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Theorem frgpuplem 15083
Description: Any assignment of the generators to target elements can be extended (uniquely) to a homomorphism from a free monoid to an arbitrary other monoid. (Contributed by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
frgpup.b  |-  B  =  ( Base `  H
)
frgpup.n  |-  N  =  ( inv g `  H )
frgpup.t  |-  T  =  ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `  y
) ,  ( N `
 ( F `  y ) ) ) )
frgpup.h  |-  ( ph  ->  H  e.  Grp )
frgpup.i  |-  ( ph  ->  I  e.  V )
frgpup.a  |-  ( ph  ->  F : I --> B )
frgpup.w  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
frgpup.r  |-  .~  =  ( ~FG  `  I )
Assertion
Ref Expression
frgpuplem  |-  ( (
ph  /\  A  .~  C )  ->  ( H  gsumg  ( T  o.  A
) )  =  ( H  gsumg  ( T  o.  C
) ) )
Distinct variable groups:    y, z, A    y, F, z    y, N, z    y, B, z    ph, y, z    y, I, z
Allowed substitution hints:    C( y, z)    .~ ( y, z)    T( y, z)    H( y, z)    V( y, z)    W( y, z)

Proof of Theorem frgpuplem
Dummy variables  a 
b  u  v  n  r  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frgpup.w . . . . . . 7  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
2 frgpup.r . . . . . . 7  |-  .~  =  ( ~FG  `  I )
31, 2efgval 15028 . . . . . 6  |-  .~  =  |^| { r  |  ( r  Er  W  /\  A. x  e.  W  A. n  e.  ( 0 ... ( # `  x
) ) A. a  e.  I  A. b  e.  2o  x r ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) ) }
4 coeq2 4844 . . . . . . . . . . . . 13  |-  ( u  =  v  ->  ( T  o.  u )  =  ( T  o.  v ) )
54oveq2d 5876 . . . . . . . . . . . 12  |-  ( u  =  v  ->  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) )
6 eqid 2285 . . . . . . . . . . . 12  |-  { <. u ,  v >.  |  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) }  =  { <. u ,  v
>.  |  ( H  gsumg  ( T  o.  u ) )  =  ( H 
gsumg  ( T  o.  v
) ) }
75, 6eqer 6695 . . . . . . . . . . 11  |-  { <. u ,  v >.  |  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) }  Er  _V
87a1i 10 . . . . . . . . . 10  |-  ( ph  ->  { <. u ,  v
>.  |  ( H  gsumg  ( T  o.  u ) )  =  ( H 
gsumg  ( T  o.  v
) ) }  Er  _V )
9 ssv 3200 . . . . . . . . . . 11  |-  W  C_  _V
109a1i 10 . . . . . . . . . 10  |-  ( ph  ->  W  C_  _V )
118, 10erinxp 6735 . . . . . . . . 9  |-  ( ph  ->  ( { <. u ,  v >.  |  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) }  i^i  ( W  X.  W
) )  Er  W
)
12 df-xp 4697 . . . . . . . . . . . . 13  |-  ( W  X.  W )  =  { <. u ,  v
>.  |  ( u  e.  W  /\  v  e.  W ) }
1312ineq1i 3368 . . . . . . . . . . . 12  |-  ( ( W  X.  W )  i^i  { <. u ,  v >.  |  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) } )  =  ( { <. u ,  v >.  |  ( u  e.  W  /\  v  e.  W ) }  i^i  { <. u ,  v >.  |  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) } )
14 incom 3363 . . . . . . . . . . . 12  |-  ( ( W  X.  W )  i^i  { <. u ,  v >.  |  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) } )  =  ( { <. u ,  v >.  |  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) }  i^i  ( W  X.  W
) )
15 inopab 4818 . . . . . . . . . . . 12  |-  ( {
<. u ,  v >.  |  ( u  e.  W  /\  v  e.  W ) }  i^i  {
<. u ,  v >.  |  ( H  gsumg  ( T  o.  u ) )  =  ( H  gsumg  ( T  o.  v ) ) } )  =  { <. u ,  v >.  |  ( ( u  e.  W  /\  v  e.  W )  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }
1613, 14, 153eqtr3i 2313 . . . . . . . . . . 11  |-  ( {
<. u ,  v >.  |  ( H  gsumg  ( T  o.  u ) )  =  ( H  gsumg  ( T  o.  v ) ) }  i^i  ( W  X.  W ) )  =  { <. u ,  v >.  |  ( ( u  e.  W  /\  v  e.  W
)  /\  ( H  gsumg  ( T  o.  u ) )  =  ( H 
gsumg  ( T  o.  v
) ) ) }
17 vex 2793 . . . . . . . . . . . . . 14  |-  u  e. 
_V
18 vex 2793 . . . . . . . . . . . . . 14  |-  v  e. 
_V
1917, 18prss 3771 . . . . . . . . . . . . 13  |-  ( ( u  e.  W  /\  v  e.  W )  <->  { u ,  v } 
C_  W )
2019anbi1i 676 . . . . . . . . . . . 12  |-  ( ( ( u  e.  W  /\  v  e.  W
)  /\  ( H  gsumg  ( T  o.  u ) )  =  ( H 
gsumg  ( T  o.  v
) ) )  <->  ( {
u ,  v } 
C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) )
2120opabbii 4085 . . . . . . . . . . 11  |-  { <. u ,  v >.  |  ( ( u  e.  W  /\  v  e.  W
)  /\  ( H  gsumg  ( T  o.  u ) )  =  ( H 
gsumg  ( T  o.  v
) ) ) }  =  { <. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }
2216, 21eqtri 2305 . . . . . . . . . 10  |-  ( {
<. u ,  v >.  |  ( H  gsumg  ( T  o.  u ) )  =  ( H  gsumg  ( T  o.  v ) ) }  i^i  ( W  X.  W ) )  =  { <. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }
23 ereq1 6669 . . . . . . . . . 10  |-  ( ( { <. u ,  v
>.  |  ( H  gsumg  ( T  o.  u ) )  =  ( H 
gsumg  ( T  o.  v
) ) }  i^i  ( W  X.  W
) )  =  { <. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u ) )  =  ( H  gsumg  ( T  o.  v ) ) ) }  ->  (
( { <. u ,  v >.  |  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) }  i^i  ( W  X.  W
) )  Er  W  <->  {
<. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u ) )  =  ( H  gsumg  ( T  o.  v ) ) ) }  Er  W
) )
2422, 23ax-mp 8 . . . . . . . . 9  |-  ( ( { <. u ,  v
>.  |  ( H  gsumg  ( T  o.  u ) )  =  ( H 
gsumg  ( T  o.  v
) ) }  i^i  ( W  X.  W
) )  Er  W  <->  {
<. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u ) )  =  ( H  gsumg  ( T  o.  v ) ) ) }  Er  W
)
2511, 24sylib 188 . . . . . . . 8  |-  ( ph  ->  { <. u ,  v
>.  |  ( {
u ,  v } 
C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }  Er  W )
26 simplrl 736 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  ->  x  e.  W )
27 fviss 5582 . . . . . . . . . . . . . . . 16  |-  (  _I 
` Word  ( I  X.  2o ) )  C_ Word  ( I  X.  2o )
281, 27eqsstri 3210 . . . . . . . . . . . . . . 15  |-  W  C_ Word  ( I  X.  2o )
2928, 26sseldi 3180 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  ->  x  e. Word  ( I  X.  2o ) )
30 opelxpi 4723 . . . . . . . . . . . . . . . 16  |-  ( ( a  e.  I  /\  b  e.  2o )  -> 
<. a ,  b >.  e.  ( I  X.  2o ) )
3130adantl 452 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  ->  <. a ,  b >.  e.  ( I  X.  2o ) )
32 simprl 732 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
a  e.  I )
33 2oconcl 6504 . . . . . . . . . . . . . . . . 17  |-  ( b  e.  2o  ->  ( 1o  \  b )  e.  2o )
3433ad2antll 709 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( 1o  \  b
)  e.  2o )
35 opelxpi 4723 . . . . . . . . . . . . . . . 16  |-  ( ( a  e.  I  /\  ( 1o  \  b
)  e.  2o )  ->  <. a ,  ( 1o  \  b )
>.  e.  ( I  X.  2o ) )
3632, 34, 35syl2anc 642 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  ->  <. a ,  ( 1o 
\  b ) >.  e.  ( I  X.  2o ) )
3731, 36s2cld 11521 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  ->  <" <. a ,  b
>. <. a ,  ( 1o  \  b )
>. ">  e. Word  (
I  X.  2o ) )
38 splcl 11469 . . . . . . . . . . . . . 14  |-  ( ( x  e. Word  ( I  X.  2o )  /\  <" <. a ,  b
>. <. a ,  ( 1o  \  b )
>. ">  e. Word  (
I  X.  2o ) )  ->  ( x splice  <.
n ,  n , 
<" <. a ,  b
>. <. a ,  ( 1o  \  b )
>. "> >. )  e. Word  ( I  X.  2o ) )
3929, 37, 38syl2anc 642 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
)  e. Word  ( I  X.  2o ) )
401efgrcl 15026 . . . . . . . . . . . . . . 15  |-  ( x  e.  W  ->  (
I  e.  _V  /\  W  = Word  ( I  X.  2o ) ) )
4126, 40syl 15 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( I  e.  _V  /\  W  = Word  ( I  X.  2o ) ) )
4241simprd 449 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  ->  W  = Word  ( I  X.  2o ) )
4339, 42eleqtrrd 2362 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
)  e.  W )
4426, 43jca 518 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( x  e.  W  /\  ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
)  e.  W ) )
45 swrdcl 11454 . . . . . . . . . . . . . . . . . 18  |-  ( x  e. Word  ( I  X.  2o )  ->  ( x substr  <. 0 ,  n >. )  e. Word  ( I  X.  2o ) )
4629, 45syl 15 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( x substr  <. 0 ,  n >. )  e. Word  (
I  X.  2o ) )
47 frgpup.b . . . . . . . . . . . . . . . . . . 19  |-  B  =  ( Base `  H
)
48 frgpup.n . . . . . . . . . . . . . . . . . . 19  |-  N  =  ( inv g `  H )
49 frgpup.t . . . . . . . . . . . . . . . . . . 19  |-  T  =  ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `  y
) ,  ( N `
 ( F `  y ) ) ) )
50 frgpup.h . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  H  e.  Grp )
51 frgpup.i . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  I  e.  V )
52 frgpup.a . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  F : I --> B )
5347, 48, 49, 50, 51, 52frgpuptf 15081 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  T : ( I  X.  2o ) --> B )
5453ad2antrr 706 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  ->  T : ( I  X.  2o ) --> B )
55 ccatco 11492 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x substr  <. 0 ,  n >. )  e. Word  (
I  X.  2o )  /\  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. ">  e. Word  ( I  X.  2o )  /\  T : ( I  X.  2o ) --> B )  ->  ( T  o.  ( (
x substr  <. 0 ,  n >. ) concat  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) )  =  ( ( T  o.  (
x substr  <. 0 ,  n >. ) ) concat  ( T  o.  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) ) )
5646, 37, 54, 55syl3anc 1182 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( T  o.  (
( x substr  <. 0 ,  n >. ) concat  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) )  =  ( ( T  o.  (
x substr  <. 0 ,  n >. ) ) concat  ( T  o.  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) ) )
5756oveq2d 5876 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( H  gsumg  ( T  o.  (
( x substr  <. 0 ,  n >. ) concat  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) ) )  =  ( H  gsumg  ( ( T  o.  ( x substr  <. 0 ,  n >. ) ) concat  ( T  o.  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) ) ) )
5850ad2antrr 706 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  ->  H  e.  Grp )
59 grpmnd 14496 . . . . . . . . . . . . . . . . 17  |-  ( H  e.  Grp  ->  H  e.  Mnd )
6058, 59syl 15 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  ->  H  e.  Mnd )
61 wrdco 11488 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x substr  <. 0 ,  n >. )  e. Word  (
I  X.  2o )  /\  T : ( I  X.  2o ) --> B )  ->  ( T  o.  ( x substr  <.
0 ,  n >. ) )  e. Word  B )
6246, 54, 61syl2anc 642 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( T  o.  (
x substr  <. 0 ,  n >. ) )  e. Word  B
)
63 wrdco 11488 . . . . . . . . . . . . . . . . 17  |-  ( (
<" <. a ,  b
>. <. a ,  ( 1o  \  b )
>. ">  e. Word  (
I  X.  2o )  /\  T : ( I  X.  2o ) --> B )  ->  ( T  o.  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> )  e. Word  B )
6437, 54, 63syl2anc 642 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( T  o.  <"
<. a ,  b >. <. a ,  ( 1o 
\  b ) >. "> )  e. Word  B
)
65 eqid 2285 . . . . . . . . . . . . . . . . 17  |-  ( +g  `  H )  =  ( +g  `  H )
6647, 65gsumccat 14466 . . . . . . . . . . . . . . . 16  |-  ( ( H  e.  Mnd  /\  ( T  o.  (
x substr  <. 0 ,  n >. ) )  e. Word  B  /\  ( T  o.  <"
<. a ,  b >. <. a ,  ( 1o 
\  b ) >. "> )  e. Word  B
)  ->  ( H  gsumg  ( ( T  o.  (
x substr  <. 0 ,  n >. ) ) concat  ( T  o.  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) ) )  =  ( ( H  gsumg  ( T  o.  ( x substr  <. 0 ,  n >. ) ) ) ( +g  `  H
) ( H  gsumg  ( T  o.  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) ) ) )
6760, 62, 64, 66syl3anc 1182 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( H  gsumg  ( ( T  o.  ( x substr  <. 0 ,  n >. ) ) concat  ( T  o.  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) ) )  =  ( ( H  gsumg  ( T  o.  ( x substr  <. 0 ,  n >. ) ) ) ( +g  `  H
) ( H  gsumg  ( T  o.  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) ) ) )
6854, 31, 36s2co 11549 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( T  o.  <"
<. a ,  b >. <. a ,  ( 1o 
\  b ) >. "> )  =  <" ( T `  <. a ,  b >. )
( T `  <. a ,  ( 1o  \ 
b ) >. ) "> )
69 df-ov 5863 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( a T b )  =  ( T `  <. a ,  b >. )
7069a1i 10 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( a T b )  =  ( T `
 <. a ,  b
>. ) )
71 df-ov 5863 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( a ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
) b )  =  ( ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \  z )
>. ) `  <. a ,  b >. )
72 eqid 2285 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \ 
z ) >. )  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \  z )
>. )
7372efgmval 15023 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  ( a ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \ 
z ) >. )
b )  =  <. a ,  ( 1o  \ 
b ) >. )
7471, 73syl5eqr 2331 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  ( ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \  z )
>. ) `  <. a ,  b >. )  =  <. a ,  ( 1o  \  b )
>. )
7574adantl 452 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \  z )
>. ) `  <. a ,  b >. )  =  <. a ,  ( 1o  \  b )
>. )
7675fveq2d 5531 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( T `  (
( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
) `  <. a ,  b >. ) )  =  ( T `  <. a ,  ( 1o  \ 
b ) >. )
)
7747, 48, 49, 50, 51, 52, 72frgpuptinv 15082 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( (
ph  /\  <. a ,  b >.  e.  (
I  X.  2o ) )  ->  ( T `  ( ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \  z )
>. ) `  <. a ,  b >. )
)  =  ( N `
 ( T `  <. a ,  b >.
) ) )
7830, 77sylan2 460 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( (
ph  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( T `  (
( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
) `  <. a ,  b >. ) )  =  ( N `  ( T `  <. a ,  b >. ) ) )
7978adantlr 695 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( T `  (
( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
) `  <. a ,  b >. ) )  =  ( N `  ( T `  <. a ,  b >. ) ) )
8076, 79eqtr3d 2319 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( T `  <. a ,  ( 1o  \ 
b ) >. )  =  ( N `  ( T `  <. a ,  b >. )
) )
8169fveq2i 5530 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( N `
 ( a T b ) )  =  ( N `  ( T `  <. a ,  b >. ) )
8280, 81syl6reqr 2336 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( N `  (
a T b ) )  =  ( T `
 <. a ,  ( 1o  \  b )
>. ) )
8370, 82s2eqd 11514 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  ->  <" ( a T b ) ( N `
 ( a T b ) ) ">  =  <" ( T `  <. a ,  b >. ) ( T `
 <. a ,  ( 1o  \  b )
>. ) "> )
8468, 83eqtr4d 2320 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( T  o.  <"
<. a ,  b >. <. a ,  ( 1o 
\  b ) >. "> )  =  <" ( a T b ) ( N `  ( a T b ) ) "> )
8584oveq2d 5876 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( H  gsumg  ( T  o.  <"
<. a ,  b >. <. a ,  ( 1o 
\  b ) >. "> ) )  =  ( H  gsumg 
<" ( a T b ) ( N `
 ( a T b ) ) "> ) )
86 simprr 733 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
b  e.  2o )
87 fovrn 5992 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( T : ( I  X.  2o ) --> B  /\  a  e.  I  /\  b  e.  2o )  ->  ( a T b )  e.  B
)
8854, 32, 86, 87syl3anc 1182 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( a T b )  e.  B )
8947, 48grpinvcl 14529 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( H  e.  Grp  /\  ( a T b )  e.  B )  ->  ( N `  ( a T b ) )  e.  B
)
9058, 88, 89syl2anc 642 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( N `  (
a T b ) )  e.  B )
9147, 65gsumws2 14467 . . . . . . . . . . . . . . . . . . 19  |-  ( ( H  e.  Mnd  /\  ( a T b )  e.  B  /\  ( N `  ( a T b ) )  e.  B )  -> 
( H  gsumg 
<" ( a T b ) ( N `
 ( a T b ) ) "> )  =  ( ( a T b ) ( +g  `  H
) ( N `  ( a T b ) ) ) )
9260, 88, 90, 91syl3anc 1182 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( H  gsumg 
<" ( a T b ) ( N `
 ( a T b ) ) "> )  =  ( ( a T b ) ( +g  `  H
) ( N `  ( a T b ) ) ) )
93 eqid 2285 . . . . . . . . . . . . . . . . . . . 20  |-  ( 0g
`  H )  =  ( 0g `  H
)
9447, 65, 93, 48grprinv 14531 . . . . . . . . . . . . . . . . . . 19  |-  ( ( H  e.  Grp  /\  ( a T b )  e.  B )  ->  ( ( a T b ) ( +g  `  H ) ( N `  (
a T b ) ) )  =  ( 0g `  H ) )
9558, 88, 94syl2anc 642 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( ( a T b ) ( +g  `  H ) ( N `
 ( a T b ) ) )  =  ( 0g `  H ) )
9685, 92, 953eqtrd 2321 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( H  gsumg  ( T  o.  <"
<. a ,  b >. <. a ,  ( 1o 
\  b ) >. "> ) )  =  ( 0g `  H
) )
9796oveq2d 5876 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( ( H  gsumg  ( T  o.  ( x substr  <. 0 ,  n >. ) ) ) ( +g  `  H
) ( H  gsumg  ( T  o.  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) ) )  =  ( ( H  gsumg  ( T  o.  ( x substr  <. 0 ,  n >. ) ) ) ( +g  `  H
) ( 0g `  H ) ) )
9847gsumwcl 14465 . . . . . . . . . . . . . . . . . 18  |-  ( ( H  e.  Mnd  /\  ( T  o.  (
x substr  <. 0 ,  n >. ) )  e. Word  B
)  ->  ( H  gsumg  ( T  o.  ( x substr  <. 0 ,  n >. ) ) )  e.  B
)
9960, 62, 98syl2anc 642 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( H  gsumg  ( T  o.  (
x substr  <. 0 ,  n >. ) ) )  e.  B )
10047, 65, 93grprid 14515 . . . . . . . . . . . . . . . . 17  |-  ( ( H  e.  Grp  /\  ( H  gsumg  ( T  o.  (
x substr  <. 0 ,  n >. ) ) )  e.  B )  ->  (
( H  gsumg  ( T  o.  (
x substr  <. 0 ,  n >. ) ) ) ( +g  `  H ) ( 0g `  H
) )  =  ( H  gsumg  ( T  o.  (
x substr  <. 0 ,  n >. ) ) ) )
10158, 99, 100syl2anc 642 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( ( H  gsumg  ( T  o.  ( x substr  <. 0 ,  n >. ) ) ) ( +g  `  H
) ( 0g `  H ) )  =  ( H  gsumg  ( T  o.  (
x substr  <. 0 ,  n >. ) ) ) )
10297, 101eqtrd 2317 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( ( H  gsumg  ( T  o.  ( x substr  <. 0 ,  n >. ) ) ) ( +g  `  H
) ( H  gsumg  ( T  o.  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) ) )  =  ( H  gsumg  ( T  o.  (
x substr  <. 0 ,  n >. ) ) ) )
10357, 67, 1023eqtrrd 2322 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( H  gsumg  ( T  o.  (
x substr  <. 0 ,  n >. ) ) )  =  ( H  gsumg  ( T  o.  (
( x substr  <. 0 ,  n >. ) concat  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) ) ) )
104103oveq1d 5875 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( ( H  gsumg  ( T  o.  ( x substr  <. 0 ,  n >. ) ) ) ( +g  `  H
) ( H  gsumg  ( T  o.  ( x substr  <. n ,  ( # `  x
) >. ) ) ) )  =  ( ( H  gsumg  ( T  o.  (
( x substr  <. 0 ,  n >. ) concat  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) ) ) ( +g  `  H ) ( H  gsumg  ( T  o.  (
x substr  <. n ,  (
# `  x ) >. ) ) ) ) )
105 swrdcl 11454 . . . . . . . . . . . . . . . 16  |-  ( x  e. Word  ( I  X.  2o )  ->  ( x substr  <. n ,  ( # `  x ) >. )  e. Word  ( I  X.  2o ) )
10629, 105syl 15 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( x substr  <. n ,  ( # `  x
) >. )  e. Word  (
I  X.  2o ) )
107 wrdco 11488 . . . . . . . . . . . . . . 15  |-  ( ( ( x substr  <. n ,  ( # `  x
) >. )  e. Word  (
I  X.  2o )  /\  T : ( I  X.  2o ) --> B )  ->  ( T  o.  ( x substr  <.
n ,  ( # `  x ) >. )
)  e. Word  B )
108106, 54, 107syl2anc 642 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( T  o.  (
x substr  <. n ,  (
# `  x ) >. ) )  e. Word  B
)
10947, 65gsumccat 14466 . . . . . . . . . . . . . 14  |-  ( ( H  e.  Mnd  /\  ( T  o.  (
x substr  <. 0 ,  n >. ) )  e. Word  B  /\  ( T  o.  (
x substr  <. n ,  (
# `  x ) >. ) )  e. Word  B
)  ->  ( H  gsumg  ( ( T  o.  (
x substr  <. 0 ,  n >. ) ) concat  ( T  o.  ( x substr  <. n ,  ( # `  x
) >. ) ) ) )  =  ( ( H  gsumg  ( T  o.  (
x substr  <. 0 ,  n >. ) ) ) ( +g  `  H ) ( H  gsumg  ( T  o.  (
x substr  <. n ,  (
# `  x ) >. ) ) ) ) )
11060, 62, 108, 109syl3anc 1182 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( H  gsumg  ( ( T  o.  ( x substr  <. 0 ,  n >. ) ) concat  ( T  o.  ( x substr  <.
n ,  ( # `  x ) >. )
) ) )  =  ( ( H  gsumg  ( T  o.  ( x substr  <. 0 ,  n >. ) ) ) ( +g  `  H
) ( H  gsumg  ( T  o.  ( x substr  <. n ,  ( # `  x
) >. ) ) ) ) )
111 ccatcl 11431 . . . . . . . . . . . . . . . 16  |-  ( ( ( x substr  <. 0 ,  n >. )  e. Word  (
I  X.  2o )  /\  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. ">  e. Word  ( I  X.  2o ) )  ->  (
( x substr  <. 0 ,  n >. ) concat  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> )  e. Word  ( I  X.  2o ) )
11246, 37, 111syl2anc 642 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( ( x substr  <. 0 ,  n >. ) concat  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> )  e. Word  ( I  X.  2o ) )
113 wrdco 11488 . . . . . . . . . . . . . . 15  |-  ( ( ( ( x substr  <. 0 ,  n >. ) concat  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> )  e. Word  ( I  X.  2o )  /\  T : ( I  X.  2o ) --> B )  -> 
( T  o.  (
( x substr  <. 0 ,  n >. ) concat  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) )  e. Word  B
)
114112, 54, 113syl2anc 642 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( T  o.  (
( x substr  <. 0 ,  n >. ) concat  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) )  e. Word  B
)
11547, 65gsumccat 14466 . . . . . . . . . . . . . 14  |-  ( ( H  e.  Mnd  /\  ( T  o.  (
( x substr  <. 0 ,  n >. ) concat  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) )  e. Word  B  /\  ( T  o.  (
x substr  <. n ,  (
# `  x ) >. ) )  e. Word  B
)  ->  ( H  gsumg  ( ( T  o.  (
( x substr  <. 0 ,  n >. ) concat  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) ) concat  ( T  o.  ( x substr  <. n ,  ( # `  x
) >. ) ) ) )  =  ( ( H  gsumg  ( T  o.  (
( x substr  <. 0 ,  n >. ) concat  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) ) ) ( +g  `  H ) ( H  gsumg  ( T  o.  (
x substr  <. n ,  (
# `  x ) >. ) ) ) ) )
11660, 114, 108, 115syl3anc 1182 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( H  gsumg  ( ( T  o.  ( ( x substr  <. 0 ,  n >. ) concat  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) ) concat  ( T  o.  ( x substr  <. n ,  ( # `  x
) >. ) ) ) )  =  ( ( H  gsumg  ( T  o.  (
( x substr  <. 0 ,  n >. ) concat  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) ) ) ( +g  `  H ) ( H  gsumg  ( T  o.  (
x substr  <. n ,  (
# `  x ) >. ) ) ) ) )
117104, 110, 1163eqtr4d 2327 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( H  gsumg  ( ( T  o.  ( x substr  <. 0 ,  n >. ) ) concat  ( T  o.  ( x substr  <.
n ,  ( # `  x ) >. )
) ) )  =  ( H  gsumg  ( ( T  o.  ( ( x substr  <. 0 ,  n >. ) concat  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) ) concat  ( T  o.  ( x substr  <. n ,  ( # `  x
) >. ) ) ) ) )
118 simplrr 737 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  ->  n  e.  ( 0 ... ( # `  x
) ) )
119 elfzuz 10796 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  ( 0 ... ( # `  x
) )  ->  n  e.  ( ZZ>= `  0 )
)
120 eluzfz1 10805 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... n
) )
121118, 119, 1203syl 18 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
0  e.  ( 0 ... n ) )
122 lencl 11423 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  e. Word  ( I  X.  2o )  ->  ( # `  x )  e.  NN0 )
12329, 122syl 15 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( # `  x )  e.  NN0 )
124 nn0uz 10264 . . . . . . . . . . . . . . . . . . 19  |-  NN0  =  ( ZZ>= `  0 )
125123, 124syl6eleq 2375 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( # `  x )  e.  ( ZZ>= `  0
) )
126 eluzfz2 10806 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  x )  e.  ( ZZ>= `  0 )  ->  ( # `  x
)  e.  ( 0 ... ( # `  x
) ) )
127125, 126syl 15 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( # `  x )  e.  ( 0 ... ( # `  x
) ) )
128 ccatswrd 11461 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e. Word  ( I  X.  2o )  /\  ( 0  e.  ( 0 ... n )  /\  n  e.  ( 0 ... ( # `  x ) )  /\  ( # `  x )  e.  ( 0 ... ( # `  x
) ) ) )  ->  ( ( x substr  <. 0 ,  n >. ) concat 
( x substr  <. n ,  ( # `  x
) >. ) )  =  ( x substr  <. 0 ,  ( # `  x
) >. ) )
12929, 121, 118, 127, 128syl13anc 1184 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( ( x substr  <. 0 ,  n >. ) concat  ( x substr  <.
n ,  ( # `  x ) >. )
)  =  ( x substr  <. 0 ,  ( # `  x ) >. )
)
130 swrdid 11460 . . . . . . . . . . . . . . . . 17  |-  ( x  e. Word  ( I  X.  2o )  ->  ( x substr  <. 0 ,  ( # `  x ) >. )  =  x )
13129, 130syl 15 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( x substr  <. 0 ,  ( # `  x
) >. )  =  x )
132129, 131eqtrd 2317 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( ( x substr  <. 0 ,  n >. ) concat  ( x substr  <.
n ,  ( # `  x ) >. )
)  =  x )
133132coeq2d 4848 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( T  o.  (
( x substr  <. 0 ,  n >. ) concat  ( x substr  <.
n ,  ( # `  x ) >. )
) )  =  ( T  o.  x ) )
134 ccatco 11492 . . . . . . . . . . . . . . 15  |-  ( ( ( x substr  <. 0 ,  n >. )  e. Word  (
I  X.  2o )  /\  ( x substr  <. n ,  ( # `  x
) >. )  e. Word  (
I  X.  2o )  /\  T : ( I  X.  2o ) --> B )  ->  ( T  o.  ( (
x substr  <. 0 ,  n >. ) concat  ( x substr  <. n ,  ( # `  x
) >. ) ) )  =  ( ( T  o.  ( x substr  <. 0 ,  n >. ) ) concat  ( T  o.  ( x substr  <.
n ,  ( # `  x ) >. )
) ) )
13546, 106, 54, 134syl3anc 1182 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( T  o.  (
( x substr  <. 0 ,  n >. ) concat  ( x substr  <.
n ,  ( # `  x ) >. )
) )  =  ( ( T  o.  (
x substr  <. 0 ,  n >. ) ) concat  ( T  o.  ( x substr  <. n ,  ( # `  x
) >. ) ) ) )
136133, 135eqtr3d 2319 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( T  o.  x
)  =  ( ( T  o.  ( x substr  <. 0 ,  n >. ) ) concat  ( T  o.  ( x substr  <. n ,  ( # `  x
) >. ) ) ) )
137136oveq2d 5876 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( H  gsumg  ( T  o.  x
) )  =  ( H  gsumg  ( ( T  o.  ( x substr  <. 0 ,  n >. ) ) concat  ( T  o.  ( x substr  <.
n ,  ( # `  x ) >. )
) ) ) )
138 splval 11468 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  W  /\  ( n  e.  (
0 ... ( # `  x
) )  /\  n  e.  ( 0 ... ( # `
 x ) )  /\  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. ">  e. Word  ( I  X.  2o ) ) )  -> 
( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
)  =  ( ( ( x substr  <. 0 ,  n >. ) concat  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) concat  ( x substr  <. n ,  ( # `  x
) >. ) ) )
13926, 118, 118, 37, 138syl13anc 1184 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
)  =  ( ( ( x substr  <. 0 ,  n >. ) concat  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) concat  ( x substr  <. n ,  ( # `  x
) >. ) ) )
140139coeq2d 4848 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( T  o.  (
x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) )  =  ( T  o.  ( ( ( x substr  <. 0 ,  n >. ) concat  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) concat  ( x substr  <. n ,  ( # `  x
) >. ) ) ) )
141 ccatco 11492 . . . . . . . . . . . . . . 15  |-  ( ( ( ( x substr  <. 0 ,  n >. ) concat  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> )  e. Word  ( I  X.  2o )  /\  (
x substr  <. n ,  (
# `  x ) >. )  e. Word  ( I  X.  2o )  /\  T : ( I  X.  2o ) --> B )  -> 
( T  o.  (
( ( x substr  <. 0 ,  n >. ) concat  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) concat  ( x substr  <. n ,  ( # `  x
) >. ) ) )  =  ( ( T  o.  ( ( x substr  <. 0 ,  n >. ) concat  <" <. a ,  b
>. <. a ,  ( 1o  \  b )
>. "> ) ) concat 
( T  o.  (
x substr  <. n ,  (
# `  x ) >. ) ) ) )
142112, 106, 54, 141syl3anc 1182 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( T  o.  (
( ( x substr  <. 0 ,  n >. ) concat  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) concat  ( x substr  <. n ,  ( # `  x
) >. ) ) )  =  ( ( T  o.  ( ( x substr  <. 0 ,  n >. ) concat  <" <. a ,  b
>. <. a ,  ( 1o  \  b )
>. "> ) ) concat 
( T  o.  (
x substr  <. n ,  (
# `  x ) >. ) ) ) )
143140, 142eqtrd 2317 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( T  o.  (
x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) )  =  ( ( T  o.  (
( x substr  <. 0 ,  n >. ) concat  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) ) concat  ( T  o.  ( x substr  <. n ,  ( # `  x
) >. ) ) ) )
144143oveq2d 5876 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( H  gsumg  ( T  o.  (
x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) ) )  =  ( H  gsumg  ( ( T  o.  ( ( x substr  <. 0 ,  n >. ) concat  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) ) concat  ( T  o.  ( x substr  <. n ,  ( # `  x
) >. ) ) ) ) )
145117, 137, 1443eqtr4d 2327 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( H  gsumg  ( T  o.  x
) )  =  ( H  gsumg  ( T  o.  (
x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) ) ) )
146 vex 2793 . . . . . . . . . . . 12  |-  x  e. 
_V
147 ovex 5885 . . . . . . . . . . . 12  |-  ( x splice  <. n ,  n , 
<" <. a ,  b
>. <. a ,  ( 1o  \  b )
>. "> >. )  e.  _V
148 eleq1 2345 . . . . . . . . . . . . . . 15  |-  ( u  =  x  ->  (
u  e.  W  <->  x  e.  W ) )
149 eleq1 2345 . . . . . . . . . . . . . . 15  |-  ( v  =  ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
)  ->  ( v  e.  W  <->  ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
)  e.  W ) )
150148, 149bi2anan9 843 . . . . . . . . . . . . . 14  |-  ( ( u  =  x  /\  v  =  ( x splice  <.
n ,  n , 
<" <. a ,  b
>. <. a ,  ( 1o  \  b )
>. "> >. )
)  ->  ( (
u  e.  W  /\  v  e.  W )  <->  ( x  e.  W  /\  ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
)  e.  W ) ) )
15119, 150syl5bbr 250 . . . . . . . . . . . . 13  |-  ( ( u  =  x  /\  v  =  ( x splice  <.
n ,  n , 
<" <. a ,  b
>. <. a ,  ( 1o  \  b )
>. "> >. )
)  ->  ( {
u ,  v } 
C_  W  <->  ( x  e.  W  /\  (
x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
)  e.  W ) ) )
152 coeq2 4844 . . . . . . . . . . . . . . 15  |-  ( u  =  x  ->  ( T  o.  u )  =  ( T  o.  x ) )
153152oveq2d 5876 . . . . . . . . . . . . . 14  |-  ( u  =  x  ->  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  x
) ) )
154 coeq2 4844 . . . . . . . . . . . . . . 15  |-  ( v  =  ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
)  ->  ( T  o.  v )  =  ( T  o.  ( x splice  <. n ,  n , 
<" <. a ,  b
>. <. a ,  ( 1o  \  b )
>. "> >. )
) )
155154oveq2d 5876 . . . . . . . . . . . . . 14  |-  ( v  =  ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
)  ->  ( H  gsumg  ( T  o.  v ) )  =  ( H 
gsumg  ( T  o.  (
x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) ) ) )
156153, 155eqeqan12d 2300 . . . . . . . . . . . . 13  |-  ( ( u  =  x  /\  v  =  ( x splice  <.
n ,  n , 
<" <. a ,  b
>. <. a ,  ( 1o  \  b )
>. "> >. )
)  ->  ( ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) )  <->  ( H  gsumg  ( T  o.  x ) )  =  ( H 
gsumg  ( T  o.  (
x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) ) ) ) )
157151, 156anbi12d 691 . . . . . . . . . . . 12  |-  ( ( u  =  x  /\  v  =  ( x splice  <.
n ,  n , 
<" <. a ,  b
>. <. a ,  ( 1o  \  b )
>. "> >. )
)  ->  ( ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) )  <->  ( (
x  e.  W  /\  ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
)  e.  W )  /\  ( H  gsumg  ( T  o.  x ) )  =  ( H  gsumg  ( T  o.  ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) ) ) ) ) )
158 eqid 2285 . . . . . . . . . . . 12  |-  { <. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }  =  { <. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }
159146, 147, 157, 158braba 4284 . . . . . . . . . . 11  |-  ( x { <. u ,  v
>.  |  ( {
u ,  v } 
C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }  ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
)  <->  ( ( x  e.  W  /\  (
x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
)  e.  W )  /\  ( H  gsumg  ( T  o.  x ) )  =  ( H  gsumg  ( T  o.  ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) ) ) ) )
16044, 145, 159sylanbrc 645 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  ->  x { <. u ,  v
>.  |  ( {
u ,  v } 
C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }  ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) )
161160ralrimivva 2637 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  W  /\  n  e.  ( 0 ... ( # `
 x ) ) ) )  ->  A. a  e.  I  A. b  e.  2o  x { <. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }  ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) )
162161ralrimivva 2637 . . . . . . . 8  |-  ( ph  ->  A. x  e.  W  A. n  e.  (
0 ... ( # `  x
) ) A. a  e.  I  A. b  e.  2o  x { <. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }  ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) )
163 fvex 5541 . . . . . . . . . . 11  |-  (  _I 
` Word  ( I  X.  2o ) )  e.  _V
1641, 163eqeltri 2355 . . . . . . . . . 10  |-  W  e. 
_V
165 erex 6686 . . . . . . . . . 10  |-  ( {
<. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u ) )  =  ( H  gsumg  ( T  o.  v ) ) ) }  Er  W  ->  ( W  e.  _V  ->  { <. u ,  v
>.  |  ( {
u ,  v } 
C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }  e.  _V ) )
16625, 164, 165ee10 1366 . . . . . . . . 9  |-  ( ph  ->  { <. u ,  v
>.  |  ( {
u ,  v } 
C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }  e.  _V )
167 ereq1 6669 . . . . . . . . . . 11  |-  ( r  =  { <. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }  ->  ( r  Er  W  <->  { <. u ,  v
>.  |  ( {
u ,  v } 
C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }  Er  W ) )
168 breq 4027 . . . . . . . . . . . . 13  |-  ( r  =  { <. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }  ->  ( x r ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
)  <->  x { <. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }  ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) ) )
1691682ralbidv 2587 . . . . . . . . . . . 12  |-  ( r  =  { <. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }  ->  ( A. a  e.  I  A. b  e.  2o  x r ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
)  <->  A. a  e.  I  A. b  e.  2o  x { <. u ,  v
>.  |  ( {
u ,  v } 
C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }  ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) ) )
1701692ralbidv 2587 . . . . . . . . . . 11  |-  ( r  =  { <. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }  ->  ( A. x  e.  W  A. n  e.  ( 0 ... ( # `
 x ) ) A. a  e.  I  A. b  e.  2o  x r ( x splice  <. n ,  n , 
<" <. a ,  b
>. <. a ,  ( 1o  \  b )
>. "> >. )  <->  A. x  e.  W  A. n  e.  ( 0 ... ( # `  x
) ) A. a  e.  I  A. b  e.  2o  x { <. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }  ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) ) )
171167, 170anbi12d 691 . . . . . . . . . 10  |-  ( r  =  { <. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }  ->  ( ( r  Er  W  /\  A. x  e.  W  A. n  e.  ( 0 ... ( # `  x
) ) A. a  e.  I  A. b  e.  2o  x r ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) )  <->  ( { <. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u ) )  =  ( H  gsumg  ( T  o.  v ) ) ) }  Er  W  /\  A. x  e.  W  A. n  e.  (
0 ... ( # `  x
) ) A. a  e.  I  A. b  e.  2o  x { <. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }  ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) ) ) )
172171elabg 2917 . . . . . . . . 9  |-  ( {
<. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u ) )  =  ( H  gsumg  ( T  o.  v ) ) ) }  e.  _V  ->  ( { <. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }  e.  { r  |  ( r  Er  W  /\  A. x  e.  W  A. n  e.  (
0 ... ( # `  x
) ) A. a  e.  I  A. b  e.  2o  x r ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) ) }  <->  ( { <. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u ) )  =  ( H  gsumg  ( T  o.  v ) ) ) }  Er  W  /\  A. x  e.  W  A. n  e.  (
0 ... ( # `  x
) ) A. a  e.  I  A. b  e.  2o  x { <. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }  ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) ) ) )
173166, 172syl 15 . . . . . . . 8  |-  ( ph  ->  ( { <. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }  e.  { r  |  ( r  Er  W  /\  A. x  e.  W  A. n  e.  (
0 ... ( # `  x
) ) A. a  e.  I  A. b  e.  2o  x r ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) ) }  <->  ( { <. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u ) )  =  ( H  gsumg  ( T  o.  v ) ) ) }  Er  W  /\  A. x  e.  W  A. n  e.  (
0 ... ( # `  x
) ) A. a  e.  I  A. b  e.  2o  x { <. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }  ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) ) ) )
17425, 162, 173mpbir2and 888 . . . . . . 7  |-  ( ph  ->  { <. u ,  v
>.  |  ( {
u ,  v } 
C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }  e.  { r  |  ( r  Er  W  /\  A. x  e.  W  A. n  e.  (
0 ... ( # `  x
) ) A. a  e.  I  A. b  e.  2o  x r ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) ) } )
175 intss1 3879 . . . . . . 7  |-  ( {
<. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u ) )  =  ( H  gsumg  ( T  o.  v ) ) ) }  e.  {
r  |  ( r  Er  W  /\  A. x  e.  W  A. n  e.  ( 0 ... ( # `  x
) ) A. a  e.  I  A. b  e.  2o  x r ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) ) }  ->  |^|
{ r  |  ( r  Er  W  /\  A. x  e.  W  A. n  e.  ( 0 ... ( # `  x
) ) A. a  e.  I  A. b  e.  2o  x r ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) ) }  C_  {
<. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u ) )  =  ( H  gsumg  ( T  o.  v ) ) ) } )
176174, 175syl 15 . . . . . 6  |-  ( ph  ->  |^| { r  |  ( r  Er  W  /\  A. x  e.  W  A. n  e.  (
0 ... ( # `  x
) ) A. a  e.  I  A. b  e.  2o  x r ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) ) }  C_  {
<. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u ) )  =  ( H  gsumg  ( T  o.  v ) ) ) } )
1773, 176syl5eqss 3224 . . . . 5  |-  ( ph  ->  .~  C_  { <. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) } )
178177ssbrd 4066 . . . 4  |-  ( ph  ->  ( A  .~  C  ->  A { <. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) } C ) )
179178imp 418 . . 3  |-  ( (
ph  /\  A  .~  C )  ->  A { <. u ,  v
>.  |  ( {
u ,  v } 
C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) } C )
1801, 2efger 15029 . . . . . 6  |-  .~  Er  W
181 errel 6671 . . . . . 6  |-  (  .~  Er  W  ->  Rel  .~  )
182180, 181mp1i 11 . . . . 5  |-  ( ph  ->  Rel  .~  )
183 brrelex12 4728 . . . . 5  |-  ( ( Rel  .~  /\  A  .~  C )  ->  ( A  e.  _V  /\  C  e.  _V ) )
184182, 183sylan 457 . . . 4  |-  ( (
ph  /\  A  .~  C )  ->  ( A  e.  _V  /\  C  e.  _V ) )
185 preq12 3710 . . . . . . 7  |-  ( ( u  =  A  /\  v  =  C )  ->  { u ,  v }  =  { A ,  C } )
186185sseq1d 3207 . . . . . 6  |-  ( ( u  =  A  /\  v  =  C )  ->  ( { u ,  v }  C_  W  <->  { A ,  C }  C_  W ) )
187 coeq2 4844 . . . . . . . 8  |-  ( u  =  A  ->  ( T  o.  u )  =  ( T  o.  A ) )
188187oveq2d 5876 . . . . . . 7  |-  ( u  =  A  ->  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  A
) ) )
189 coeq2 4844 . . . . . . . 8  |-  ( v  =  C  ->  ( T  o.  v )  =  ( T  o.  C ) )
190189oveq2d 5876 . . . . . . 7  |-  ( v  =  C  ->  ( H  gsumg  ( T  o.  v
) )  =  ( H  gsumg  ( T  o.  C
) ) )
191188, 190eqeqan12d 2300 . . . . . 6  |-  ( ( u  =  A  /\  v  =  C )  ->  ( ( H  gsumg  ( T  o.  u ) )  =  ( H  gsumg  ( T  o.  v ) )  <-> 
( H  gsumg  ( T  o.  A
) )  =  ( H  gsumg  ( T  o.  C
) ) ) )
192186, 191anbi12d 691 . . . . 5  |-  ( ( u  =  A  /\  v  =  C )  ->  ( ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u ) )  =  ( H  gsumg  ( T  o.  v ) ) )  <->  ( { A ,  C }  C_  W  /\  ( H  gsumg  ( T  o.  A
) )  =  ( H  gsumg  ( T  o.  C
) ) ) ) )
193192, 158brabga 4281 . . . 4  |-  ( ( A  e.  _V  /\  C  e.  _V )  ->  ( A { <. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) } C  <->  ( { A ,  C }  C_  W  /\  ( H  gsumg  ( T  o.  A
) )  =  ( H  gsumg  ( T  o.  C
) ) ) ) )
194184, 193syl 15 . . 3  |-  ( (
ph  /\  A  .~  C )  ->  ( A { <. u ,  v
>.  |  ( {
u ,  v } 
C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) } C  <->  ( { A ,  C }  C_  W  /\  ( H  gsumg  ( T  o.  A
) )  =  ( H  gsumg  ( T  o.  C
) ) ) ) )
195179, 194mpbid 201 . 2  |-  ( (
ph  /\  A  .~  C )  ->  ( { A ,  C }  C_  W  /\  ( H 
gsumg  ( T  o.  A
) )  =  ( H  gsumg  ( T  o.  C
) ) ) )
196195simprd 449 1  |-  ( (
ph  /\  A  .~  C )  ->  ( H  gsumg  ( T  o.  A
) )  =  ( H  gsumg  ( T  o.  C
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1625    e. wcel 1686   {cab 2271   A.wral 2545   _Vcvv 2790    \ cdif 3151    i^i cin 3153    C_ wss 3154   (/)c0 3457   ifcif 3567   {cpr 3643   <.cop 3645   <.cotp 3646   |^|cint 3864   class class class wbr 4025   {copab 4078    _I cid 4306    X. cxp 4689    o. ccom 4695   Rel wrel 4696   -->wf 5253   ` cfv 5257  (class class class)co 5860    e. cmpt2 5862   1oc1o 6474   2oc2o 6475    Er wer 6659   0cc0 8739   NN0cn0 9967   ZZ>=cuz 10232   ...cfz 10784   #chash 11339  Word cword 11405   concat cconcat 11406   substr csubstr 11408   splice csplice 11409   <"cs2 11493   Basecbs 13150   +g cplusg 13210   0gc0g 13402    gsumg cgsu 13403   Mndcmnd 14363   Grpcgrp 14364   inv gcminusg 14365   ~FG cefg 15017
This theorem is referenced by:  frgpupf  15084
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-ot 3652  df-uni 3830  df-int 3865  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-2o 6482  df-oadd 6485  df-er 6662  df-map 6776  df-pm 6777  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-card 7574  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-nn 9749  df-2 9806  df-n0 9968  df-z 10027  df-uz 10233  df-fz 10785  df-fzo 10873  df-seq 11049  df-hash 11340  df-word 11411  df-concat 11412  df-s1 11413  df-substr 11414  df-splice 11415  df-s2 11500  df-ndx 13153  df-slot 13154  df-base 13155  df-sets 13156  df-ress 13157  df-plusg 13223  df-0g 13406  df-gsum 13407  df-mnd 14369  df-submnd 14418  df-grp 14491  df-minusg 14492  df-efg 15020
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