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Theorem efginvrel2 17365
Description: The inverse of the reverse of a word composed with the word relates to the identity. (This provides an explicit expression for the representation of the group inverse, given a representative of the free group equivalence class.) (Contributed by Mario Carneiro, 1-Oct-2015.)
Hypotheses
Ref Expression
efgval.w  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
efgval.r  |-  .~  =  ( ~FG  `  I )
efgval2.m  |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
)
efgval2.t  |-  T  =  ( v  e.  W  |->  ( n  e.  ( 0 ... ( # `  v ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. )
) )
Assertion
Ref Expression
efginvrel2  |-  ( A  e.  W  ->  ( A ++  ( M  o.  (reverse `  A ) ) )  .~  (/) )
Distinct variable groups:    y, z    v, n, w, y, z   
n, M, v, w   
n, W, v, w, y, z    y,  .~ , z    n, I, v, w, y, z
Allowed substitution hints:    A( y, z, w, v, n)    .~ ( w, v, n)    T( y, z, w, v, n)    M( y, z)

Proof of Theorem efginvrel2
Dummy variables  a 
b  c  u  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 efgval.w . . . 4  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
2 fviss 5936 . . . 4  |-  (  _I 
` Word  ( I  X.  2o ) )  C_ Word  ( I  X.  2o )
31, 2eqsstri 3494 . . 3  |-  W  C_ Word  ( I  X.  2o )
43sseli 3460 . 2  |-  ( A  e.  W  ->  A  e. Word  ( I  X.  2o ) )
5 id 23 . . . . . 6  |-  ( c  =  (/)  ->  c  =  (/) )
6 fveq2 5878 . . . . . . . . 9  |-  ( c  =  (/)  ->  (reverse `  c
)  =  (reverse `  (/) ) )
7 rev0 12860 . . . . . . . . 9  |-  (reverse `  (/) )  =  (/)
86, 7syl6eq 2479 . . . . . . . 8  |-  ( c  =  (/)  ->  (reverse `  c
)  =  (/) )
98coeq2d 5013 . . . . . . 7  |-  ( c  =  (/)  ->  ( M  o.  (reverse `  c
) )  =  ( M  o.  (/) ) )
10 co02 5365 . . . . . . 7  |-  ( M  o.  (/) )  =  (/)
119, 10syl6eq 2479 . . . . . 6  |-  ( c  =  (/)  ->  ( M  o.  (reverse `  c
) )  =  (/) )
125, 11oveq12d 6320 . . . . 5  |-  ( c  =  (/)  ->  ( c ++  ( M  o.  (reverse `  c ) ) )  =  ( (/) ++  (/) ) )
1312breq1d 4430 . . . 4  |-  ( c  =  (/)  ->  ( ( c ++  ( M  o.  (reverse `  c ) ) )  .~  (/)  <->  ( (/) ++  (/) )  .~  (/) ) )
1413imbi2d 317 . . 3  |-  ( c  =  (/)  ->  ( ( A  e.  W  -> 
( c ++  ( M  o.  (reverse `  c
) ) )  .~  (/) )  <->  ( A  e.  W  ->  ( (/) ++  (/) )  .~  (/) ) ) )
15 id 23 . . . . . 6  |-  ( c  =  a  ->  c  =  a )
16 fveq2 5878 . . . . . . 7  |-  ( c  =  a  ->  (reverse `  c )  =  (reverse `  a ) )
1716coeq2d 5013 . . . . . 6  |-  ( c  =  a  ->  ( M  o.  (reverse `  c
) )  =  ( M  o.  (reverse `  a
) ) )
1815, 17oveq12d 6320 . . . . 5  |-  ( c  =  a  ->  (
c ++  ( M  o.  (reverse `  c ) ) )  =  ( a ++  ( M  o.  (reverse `  a ) ) ) )
1918breq1d 4430 . . . 4  |-  ( c  =  a  ->  (
( c ++  ( M  o.  (reverse `  c
) ) )  .~  (/)  <->  ( a ++  ( M  o.  (reverse `  a ) ) )  .~  (/) ) )
2019imbi2d 317 . . 3  |-  ( c  =  a  ->  (
( A  e.  W  ->  ( c ++  ( M  o.  (reverse `  c
) ) )  .~  (/) )  <->  ( A  e.  W  ->  ( a ++  ( M  o.  (reverse `  a ) ) )  .~  (/) ) ) )
21 id 23 . . . . . 6  |-  ( c  =  ( a ++  <" b "> )  ->  c  =  ( a ++ 
<" b "> ) )
22 fveq2 5878 . . . . . . 7  |-  ( c  =  ( a ++  <" b "> )  ->  (reverse `  c )  =  (reverse `  ( a ++  <" b "> ) ) )
2322coeq2d 5013 . . . . . 6  |-  ( c  =  ( a ++  <" b "> )  ->  ( M  o.  (reverse `  c ) )  =  ( M  o.  (reverse `  ( a ++  <" b "> ) ) ) )
2421, 23oveq12d 6320 . . . . 5  |-  ( c  =  ( a ++  <" b "> )  ->  ( c ++  ( M  o.  (reverse `  c
) ) )  =  ( ( a ++  <" b "> ) ++  ( M  o.  (reverse `  ( a ++  <" b "> ) ) ) ) )
2524breq1d 4430 . . . 4  |-  ( c  =  ( a ++  <" b "> )  ->  ( ( c ++  ( M  o.  (reverse `  c
) ) )  .~  (/)  <->  ( ( a ++  <" b "> ) ++  ( M  o.  (reverse `  (
a ++  <" b "> ) ) ) )  .~  (/) ) )
2625imbi2d 317 . . 3  |-  ( c  =  ( a ++  <" b "> )  ->  ( ( A  e.  W  ->  ( c ++  ( M  o.  (reverse `  c ) ) )  .~  (/) )  <->  ( A  e.  W  ->  ( ( a ++  <" b "> ) ++  ( M  o.  (reverse `  (
a ++  <" b "> ) ) ) )  .~  (/) ) ) )
27 id 23 . . . . . 6  |-  ( c  =  A  ->  c  =  A )
28 fveq2 5878 . . . . . . 7  |-  ( c  =  A  ->  (reverse `  c )  =  (reverse `  A ) )
2928coeq2d 5013 . . . . . 6  |-  ( c  =  A  ->  ( M  o.  (reverse `  c
) )  =  ( M  o.  (reverse `  A
) ) )
3027, 29oveq12d 6320 . . . . 5  |-  ( c  =  A  ->  (
c ++  ( M  o.  (reverse `  c ) ) )  =  ( A ++  ( M  o.  (reverse `  A ) ) ) )
3130breq1d 4430 . . . 4  |-  ( c  =  A  ->  (
( c ++  ( M  o.  (reverse `  c
) ) )  .~  (/)  <->  ( A ++  ( M  o.  (reverse `  A ) ) )  .~  (/) ) )
3231imbi2d 317 . . 3  |-  ( c  =  A  ->  (
( A  e.  W  ->  ( c ++  ( M  o.  (reverse `  c
) ) )  .~  (/) )  <->  ( A  e.  W  ->  ( A ++  ( M  o.  (reverse `  A ) ) )  .~  (/) ) ) )
33 wrd0 12684 . . . . 5  |-  (/)  e. Word  (
I  X.  2o )
34 ccatlid 12723 . . . . 5  |-  ( (/)  e. Word  ( I  X.  2o )  ->  ( (/) ++  (/) )  =  (/) )
3533, 34ax-mp 5 . . . 4  |-  ( (/) ++  (/) )  =  (/)
36 efgval.r . . . . . . 7  |-  .~  =  ( ~FG  `  I )
371, 36efger 17356 . . . . . 6  |-  .~  Er  W
3837a1i 11 . . . . 5  |-  ( A  e.  W  ->  .~  Er  W )
391efgrcl 17353 . . . . . . 7  |-  ( A  e.  W  ->  (
I  e.  _V  /\  W  = Word  ( I  X.  2o ) ) )
4039simprd 464 . . . . . 6  |-  ( A  e.  W  ->  W  = Word  ( I  X.  2o ) )
4133, 40syl5eleqr 2517 . . . . 5  |-  ( A  e.  W  ->  (/)  e.  W
)
4238, 41erref 7388 . . . 4  |-  ( A  e.  W  ->  (/)  .~  (/) )
4335, 42syl5eqbr 4454 . . 3  |-  ( A  e.  W  ->  ( (/) ++  (/) )  .~  (/) )
4437a1i 11 . . . . . . 7  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  .~  Er  W )
45 simprl 762 . . . . . . . . . 10  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  a  e. Word  ( I  X.  2o ) )
46 revcl 12857 . . . . . . . . . . . 12  |-  ( a  e. Word  ( I  X.  2o )  ->  (reverse `  a
)  e. Word  ( I  X.  2o ) )
4746ad2antrl 732 . . . . . . . . . . 11  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (reverse `  a )  e. Word  (
I  X.  2o ) )
48 efgval2.m . . . . . . . . . . . 12  |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
)
4948efgmf 17351 . . . . . . . . . . 11  |-  M :
( I  X.  2o )
--> ( I  X.  2o )
50 wrdco 12919 . . . . . . . . . . 11  |-  ( ( (reverse `  a )  e. Word  ( I  X.  2o )  /\  M : ( I  X.  2o ) --> ( I  X.  2o ) )  ->  ( M  o.  (reverse `  a
) )  e. Word  (
I  X.  2o ) )
5147, 49, 50sylancl 666 . . . . . . . . . 10  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( M  o.  (reverse `  a
) )  e. Word  (
I  X.  2o ) )
52 ccatcl 12713 . . . . . . . . . 10  |-  ( ( a  e. Word  ( I  X.  2o )  /\  ( M  o.  (reverse `  a ) )  e. Word 
( I  X.  2o ) )  ->  (
a ++  ( M  o.  (reverse `  a ) ) )  e. Word  ( I  X.  2o ) )
5345, 51, 52syl2anc 665 . . . . . . . . 9  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
a ++  ( M  o.  (reverse `  a ) ) )  e. Word  ( I  X.  2o ) )
5440adantr 466 . . . . . . . . 9  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  W  = Word  ( I  X.  2o ) )
5553, 54eleqtrrd 2513 . . . . . . . 8  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
a ++  ( M  o.  (reverse `  a ) ) )  e.  W )
56 lencl 12680 . . . . . . . . . . . . . 14  |-  ( a  e. Word  ( I  X.  2o )  ->  ( # `  a )  e.  NN0 )
5756ad2antrl 732 . . . . . . . . . . . . 13  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 a )  e. 
NN0 )
58 nn0uz 11194 . . . . . . . . . . . . 13  |-  NN0  =  ( ZZ>= `  0 )
5957, 58syl6eleq 2520 . . . . . . . . . . . 12  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 a )  e.  ( ZZ>= `  0 )
)
60 ccatlen 12714 . . . . . . . . . . . . . 14  |-  ( ( a  e. Word  ( I  X.  2o )  /\  ( M  o.  (reverse `  a ) )  e. Word 
( I  X.  2o ) )  ->  ( # `
 ( a ++  ( M  o.  (reverse `  a
) ) ) )  =  ( ( # `  a )  +  (
# `  ( M  o.  (reverse `  a )
) ) ) )
6145, 51, 60syl2anc 665 . . . . . . . . . . . . 13  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 ( a ++  ( M  o.  (reverse `  a
) ) ) )  =  ( ( # `  a )  +  (
# `  ( M  o.  (reverse `  a )
) ) ) )
6257nn0zd 11039 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 a )  e.  ZZ )
63 uzid 11174 . . . . . . . . . . . . . . 15  |-  ( (
# `  a )  e.  ZZ  ->  ( # `  a
)  e.  ( ZZ>= `  ( # `  a ) ) )
6462, 63syl 17 . . . . . . . . . . . . . 14  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 a )  e.  ( ZZ>= `  ( # `  a
) ) )
65 lencl 12680 . . . . . . . . . . . . . . 15  |-  ( ( M  o.  (reverse `  a
) )  e. Word  (
I  X.  2o )  ->  ( # `  ( M  o.  (reverse `  a
) ) )  e. 
NN0 )
6651, 65syl 17 . . . . . . . . . . . . . 14  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 ( M  o.  (reverse `  a ) ) )  e.  NN0 )
67 uzaddcl 11216 . . . . . . . . . . . . . 14  |-  ( ( ( # `  a
)  e.  ( ZZ>= `  ( # `  a ) )  /\  ( # `  ( M  o.  (reverse `  a ) ) )  e.  NN0 )  -> 
( ( # `  a
)  +  ( # `  ( M  o.  (reverse `  a ) ) ) )  e.  ( ZZ>= `  ( # `  a ) ) )
6864, 66, 67syl2anc 665 . . . . . . . . . . . . 13  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( # `  a )  +  ( # `  ( M  o.  (reverse `  a
) ) ) )  e.  ( ZZ>= `  ( # `
 a ) ) )
6961, 68eqeltrd 2510 . . . . . . . . . . . 12  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 ( a ++  ( M  o.  (reverse `  a
) ) ) )  e.  ( ZZ>= `  ( # `
 a ) ) )
70 elfzuzb 11795 . . . . . . . . . . . 12  |-  ( (
# `  a )  e.  ( 0 ... ( # `
 ( a ++  ( M  o.  (reverse `  a
) ) ) ) )  <->  ( ( # `  a )  e.  (
ZZ>= `  0 )  /\  ( # `  ( a ++  ( M  o.  (reverse `  a ) ) ) )  e.  ( ZZ>= `  ( # `  a ) ) ) )
7159, 69, 70sylanbrc 668 . . . . . . . . . . 11  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 a )  e.  ( 0 ... ( # `
 ( a ++  ( M  o.  (reverse `  a
) ) ) ) ) )
72 simprr 764 . . . . . . . . . . 11  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  b  e.  ( I  X.  2o ) )
73 efgval2.t . . . . . . . . . . . 12  |-  T  =  ( v  e.  W  |->  ( n  e.  ( 0 ... ( # `  v ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. )
) )
741, 36, 48, 73efgtval 17361 . . . . . . . . . . 11  |-  ( ( ( a ++  ( M  o.  (reverse `  a
) ) )  e.  W  /\  ( # `  a )  e.  ( 0 ... ( # `  ( a ++  ( M  o.  (reverse `  a
) ) ) ) )  /\  b  e.  ( I  X.  2o ) )  ->  (
( # `  a ) ( T `  (
a ++  ( M  o.  (reverse `  a ) ) ) ) b )  =  ( ( a ++  ( M  o.  (reverse `  a ) ) ) splice  <. ( # `  a
) ,  ( # `  a ) ,  <" b ( M `  b ) "> >.
) )
7555, 71, 72, 74syl3anc 1264 . . . . . . . . . 10  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( # `  a ) ( T `  (
a ++  ( M  o.  (reverse `  a ) ) ) ) b )  =  ( ( a ++  ( M  o.  (reverse `  a ) ) ) splice  <. ( # `  a
) ,  ( # `  a ) ,  <" b ( M `  b ) "> >.
) )
7633a1i 11 . . . . . . . . . . 11  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (/)  e. Word  (
I  X.  2o ) )
7749ffvelrni 6033 . . . . . . . . . . . . 13  |-  ( b  e.  ( I  X.  2o )  ->  ( M `
 b )  e.  ( I  X.  2o ) )
7872, 77syl 17 . . . . . . . . . . . 12  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( M `  b )  e.  ( I  X.  2o ) )
7972, 78s2cld 12956 . . . . . . . . . . 11  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  <" b
( M `  b
) ">  e. Word  ( I  X.  2o ) )
80 ccatrid 12724 . . . . . . . . . . . . . 14  |-  ( a  e. Word  ( I  X.  2o )  ->  ( a ++  (/) )  =  a
)
8180ad2antrl 732 . . . . . . . . . . . . 13  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
a ++  (/) )  =  a )
8281eqcomd 2430 . . . . . . . . . . . 12  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  a  =  ( a ++  (/) ) )
8382oveq1d 6317 . . . . . . . . . . 11  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
a ++  ( M  o.  (reverse `  a ) ) )  =  ( ( a ++  (/) ) ++  ( M  o.  (reverse `  a
) ) ) )
84 eqidd 2423 . . . . . . . . . . 11  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 a )  =  ( # `  a
) )
85 hash0 12548 . . . . . . . . . . . . 13  |-  ( # `  (/) )  =  0
8685oveq2i 6313 . . . . . . . . . . . 12  |-  ( (
# `  a )  +  ( # `  (/) ) )  =  ( ( # `  a )  +  0 )
8757nn0cnd 10928 . . . . . . . . . . . . 13  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 a )  e.  CC )
8887addid1d 9834 . . . . . . . . . . . 12  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( # `  a )  +  0 )  =  ( # `  a
) )
8986, 88syl5req 2476 . . . . . . . . . . 11  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 a )  =  ( ( # `  a
)  +  ( # `  (/) ) ) )
9045, 76, 51, 79, 83, 84, 89splval2 12855 . . . . . . . . . 10  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( a ++  ( M  o.  (reverse `  a
) ) ) splice  <. (
# `  a ) ,  ( # `  a
) ,  <" b
( M `  b
) "> >. )  =  ( ( a ++ 
<" b ( M `
 b ) "> ) ++  ( M  o.  (reverse `  a
) ) ) )
9172s1cld 12735 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  <" b ">  e. Word  ( I  X.  2o ) )
92 revccat 12862 . . . . . . . . . . . . . . . 16  |-  ( ( a  e. Word  ( I  X.  2o )  /\  <" b ">  e. Word  ( I  X.  2o ) )  ->  (reverse `  ( a ++  <" b "> ) )  =  ( (reverse `  <" b "> ) ++  (reverse `  a ) ) )
9345, 91, 92syl2anc 665 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (reverse `  ( a ++  <" b "> ) )  =  ( (reverse `  <" b "> ) ++  (reverse `  a ) ) )
94 revs1 12861 . . . . . . . . . . . . . . . 16  |-  (reverse `  <" b "> )  =  <" b ">
9594oveq1i 6312 . . . . . . . . . . . . . . 15  |-  ( (reverse `  <" b "> ) ++  (reverse `  a
) )  =  (
<" b "> ++  (reverse `  a ) )
9693, 95syl6eq 2479 . . . . . . . . . . . . . 14  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (reverse `  ( a ++  <" b "> ) )  =  ( <" b "> ++  (reverse `  a )
) )
9796coeq2d 5013 . . . . . . . . . . . . 13  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( M  o.  (reverse `  (
a ++  <" b "> ) ) )  =  ( M  o.  ( <" b "> ++  (reverse `  a )
) ) )
9849a1i 11 . . . . . . . . . . . . . 14  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  M : ( I  X.  2o ) --> ( I  X.  2o ) )
99 ccatco 12923 . . . . . . . . . . . . . 14  |-  ( (
<" b ">  e. Word  ( I  X.  2o )  /\  (reverse `  a
)  e. Word  ( I  X.  2o )  /\  M : ( I  X.  2o ) --> ( I  X.  2o ) )  ->  ( M  o.  ( <" b "> ++  (reverse `  a
) ) )  =  ( ( M  o.  <" b "> ) ++  ( M  o.  (reverse `  a ) ) ) )
10091, 47, 98, 99syl3anc 1264 . . . . . . . . . . . . 13  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( M  o.  ( <" b "> ++  (reverse `  a
) ) )  =  ( ( M  o.  <" b "> ) ++  ( M  o.  (reverse `  a ) ) ) )
101 s1co 12921 . . . . . . . . . . . . . . 15  |-  ( ( b  e.  ( I  X.  2o )  /\  M : ( I  X.  2o ) --> ( I  X.  2o ) )  ->  ( M  o.  <" b "> )  =  <" ( M `  b
) "> )
10272, 49, 101sylancl 666 . . . . . . . . . . . . . 14  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( M  o.  <" b "> )  =  <" ( M `  b
) "> )
103102oveq1d 6317 . . . . . . . . . . . . 13  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( M  o.  <" b "> ) ++  ( M  o.  (reverse `  a ) ) )  =  ( <" ( M `  b ) "> ++  ( M  o.  (reverse `  a ) ) ) )
10497, 100, 1033eqtrd 2467 . . . . . . . . . . . 12  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( M  o.  (reverse `  (
a ++  <" b "> ) ) )  =  ( <" ( M `  b ) "> ++  ( M  o.  (reverse `  a ) ) ) )
105104oveq2d 6318 . . . . . . . . . . 11  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( a ++  <" b "> ) ++  ( M  o.  (reverse `  (
a ++  <" b "> ) ) ) )  =  ( ( a ++  <" b "> ) ++  ( <" ( M `  b ) "> ++  ( M  o.  (reverse `  a
) ) ) ) )
106 ccatcl 12713 . . . . . . . . . . . . 13  |-  ( ( a  e. Word  ( I  X.  2o )  /\  <" b ">  e. Word  ( I  X.  2o ) )  ->  (
a ++  <" b "> )  e. Word  (
I  X.  2o ) )
10745, 91, 106syl2anc 665 . . . . . . . . . . . 12  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
a ++  <" b "> )  e. Word  (
I  X.  2o ) )
10878s1cld 12735 . . . . . . . . . . . 12  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  <" ( M `  b ) ">  e. Word  ( I  X.  2o ) )
109 ccatass 12725 . . . . . . . . . . . 12  |-  ( ( ( a ++  <" b "> )  e. Word  (
I  X.  2o )  /\  <" ( M `
 b ) ">  e. Word  ( I  X.  2o )  /\  ( M  o.  (reverse `  a
) )  e. Word  (
I  X.  2o ) )  ->  ( (
( a ++  <" b "> ) ++  <" ( M `  b ) "> ) ++  ( M  o.  (reverse `  a
) ) )  =  ( ( a ++  <" b "> ) ++  ( <" ( M `
 b ) "> ++  ( M  o.  (reverse `  a ) ) ) ) )
110107, 108, 51, 109syl3anc 1264 . . . . . . . . . . 11  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( ( a ++  <" b "> ) ++  <" ( M `  b ) "> ) ++  ( M  o.  (reverse `  a ) ) )  =  ( ( a ++ 
<" b "> ) ++  ( <" ( M `  b ) "> ++  ( M  o.  (reverse `  a ) ) ) ) )
111 ccatass 12725 . . . . . . . . . . . . . 14  |-  ( ( a  e. Word  ( I  X.  2o )  /\  <" b ">  e. Word  ( I  X.  2o )  /\  <" ( M `
 b ) ">  e. Word  ( I  X.  2o ) )  -> 
( ( a ++  <" b "> ) ++  <" ( M `  b ) "> )  =  ( a ++  ( <" b "> ++  <" ( M `
 b ) "> ) ) )
11245, 91, 108, 111syl3anc 1264 . . . . . . . . . . . . 13  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( a ++  <" b "> ) ++  <" ( M `  b ) "> )  =  ( a ++  ( <" b "> ++  <" ( M `
 b ) "> ) ) )
113 df-s2 12935 . . . . . . . . . . . . . 14  |-  <" b
( M `  b
) ">  =  ( <" b "> ++  <" ( M `
 b ) "> )
114113oveq2i 6313 . . . . . . . . . . . . 13  |-  ( a ++ 
<" b ( M `
 b ) "> )  =  ( a ++  ( <" b "> ++  <" ( M `
 b ) "> ) )
115112, 114syl6eqr 2481 . . . . . . . . . . . 12  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( a ++  <" b "> ) ++  <" ( M `  b ) "> )  =  ( a ++  <" b ( M `  b ) "> ) )
116115oveq1d 6317 . . . . . . . . . . 11  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( ( a ++  <" b "> ) ++  <" ( M `  b ) "> ) ++  ( M  o.  (reverse `  a ) ) )  =  ( ( a ++ 
<" b ( M `
 b ) "> ) ++  ( M  o.  (reverse `  a
) ) ) )
117105, 110, 1163eqtr2rd 2470 . . . . . . . . . 10  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( a ++  <" b
( M `  b
) "> ) ++  ( M  o.  (reverse `  a ) ) )  =  ( ( a ++ 
<" b "> ) ++  ( M  o.  (reverse `  ( a ++  <" b "> ) ) ) ) )
11875, 90, 1173eqtrd 2467 . . . . . . . . 9  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( # `  a ) ( T `  (
a ++  ( M  o.  (reverse `  a ) ) ) ) b )  =  ( ( a ++ 
<" b "> ) ++  ( M  o.  (reverse `  ( a ++  <" b "> ) ) ) ) )
1191, 36, 48, 73efgtf 17360 . . . . . . . . . . . 12  |-  ( ( a ++  ( M  o.  (reverse `  a ) ) )  e.  W  -> 
( ( T `  ( a ++  ( M  o.  (reverse `  a )
) ) )  =  ( m  e.  ( 0 ... ( # `  ( a ++  ( M  o.  (reverse `  a
) ) ) ) ) ,  u  e.  ( I  X.  2o )  |->  ( ( a ++  ( M  o.  (reverse `  a ) ) ) splice  <. m ,  m , 
<" u ( M `
 u ) "> >. ) )  /\  ( T `  ( a ++  ( M  o.  (reverse `  a ) ) ) ) : ( ( 0 ... ( # `  ( a ++  ( M  o.  (reverse `  a
) ) ) ) )  X.  ( I  X.  2o ) ) --> W ) )
120119simprd 464 . . . . . . . . . . 11  |-  ( ( a ++  ( M  o.  (reverse `  a ) ) )  e.  W  -> 
( T `  (
a ++  ( M  o.  (reverse `  a ) ) ) ) : ( ( 0 ... ( # `
 ( a ++  ( M  o.  (reverse `  a
) ) ) ) )  X.  ( I  X.  2o ) ) --> W )
121 ffn 5743 . . . . . . . . . . 11  |-  ( ( T `  ( a ++  ( M  o.  (reverse `  a ) ) ) ) : ( ( 0 ... ( # `  ( a ++  ( M  o.  (reverse `  a
) ) ) ) )  X.  ( I  X.  2o ) ) --> W  ->  ( T `  ( a ++  ( M  o.  (reverse `  a
) ) ) )  Fn  ( ( 0 ... ( # `  (
a ++  ( M  o.  (reverse `  a ) ) ) ) )  X.  ( I  X.  2o ) ) )
12255, 120, 1213syl 18 . . . . . . . . . 10  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( T `  ( a ++  ( M  o.  (reverse `  a ) ) ) )  Fn  ( ( 0 ... ( # `  ( a ++  ( M  o.  (reverse `  a
) ) ) ) )  X.  ( I  X.  2o ) ) )
123 fnovrn 6455 . . . . . . . . . 10  |-  ( ( ( T `  (
a ++  ( M  o.  (reverse `  a ) ) ) )  Fn  (
( 0 ... ( # `
 ( a ++  ( M  o.  (reverse `  a
) ) ) ) )  X.  ( I  X.  2o ) )  /\  ( # `  a
)  e.  ( 0 ... ( # `  (
a ++  ( M  o.  (reverse `  a ) ) ) ) )  /\  b  e.  ( I  X.  2o ) )  -> 
( ( # `  a
) ( T `  ( a ++  ( M  o.  (reverse `  a )
) ) ) b )  e.  ran  ( T `  ( a ++  ( M  o.  (reverse `  a ) ) ) ) )
124122, 71, 72, 123syl3anc 1264 . . . . . . . . 9  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( # `  a ) ( T `  (
a ++  ( M  o.  (reverse `  a ) ) ) ) b )  e.  ran  ( T `
 ( a ++  ( M  o.  (reverse `  a
) ) ) ) )
125118, 124eqeltrrd 2511 . . . . . . . 8  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( a ++  <" b "> ) ++  ( M  o.  (reverse `  (
a ++  <" b "> ) ) ) )  e.  ran  ( T `  ( a ++  ( M  o.  (reverse `  a ) ) ) ) )
1261, 36, 48, 73efgi2 17363 . . . . . . . 8  |-  ( ( ( a ++  ( M  o.  (reverse `  a
) ) )  e.  W  /\  ( ( a ++  <" b "> ) ++  ( M  o.  (reverse `  (
a ++  <" b "> ) ) ) )  e.  ran  ( T `  ( a ++  ( M  o.  (reverse `  a ) ) ) ) )  ->  (
a ++  ( M  o.  (reverse `  a ) ) )  .~  ( ( a ++  <" b "> ) ++  ( M  o.  (reverse `  (
a ++  <" b "> ) ) ) ) )
12755, 125, 126syl2anc 665 . . . . . . 7  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
a ++  ( M  o.  (reverse `  a ) ) )  .~  ( ( a ++  <" b "> ) ++  ( M  o.  (reverse `  (
a ++  <" b "> ) ) ) ) )
12844, 127ersym 7380 . . . . . 6  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( a ++  <" b "> ) ++  ( M  o.  (reverse `  (
a ++  <" b "> ) ) ) )  .~  ( a ++  ( M  o.  (reverse `  a ) ) ) )
12944ertr 7383 . . . . . 6  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( ( ( a ++ 
<" b "> ) ++  ( M  o.  (reverse `  ( a ++  <" b "> ) ) ) )  .~  ( a ++  ( M  o.  (reverse `  a ) ) )  /\  ( a ++  ( M  o.  (reverse `  a
) ) )  .~  (/) )  ->  ( (
a ++  <" b "> ) ++  ( M  o.  (reverse `  (
a ++  <" b "> ) ) ) )  .~  (/) ) )
130128, 129mpand 679 . . . . 5  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( a ++  ( M  o.  (reverse `  a
) ) )  .~  (/) 
->  ( ( a ++  <" b "> ) ++  ( M  o.  (reverse `  ( a ++  <" b "> ) ) ) )  .~  (/) ) )
131130expcom 436 . . . 4  |-  ( ( a  e. Word  ( I  X.  2o )  /\  b  e.  ( I  X.  2o ) )  -> 
( A  e.  W  ->  ( ( a ++  ( M  o.  (reverse `  a
) ) )  .~  (/) 
->  ( ( a ++  <" b "> ) ++  ( M  o.  (reverse `  ( a ++  <" b "> ) ) ) )  .~  (/) ) ) )
132131a2d 29 . . 3  |-  ( ( a  e. Word  ( I  X.  2o )  /\  b  e.  ( I  X.  2o ) )  -> 
( ( A  e.  W  ->  ( a ++  ( M  o.  (reverse `  a ) ) )  .~  (/) )  ->  ( A  e.  W  ->  ( ( a ++  <" b "> ) ++  ( M  o.  (reverse `  (
a ++  <" b "> ) ) ) )  .~  (/) ) ) )
13314, 20, 26, 32, 43, 132wrdind 12824 . 2  |-  ( A  e. Word  ( I  X.  2o )  ->  ( A  e.  W  ->  ( A ++  ( M  o.  (reverse `  A ) ) )  .~  (/) ) )
1344, 133mpcom 37 1  |-  ( A  e.  W  ->  ( A ++  ( M  o.  (reverse `  A ) ) )  .~  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1868   _Vcvv 3081    \ cdif 3433   (/)c0 3761   <.cop 4002   <.cotp 4004   class class class wbr 4420    |-> cmpt 4479    _I cid 4760    X. cxp 4848   ran crn 4851    o. ccom 4854    Fn wfn 5593   -->wf 5594   ` cfv 5598  (class class class)co 6302    |-> cmpt2 6304   1oc1o 7180   2oc2o 7181    Er wer 7365   0cc0 9540    + caddc 9543   NN0cn0 10870   ZZcz 10938   ZZ>=cuz 11160   ...cfz 11785   #chash 12515  Word cword 12649   ++ cconcat 12651   <"cs1 12652   splice csplice 12654  reversecreverse 12655   <"cs2 12928   ~FG cefg 17344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-cnex 9596  ax-resscn 9597  ax-1cn 9598  ax-icn 9599  ax-addcl 9600  ax-addrcl 9601  ax-mulcl 9602  ax-mulrcl 9603  ax-mulcom 9604  ax-addass 9605  ax-mulass 9606  ax-distr 9607  ax-i2m1 9608  ax-1ne0 9609  ax-1rid 9610  ax-rnegex 9611  ax-rrecex 9612  ax-cnre 9613  ax-pre-lttri 9614  ax-pre-lttrn 9615  ax-pre-ltadd 9616  ax-pre-mulgt0 9617
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-ot 4005  df-uni 4217  df-int 4253  df-iun 4298  df-iin 4299  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-riota 6264  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-om 6704  df-1st 6804  df-2nd 6805  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-2o 7188  df-oadd 7191  df-er 7368  df-ec 7370  df-map 7479  df-pm 7480  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-card 8375  df-cda 8599  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-sub 9863  df-neg 9864  df-nn 10611  df-2 10669  df-n0 10871  df-z 10939  df-uz 11161  df-fz 11786  df-fzo 11917  df-hash 12516  df-word 12657  df-lsw 12658  df-concat 12659  df-s1 12660  df-substr 12661  df-splice 12662  df-reverse 12663  df-s2 12935  df-efg 17347
This theorem is referenced by:  efginvrel1  17366  frgpinv  17402
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