3d 2503 . . . . . . . . 9  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  ( G  gsumg  ( y  e.  I  |->  ( if ( y  =  x ,  ( F `  x ) ,  .0.  ) (
-g `  G )
( F `  y
) ) ) )  =  ( F `  x ) )
46 eqid 2454 . . . . . . . . . 10  |-  (Cntz `  G )  =  (Cntz `  G )
47 grpmnd 16261 . . . . . . . . . . . 12  |-  ( G  e.  Grp  ->  G  e.  Mnd )
482, 39, 473syl 20 . . . . . . . . . . 11  |-  ( ph  ->  G  e.  Mnd )
4948ad2antrr 723 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  G  e.  Mnd )
505subgacs 16435 . . . . . . . . . . . . 13  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) ) )
51 acsmre 15141 . . . . . . . . . . . . 13  |-  ( (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
5240, 50, 513syl 20 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
53 imassrn 5336 . . . . . . . . . . . . . 14  |-  ( S
" ( I  \  { x } ) )  C_  ran  S
542, 3dprdf2 17235 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  S : I --> (SubGrp `  G ) )
5554ad2antrr 723 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  S : I --> (SubGrp `  G ) )
56 frn 5719 . . . . . . . . . . . . . . . 16  |-  ( S : I --> (SubGrp `  G )  ->  ran  S 
C_  (SubGrp `  G )
)
5755, 56syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  ran  S 
C_  (SubGrp `  G )
)
58 mresspw 15081 . . . . . . . . . . . . . . . 16  |-  ( (SubGrp `  G )  e.  (Moore `  ( Base `  G
) )  ->  (SubGrp `  G )  C_  ~P ( Base `  G )
)
5952, 58syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  (SubGrp `  G )  C_  ~P ( Base `  G )
)
6057, 59sstrd 3499 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  ran  S 
C_  ~P ( Base `  G
) )
6153, 60syl5ss 3500 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  ( S " ( I  \  { x } ) )  C_  ~P ( Base `  G ) )
62 sspwuni 4404 . . . . . . . . . . . . 13  |-  ( ( S " ( I 
\  { x }
) )  C_  ~P ( Base `  G )  <->  U. ( S " (
I  \  { x } ) )  C_  ( Base `  G )
)
6361, 62sylib 196 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  U. ( S " ( I  \  { x } ) )  C_  ( Base `  G ) )
64 eqid 2454 . . . . . . . . . . . . 13  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
6564mrccl 15100 . . . . . . . . . . . 12  |-  ( ( (SubGrp `  G )  e.  (Moore `  ( Base `  G ) )  /\  U. ( S " (
I  \  { x } ) )  C_  ( Base `  G )
)  ->  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( I  \  { x } ) ) )  e.  (SubGrp `  G ) )
6652, 63, 65syl2anc 659 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  (
(mrCls `  (SubGrp `  G
) ) `  U. ( S " ( I 
\  { x }
) ) )  e.  (SubGrp `  G )
)
67 subgsubm 16422 . . . . . . . . . . 11  |-  ( ( (mrCls `  (SubGrp `  G
) ) `  U. ( S " ( I 
\  { x }
) ) )  e.  (SubGrp `  G )  ->  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " (
I  \  { x } ) ) )  e.  (SubMnd `  G
) )
6866, 67syl 16 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  (
(mrCls `  (SubGrp `  G
) ) `  U. ( S " ( I 
\  { x }
) ) )  e.  (SubMnd `  G )
)
69 oveq1 6277 . . . . . . . . . . . . 13  |-  ( ( F `  x )  =  if ( y  =  x ,  ( F `  x ) ,  .0.  )  -> 
( ( F `  x ) ( -g `  G ) ( F `
 y ) )  =  ( if ( y  =  x ,  ( F `  x
) ,  .0.  )
( -g `  G ) ( F `  y
) ) )
7069eleq1d 2523 . . . . . . . . . . . 12  |-  ( ( F `  x )  =  if ( y  =  x ,  ( F `  x ) ,  .0.  )  -> 
( ( ( F `
 x ) (
-g `  G )
( F `  y
) )  e.  ( (mrCls `  (SubGrp `  G
) ) `  U. ( S " ( I 
\  { x }
) ) )  <->  ( if ( y  =  x ,  ( F `  x ) ,  .0.  ) ( -g `  G
) ( F `  y ) )  e.  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " (
I  \  { x } ) ) ) ) )
71 oveq1 6277 . . . . . . . . . . . . 13  |-  (  .0.  =  if ( y  =  x ,  ( F `  x ) ,  .0.  )  -> 
(  .0.  ( -g `  G ) ( F `
 y ) )  =  ( if ( y  =  x ,  ( F `  x
) ,  .0.  )
( -g `  G ) ( F `  y
) ) )
7271eleq1d 2523 . . . . . . . . . . . 12  |-  (  .0.  =  if ( y  =  x ,  ( F `  x ) ,  .0.  )  -> 
( (  .0.  ( -g `  G ) ( F `  y ) )  e.  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( I  \  { x } ) ) )  <->  ( if ( y  =  x ,  ( F `  x ) ,  .0.  ) ( -g `  G
) ( F `  y ) )  e.  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " (
I  \  { x } ) ) ) ) )
73 simpr 459 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  /\  y  =  x )  ->  y  =  x )
7473fveq2d 5852 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  /\  y  =  x )  ->  ( F `  y )  =  ( F `  x ) )
7574oveq2d 6286 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  /\  y  =  x )  ->  (
( F `  x
) ( -g `  G
) ( F `  y ) )  =  ( ( F `  x ) ( -g `  G ) ( F `
 x ) ) )
765, 11, 19grpsubid 16321 . . . . . . . . . . . . . . . 16  |-  ( ( G  e.  Grp  /\  ( F `  x )  e.  ( Base `  G
) )  ->  (
( F `  x
) ( -g `  G
) ( F `  x ) )  =  .0.  )
7740, 41, 76syl2anc 659 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  (
( F `  x
) ( -g `  G
) ( F `  x ) )  =  .0.  )
7811subg0cl 16408 . . . . . . . . . . . . . . . 16  |-  ( ( (mrCls `  (SubGrp `  G
) ) `  U. ( S " ( I 
\  { x }
) ) )  e.  (SubGrp `  G )  ->  .0.  e.  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( I  \  { x } ) ) ) )
7966, 78syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  .0.  e.  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " (
I  \  { x } ) ) ) )
8077, 79eqeltrd 2542 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  (
( F `  x
) ( -g `  G
) ( F `  x ) )  e.  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " (
I  \  { x } ) ) ) )
8180ad2antrr 723 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  /\  y  =  x )  ->  (
( F `  x
) ( -g `  G
) ( F `  x ) )  e.  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " (
I  \  { x } ) ) ) )
8275, 81eqeltrd 2542 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  /\  y  =  x )  ->  (
( F `  x
) ( -g `  G
) ( F `  y ) )  e.  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " (
I  \  { x } ) ) ) )
8366ad2antrr 723 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  /\  -.  y  =  x )  ->  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " (
I  \  { x } ) ) )  e.  (SubGrp `  G
) )
8483, 78syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  /\  -.  y  =  x )  ->  .0.  e.  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( I  \  { x } ) ) ) )
8552, 64, 63mrcssidd 15114 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  U. ( S " ( I  \  { x } ) )  C_  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( I  \  { x } ) ) ) )
8685ad2antrr 723 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  /\  -.  y  =  x )  ->  U. ( S "
( I  \  {
x } ) ) 
C_  ( (mrCls `  (SubGrp `  G ) ) `
 U. ( S
" ( I  \  { x } ) ) ) )
871, 12, 13, 18dprdfcl 17242 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  ->  ( F `  y )  e.  ( S `  y
) )
8887adantr 463 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  /\  -.  y  =  x )  ->  ( F `  y
)  e.  ( S `
 y ) )
89 ffn 5713 . . . . . . . . . . . . . . . . . 18  |-  ( S : I --> (SubGrp `  G )  ->  S  Fn  I )
9055, 89syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  S  Fn  I )
9190ad2antrr 723 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  /\  -.  y  =  x )  ->  S  Fn  I )
92 difssd 3618 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  /\  -.  y  =  x )  ->  ( I  \  {
x } )  C_  I )
93 df-ne 2651 . . . . . . . . . . . . . . . . . 18  |-  ( y  =/=  x  <->  -.  y  =  x )
94 eldifsn 4141 . . . . . . . . . . . . . . . . . . 19  |-  ( y  e.  ( I  \  { x } )  <-> 
( y  e.  I  /\  y  =/=  x
) )
9594biimpri 206 . . . . . . . . . . . . . . . . . 18  |-  ( ( y  e.  I  /\  y  =/=  x )  -> 
y  e.  ( I 
\  { x }
) )
9693, 95sylan2br 474 . . . . . . . . . . . . . . . . 17  |-  ( ( y  e.  I  /\  -.  y  =  x
)  ->  y  e.  ( I  \  { x } ) )
9796adantll 711 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  /\  -.  y  =  x )  ->  y  e.  ( I 
\  { x }
) )
98 fnfvima 6125 . . . . . . . . . . . . . . . 16  |-  ( ( S  Fn  I  /\  ( I  \  { x } )  C_  I  /\  y  e.  (
I  \  { x } ) )  -> 
( S `  y
)  e.  ( S
" ( I  \  { x } ) ) )
9991, 92, 97, 98syl3anc 1226 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  /\  -.  y  =  x )  ->  ( S `  y
)  e.  ( S
" ( I  \  { x } ) ) )
100 elunii 4240 . . . . . . . . . . . . . . 15  |-  ( ( ( F `  y
)  e.  ( S `
 y )  /\  ( S `  y )  e.  ( S "
( I  \  {
x } ) ) )  ->  ( F `  y )  e.  U. ( S " ( I 
\  { x }
) ) )
10188, 99, 100syl2anc 659 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  /\  -.  y  =  x )  ->  ( F `  y
)  e.  U. ( S " ( I  \  { x } ) ) )
10286, 101sseldd 3490 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  /\  -.  y  =  x )  ->  ( F `  y
)  e.  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( I  \  { x } ) ) ) )
10319subgsubcl 16411 . . . . . . . . . . . . 13  |-  ( ( ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " (
I  \  { x } ) ) )  e.  (SubGrp `  G
)  /\  .0.  e.  ( (mrCls `  (SubGrp `  G
) ) `  U. ( S " ( I 
\  { x }
) ) )  /\  ( F `  y )  e.  ( (mrCls `  (SubGrp `  G ) ) `
 U. ( S
" ( I  \  { x } ) ) ) )  -> 
(  .0.  ( -g `  G ) ( F `
 y ) )  e.  ( (mrCls `  (SubGrp `  G ) ) `
 U. ( S
" ( I  \  { x } ) ) ) )
10483, 84, 102, 103syl3anc 1226 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  /\  -.  y  =  x )  ->  (  .0.  ( -g `  G ) ( F `
 y ) )  e.  ( (mrCls `  (SubGrp `  G ) ) `
 U. ( S
" ( I  \  { x } ) ) ) )
10570, 72, 82, 104ifbothda 3964 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  ->  ( if ( y  =  x ,  ( F `  x ) ,  .0.  ) ( -g `  G
) ( F `  y ) )  e.  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " (
I  \  { x } ) ) ) )
106 eqid 2454 . . . . . . . . . . 11  |-  ( y  e.  I  |->  ( if ( y  =  x ,  ( F `  x ) ,  .0.  ) ( -g `  G
) ( F `  y ) ) )  =  ( y  e.  I  |->  ( if ( y  =  x ,  ( F `  x
) ,  .0.  )
( -g `  G ) ( F `  y
) ) )
107105, 106fmptd 6031 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  (
y  e.  I  |->  ( if ( y  =  x ,  ( F `
 x ) ,  .0.  ) ( -g `  G ) ( F `
 y ) ) ) : I --> ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( I  \  { x } ) ) ) )
10820simpld 457 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  (
( y  e.  I  |->  if ( y  =  x ,  ( F `
 x ) ,  .0.  ) )  oF ( -g `  G
) F )  e.  W )
10934, 108eqeltrrd 2543 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  (
y  e.  I  |->  ( if ( y  =  x ,  ( F `
 x ) ,  .0.  ) ( -g `  G ) ( F `
 y ) ) )  e.  W )
1101, 12, 13, 109, 46dprdfcntz 17244 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  ran  ( y  e.  I  |->  ( if ( y  =  x ,  ( F `  x ) ,  .0.  ) (
-g `  G )
( F `  y
) ) )  C_  ( (Cntz `  G ) `  ran  ( y  e.  I  |->  ( if ( y  =  x ,  ( F `  x
) ,  .0.  )
( -g `  G ) ( F `  y
) ) ) ) )
1111, 12, 13, 109dprdffsupp 17243 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  (
y  e.  I  |->  ( if ( y  =  x ,  ( F `
 x ) ,  .0.  ) ( -g `  G ) ( F `
 y ) ) ) finSupp  .0.  )
11211, 46, 49, 23, 68, 107, 110, 111gsumzsubmcl 17127 . . . . . . . . 9  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  ( G  gsumg  ( y  e.  I  |->  ( if ( y  =  x ,  ( F `  x ) ,  .0.  ) (
-g `  G )
( F `  y
) ) ) )  e.  ( (mrCls `  (SubGrp `  G ) ) `
 U. ( S
" ( I  \  { x } ) ) ) )
11345, 112eqeltrrd 2543 . . . . . . . 8  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  ( F `  x )  e.  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " (
I  \  { x } ) ) ) )
11410, 113elind 3674 . . . . . . 7  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  ( F `  x )  e.  ( ( S `  x )  i^i  (
(mrCls `  (SubGrp `  G
) ) `  U. ( S " ( I 
\  { x }
) ) ) ) )
11512, 13, 14, 11, 64dprddisj 17237 . . . . . . 7  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  (
( S `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( I  \  { x } ) ) ) )  =  {  .0.  } )
116114, 115eleqtrd 2544 . . . . . 6  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  ( F `  x )  e.  {  .0.  } )
117 elsni 4041 . . . . . 6  |-  ( ( F `  x )  e.  {  .0.  }  ->  ( F `  x
)  =  .0.  )
118116, 117syl 16 . . . . 5  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  ( F `  x )  =  .0.  )
119118mpteq2dva 4525 . . . 4  |-  ( (
ph  /\  ( G  gsumg  F )  =  .0.  )  ->  ( x  e.  I  |->  ( F `  x
) )  =  ( x  e.  I  |->  .0.  ) )
1208, 119eqtrd 2495 . . 3  |-  ( (
ph  /\  ( G  gsumg  F )  =  .0.  )  ->  F  =  ( x  e.  I  |->  .0.  )
)
121120ex 432 . 2  |-  ( ph  ->  ( ( G  gsumg  F )  =  .0.  ->  F  =  ( x  e.  I  |->  .0.  ) )
)
12211gsumz 16204 . . . 4  |-  ( ( G  e.  Mnd  /\  I  e.  _V )  ->  ( G  gsumg  ( x  e.  I  |->  .0.  ) )  =  .0.  )
12348, 22, 122syl2anc 659 . . 3  |-  ( ph  ->  ( G  gsumg  ( x  e.  I  |->  .0.  ) )  =  .0.  )
124 oveq2 6278 . . . 4  |-  ( F  =  ( x  e.  I  |->  .0.  )  ->  ( G  gsumg  F )  =  ( G  gsumg  ( x  e.  I  |->  .0.  ) ) )
125124eqeq1d 2456 . . 3  |-  ( F  =  ( x  e.  I  |->  .0.  )  ->  ( ( G  gsumg  F )  =  .0.  <->  ( G  gsumg  ( x  e.  I  |->  .0.  ) )  =  .0.  ) )
126123, 125syl5ibrcom 222 . 2  |-  ( ph  ->  ( F  =  ( x  e.  I  |->  .0.  )  ->  ( G  gsumg  F )  =  .0.  )
)
127121, 126impbid 191 1  |-  ( ph  ->  ( ( G  gsumg  F )  =  .0.  <->  F  =  ( x  e.  I  |->  .0.  ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   {crab 2808   _Vcvv 3106    \ cdif 3458    i^i cin 3460    C_ wss 3461   ifcif 3929   ~Pcpw 3999   {csn 4016   U.cuni 4235   class class class wbr 4439    |-> cmpt 4497   dom cdm 4988   ran crn 4989   "cima 4991    Fn wfn 5565   -->wf 5566   ` cfv 5570  (class class class)co 6270    oFcof 6511   X_cixp 7462   finSupp cfsupp 7821   Basecbs 14716   0gc0g 14929    gsumg cgsu 14930  Moorecmre 15071  mrClscmrc 15072  ACScacs 15074   Mndcmnd 16118  SubMndcsubmnd 16164   Grpcgrp 16252   -gcsg 16254  SubGrpcsubg 16394  Cntzccntz 16552   DProd cdprd 17219
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-tpos 6947  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-oi 7927  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-seq 12090  df-hash 12388  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-0g 14931  df-gsum 14932  df-mre 15075  df-mrc 15076  df-acs 15078  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-mhm 16165  df-submnd 16166  df-grp 16256  df-minusg 16257  df-sbg 16258  df-mulg 16259  df-subg 16397  df-ghm 16464  df-gim 16506  df-cntz 16554  df-oppg 16580  df-cmn 16999  df-dprd 17221
This theorem is referenced by:  dprdf11  17258
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