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Theorem frgpuplem 15359
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 15304 . . . . . 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 4990 . . . . . . . . . . . . 13  |-  ( u  =  v  ->  ( T  o.  u )  =  ( T  o.  v ) )
54oveq2d 6056 . . . . . . . . . . . 12  |-  ( u  =  v  ->  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) )
6 eqid 2404 . . . . . . . . . . . 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 6897 . . . . . . . . . . 11  |-  { <. u ,  v >.  |  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) }  Er  _V
87a1i 11 . . . . . . . . . 10  |-  ( ph  ->  { <. u ,  v
>.  |  ( H  gsumg  ( T  o.  u ) )  =  ( H 
gsumg  ( T  o.  v
) ) }  Er  _V )
9 ssv 3328 . . . . . . . . . . 11  |-  W  C_  _V
109a1i 11 . . . . . . . . . 10  |-  ( ph  ->  W  C_  _V )
118, 10erinxp 6937 . . . . . . . . 9  |-  ( ph  ->  ( { <. u ,  v >.  |  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) }  i^i  ( W  X.  W
) )  Er  W
)
12 df-xp 4843 . . . . . . . . . . . . 13  |-  ( W  X.  W )  =  { <. u ,  v
>.  |  ( u  e.  W  /\  v  e.  W ) }
1312ineq1i 3498 . . . . . . . . . . . 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 3493 . . . . . . . . . . . 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 4964 . . . . . . . . . . . 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 2432 . . . . . . . . . . 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 2919 . . . . . . . . . . . . . 14  |-  u  e. 
_V
18 vex 2919 . . . . . . . . . . . . . 14  |-  v  e. 
_V
1917, 18prss 3912 . . . . . . . . . . . . 13  |-  ( ( u  e.  W  /\  v  e.  W )  <->  { u ,  v } 
C_  W )
2019anbi1i 677 . . . . . . . . . . . 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 4232 . . . . . . . . . . 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 2424 . . . . . . . . . 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 6871 . . . . . . . . . 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 189 . . . . . . . 8  |-  ( ph  ->  { <. u ,  v
>.  |  ( {
u ,  v } 
C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }  Er  W )
26 simplrl 737 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  ->  x  e.  W )
27 fviss 5743 . . . . . . . . . . . . . . . 16  |-  (  _I 
` Word  ( I  X.  2o ) )  C_ Word  ( I  X.  2o )
281, 27eqsstri 3338 . . . . . . . . . . . . . . 15  |-  W  C_ Word  ( I  X.  2o )
2928, 26sseldi 3306 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  ->  x  e. Word  ( I  X.  2o ) )
30 opelxpi 4869 . . . . . . . . . . . . . . . 16  |-  ( ( a  e.  I  /\  b  e.  2o )  -> 
<. a ,  b >.  e.  ( I  X.  2o ) )
3130adantl 453 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  ->  <. a ,  b >.  e.  ( I  X.  2o ) )
32 simprl 733 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
a  e.  I )
33 2oconcl 6706 . . . . . . . . . . . . . . . . 17  |-  ( b  e.  2o  ->  ( 1o  \  b )  e.  2o )
3433ad2antll 710 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( 1o  \  b
)  e.  2o )
35 opelxpi 4869 . . . . . . . . . . . . . . . 16  |-  ( ( a  e.  I  /\  ( 1o  \  b
)  e.  2o )  ->  <. a ,  ( 1o  \  b )
>.  e.  ( I  X.  2o ) )
3632, 34, 35syl2anc 643 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  ->  <. a ,  ( 1o 
\  b ) >.  e.  ( I  X.  2o ) )
3731, 36s2cld 11788 . . . . . . . . . . . . . 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 11736 . . . . . . . . . . . . . 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 643 . . . . . . . . . . . . 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 15302 . . . . . . . . . . . . . . 15  |-  ( x  e.  W  ->  (
I  e.  _V  /\  W  = Word  ( I  X.  2o ) ) )
4126, 40syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( I  e.  _V  /\  W  = Word  ( I  X.  2o ) ) )
4241simprd 450 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  ->  W  = Word  ( I  X.  2o ) )
4339, 42eleqtrrd 2481 . . . . . . . . . . . 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 519 . . . . . . . . . . 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 11721 . . . . . . . . . . . . . . . . . 18  |-  ( x  e. Word  ( I  X.  2o )  ->  ( x substr  <. 0 ,  n >. )  e. Word  ( I  X.  2o ) )
4629, 45syl 16 . . . . . . . . . . . . . . . . 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 15357 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  T : ( I  X.  2o ) --> B )
5453ad2antrr 707 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  ->  T : ( I  X.  2o ) --> B )
55 ccatco 11759 . . . . . . . . . . . . . . . . 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 1184 . . . . . . . . . . . . . . . 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 6056 . . . . . . . . . . . . . . 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 707 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  ->  H  e.  Grp )
59 grpmnd 14772 . . . . . . . . . . . . . . . . 17  |-  ( H  e.  Grp  ->  H  e.  Mnd )
6058, 59syl 16 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  ->  H  e.  Mnd )
61 wrdco 11755 . . . . . . . . . . . . . . . . 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 643 . . . . . . . . . . . . . . . 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 11755 . . . . . . . . . . . . . . . . 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 643 . . . . . . . . . . . . . . . 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 2404 . . . . . . . . . . . . . . . . 17  |-  ( +g  `  H )  =  ( +g  `  H )
6647, 65gsumccat 14742 . . . . . . . . . . . . . . . 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 1184 . . . . . . . . . . . . . . 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 11822 . . . . . . . . . . . . . . . . . . . 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 6043 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( a T b )  =  ( T `  <. a ,  b >. )
7069a1i 11 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( a T b )  =  ( T `
 <. a ,  b
>. ) )
71 df-ov 6043 . . . . . . . . . . . . . . . . . . . . . . . . . 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 2404 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \ 
z ) >. )  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \  z )
>. )
7372efgmval 15299 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  ( a ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \ 
z ) >. )
b )  =  <. a ,  ( 1o  \ 
b ) >. )
7471, 73syl5eqr 2450 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  ( ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \  z )
>. ) `  <. a ,  b >. )  =  <. a ,  ( 1o  \  b )
>. )
7574adantl 453 . . . . . . . . . . . . . . . . . . . . . . . 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 5691 . . . . . . . . . . . . . . . . . . . . . . 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 15358 . . . . . . . . . . . . . . . . . . . . . . . . 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 461 . . . . . . . . . . . . . . . . . . . . . . . 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 696 . . . . . . . . . . . . . . . . . . . . . . 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 2438 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( T `  <. a ,  ( 1o  \ 
b ) >. )  =  ( N `  ( T `  <. a ,  b >. )
) )
8169fveq2i 5690 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( N `
 ( a T b ) )  =  ( N `  ( T `  <. a ,  b >. ) )
8280, 81syl6reqr 2455 . . . . . . . . . . . . . . . . . . . . 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 11781 . . . . . . . . . . . . . . . . . . . 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 2439 . . . . . . . . . . . . . . . . . . 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 6056 . . . . . . . . . . . . . . . . . 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 734 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
b  e.  2o )
8754, 32, 86fovrnd 6177 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( a T b )  e.  B )
8847, 48grpinvcl 14805 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( H  e.  Grp  /\  ( a T b )  e.  B )  ->  ( N `  ( a T b ) )  e.  B
)
8958, 87, 88syl2anc 643 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( N `  (
a T b ) )  e.  B )
9047, 65gsumws2 14743 . . . . . . . . . . . . . . . . . . 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 ) ) ) )
9160, 87, 89, 90syl3anc 1184 . . . . . . . . . . . . . . . . . 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 ) ) ) )
92 eqid 2404 . . . . . . . . . . . . . . . . . . . 20  |-  ( 0g
`  H )  =  ( 0g `  H
)
9347, 65, 92, 48grprinv 14807 . . . . . . . . . . . . . . . . . . 19  |-  ( ( H  e.  Grp  /\  ( a T b )  e.  B )  ->  ( ( a T b ) ( +g  `  H ) ( N `  (
a T b ) ) )  =  ( 0g `  H ) )
9458, 87, 93syl2anc 643 . . . . . . . . . . . . . . . . . 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 ) )
9585, 91, 943eqtrd 2440 . . . . . . . . . . . . . . . . 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
) )
9695oveq2d 6056 . . . . . . . . . . . . . . . 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 ) ) )
9747gsumwcl 14741 . . . . . . . . . . . . . . . . . 18  |-  ( ( H  e.  Mnd  /\  ( T  o.  (
x substr  <. 0 ,  n >. ) )  e. Word  B
)  ->  ( H  gsumg  ( T  o.  ( x substr  <. 0 ,  n >. ) ) )  e.  B
)
9860, 62, 97syl2anc 643 . . . . . . . . . . . . . . . . 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 )
9947, 65, 92grprid 14791 . . . . . . . . . . . . . . . . 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 >. ) ) ) )
10058, 98, 99syl2anc 643 . . . . . . . . . . . . . . . 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 >. ) ) ) )
10196, 100eqtrd 2436 . . . . . . . . . . . . . . 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 >. ) ) ) )
10257, 67, 1013eqtrrd 2441 . . . . . . . . . . . . . 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 ) >. "> ) ) ) )
103102oveq1d 6055 . . . . . . . . . . . . 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 ) >. ) ) ) ) )
104 swrdcl 11721 . . . . . . . . . . . . . . . 16  |-  ( x  e. Word  ( I  X.  2o )  ->  ( x substr  <. n ,  ( # `  x ) >. )  e. Word  ( I  X.  2o ) )
10529, 104syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( x substr  <. n ,  ( # `  x
) >. )  e. Word  (
I  X.  2o ) )
106 wrdco 11755 . . . . . . . . . . . . . . 15  |-  ( ( ( x substr  <. n ,  ( # `  x
) >. )  e. Word  (
I  X.  2o )  /\  T : ( I  X.  2o ) --> B )  ->  ( T  o.  ( x substr  <.
n ,  ( # `  x ) >. )
)  e. Word  B )
107105, 54, 106syl2anc 643 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( T  o.  (
x substr  <. n ,  (
# `  x ) >. ) )  e. Word  B
)
10847, 65gsumccat 14742 . . . . . . . . . . . . . 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 ) >. ) ) ) ) )
10960, 62, 107, 108syl3anc 1184 . . . . . . . . . . . . 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
) >. ) ) ) ) )
110 ccatcl 11698 . . . . . . . . . . . . . . . 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 ) )
11146, 37, 110syl2anc 643 . . . . . . . . . . . . . . 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 ) )
112 wrdco 11755 . . . . . . . . . . . . . . 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
)
113111, 54, 112syl2anc 643 . . . . . . . . . . . . . 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
)
11447, 65gsumccat 14742 . . . . . . . . . . . . . 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 ) >. ) ) ) ) )
11560, 113, 107, 114syl3anc 1184 . . . . . . . . . . . . 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 ) >. ) ) ) ) )
116103, 109, 1153eqtr4d 2446 . . . . . . . . . . . 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
) >. ) ) ) ) )
117 simplrr 738 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  ->  n  e.  ( 0 ... ( # `  x
) ) )
118 elfzuz 11011 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  ( 0 ... ( # `  x
) )  ->  n  e.  ( ZZ>= `  0 )
)
119 eluzfz1 11020 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... n
) )
120117, 118, 1193syl 19 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
0  e.  ( 0 ... n ) )
121 lencl 11690 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  e. Word  ( I  X.  2o )  ->  ( # `  x )  e.  NN0 )
12229, 121syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( # `  x )  e.  NN0 )
123 nn0uz 10476 . . . . . . . . . . . . . . . . . . 19  |-  NN0  =  ( ZZ>= `  0 )
124122, 123syl6eleq 2494 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( # `  x )  e.  ( ZZ>= `  0
) )
125 eluzfz2 11021 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  x )  e.  ( ZZ>= `  0 )  ->  ( # `  x
)  e.  ( 0 ... ( # `  x
) ) )
126124, 125syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( # `  x )  e.  ( 0 ... ( # `  x
) ) )
127 ccatswrd 11728 . . . . . . . . . . . . . . . . 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
) >. ) )
12829, 120, 117, 126, 127syl13anc 1186 . . . . . . . . . . . . . . . 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 ) >. )
)
129 swrdid 11727 . . . . . . . . . . . . . . . . 17  |-  ( x  e. Word  ( I  X.  2o )  ->  ( x substr  <. 0 ,  ( # `  x ) >. )  =  x )
13029, 129syl 16 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( x substr  <. 0 ,  ( # `  x
) >. )  =  x )
131128, 130eqtrd 2436 . . . . . . . . . . . . . . 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 )
132131coeq2d 4994 . . . . . . . . . . . . . 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 ) )
133 ccatco 11759 . . . . . . . . . . . . . . 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 ) >. )
) ) )
13446, 105, 54, 133syl3anc 1184 . . . . . . . . . . . . . 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
) >. ) ) ) )
135132, 134eqtr3d 2438 . . . . . . . . . . . . 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
) >. ) ) ) )
136135oveq2d 6056 . . . . . . . . . . . 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 ) >. )
) ) ) )
137 splval 11735 . . . . . . . . . . . . . . . 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
) >. ) ) )
13826, 117, 117, 37, 137syl13anc 1186 . . . . . . . . . . . . . . 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
) >. ) ) )
139138coeq2d 4994 . . . . . . . . . . . . . 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
) >. ) ) ) )
140 ccatco 11759 . . . . . . . . . . . . . . 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 ) >. ) ) ) )
141111, 105, 54, 140syl3anc 1184 . . . . . . . . . . . . . 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 ) >. ) ) ) )
142139, 141eqtrd 2436 . . . . . . . . . . . . 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
) >. ) ) ) )
143142oveq2d 6056 . . . . . . . . . . . 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
) >. ) ) ) ) )
144116, 136, 1433eqtr4d 2446 . . . . . . . . . . 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 ) >. "> >.
) ) ) )
145 vex 2919 . . . . . . . . . . . 12  |-  x  e. 
_V
146 ovex 6065 . . . . . . . . . . . 12  |-  ( x splice  <. n ,  n , 
<" <. a ,  b
>. <. a ,  ( 1o  \  b )
>. "> >. )  e.  _V
147 eleq1 2464 . . . . . . . . . . . . . . 15  |-  ( u  =  x  ->  (
u  e.  W  <->  x  e.  W ) )
148 eleq1 2464 . . . . . . . . . . . . . . 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 ) )
149147, 148bi2anan9 844 . . . . . . . . . . . . . 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 ) ) )
15019, 149syl5bbr 251 . . . . . . . . . . . . 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 ) ) )
151 coeq2 4990 . . . . . . . . . . . . . . 15  |-  ( u  =  x  ->  ( T  o.  u )  =  ( T  o.  x ) )
152151oveq2d 6056 . . . . . . . . . . . . . 14  |-  ( u  =  x  ->  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  x
) ) )
153 coeq2 4990 . . . . . . . . . . . . . . 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 )
>. "> >. )
) )
154153oveq2d 6056 . . . . . . . . . . . . . 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 ) >. "> >.
) ) ) )
155152, 154eqeqan12d 2419 . . . . . . . . . . . . 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 ) >. "> >.
) ) ) ) )
156150, 155anbi12d 692 . . . . . . . . . . . 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 ) >. "> >.
) ) ) ) ) )
157 eqid 2404 . . . . . . . . . . . 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
) ) ) }
158145, 146, 156, 157braba 4432 . . . . . . . . . . 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 ) >. "> >.
) ) ) ) )
15944, 144, 158sylanbrc 646 . . . . . . . . . 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 ) >. "> >.
) )
160159ralrimivva 2758 . . . . . . . . 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 ) >. "> >.
) )
161160ralrimivva 2758 . . . . . . . 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 ) >. "> >.
) )
162 fvex 5701 . . . . . . . . . . 11  |-  (  _I 
` Word  ( I  X.  2o ) )  e.  _V
1631, 162eqeltri 2474 . . . . . . . . . 10  |-  W  e. 
_V
164 erex 6888 . . . . . . . . . 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 ) )
16525, 163, 164ee10 1382 . . . . . . . . 9  |-  ( ph  ->  { <. u ,  v
>.  |  ( {
u ,  v } 
C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }  e.  _V )
166 ereq1 6871 . . . . . . . . . . 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 ) )
167 breq 4174 . . . . . . . . . . . . 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 ) >. "> >.
) ) )
1681672ralbidv 2708 . . . . . . . . . . . 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 ) >. "> >.
) ) )
1691682ralbidv 2708 . . . . . . . . . . 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 ) >. "> >.
) ) )
170166, 169anbi12d 692 . . . . . . . . . 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 ) >. "> >.
) ) ) )
171170elabg 3043 . . . . . . . . 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 ) >. "> >.
) ) ) )
172165, 171syl 16 . . . . . . . 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 ) >. "> >.
) ) ) )
17325, 161, 172mpbir2and 889 . . . . . . 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 ) >. "> >.
) ) } )
174 intss1 4025 . . . . . . 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 ) ) ) } )
175173, 174syl 16 . . . . . 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 ) ) ) } )
1763, 175syl5eqss 3352 . . . . 5  |-  ( ph  ->  .~  C_  { <. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) } )
177176ssbrd 4213 . . . 4  |-  ( ph  ->  ( A  .~  C  ->  A { <. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) } C ) )
178177imp 419 . . 3  |-  ( (
ph  /\  A  .~  C )  ->  A { <. u ,  v
>.  |  ( {
u ,  v } 
C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) } C )
1791, 2efger 15305 . . . . . 6  |-  .~  Er  W
180 errel 6873 . . . . . 6  |-  (  .~  Er  W  ->  Rel  .~  )
181179, 180mp1i 12 . . . . 5  |-  ( ph  ->  Rel  .~  )
182 brrelex12 4874 . . . . 5  |-  ( ( Rel  .~  /\  A  .~  C )  ->  ( A  e.  _V  /\  C  e.  _V ) )
183181, 182sylan 458 . . . 4  |-  ( (
ph  /\  A  .~  C )  ->  ( A  e.  _V  /\  C  e.  _V ) )
184 preq12 3845 . . . . . . 7  |-  ( ( u  =  A  /\  v  =  C )  ->  { u ,  v }  =  { A ,  C } )
185184sseq1d 3335 . . . . . 6  |-  ( ( u  =  A  /\  v  =  C )  ->  ( { u ,  v }  C_  W  <->  { A ,  C }  C_  W ) )
186 coeq2 4990 . . . . . . . 8  |-  ( u  =  A  ->  ( T  o.  u )  =  ( T  o.  A ) )
187186oveq2d 6056 . . . . . . 7  |-  ( u  =  A  ->  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  A
) ) )
188 coeq2 4990 . . . . . . . 8  |-  ( v  =  C  ->  ( T  o.  v )  =  ( T  o.  C ) )
189188oveq2d 6056 . . . . . . 7  |-  ( v  =  C  ->  ( H  gsumg  ( T  o.  v
) )  =  ( H  gsumg  ( T  o.  C
) ) )
190187, 189eqeqan12d 2419 . . . . . 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
) ) ) )
191185, 190anbi12d 692 . . . . 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
) ) ) ) )
192191, 157brabga 4429 . . . 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
) ) ) ) )
193183, 192syl 16 . . 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
) ) ) ) )
194178, 193mpbid 202 . 2  |-  ( (
ph  /\  A  .~  C )  ->  ( { A ,  C }  C_  W  /\  ( H 
gsumg  ( T  o.  A
) )  =  ( H  gsumg  ( T  o.  C
) ) ) )
195194simprd 450 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 177    /\ wa 359    = wceq 1649    e. wcel 1721   {cab 2390   A.wral 2666   _Vcvv 2916    \ cdif 3277    i^i cin 3279    C_ wss 3280   (/)c0 3588   ifcif 3699   {cpr 3775   <.cop 3777   <.cotp 3778   |^|cint 4010   class class class wbr 4172   {copab 4225    _I cid 4453    X. cxp 4835    o. ccom 4841   Rel wrel 4842   -->wf 5409   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   1oc1o 6676   2oc2o 6677    Er wer 6861   0cc0 8946   NN0cn0 10177   ZZ>=cuz 10444   ...cfz 10999   #chash 11573  Word cword 11672   concat cconcat 11673   substr csubstr 11675   splice csplice 11676   <"cs2 11760   Basecbs 13424   +g cplusg 13484   0gc0g 13678    gsumg cgsu 13679   Mndcmnd 14639   Grpcgrp 14640   inv gcminusg 14641   ~FG cefg 15293
This theorem is referenced by:  frgpupf  15360
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-ot 3784  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-fzo 11091  df-seq 11279  df-hash 11574  df-word 11678  df-concat 11679  df-s1 11680  df-substr 11681  df-splice 11682  df-s2 11767  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-0g 13682  df-gsum 13683  df-mnd 14645  df-submnd 14694  df-grp 14767  df-minusg 14768  df-efg 15296
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