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Theorem dprdfeq0OLD 17196
Description: The zero function is the only function that sums two zero in a direct product. (Contributed by Mario Carneiro, 25-Apr-2016.) Obsolete version of dprdfeq0 17189 as of 14-Jul-2019. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
eldprdiOLD.0  |-  .0.  =  ( 0g `  G )
eldprdiOLD.w  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }
eldprdiOLD.1  |-  ( ph  ->  G dom DProd  S )
eldprdiOLD.2  |-  ( ph  ->  dom  S  =  I )
eldprdiOLD.3  |-  ( ph  ->  F  e.  W )
Assertion
Ref Expression
dprdfeq0OLD  |-  ( 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 dprdfeq0OLD
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eldprdiOLD.w . . . . . . 7  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }
2 eldprdiOLD.1 . . . . . . 7  |-  ( ph  ->  G dom DProd  S )
3 eldprdiOLD.2 . . . . . . 7  |-  ( ph  ->  dom  S  =  I )
4 eldprdiOLD.3 . . . . . . 7  |-  ( ph  ->  F  e.  W )
5 eqid 2457 . . . . . . 7  |-  ( Base `  G )  =  (
Base `  G )
61, 2, 3, 4, 5dprdffOLD 17179 . . . . . 6  |-  ( ph  ->  F : I --> ( Base `  G ) )
76feqmptd 5926 . . . . 5  |-  ( ph  ->  F  =  ( x  e.  I  |->  ( F `
 x ) ) )
87adantr 465 . . . 4  |-  ( (
ph  /\  ( G  gsumg  F )  =  .0.  )  ->  F  =  ( x  e.  I  |->  ( F `
 x ) ) )
91, 2, 3, 4dprdfclOLD 17180 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  I )  ->  ( F `  x )  e.  ( S `  x
) )
109adantlr 714 . . . . . . . 8  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  ( F `  x )  e.  ( S `  x
) )
11 eldprdiOLD.0 . . . . . . . . . . . 12  |-  .0.  =  ( 0g `  G )
122ad2antrr 725 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  G dom DProd  S )
133ad2antrr 725 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  dom  S  =  I )
14 simpr 461 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  x  e.  I )
15 eqid 2457 . . . . . . . . . . . . . 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, 15dprdfidOLD 17191 . . . . . . . . . . . . 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 459 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  (
y  e.  I  |->  if ( y  =  x ,  ( F `  x ) ,  .0.  ) )  e.  W
)
184ad2antrr 725 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  F  e.  W )
19 eqid 2457 . . . . . . . . . . . 12  |-  ( -g `  G )  =  (
-g `  G )
2011, 1, 12, 13, 17, 18, 19dprdfsubOLD 17195 . . . . . . . . . . 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 463 . . . . . . . . . 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 ) ) )
22 reldmdprd 17155 . . . . . . . . . . . . . . . 16  |-  Rel  dom DProd
2322brrelex2i 5050 . . . . . . . . . . . . . . 15  |-  ( G dom DProd  S  ->  S  e. 
_V )
24 dmexg 6730 . . . . . . . . . . . . . . 15  |-  ( S  e.  _V  ->  dom  S  e.  _V )
252, 23, 243syl 20 . . . . . . . . . . . . . 14  |-  ( ph  ->  dom  S  e.  _V )
263, 25eqeltrrd 2546 . . . . . . . . . . . . 13  |-  ( ph  ->  I  e.  _V )
2726ad2antrr 725 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  I  e.  _V )
28 fvex 5882 . . . . . . . . . . . . . 14  |-  ( F `
 x )  e. 
_V
29 fvex 5882 . . . . . . . . . . . . . . 15  |-  ( 0g
`  G )  e. 
_V
3011, 29eqeltri 2541 . . . . . . . . . . . . . 14  |-  .0.  e.  _V
3128, 30ifex 4013 . . . . . . . . . . . . 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 5882 . . . . . . . . . . . . 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 2458 . . . . . . . . . . . 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 725 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  F : I --> ( Base `  G ) )
3736feqmptd 5926 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  F  =  ( y  e.  I  |->  ( F `  y ) ) )
3827, 32, 34, 35, 37offval2 6555 . . . . . . . . . . 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
) ) ) )
3938oveq2d 6312 . . . . . . . . . 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
) ) ) ) )
4016simprd 463 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  ( G  gsumg  ( y  e.  I  |->  if ( y  =  x ,  ( F `
 x ) ,  .0.  ) ) )  =  ( F `  x ) )
41 simplr 755 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  ( G  gsumg  F )  =  .0.  )
4240, 41oveq12d 6314 . . . . . . . . . . 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 17165 . . . . . . . . . . . . 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 6033 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  ( F `  x )  e.  ( Base `  G
) )
465, 11, 19grpsubid1 16250 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  ( F `  x )  e.  ( Base `  G
) )  ->  (
( F `  x
) ( -g `  G
)  .0.  )  =  ( F `  x
) )
4744, 45, 46syl2anc 661 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  (
( F `  x
) ( -g `  G
)  .0.  )  =  ( F `  x
) )
4842, 47eqtrd 2498 . . . . . . . . . 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 2506 . . . . . . . . 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 2457 . . . . . . . . . 10  |-  (Cntz `  G )  =  (Cntz `  G )
51 grpmnd 16189 . . . . . . . . . . . 12  |-  ( G  e.  Grp  ->  G  e.  Mnd )
522, 43, 513syl 20 . . . . . . . . . . 11  |-  ( ph  ->  G  e.  Mnd )
5352ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  G  e.  Mnd )
545subgacs 16363 . . . . . . . . . . . . 13  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) ) )
55 acsmre 15069 . . . . . . . . . . . . 13  |-  ( (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
5644, 54, 553syl 20 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
57 imassrn 5358 . . . . . . . . . . . . . 14  |-  ( S
" ( I  \  { x } ) )  C_  ran  S
582, 3dprdf2 17167 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  S : I --> (SubGrp `  G ) )
5958ad2antrr 725 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  S : I --> (SubGrp `  G ) )
60 frn 5743 . . . . . . . . . . . . . . . 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 15009 . . . . . . . . . . . . . . . 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 3509 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  ran  S 
C_  ~P ( Base `  G
) )
6557, 64syl5ss 3510 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  ( S " ( I  \  { x } ) )  C_  ~P ( Base `  G ) )
66 sspwuni 4421 . . . . . . . . . . . . 13  |-  ( ( S " ( I 
\  { x }
) )  C_  ~P ( Base `  G )  <->  U. ( S " (
I  \  { x } ) )  C_  ( Base `  G )
)
6765, 66sylib 196 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  U. ( S " ( I  \  { x } ) )  C_  ( Base `  G ) )
68 eqid 2457 . . . . . . . . . . . . 13  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
6968mrccl 15028 . . . . . . . . . . . 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 661 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  (
(mrCls `  (SubGrp `  G
) ) `  U. ( S " ( I 
\  { x }
) ) )  e.  (SubGrp `  G )
)
71 subgsubm 16350 . . . . . . . . . . 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 6303 . . . . . . . . . . . . 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 2526 . . . . . . . . . . . 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 6303 . . . . . . . . . . . . 13  |-  (  .0.  =  if ( y  =  x ,  ( F `  x ) ,  .0.  )  -> 
(  .0.  ( -g `  G ) ( F `
 y ) )  =  ( if ( y  =  x ,  ( F `  x
) ,  .0.  )
( -g `  G ) ( F `  y
) ) )
7675eleq1d 2526 . . . . . . . . . . . 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 461 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  /\  y  =  x )  ->  y  =  x )
7877fveq2d 5876 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  /\  y  =  x )  ->  ( F `  y )  =  ( F `  x ) )
7978oveq2d 6312 . . . . . . . . . . . . 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 16249 . . . . . . . . . . . . . . . 16  |-  ( ( G  e.  Grp  /\  ( F `  x )  e.  ( Base `  G
) )  ->  (
( F `  x
) ( -g `  G
) ( F `  x ) )  =  .0.  )
8144, 45, 80syl2anc 661 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  (
( F `  x
) ( -g `  G
) ( F `  x ) )  =  .0.  )
8211subg0cl 16336 . . . . . . . . . . . . . . . 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 2545 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  (
( F `  x
) ( -g `  G
) ( F `  x ) )  e.  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " (
I  \  { x } ) ) ) )
8584ad2antrr 725 . . . . . . . . . . . . 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 2545 . . . . . . . . . . . 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 725 . . . . . . . . . . . . 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 15042 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  U. ( S " ( I  \  { x } ) )  C_  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( I  \  { x } ) ) ) )
9089ad2antrr 725 . . . . . . . . . . . . . 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, 18dprdfclOLD 17180 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  ->  ( F `  y )  e.  ( S `  y
) )
9291adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  /\  -.  y  =  x )  ->  ( F `  y
)  e.  ( S `
 y ) )
93 ffn 5737 . . . . . . . . . . . . . . . . . 18  |-  ( S : I --> (SubGrp `  G )  ->  S  Fn  I )
9459, 93syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  S  Fn  I )
9594ad2antrr 725 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  /\  -.  y  =  x )  ->  S  Fn  I )
96 difssd 3628 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  /\  -.  y  =  x )  ->  ( I  \  {
x } )  C_  I )
97 df-ne 2654 . . . . . . . . . . . . . . . . . 18  |-  ( y  =/=  x  <->  -.  y  =  x )
98 eldifsn 4157 . . . . . . . . . . . . . . . . . . 19  |-  ( y  e.  ( I  \  { x } )  <-> 
( y  e.  I  /\  y  =/=  x
) )
9998biimpri 206 . . . . . . . . . . . . . . . . . 18  |-  ( ( y  e.  I  /\  y  =/=  x )  -> 
y  e.  ( I 
\  { x }
) )
10097, 99sylan2br 476 . . . . . . . . . . . . . . . . 17  |-  ( ( y  e.  I  /\  -.  y  =  x
)  ->  y  e.  ( I  \  { x } ) )
101100adantll 713 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  /\  -.  y  =  x )  ->  y  e.  ( I 
\  { x }
) )
102 fnfvima 6151 . . . . . . . . . . . . . . . 16  |-  ( ( S  Fn  I  /\  ( I  \  { x } )  C_  I  /\  y  e.  (
I  \  { x } ) )  -> 
( S `  y
)  e.  ( S
" ( I  \  { x } ) ) )
10395, 96, 101, 102syl3anc 1228 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  /\  -.  y  =  x )  ->  ( S `  y
)  e.  ( S
" ( I  \  { x } ) ) )
104 elunii 4256 . . . . . . . . . . . . . . 15  |-  ( ( ( F `  y
)  e.  ( S `
 y )  /\  ( S `  y )  e.  ( S "
( I  \  {
x } ) ) )  ->  ( F `  y )  e.  U. ( S " ( I 
\  { x }
) ) )
10592, 103, 104syl2anc 661 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  /\  y  e.  I )  /\  -.  y  =  x )  ->  ( F `  y
)  e.  U. ( S " ( I  \  { x } ) ) )
10690, 105sseldd 3500 . . . . . . . . . . . . 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 16339 . . . . . . . . . . . . 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 1228 . . . . . . . . . . . 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 3979 . . . . . . . . . . 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 2457 . . . . . . . . . . 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 6056 . . . . . . . . . 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 459 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  (
( y  e.  I  |->  if ( y  =  x ,  ( F `
 x ) ,  .0.  ) )  oF ( -g `  G
) F )  e.  W )
11338, 112eqeltrrd 2546 . . . . . . . . . . 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, 50dprdfcntzOLD 17182 . . . . . . . . . 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, 113dprdffiOLD 17181 . . . . . . . . . 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, 115gsumzsubmclOLD 17056 . . . . . . . . 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 2546 . . . . . . . 8  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  ( F `  x )  e.  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " (
I  \  { x } ) ) ) )
11810, 117elind 3684 . . . . . . 7  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  ( F `  x )  e.  ( ( S `  x )  i^i  (
(mrCls `  (SubGrp `  G
) ) `  U. ( S " ( I 
\  { x }
) ) ) ) )
11912, 13, 14, 11, 68dprddisj 17169 . . . . . . 7  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  (
( S `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( I  \  { x } ) ) ) )  =  {  .0.  } )
120118, 119eleqtrd 2547 . . . . . 6  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  ( F `  x )  e.  {  .0.  } )
121 elsni 4057 . . . . . 6  |-  ( ( F `  x )  e.  {  .0.  }  ->  ( F `  x
)  =  .0.  )
122120, 121syl 16 . . . . 5  |-  ( ( ( ph  /\  ( G  gsumg  F )  =  .0.  )  /\  x  e.  I )  ->  ( F `  x )  =  .0.  )
123122mpteq2dva 4543 . . . 4  |-  ( (
ph  /\  ( G  gsumg  F )  =  .0.  )  ->  ( x  e.  I  |->  ( F `  x
) )  =  ( x  e.  I  |->  .0.  ) )
1248, 123eqtrd 2498 . . 3  |-  ( (
ph  /\  ( G  gsumg  F )  =  .0.  )  ->  F  =  ( x  e.  I  |->  .0.  )
)
125124ex 434 . 2  |-  ( ph  ->  ( ( G  gsumg  F )  =  .0.  ->  F  =  ( x  e.  I  |->  .0.  ) )
)
12611gsumz 16132 . . . 4  |-  ( ( G  e.  Mnd  /\  I  e.  _V )  ->  ( G  gsumg  ( x  e.  I  |->  .0.  ) )  =  .0.  )
12752, 26, 126syl2anc 661 . . 3  |-  ( ph  ->  ( G  gsumg  ( x  e.  I  |->  .0.  ) )  =  .0.  )
128 oveq2 6304 . . . 4  |-  ( F  =  ( x  e.  I  |->  .0.  )  ->  ( G  gsumg  F )  =  ( G  gsumg  ( x  e.  I  |->  .0.  ) ) )
129128eqeq1d 2459 . . 3  |-  ( F  =  ( x  e.  I  |->  .0.  )  ->  ( ( G  gsumg  F )  =  .0.  <->  ( G  gsumg  ( x  e.  I  |->  .0.  ) )  =  .0.  ) )
130127, 129syl5ibrcom 222 . 2  |-  ( ph  ->  ( F  =  ( x  e.  I  |->  .0.  )  ->  ( G  gsumg  F )  =  .0.  )
)
131125, 130impbid 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 369    = wceq 1395    e. wcel 1819    =/= wne 2652   {crab 2811   _Vcvv 3109    \ cdif 3468    i^i cin 3470    C_ wss 3471   ifcif 3944   ~Pcpw 4015   {csn 4032   U.cuni 4251   class class class wbr 4456    |-> cmpt 4515   `'ccnv 5007   dom cdm 5008   ran crn 5009   "cima 5011    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6296    oFcof 6537   X_cixp 7488   Fincfn 7535   Basecbs 14644   0gc0g 14857    gsumg cgsu 14858  Moorecmre 14999  mrClscmrc 15000  ACScacs 15002   Mndcmnd 16046  SubMndcsubmnd 16092   Grpcgrp 16180   -gcsg 16182  SubGrpcsubg 16322  Cntzccntz 16480   DProd cdprd 17151
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-tpos 6973  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-oi 7953  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11822  df-seq 12111  df-hash 12409  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-0g 14859  df-gsum 14860  df-mre 15003  df-mrc 15004  df-acs 15006  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-mhm 16093  df-submnd 16094  df-grp 16184  df-minusg 16185  df-sbg 16186  df-mulg 16187  df-subg 16325  df-ghm 16392  df-gim 16434  df-cntz 16482  df-oppg 16508  df-cmn 16927  df-dprd 17153
This theorem is referenced by:  dprdf11OLD  17197
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