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Theorem dprdfeq0 15572
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 2435 . . . . . . 7  |-  ( Base `  G )  =  (
Base `  G )
61, 2, 3, 4, 5dprdff 15562 . . . . . 6  |-  ( ph  ->  F : I --> ( Base `  G ) )
76feqmptd 5771 . . . . 5  |-  ( ph  ->  F  =  ( x  e.  I  |->  ( F `
 x ) ) )
87adantr 452 . . . 4  |-  ( (
ph  /\  ( G  gsumg  F )  =  .0.  )  ->  F  =  ( x  e.  I  |->  ( F `
 x ) ) )
91, 2, 3, 4dprdfcl 15563 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  I )  ->  ( F `  x )  e.  ( S `  x
) )
109adantlr 696 . . . . . . . 8  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  ( F `  x )  e.  ( S `  x
) )
11 eldprdi.0 . . . . . . . . . . . 12  |-  .0.  =  ( 0g `  G )
122ad2antrr 707 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  G dom DProd  S )
133ad2antrr 707 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  dom  S  =  I )
14 simpr 448 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  x  e.  I )
15 eqid 2435 . . . . . . . . . . . . . 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 15567 . . . . . . . . . . . . 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 446 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  (
y  e.  I  |->  if ( y  =  x ,  ( F `  x ) ,  .0.  ) )  e.  W
)
184ad2antrr 707 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  F  e.  W )
19 eqid 2435 . . . . . . . . . . . 12  |-  ( -g `  G )  =  (
-g `  G )
2011, 1, 12, 13, 17, 18, 19dprdfsub 15571 . . . . . . . . . . 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 450 . . . . . . . . . 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 15550 . . . . . . . . . . . . . . . 16  |-  Rel  dom DProd
2322brrelex2i 4911 . . . . . . . . . . . . . . 15  |-  ( G dom DProd  S  ->  S  e. 
_V )
24 dmexg 5122 . . . . . . . . . . . . . . 15  |-  ( S  e.  _V  ->  dom  S  e.  _V )
252, 23, 243syl 19 . . . . . . . . . . . . . 14  |-  ( ph  ->  dom  S  e.  _V )
263, 25eqeltrrd 2510 . . . . . . . . . . . . 13  |-  ( ph  ->  I  e.  _V )
2726ad2antrr 707 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  I  e.  _V )
28 fvex 5734 . . . . . . . . . . . . . 14  |-  ( F `
 x )  e. 
_V
29 fvex 5734 . . . . . . . . . . . . . . 15  |-  ( 0g
`  G )  e. 
_V
3011, 29eqeltri 2505 . . . . . . . . . . . . . 14  |-  .0.  e.  _V
3128, 30ifex 3789 . . . . . . . . . . . . 13  |-  if ( y  =  x ,  ( F `  x
) ,  .0.  )  e.  _V
3231a1i 11 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  ->  if ( y  =  x ,  ( F `  x ) ,  .0.  )  e.  _V )
33 fvex 5734 . . . . . . . . . . . . 13  |-  ( F `
 y )  e. 
_V
3433a1i 11 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  ->  ( F `  y )  e.  _V )
35 eqidd 2436 . . . . . . . . . . . 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 707 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  F : I --> ( Base `  G ) )
3736feqmptd 5771 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  F  =  ( y  e.  I  |->  ( F `  y ) ) )
3827, 32, 34, 35, 37offval2 6314 . . . . . . . . . . 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 6089 . . . . . . . . . 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 450 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  ( G  gsumg  ( y  e.  I  |->  if ( y  =  x ,  ( F `
 x ) ,  .0.  ) ) )  =  ( F `  x ) )
41 simplr 732 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  ( G  gsumg  F )  =  .0.  )
4240, 41oveq12d 6091 . . . . . . . . . . 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 15555 . . . . . . . . . . . . 13  |-  ( G dom DProd  S  ->  G  e. 
Grp )
4412, 43syl 16 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  G  e.  Grp )
4536, 14ffvelrnd 5863 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  ( F `  x )  e.  ( Base `  G
) )
465, 11, 19grpsubid1 14866 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  ( F `  x )  e.  ( Base `  G
) )  ->  (
( F `  x
) ( -g `  G
)  .0.  )  =  ( F `  x
) )
4744, 45, 46syl2anc 643 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  (
( F `  x
) ( -g `  G
)  .0.  )  =  ( F `  x
) )
4842, 47eqtrd 2467 . . . . . . . . . 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 ) )
4921, 39, 483eqtr3d 2475 . . . . . . . . 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 ) )
50 eqid 2435 . . . . . . . . . 10  |-  (Cntz `  G )  =  (Cntz `  G )
51 grpmnd 14809 . . . . . . . . . . . 12  |-  ( G  e.  Grp  ->  G  e.  Mnd )
522, 43, 513syl 19 . . . . . . . . . . 11  |-  ( ph  ->  G  e.  Mnd )
5352ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  G  e.  Mnd )
545subgacs 14967 . . . . . . . . . . . . 13  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) ) )
55 acsmre 13869 . . . . . . . . . . . . 13  |-  ( (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
5644, 54, 553syl 19 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
57 imassrn 5208 . . . . . . . . . . . . . 14  |-  ( S
" ( I  \  { x } ) )  C_  ran  S
582, 3dprdf2 15557 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  S : I --> (SubGrp `  G ) )
5958ad2antrr 707 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  S : I --> (SubGrp `  G ) )
60 frn 5589 . . . . . . . . . . . . . . . 16  |-  ( S : I --> (SubGrp `  G )  ->  ran  S 
C_  (SubGrp `  G )
)
6159, 60syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  ran  S 
C_  (SubGrp `  G )
)
62 mresspw 13809 . . . . . . . . . . . . . . . 16  |-  ( (SubGrp `  G )  e.  (Moore `  ( Base `  G
) )  ->  (SubGrp `  G )  C_  ~P ( Base `  G )
)
6356, 62syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  (SubGrp `  G )  C_  ~P ( Base `  G )
)
6461, 63sstrd 3350 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  ran  S 
C_  ~P ( Base `  G
) )
6557, 64syl5ss 3351 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  ( S " ( I  \  { x } ) )  C_  ~P ( Base `  G ) )
66 sspwuni 4168 . . . . . . . . . . . . 13  |-  ( ( S " ( I 
\  { x }
) )  C_  ~P ( Base `  G )  <->  U. ( S " (
I  \  { x } ) )  C_  ( Base `  G )
)
6765, 66sylib 189 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  U. ( S " ( I  \  { x } ) )  C_  ( Base `  G ) )
68 eqid 2435 . . . . . . . . . . . . 13  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
6968mrccl 13828 . . . . . . . . . . . 12  |-  ( ( (SubGrp `  G )  e.  (Moore `  ( Base `  G ) )  /\  U. ( S " (
I  \  { x } ) )  C_  ( Base `  G )
)  ->  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( I  \  { x } ) ) )  e.  (SubGrp `  G ) )
7056, 67, 69syl2anc 643 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  (
(mrCls `  (SubGrp `  G
) ) `  U. ( S " ( I 
\  { x }
) ) )  e.  (SubGrp `  G )
)
71 subgsubm 14954 . . . . . . . . . . 11  |-  ( ( (mrCls `  (SubGrp `  G
) ) `  U. ( S " ( I 
\  { x }
) ) )  e.  (SubGrp `  G )  ->  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " (
I  \  { x } ) ) )  e.  (SubMnd `  G
) )
7270, 71syl 16 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  (
(mrCls `  (SubGrp `  G
) ) `  U. ( S " ( I 
\  { x }
) ) )  e.  (SubMnd `  G )
)
73 oveq1 6080 . . . . . . . . . . . . 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
) ) )
7473eleq1d 2501 . . . . . . . . . . . 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 } ) ) ) ) )
75 oveq1 6080 . . . . . . . . . . . . 13  |-  (  .0.  =  if ( y  =  x ,  ( F `  x ) ,  .0.  )  -> 
(  .0.  ( -g `  G ) ( F `
 y ) )  =  ( if ( y  =  x ,  ( F `  x
) ,  .0.  )
( -g `  G ) ( F `  y
) ) )
7675eleq1d 2501 . . . . . . . . . . . 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 } ) ) ) ) )
77 simpr 448 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  /\  y  =  x )  ->  y  =  x )
7877fveq2d 5724 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  /\  y  =  x )  ->  ( F `  y )  =  ( F `  x ) )
7978oveq2d 6089 . . . . . . . . . . . . 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 ) ) )
805, 11, 19grpsubid 14865 . . . . . . . . . . . . . . . 16  |-  ( ( G  e.  Grp  /\  ( F `  x )  e.  ( Base `  G
) )  ->  (
( F `  x
) ( -g `  G
) ( F `  x ) )  =  .0.  )
8144, 45, 80syl2anc 643 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  (
( F `  x
) ( -g `  G
) ( F `  x ) )  =  .0.  )
8211subg0cl 14944 . . . . . . . . . . . . . . . 16  |-  ( ( (mrCls `  (SubGrp `  G
) ) `  U. ( S " ( I 
\  { x }
) ) )  e.  (SubGrp `  G )  ->  .0.  e.  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( I  \  { x } ) ) ) )
8370, 82syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  .0.  e.  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " (
I  \  { x } ) ) ) )
8481, 83eqeltrd 2509 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  (
( F `  x
) ( -g `  G
) ( F `  x ) )  e.  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " (
I  \  { x } ) ) ) )
8584ad2antrr 707 . . . . . . . . . . . . 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 } ) ) ) )
8679, 85eqeltrd 2509 . . . . . . . . . . . 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 } ) ) ) )
8770ad2antrr 707 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  /\  -.  y  =  x )  ->  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " (
I  \  { x } ) ) )  e.  (SubGrp `  G
) )
8887, 82syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  /\  -.  y  =  x )  ->  .0.  e.  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( I  \  { x } ) ) ) )
8956, 68, 67mrcssidd 13842 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  U. ( S " ( I  \  { x } ) )  C_  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( I  \  { x } ) ) ) )
9089ad2antrr 707 . . . . . . . . . . . . . 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 } ) ) ) )
911, 12, 13, 18dprdfcl 15563 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  ->  ( F `  y )  e.  ( S `  y
) )
9291adantr 452 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  /\  -.  y  =  x )  ->  ( F `  y
)  e.  ( S `
 y ) )
93 ffn 5583 . . . . . . . . . . . . . . . . . 18  |-  ( S : I --> (SubGrp `  G )  ->  S  Fn  I )
9459, 93syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  S  Fn  I )
9594ad2antrr 707 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  /\  -.  y  =  x )  ->  S  Fn  I )
96 difssd 3467 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  /\  -.  y  =  x )  ->  ( I  \  {
x } )  C_  I )
97 df-ne 2600 . . . . . . . . . . . . . . . . . 18  |-  ( y  =/=  x  <->  -.  y  =  x )
98 eldifsn 3919 . . . . . . . . . . . . . . . . . . 19  |-  ( y  e.  ( I  \  { x } )  <-> 
( y  e.  I  /\  y  =/=  x
) )
9998biimpri 198 . . . . . . . . . . . . . . . . . 18  |-  ( ( y  e.  I  /\  y  =/=  x )  -> 
y  e.  ( I 
\  { x }
) )
10097, 99sylan2br 463 . . . . . . . . . . . . . . . . 17  |-  ( ( y  e.  I  /\  -.  y  =  x
)  ->  y  e.  ( I  \  { x } ) )
101100adantll 695 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  /\  -.  y  =  x )  ->  y  e.  ( I 
\  { x }
) )
102 fnfvima 5968 . . . . . . . . . . . . . . . 16  |-  ( ( S  Fn  I  /\  ( I  \  { x } )  C_  I  /\  y  e.  (
I  \  { x } ) )  -> 
( S `  y
)  e.  ( S
" ( I  \  { x } ) ) )
10395, 96, 101, 102syl3anc 1184 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  /\  -.  y  =  x )  ->  ( S `  y
)  e.  ( S
" ( I  \  { x } ) ) )
104 elunii 4012 . . . . . . . . . . . . . . 15  |-  ( ( ( F `  y
)  e.  ( S `
 y )  /\  ( S `  y )  e.  ( S "
( I  \  {
x } ) ) )  ->  ( F `  y )  e.  U. ( S " ( I 
\  { x }
) ) )
10592, 103, 104syl2anc 643 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  /\  -.  y  =  x )  ->  ( F `  y
)  e.  U. ( S " ( I  \  { x } ) ) )
10690, 105sseldd 3341 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  /\  -.  y  =  x )  ->  ( F `  y
)  e.  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( I  \  { x } ) ) ) )
10719subgsubcl 14947 . . . . . . . . . . . . 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 } ) ) ) )
10887, 88, 106, 107syl3anc 1184 . . . . . . . . . . . 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 } ) ) ) )
10974, 76, 86, 108ifbothda 3761 . . . . . . . . . . 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 } ) ) ) )
110 eqid 2435 . . . . . . . . . . 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
) ) )
111109, 110fmptd 5885 . . . . . . . . . 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 } ) ) ) )
11220simpld 446 . . . . . . . . . . . 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 )
11338, 112eqeltrrd 2510 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  (
y  e.  I  |->  ( if ( y  =  x ,  ( F `
 x ) ,  .0.  ) ( -g `  G ) ( F `
 y ) ) )  e.  W )
1141, 12, 13, 113, 50dprdfcntz 15565 . . . . . . . . . 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
) ) ) ) )
1151, 12, 13, 113dprdffi 15564 . . . . . . . . . 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 )
11611, 50, 53, 27, 72, 111, 114, 115gsumzsubmcl 15515 . . . . . . . . 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 } ) ) ) )
11749, 116eqeltrrd 2510 . . . . . . . 8  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  ( F `  x )  e.  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " (
I  \  { x } ) ) ) )
118 elin 3522 . . . . . . . 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 }
) ) ) ) )
11910, 117, 118sylanbrc 646 . . . . . . 7  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  ( F `  x )  e.  ( ( S `  x )  i^i  (
(mrCls `  (SubGrp `  G
) ) `  U. ( S " ( I 
\  { x }
) ) ) ) )
12012, 13, 14, 11, 68dprddisj 15559 . . . . . . 7  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  (
( S `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( I  \  { x } ) ) ) )  =  {  .0.  } )
121119, 120eleqtrd 2511 . . . . . 6  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  ( F `  x )  e.  {  .0.  } )
122 elsni 3830 . . . . . 6  |-  ( ( F `  x )  e.  {  .0.  }  ->  ( F `  x
)  =  .0.  )
123121, 122syl 16 . . . . 5  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  ( F `  x )  =  .0.  )
124123mpteq2dva 4287 . . . 4  |-  ( (
ph  /\  ( G  gsumg  F )  =  .0.  )  ->  ( x  e.  I  |->  ( F `  x
) )  =  ( x  e.  I  |->  .0.  ) )
1258, 124eqtrd 2467 . . 3  |-  ( (
ph  /\  ( G  gsumg  F )  =  .0.  )  ->  F  =  ( x  e.  I  |->  .0.  )
)
126125ex 424 . 2  |-  ( ph  ->  ( ( G  gsumg  F )  =  .0.  ->  F  =  ( x  e.  I  |->  .0.  ) )
)
12711gsumz 14773 . . . 4  |-  ( ( G  e.  Mnd  /\  I  e.  _V )  ->  ( G  gsumg  ( x  e.  I  |->  .0.  ) )  =  .0.  )
12852, 26, 127syl2anc 643 . . 3  |-  ( ph  ->  ( G  gsumg  ( x  e.  I  |->  .0.  ) )  =  .0.  )
129 oveq2 6081 . . . 4  |-  ( F  =  ( x  e.  I  |->  .0.  )  ->  ( G  gsumg  F )  =  ( G  gsumg  ( x  e.  I  |->  .0.  ) ) )
130129eqeq1d 2443 . . 3  |-  ( F  =  ( x  e.  I  |->  .0.  )  ->  ( ( G  gsumg  F )  =  .0.  <->  ( G  gsumg  ( x  e.  I  |->  .0.  ) )  =  .0.  ) )
131128, 130syl5ibrcom 214 . 2  |-  ( ph  ->  ( F  =  ( x  e.  I  |->  .0.  )  ->  ( G  gsumg  F )  =  .0.  )
)
132126, 131impbid 184 1  |-  ( ph  ->  ( ( G  gsumg  F )  =  .0.  <->  F  =  ( x  e.  I  |->  .0.  ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   {crab 2701   _Vcvv 2948    \ cdif 3309    i^i cin 3311    C_ wss 3312   ifcif 3731   ~Pcpw 3791   {csn 3806   U.cuni 4007   class class class wbr 4204    e. cmpt 4258   `'ccnv 4869   dom cdm 4870   ran crn 4871   "cima 4873    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073    o Fcof 6295   X_cixp 7055   Fincfn 7101   Basecbs 13461   0gc0g 13715    gsumg cgsu 13716  Moorecmre 13799  mrClscmrc 13800  ACScacs 13802   Mndcmnd 14676   Grpcgrp 14677   -gcsg 14680  SubMndcsubmnd 14729  SubGrpcsubg 14930  Cntzccntz 15106   DProd cdprd 15546
This theorem is referenced by:  dprdf11  15573
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-tpos 6471  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-fzo 11128  df-seq 11316  df-hash 11611  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-0g 13719  df-gsum 13720  df-mre 13803  df-mrc 13804  df-acs 13806  df-mnd 14682  df-mhm 14730  df-submnd 14731  df-grp 14804  df-minusg 14805  df-sbg 14806  df-mulg 14807  df-subg 14933  df-ghm 14996  df-gim 15038  df-cntz 15108  df-oppg 15134  df-cmn 15406  df-dprd 15548
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