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Theorem efginvrel2 17455
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 5938 . . . 4  |-  (  _I 
` Word  ( I  X.  2o ) )  C_ Word  ( I  X.  2o )
31, 2eqsstri 3448 . . 3  |-  W  C_ Word  ( I  X.  2o )
43sseli 3414 . 2  |-  ( A  e.  W  ->  A  e. Word  ( I  X.  2o ) )
5 id 22 . . . . . 6  |-  ( c  =  (/)  ->  c  =  (/) )
6 fveq2 5879 . . . . . . . . 9  |-  ( c  =  (/)  ->  (reverse `  c
)  =  (reverse `  (/) ) )
7 rev0 12923 . . . . . . . . 9  |-  (reverse `  (/) )  =  (/)
86, 7syl6eq 2521 . . . . . . . 8  |-  ( c  =  (/)  ->  (reverse `  c
)  =  (/) )
98coeq2d 5002 . . . . . . 7  |-  ( c  =  (/)  ->  ( M  o.  (reverse `  c
) )  =  ( M  o.  (/) ) )
10 co02 5356 . . . . . . 7  |-  ( M  o.  (/) )  =  (/)
119, 10syl6eq 2521 . . . . . 6  |-  ( c  =  (/)  ->  ( M  o.  (reverse `  c
) )  =  (/) )
125, 11oveq12d 6326 . . . . 5  |-  ( c  =  (/)  ->  ( c ++  ( M  o.  (reverse `  c ) ) )  =  ( (/) ++  (/) ) )
1312breq1d 4405 . . . 4  |-  ( c  =  (/)  ->  ( ( c ++  ( M  o.  (reverse `  c ) ) )  .~  (/)  <->  ( (/) ++  (/) )  .~  (/) ) )
1413imbi2d 323 . . 3  |-  ( c  =  (/)  ->  ( ( A  e.  W  -> 
( c ++  ( M  o.  (reverse `  c
) ) )  .~  (/) )  <->  ( A  e.  W  ->  ( (/) ++  (/) )  .~  (/) ) ) )
15 id 22 . . . . . 6  |-  ( c  =  a  ->  c  =  a )
16 fveq2 5879 . . . . . . 7  |-  ( c  =  a  ->  (reverse `  c )  =  (reverse `  a ) )
1716coeq2d 5002 . . . . . 6  |-  ( c  =  a  ->  ( M  o.  (reverse `  c
) )  =  ( M  o.  (reverse `  a
) ) )
1815, 17oveq12d 6326 . . . . 5  |-  ( c  =  a  ->  (
c ++  ( M  o.  (reverse `  c ) ) )  =  ( a ++  ( M  o.  (reverse `  a ) ) ) )
1918breq1d 4405 . . . 4  |-  ( c  =  a  ->  (
( c ++  ( M  o.  (reverse `  c
) ) )  .~  (/)  <->  ( a ++  ( M  o.  (reverse `  a ) ) )  .~  (/) ) )
2019imbi2d 323 . . 3  |-  ( c  =  a  ->  (
( A  e.  W  ->  ( c ++  ( M  o.  (reverse `  c
) ) )  .~  (/) )  <->  ( A  e.  W  ->  ( a ++  ( M  o.  (reverse `  a ) ) )  .~  (/) ) ) )
21 id 22 . . . . . 6  |-  ( c  =  ( a ++  <" b "> )  ->  c  =  ( a ++ 
<" b "> ) )
22 fveq2 5879 . . . . . . 7  |-  ( c  =  ( a ++  <" b "> )  ->  (reverse `  c )  =  (reverse `  ( a ++  <" b "> ) ) )
2322coeq2d 5002 . . . . . 6  |-  ( c  =  ( a ++  <" b "> )  ->  ( M  o.  (reverse `  c ) )  =  ( M  o.  (reverse `  ( a ++  <" b "> ) ) ) )
2421, 23oveq12d 6326 . . . . 5  |-  ( c  =  ( a ++  <" b "> )  ->  ( c ++  ( M  o.  (reverse `  c
) ) )  =  ( ( a ++  <" b "> ) ++  ( M  o.  (reverse `  ( a ++  <" b "> ) ) ) ) )
2524breq1d 4405 . . . 4  |-  ( c  =  ( a ++  <" b "> )  ->  ( ( c ++  ( M  o.  (reverse `  c
) ) )  .~  (/)  <->  ( ( a ++  <" b "> ) ++  ( M  o.  (reverse `  (
a ++  <" b "> ) ) ) )  .~  (/) ) )
2625imbi2d 323 . . 3  |-  ( c  =  ( a ++  <" b "> )  ->  ( ( A  e.  W  ->  ( c ++  ( M  o.  (reverse `  c ) ) )  .~  (/) )  <->  ( A  e.  W  ->  ( ( a ++  <" b "> ) ++  ( M  o.  (reverse `  (
a ++  <" b "> ) ) ) )  .~  (/) ) ) )
27 id 22 . . . . . 6  |-  ( c  =  A  ->  c  =  A )
28 fveq2 5879 . . . . . . 7  |-  ( c  =  A  ->  (reverse `  c )  =  (reverse `  A ) )
2928coeq2d 5002 . . . . . 6  |-  ( c  =  A  ->  ( M  o.  (reverse `  c
) )  =  ( M  o.  (reverse `  A
) ) )
3027, 29oveq12d 6326 . . . . 5  |-  ( c  =  A  ->  (
c ++  ( M  o.  (reverse `  c ) ) )  =  ( A ++  ( M  o.  (reverse `  A ) ) ) )
3130breq1d 4405 . . . 4  |-  ( c  =  A  ->  (
( c ++  ( M  o.  (reverse `  c
) ) )  .~  (/)  <->  ( A ++  ( M  o.  (reverse `  A ) ) )  .~  (/) ) )
3231imbi2d 323 . . 3  |-  ( c  =  A  ->  (
( A  e.  W  ->  ( c ++  ( M  o.  (reverse `  c
) ) )  .~  (/) )  <->  ( A  e.  W  ->  ( A ++  ( M  o.  (reverse `  A ) ) )  .~  (/) ) ) )
33 wrd0 12742 . . . . 5  |-  (/)  e. Word  (
I  X.  2o )
34 ccatlid 12781 . . . . 5  |-  ( (/)  e. Word  ( I  X.  2o )  ->  ( (/) ++  (/) )  =  (/) )
3533, 34ax-mp 5 . . . 4  |-  ( (/) ++  (/) )  =  (/)
36 efgval.r . . . . . . 7  |-  .~  =  ( ~FG  `  I )
371, 36efger 17446 . . . . . 6  |-  .~  Er  W
3837a1i 11 . . . . 5  |-  ( A  e.  W  ->  .~  Er  W )
391efgrcl 17443 . . . . . . 7  |-  ( A  e.  W  ->  (
I  e.  _V  /\  W  = Word  ( I  X.  2o ) ) )
4039simprd 470 . . . . . 6  |-  ( A  e.  W  ->  W  = Word  ( I  X.  2o ) )
4133, 40syl5eleqr 2556 . . . . 5  |-  ( A  e.  W  ->  (/)  e.  W
)
4238, 41erref 7401 . . . 4  |-  ( A  e.  W  ->  (/)  .~  (/) )
4335, 42syl5eqbr 4429 . . 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 772 . . . . . . . . . 10  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  a  e. Word  ( I  X.  2o ) )
46 revcl 12920 . . . . . . . . . . . 12  |-  ( a  e. Word  ( I  X.  2o )  ->  (reverse `  a
)  e. Word  ( I  X.  2o ) )
4746ad2antrl 742 . . . . . . . . . . 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 17441 . . . . . . . . . . 11  |-  M :
( I  X.  2o )
--> ( I  X.  2o )
50 wrdco 12987 . . . . . . . . . . 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 675 . . . . . . . . . 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 12771 . . . . . . . . . 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 673 . . . . . . . . 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 472 . . . . . . . . 9  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  W  = Word  ( I  X.  2o ) )
5553, 54eleqtrrd 2552 . . . . . . . 8  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
a ++  ( M  o.  (reverse `  a ) ) )  e.  W )
56 lencl 12737 . . . . . . . . . . . . . 14  |-  ( a  e. Word  ( I  X.  2o )  ->  ( # `  a )  e.  NN0 )
5756ad2antrl 742 . . . . . . . . . . . . 13  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 a )  e. 
NN0 )
58 nn0uz 11217 . . . . . . . . . . . . 13  |-  NN0  =  ( ZZ>= `  0 )
5957, 58syl6eleq 2559 . . . . . . . . . . . 12  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 a )  e.  ( ZZ>= `  0 )
)
60 ccatlen 12772 . . . . . . . . . . . . . 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 673 . . . . . . . . . . . . 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 11061 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 a )  e.  ZZ )
63 uzid 11197 . . . . . . . . . . . . . . 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 12737 . . . . . . . . . . . . . . 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 11238 . . . . . . . . . . . . . 14  |-  ( ( ( # `  a
)  e.  ( ZZ>= `  ( # `  a ) )  /\  ( # `  ( M  o.  (reverse `  a ) ) )  e.  NN0 )  -> 
( ( # `  a
)  +  ( # `  ( M  o.  (reverse `  a ) ) ) )  e.  ( ZZ>= `  ( # `  a ) ) )
6864, 66, 67syl2anc 673 . . . . . . . . . . . . 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 2549 . . . . . . . . . . . 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 11820 . . . . . . . . . . . 12  |-  ( (
# `  a )  e.  ( 0 ... ( # `
 ( a ++  ( M  o.  (reverse `  a
) ) ) ) )  <->  ( ( # `  a )  e.  (
ZZ>= `  0 )  /\  ( # `  ( a ++  ( M  o.  (reverse `  a ) ) ) )  e.  ( ZZ>= `  ( # `  a ) ) ) )
7159, 69, 70sylanbrc 677 . . . . . . . . . . 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 774 . . . . . . . . . . 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 17451 . . . . . . . . . . 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 1292 . . . . . . . . . 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 6036 . . . . . . . . . . . . 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 13025 . . . . . . . . . . 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 12782 . . . . . . . . . . . . . 14  |-  ( a  e. Word  ( I  X.  2o )  ->  ( a ++  (/) )  =  a
)
8180ad2antrl 742 . . . . . . . . . . . . 13  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
a ++  (/) )  =  a )
8281eqcomd 2477 . . . . . . . . . . . 12  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  a  =  ( a ++  (/) ) )
8382oveq1d 6323 . . . . . . . . . . 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 2472 . . . . . . . . . . 11  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 a )  =  ( # `  a
) )
85 hash0 12586 . . . . . . . . . . . . 13  |-  ( # `  (/) )  =  0
8685oveq2i 6319 . . . . . . . . . . . 12  |-  ( (
# `  a )  +  ( # `  (/) ) )  =  ( ( # `  a )  +  0 )
8757nn0cnd 10951 . . . . . . . . . . . . 13  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 a )  e.  CC )
8887addid1d 9851 . . . . . . . . . . . 12  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( # `  a )  +  0 )  =  ( # `  a
) )
8986, 88syl5req 2518 . . . . . . . . . . 11  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 a )  =  ( ( # `  a
)  +  ( # `  (/) ) ) )
9045, 76, 51, 79, 83, 84, 89splval2 12918 . . . . . . . . . 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 12795 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  <" b ">  e. Word  ( I  X.  2o ) )
92 revccat 12925 . . . . . . . . . . . . . . . 16  |-  ( ( a  e. Word  ( I  X.  2o )  /\  <" b ">  e. Word  ( I  X.  2o ) )  ->  (reverse `  ( a ++  <" b "> ) )  =  ( (reverse `  <" b "> ) ++  (reverse `  a ) ) )
9345, 91, 92syl2anc 673 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (reverse `  ( a ++  <" b "> ) )  =  ( (reverse `  <" b "> ) ++  (reverse `  a ) ) )
94 revs1 12924 . . . . . . . . . . . . . . . 16  |-  (reverse `  <" b "> )  =  <" b ">
9594oveq1i 6318 . . . . . . . . . . . . . . 15  |-  ( (reverse `  <" b "> ) ++  (reverse `  a
) )  =  (
<" b "> ++  (reverse `  a ) )
9693, 95syl6eq 2521 . . . . . . . . . . . . . 14  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (reverse `  ( a ++  <" b "> ) )  =  ( <" b "> ++  (reverse `  a )
) )
9796coeq2d 5002 . . . . . . . . . . . . 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 12991 . . . . . . . . . . . . . 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 1292 . . . . . . . . . . . . 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 12989 . . . . . . . . . . . . . . 15  |-  ( ( b  e.  ( I  X.  2o )  /\  M : ( I  X.  2o ) --> ( I  X.  2o ) )  ->  ( M  o.  <" b "> )  =  <" ( M `  b
) "> )
10272, 49, 101sylancl 675 . . . . . . . . . . . . . 14  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( M  o.  <" b "> )  =  <" ( M `  b
) "> )
103102oveq1d 6323 . . . . . . . . . . . . 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 2509 . . . . . . . . . . . 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 6324 . . . . . . . . . . 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 12771 . . . . . . . . . . . . 13  |-  ( ( a  e. Word  ( I  X.  2o )  /\  <" b ">  e. Word  ( I  X.  2o ) )  ->  (
a ++  <" b "> )  e. Word  (
I  X.  2o ) )
10745, 91, 106syl2anc 673 . . . . . . . . . . . 12  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
a ++  <" b "> )  e. Word  (
I  X.  2o ) )
10878s1cld 12795 . . . . . . . . . . . 12  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  <" ( M `  b ) ">  e. Word  ( I  X.  2o ) )
109 ccatass 12783 . . . . . . . . . . . 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 1292 . . . . . . . . . . 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 12783 . . . . . . . . . . . . . 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 1292 . . . . . . . . . . . . 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 13003 . . . . . . . . . . . . . 14  |-  <" b
( M `  b
) ">  =  ( <" b "> ++  <" ( M `
 b ) "> )
114113oveq2i 6319 . . . . . . . . . . . . 13  |-  ( a ++ 
<" b ( M `
 b ) "> )  =  ( a ++  ( <" b "> ++  <" ( M `
 b ) "> ) )
115112, 114syl6eqr 2523 . . . . . . . . . . . 12  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( a ++  <" b "> ) ++  <" ( M `  b ) "> )  =  ( a ++  <" b ( M `  b ) "> ) )
116115oveq1d 6323 . . . . . . . . . . 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 2512 . . . . . . . . . 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 2509 . . . . . . . . 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 17450 . . . . . . . . . . . 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 470 . . . . . . . . . . 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 5739 . . . . . . . . . . 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 6463 . . . . . . . . . 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 1292 . . . . . . . . 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 2550 . . . . . . . 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 17453 . . . . . . . 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 673 . . . . . . 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 7393 . . . . . 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 7396 . . . . . 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 689 . . . . 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 442 . . . 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 28 . . 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 12887 . 2  |-  ( A  e. Word  ( I  X.  2o )  ->  ( A  e.  W  ->  ( A ++  ( M  o.  (reverse `  A ) ) )  .~  (/) ) )
1344, 133mpcom 36 1  |-  ( A  e.  W  ->  ( A ++  ( M  o.  (reverse `  A ) ) )  .~  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904   _Vcvv 3031    \ cdif 3387   (/)c0 3722   <.cop 3965   <.cotp 3967   class class class wbr 4395    |-> cmpt 4454    _I cid 4749    X. cxp 4837   ran crn 4840    o. ccom 4843    Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308    |-> cmpt2 6310   1oc1o 7193   2oc2o 7194    Er wer 7378   0cc0 9557    + caddc 9560   NN0cn0 10893   ZZcz 10961   ZZ>=cuz 11182   ...cfz 11810   #chash 12553  Word cword 12703   ++ cconcat 12705   <"cs1 12706   splice csplice 12708  reversecreverse 12709   <"cs2 12996   ~FG cefg 17434
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-ot 3968  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-ec 7383  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-fzo 11943  df-hash 12554  df-word 12711  df-lsw 12712  df-concat 12713  df-s1 12714  df-substr 12715  df-splice 12716  df-reverse 12717  df-s2 13003  df-efg 17437
This theorem is referenced by:  efginvrel1  17456  frgpinv  17492
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