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Theorem dprdfeq0 15273
Description: The zero function is the only function that sums two zero in a direct product. (Contributed by Mario Carneiro, 25-Apr-2016.)
Hypotheses
Ref Expression
eldprdi.0  |-  .0.  =  ( 0g `  G )
eldprdi.w  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }
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 " ( _V  \  {  .0.  } ) )  e. 
Fin }
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 2296 . . . . . . 7  |-  ( Base `  G )  =  (
Base `  G )
61, 2, 3, 4, 5dprdff 15263 . . . . . 6  |-  ( ph  ->  F : I --> ( Base `  G ) )
76feqmptd 5591 . . . . 5  |-  ( ph  ->  F  =  ( x  e.  I  |->  ( F `
 x ) ) )
87adantr 451 . . . 4  |-  ( (
ph  /\  ( G  gsumg  F )  =  .0.  )  ->  F  =  ( x  e.  I  |->  ( F `
 x ) ) )
91, 2, 3, 4dprdfcl 15264 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  I )  ->  ( F `  x )  e.  ( S `  x
) )
109adantlr 695 . . . . . . . 8  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  ( F `  x )  e.  ( S `  x
) )
11 eldprdi.0 . . . . . . . . . . . 12  |-  .0.  =  ( 0g `  G )
122ad2antrr 706 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  G dom DProd  S )
133ad2antrr 706 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  dom  S  =  I )
14 simpr 447 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  x  e.  I )
15 eqid 2296 . . . . . . . . . . . . . 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 15268 . . . . . . . . . . . . 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 445 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  (
y  e.  I  |->  if ( y  =  x ,  ( F `  x ) ,  .0.  ) )  e.  W
)
184ad2antrr 706 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  F  e.  W )
19 eqid 2296 . . . . . . . . . . . 12  |-  ( -g `  G )  =  (
-g `  G )
2011, 1, 12, 13, 17, 18, 19dprdfsub 15272 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  (
( ( y  e.  I  |->  if ( y  =  x ,  ( F `  x ) ,  .0.  ) )  o F ( -g `  G ) F )  e.  W  /\  ( G  gsumg  ( ( y  e.  I  |->  if ( y  =  x ,  ( F `  x ) ,  .0.  ) )  o F ( -g `  G ) F ) )  =  ( ( G  gsumg  ( y  e.  I  |->  if ( y  =  x ,  ( F `
 x ) ,  .0.  ) ) ) ( -g `  G
) ( G  gsumg  F ) ) ) )
2120simprd 449 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  ( G  gsumg  ( ( y  e.  I  |->  if ( y  =  x ,  ( F `  x ) ,  .0.  ) )  o F ( -g `  G ) F ) )  =  ( ( G  gsumg  ( y  e.  I  |->  if ( y  =  x ,  ( F `
 x ) ,  .0.  ) ) ) ( -g `  G
) ( G  gsumg  F ) ) )
22 reldmdprd 15251 . . . . . . . . . . . . . . . 16  |-  Rel  dom DProd
2322brrelex2i 4746 . . . . . . . . . . . . . . 15  |-  ( G dom DProd  S  ->  S  e. 
_V )
24 dmexg 4955 . . . . . . . . . . . . . . 15  |-  ( S  e.  _V  ->  dom  S  e.  _V )
252, 23, 243syl 18 . . . . . . . . . . . . . 14  |-  ( ph  ->  dom  S  e.  _V )
263, 25eqeltrrd 2371 . . . . . . . . . . . . 13  |-  ( ph  ->  I  e.  _V )
2726ad2antrr 706 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  I  e.  _V )
28 fvex 5555 . . . . . . . . . . . . . 14  |-  ( F `
 x )  e. 
_V
29 fvex 5555 . . . . . . . . . . . . . . 15  |-  ( 0g
`  G )  e. 
_V
3011, 29eqeltri 2366 . . . . . . . . . . . . . 14  |-  .0.  e.  _V
3128, 30ifex 3636 . . . . . . . . . . . . 13  |-  if ( y  =  x ,  ( F `  x
) ,  .0.  )  e.  _V
3231a1i 10 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  ->  if ( y  =  x ,  ( F `  x ) ,  .0.  )  e.  _V )
33 fvex 5555 . . . . . . . . . . . . 13  |-  ( F `
 y )  e. 
_V
3433a1i 10 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  ->  ( F `  y )  e.  _V )
35 eqidd 2297 . . . . . . . . . . . 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.  ) ) )
366ad2antrr 706 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  F : I --> ( Base `  G ) )
3736feqmptd 5591 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  F  =  ( y  e.  I  |->  ( F `  y ) ) )
3827, 32, 34, 35, 37offval2 6111 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  (
( y  e.  I  |->  if ( y  =  x ,  ( F `
 x ) ,  .0.  ) )  o F ( -g `  G
) F )  =  ( y  e.  I  |->  ( if ( y  =  x ,  ( F `  x ) ,  .0.  ) (
-g `  G )
( F `  y
) ) ) )
3938oveq2d 5890 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  ( G  gsumg  ( ( y  e.  I  |->  if ( y  =  x ,  ( F `  x ) ,  .0.  ) )  o F ( -g `  G ) F ) )  =  ( G 
gsumg  ( y  e.  I  |->  ( if ( y  =  x ,  ( F `  x ) ,  .0.  ) (
-g `  G )
( F `  y
) ) ) ) )
4016simprd 449 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  ( G  gsumg  ( y  e.  I  |->  if ( y  =  x ,  ( F `
 x ) ,  .0.  ) ) )  =  ( F `  x ) )
41 simplr 731 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  ( G  gsumg  F )  =  .0.  )
4240, 41oveq12d 5892 . . . . . . . . . . 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.  ) )
43 dprdgrp 15256 . . . . . . . . . . . . 13  |-  ( G dom DProd  S  ->  G  e. 
Grp )
4412, 43syl 15 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  G  e.  Grp )
45 ffvelrn 5679 . . . . . . . . . . . . 13  |-  ( ( F : I --> ( Base `  G )  /\  x  e.  I )  ->  ( F `  x )  e.  ( Base `  G
) )
4636, 14, 45syl2anc 642 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  ( F `  x )  e.  ( Base `  G
) )
475, 11, 19grpsubid1 14567 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  ( F `  x )  e.  ( Base `  G
) )  ->  (
( F `  x
) ( -g `  G
)  .0.  )  =  ( F `  x
) )
4844, 46, 47syl2anc 642 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  (
( F `  x
) ( -g `  G
)  .0.  )  =  ( F `  x
) )
4942, 48eqtrd 2328 . . . . . . . . . 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 ) )
5021, 39, 493eqtr3d 2336 . . . . . . . . 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 ) )
51 eqid 2296 . . . . . . . . . 10  |-  (Cntz `  G )  =  (Cntz `  G )
52 grpmnd 14510 . . . . . . . . . . . 12  |-  ( G  e.  Grp  ->  G  e.  Mnd )
532, 43, 523syl 18 . . . . . . . . . . 11  |-  ( ph  ->  G  e.  Mnd )
5453ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  G  e.  Mnd )
555subgacs 14668 . . . . . . . . . . . . 13  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) ) )
56 acsmre 13570 . . . . . . . . . . . . 13  |-  ( (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
5744, 55, 563syl 18 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
58 imassrn 5041 . . . . . . . . . . . . . 14  |-  ( S
" ( I  \  { x } ) )  C_  ran  S
592, 3dprdf2 15258 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  S : I --> (SubGrp `  G ) )
6059ad2antrr 706 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  S : I --> (SubGrp `  G ) )
61 frn 5411 . . . . . . . . . . . . . . . 16  |-  ( S : I --> (SubGrp `  G )  ->  ran  S 
C_  (SubGrp `  G )
)
6260, 61syl 15 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  ran  S 
C_  (SubGrp `  G )
)
63 mresspw 13510 . . . . . . . . . . . . . . . 16  |-  ( (SubGrp `  G )  e.  (Moore `  ( Base `  G
) )  ->  (SubGrp `  G )  C_  ~P ( Base `  G )
)
6457, 63syl 15 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  (SubGrp `  G )  C_  ~P ( Base `  G )
)
6562, 64sstrd 3202 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  ran  S 
C_  ~P ( Base `  G
) )
6658, 65syl5ss 3203 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  ( S " ( I  \  { x } ) )  C_  ~P ( Base `  G ) )
67 sspwuni 4003 . . . . . . . . . . . . 13  |-  ( ( S " ( I 
\  { x }
) )  C_  ~P ( Base `  G )  <->  U. ( S " (
I  \  { x } ) )  C_  ( Base `  G )
)
6866, 67sylib 188 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  U. ( S " ( I  \  { x } ) )  C_  ( Base `  G ) )
69 eqid 2296 . . . . . . . . . . . . 13  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
7069mrccl 13529 . . . . . . . . . . . 12  |-  ( ( (SubGrp `  G )  e.  (Moore `  ( Base `  G ) )  /\  U. ( S " (
I  \  { x } ) )  C_  ( Base `  G )
)  ->  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( I  \  { x } ) ) )  e.  (SubGrp `  G ) )
7157, 68, 70syl2anc 642 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  (
(mrCls `  (SubGrp `  G
) ) `  U. ( S " ( I 
\  { x }
) ) )  e.  (SubGrp `  G )
)
72 subgsubm 14655 . . . . . . . . . . 11  |-  ( ( (mrCls `  (SubGrp `  G
) ) `  U. ( S " ( I 
\  { x }
) ) )  e.  (SubGrp `  G )  ->  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " (
I  \  { x } ) ) )  e.  (SubMnd `  G
) )
7371, 72syl 15 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  (
(mrCls `  (SubGrp `  G
) ) `  U. ( S " ( I 
\  { x }
) ) )  e.  (SubMnd `  G )
)
74 oveq1 5881 . . . . . . . . . . . . 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
) ) )
7574eleq1d 2362 . . . . . . . . . . . 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 } ) ) ) ) )
76 oveq1 5881 . . . . . . . . . . . . 13  |-  (  .0.  =  if ( y  =  x ,  ( F `  x ) ,  .0.  )  -> 
(  .0.  ( -g `  G ) ( F `
 y ) )  =  ( if ( y  =  x ,  ( F `  x
) ,  .0.  )
( -g `  G ) ( F `  y
) ) )
7776eleq1d 2362 . . . . . . . . . . . 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 } ) ) ) ) )
78 simpr 447 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  /\  y  =  x )  ->  y  =  x )
7978fveq2d 5545 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  /\  y  =  x )  ->  ( F `  y )  =  ( F `  x ) )
8079oveq2d 5890 . . . . . . . . . . . . 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 ) ) )
815, 11, 19grpsubid 14566 . . . . . . . . . . . . . . . 16  |-  ( ( G  e.  Grp  /\  ( F `  x )  e.  ( Base `  G
) )  ->  (
( F `  x
) ( -g `  G
) ( F `  x ) )  =  .0.  )
8244, 46, 81syl2anc 642 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  (
( F `  x
) ( -g `  G
) ( F `  x ) )  =  .0.  )
8311subg0cl 14645 . . . . . . . . . . . . . . . 16  |-  ( ( (mrCls `  (SubGrp `  G
) ) `  U. ( S " ( I 
\  { x }
) ) )  e.  (SubGrp `  G )  ->  .0.  e.  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( I  \  { x } ) ) ) )
8471, 83syl 15 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  .0.  e.  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " (
I  \  { x } ) ) ) )
8582, 84eqeltrd 2370 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  (
( F `  x
) ( -g `  G
) ( F `  x ) )  e.  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " (
I  \  { x } ) ) ) )
8685ad2antrr 706 . . . . . . . . . . . . 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 } ) ) ) )
8780, 86eqeltrd 2370 . . . . . . . . . . . 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 } ) ) ) )
8871ad2antrr 706 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  /\  -.  y  =  x )  ->  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " (
I  \  { x } ) ) )  e.  (SubGrp `  G
) )
8988, 83syl 15 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  /\  -.  y  =  x )  ->  .0.  e.  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( I  \  { x } ) ) ) )
9069mrcssid 13535 . . . . . . . . . . . . . . . 16  |-  ( ( (SubGrp `  G )  e.  (Moore `  ( Base `  G ) )  /\  U. ( S " (
I  \  { x } ) )  C_  ( Base `  G )
)  ->  U. ( S " ( I  \  { x } ) )  C_  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( I  \  { x } ) ) ) )
9157, 68, 90syl2anc 642 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  U. ( S " ( I  \  { x } ) )  C_  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( I  \  { x } ) ) ) )
9291ad2antrr 706 . . . . . . . . . . . . . 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 } ) ) ) )
931, 12, 13, 18dprdfcl 15264 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  ->  ( F `  y )  e.  ( S `  y
) )
9493adantr 451 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  /\  -.  y  =  x )  ->  ( F `  y
)  e.  ( S `
 y ) )
95 ffn 5405 . . . . . . . . . . . . . . . . . 18  |-  ( S : I --> (SubGrp `  G )  ->  S  Fn  I )
9660, 95syl 15 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  S  Fn  I )
9796ad2antrr 706 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  /\  -.  y  =  x )  ->  S  Fn  I )
98 difss 3316 . . . . . . . . . . . . . . . . 17  |-  ( I 
\  { x }
)  C_  I
9998a1i 10 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  /\  -.  y  =  x )  ->  ( I  \  {
x } )  C_  I )
100 df-ne 2461 . . . . . . . . . . . . . . . . . 18  |-  ( y  =/=  x  <->  -.  y  =  x )
101 eldifsn 3762 . . . . . . . . . . . . . . . . . . 19  |-  ( y  e.  ( I  \  { x } )  <-> 
( y  e.  I  /\  y  =/=  x
) )
102101biimpri 197 . . . . . . . . . . . . . . . . . 18  |-  ( ( y  e.  I  /\  y  =/=  x )  -> 
y  e.  ( I 
\  { x }
) )
103100, 102sylan2br 462 . . . . . . . . . . . . . . . . 17  |-  ( ( y  e.  I  /\  -.  y  =  x
)  ->  y  e.  ( I  \  { x } ) )
104103adantll 694 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  /\  -.  y  =  x )  ->  y  e.  ( I 
\  { x }
) )
105 fnfvima 5772 . . . . . . . . . . . . . . . 16  |-  ( ( S  Fn  I  /\  ( I  \  { x } )  C_  I  /\  y  e.  (
I  \  { x } ) )  -> 
( S `  y
)  e.  ( S
" ( I  \  { x } ) ) )
10697, 99, 104, 105syl3anc 1182 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  /\  -.  y  =  x )  ->  ( S `  y
)  e.  ( S
" ( I  \  { x } ) ) )
107 elunii 3848 . . . . . . . . . . . . . . 15  |-  ( ( ( F `  y
)  e.  ( S `
 y )  /\  ( S `  y )  e.  ( S "
( I  \  {
x } ) ) )  ->  ( F `  y )  e.  U. ( S " ( I 
\  { x }
) ) )
10894, 106, 107syl2anc 642 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  /\  -.  y  =  x )  ->  ( F `  y
)  e.  U. ( S " ( I  \  { x } ) ) )
10992, 108sseldd 3194 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  /\  -.  y  =  x )  ->  ( F `  y
)  e.  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( I  \  { x } ) ) ) )
11019subgsubcl 14648 . . . . . . . . . . . . 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 } ) ) ) )
11188, 89, 109, 110syl3anc 1182 . . . . . . . . . . . 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 } ) ) ) )
11275, 77, 87, 111ifbothda 3608 . . . . . . . . . . 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 } ) ) ) )
113 eqid 2296 . . . . . . . . . . 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
) ) )
114112, 113fmptd 5700 . . . . . . . . . 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 } ) ) ) )
11520simpld 445 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  (
( y  e.  I  |->  if ( y  =  x ,  ( F `
 x ) ,  .0.  ) )  o F ( -g `  G
) F )  e.  W )
11638, 115eqeltrrd 2371 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  (
y  e.  I  |->  ( if ( y  =  x ,  ( F `
 x ) ,  .0.  ) ( -g `  G ) ( F `
 y ) ) )  e.  W )
1171, 12, 13, 116, 51dprdfcntz 15266 . . . . . . . . . 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
) ) ) ) )
1181, 12, 13, 116dprdffi 15265 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  ( `' ( y  e.  I  |->  ( if ( y  =  x ,  ( F `  x
) ,  .0.  )
( -g `  G ) ( F `  y
) ) ) "
( _V  \  {  .0.  } ) )  e. 
Fin )
11911, 51, 54, 27, 73, 114, 117, 118gsumzsubmcl 15216 . . . . . . . . 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 } ) ) ) )
12050, 119eqeltrrd 2371 . . . . . . . 8  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  ( F `  x )  e.  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " (
I  \  { x } ) ) ) )
121 elin 3371 . . . . . . . 8  |-  ( ( F `  x )  e.  ( ( S `
 x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " (
I  \  { x } ) ) ) )  <->  ( ( F `
 x )  e.  ( S `  x
)  /\  ( F `  x )  e.  ( (mrCls `  (SubGrp `  G
) ) `  U. ( S " ( I 
\  { x }
) ) ) ) )
12210, 120, 121sylanbrc 645 . . . . . . 7  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  ( F `  x )  e.  ( ( S `  x )  i^i  (
(mrCls `  (SubGrp `  G
) ) `  U. ( S " ( I 
\  { x }
) ) ) ) )
12312, 13, 14, 11, 69dprddisj 15260 . . . . . . 7  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  (
( S `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( I  \  { x } ) ) ) )  =  {  .0.  } )
124122, 123eleqtrd 2372 . . . . . 6  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  ( F `  x )  e.  {  .0.  } )
125 elsni 3677 . . . . . 6  |-  ( ( F `  x )  e.  {  .0.  }  ->  ( F `  x
)  =  .0.  )
126124, 125syl 15 . . . . 5  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  ( F `  x )  =  .0.  )
127126mpteq2dva 4122 . . . 4  |-  ( (
ph  /\  ( G  gsumg  F )  =  .0.  )  ->  ( x  e.  I  |->  ( F `  x
) )  =  ( x  e.  I  |->  .0.  ) )
1288, 127eqtrd 2328 . . 3  |-  ( (
ph  /\  ( G  gsumg  F )  =  .0.  )  ->  F  =  ( x  e.  I  |->  .0.  )
)
129128ex 423 . 2  |-  ( ph  ->  ( ( G  gsumg  F )  =  .0.  ->  F  =  ( x  e.  I  |->  .0.  ) )
)
13011gsumz 14474 . . . 4  |-  ( ( G  e.  Mnd  /\  I  e.  _V )  ->  ( G  gsumg  ( x  e.  I  |->  .0.  ) )  =  .0.  )
13153, 26, 130syl2anc 642 . . 3  |-  ( ph  ->  ( G  gsumg  ( x  e.  I  |->  .0.  ) )  =  .0.  )
132 oveq2 5882 . . . 4  |-  ( F  =  ( x  e.  I  |->  .0.  )  ->  ( G  gsumg  F )  =  ( G  gsumg  ( x  e.  I  |->  .0.  ) ) )
133132eqeq1d 2304 . . 3  |-  ( F  =  ( x  e.  I  |->  .0.  )  ->  ( ( G  gsumg  F )  =  .0.  <->  ( G  gsumg  ( x  e.  I  |->  .0.  ) )  =  .0.  ) )
134131, 133syl5ibrcom 213 . 2  |-  ( ph  ->  ( F  =  ( x  e.  I  |->  .0.  )  ->  ( G  gsumg  F )  =  .0.  )
)
135129, 134impbid 183 1  |-  ( ph  ->  ( ( G  gsumg  F )  =  .0.  <->  F  =  ( x  e.  I  |->  .0.  ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   {crab 2560   _Vcvv 2801    \ cdif 3162    i^i cin 3164    C_ wss 3165   ifcif 3578   ~Pcpw 3638   {csn 3653   U.cuni 3843   class class class wbr 4039    e. cmpt 4093   `'ccnv 4704   dom cdm 4705   ran crn 4706   "cima 4708    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874    o Fcof 6092   X_cixp 6833   Fincfn 6879   Basecbs 13164   0gc0g 13416    gsumg cgsu 13417  Moorecmre 13500  mrClscmrc 13501  ACScacs 13503   Mndcmnd 14377   Grpcgrp 14378   -gcsg 14381  SubMndcsubmnd 14430  SubGrpcsubg 14631  Cntzccntz 14807   DProd cdprd 15247
This theorem is referenced by:  dprdf11  15274
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-fzo 10887  df-seq 11063  df-hash 11354  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-0g 13420  df-gsum 13421  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-mhm 14431  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-mulg 14508  df-subg 14634  df-ghm 14697  df-gim 14739  df-cntz 14809  df-oppg 14835  df-cmn 15107  df-dprd 15249
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