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Theorem efginvrel2 15052
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 concat  ( 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 5596 . . . 4  |-  (  _I 
` Word  ( I  X.  2o ) )  C_ Word  ( I  X.  2o )
31, 2eqsstri 3221 . . 3  |-  W  C_ Word  ( I  X.  2o )
43sseli 3189 . 2  |-  ( A  e.  W  ->  A  e. Word  ( I  X.  2o ) )
5 id 19 . . . . . 6  |-  ( c  =  (/)  ->  c  =  (/) )
6 fveq2 5541 . . . . . . . . 9  |-  ( c  =  (/)  ->  (reverse `  c
)  =  (reverse `  (/) ) )
7 rev0 11498 . . . . . . . . 9  |-  (reverse `  (/) )  =  (/)
86, 7syl6eq 2344 . . . . . . . 8  |-  ( c  =  (/)  ->  (reverse `  c
)  =  (/) )
98coeq2d 4862 . . . . . . 7  |-  ( c  =  (/)  ->  ( M  o.  (reverse `  c
) )  =  ( M  o.  (/) ) )
10 co02 5202 . . . . . . 7  |-  ( M  o.  (/) )  =  (/)
119, 10syl6eq 2344 . . . . . 6  |-  ( c  =  (/)  ->  ( M  o.  (reverse `  c
) )  =  (/) )
125, 11oveq12d 5892 . . . . 5  |-  ( c  =  (/)  ->  ( c concat 
( M  o.  (reverse `  c ) ) )  =  ( (/) concat  (/) ) )
1312breq1d 4049 . . . 4  |-  ( c  =  (/)  ->  ( ( c concat  ( M  o.  (reverse `  c ) ) )  .~  (/)  <->  ( (/) concat  (/) )  .~  (/) ) )
1413imbi2d 307 . . 3  |-  ( c  =  (/)  ->  ( ( A  e.  W  -> 
( c concat  ( M  o.  (reverse `  c )
) )  .~  (/) )  <->  ( A  e.  W  ->  ( (/) concat  (/) )  .~  (/) ) ) )
15 id 19 . . . . . 6  |-  ( c  =  a  ->  c  =  a )
16 fveq2 5541 . . . . . . 7  |-  ( c  =  a  ->  (reverse `  c )  =  (reverse `  a ) )
1716coeq2d 4862 . . . . . 6  |-  ( c  =  a  ->  ( M  o.  (reverse `  c
) )  =  ( M  o.  (reverse `  a
) ) )
1815, 17oveq12d 5892 . . . . 5  |-  ( c  =  a  ->  (
c concat  ( M  o.  (reverse `  c ) ) )  =  ( a concat  ( M  o.  (reverse `  a
) ) ) )
1918breq1d 4049 . . . 4  |-  ( c  =  a  ->  (
( c concat  ( M  o.  (reverse `  c )
) )  .~  (/)  <->  ( a concat  ( M  o.  (reverse `  a
) ) )  .~  (/) ) )
2019imbi2d 307 . . 3  |-  ( c  =  a  ->  (
( A  e.  W  ->  ( c concat  ( M  o.  (reverse `  c
) ) )  .~  (/) )  <->  ( A  e.  W  ->  ( a concat  ( M  o.  (reverse `  a
) ) )  .~  (/) ) ) )
21 id 19 . . . . . 6  |-  ( c  =  ( a concat  <" b "> )  ->  c  =  ( a concat  <" b "> ) )
22 fveq2 5541 . . . . . . 7  |-  ( c  =  ( a concat  <" b "> )  ->  (reverse `  c )  =  (reverse `  ( a concat  <" b "> ) ) )
2322coeq2d 4862 . . . . . 6  |-  ( c  =  ( a concat  <" b "> )  ->  ( M  o.  (reverse `  c ) )  =  ( M  o.  (reverse `  ( a concat  <" b "> ) ) ) )
2421, 23oveq12d 5892 . . . . 5  |-  ( c  =  ( a concat  <" b "> )  ->  ( c concat  ( M  o.  (reverse `  c
) ) )  =  ( ( a concat  <" b "> ) concat  ( M  o.  (reverse `  (
a concat  <" b "> ) ) ) ) )
2524breq1d 4049 . . . 4  |-  ( c  =  ( a concat  <" b "> )  ->  ( ( c concat  ( M  o.  (reverse `  c
) ) )  .~  (/)  <->  ( ( a concat  <" b "> ) concat  ( M  o.  (reverse `  ( a concat  <" b "> ) ) ) )  .~  (/) ) )
2625imbi2d 307 . . 3  |-  ( c  =  ( a concat  <" b "> )  ->  ( ( A  e.  W  ->  ( c concat  ( M  o.  (reverse `  c
) ) )  .~  (/) )  <->  ( A  e.  W  ->  ( (
a concat  <" b "> ) concat  ( M  o.  (reverse `  ( a concat  <" b "> ) ) ) )  .~  (/) ) ) )
27 id 19 . . . . . 6  |-  ( c  =  A  ->  c  =  A )
28 fveq2 5541 . . . . . . 7  |-  ( c  =  A  ->  (reverse `  c )  =  (reverse `  A ) )
2928coeq2d 4862 . . . . . 6  |-  ( c  =  A  ->  ( M  o.  (reverse `  c
) )  =  ( M  o.  (reverse `  A
) ) )
3027, 29oveq12d 5892 . . . . 5  |-  ( c  =  A  ->  (
c concat  ( M  o.  (reverse `  c ) ) )  =  ( A concat  ( M  o.  (reverse `  A
) ) ) )
3130breq1d 4049 . . . 4  |-  ( c  =  A  ->  (
( c concat  ( M  o.  (reverse `  c )
) )  .~  (/)  <->  ( A concat  ( M  o.  (reverse `  A
) ) )  .~  (/) ) )
3231imbi2d 307 . . 3  |-  ( c  =  A  ->  (
( A  e.  W  ->  ( c concat  ( M  o.  (reverse `  c
) ) )  .~  (/) )  <->  ( A  e.  W  ->  ( A concat  ( M  o.  (reverse `  A
) ) )  .~  (/) ) ) )
33 wrd0 11434 . . . . 5  |-  (/)  e. Word  (
I  X.  2o )
34 ccatlid 11450 . . . . 5  |-  ( (/)  e. Word  ( I  X.  2o )  ->  ( (/) concat  (/) )  =  (/) )
3533, 34ax-mp 8 . . . 4  |-  ( (/) concat  (/) )  =  (/)
36 efgval.r . . . . . . 7  |-  .~  =  ( ~FG  `  I )
371, 36efger 15043 . . . . . 6  |-  .~  Er  W
3837a1i 10 . . . . 5  |-  ( A  e.  W  ->  .~  Er  W )
391efgrcl 15040 . . . . . . 7  |-  ( A  e.  W  ->  (
I  e.  _V  /\  W  = Word  ( I  X.  2o ) ) )
4039simprd 449 . . . . . 6  |-  ( A  e.  W  ->  W  = Word  ( I  X.  2o ) )
4133, 40syl5eleqr 2383 . . . . 5  |-  ( A  e.  W  ->  (/)  e.  W
)
4238, 41erref 6696 . . . 4  |-  ( A  e.  W  ->  (/)  .~  (/) )
4335, 42syl5eqbr 4072 . . 3  |-  ( A  e.  W  ->  ( (/) concat  (/) )  .~  (/) )
4437a1i 10 . . . . . . 7  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  .~  Er  W )
45 simprl 732 . . . . . . . . . 10  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  a  e. Word  ( I  X.  2o ) )
46 revcl 11495 . . . . . . . . . . . 12  |-  ( a  e. Word  ( I  X.  2o )  ->  (reverse `  a
)  e. Word  ( I  X.  2o ) )
4746ad2antrl 708 . . . . . . . . . . 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 15038 . . . . . . . . . . 11  |-  M :
( I  X.  2o )
--> ( I  X.  2o )
50 wrdco 11502 . . . . . . . . . . 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 643 . . . . . . . . . 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 11445 . . . . . . . . . 10  |-  ( ( a  e. Word  ( I  X.  2o )  /\  ( M  o.  (reverse `  a ) )  e. Word 
( I  X.  2o ) )  ->  (
a concat  ( M  o.  (reverse `  a ) ) )  e. Word  ( I  X.  2o ) )
5345, 51, 52syl2anc 642 . . . . . . . . 9  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
a concat  ( M  o.  (reverse `  a ) ) )  e. Word  ( I  X.  2o ) )
5440adantr 451 . . . . . . . . 9  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  W  = Word  ( I  X.  2o ) )
5553, 54eleqtrrd 2373 . . . . . . . 8  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
a concat  ( M  o.  (reverse `  a ) ) )  e.  W )
56 lencl 11437 . . . . . . . . . . . . . 14  |-  ( a  e. Word  ( I  X.  2o )  ->  ( # `  a )  e.  NN0 )
5756ad2antrl 708 . . . . . . . . . . . . 13  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 a )  e. 
NN0 )
58 nn0uz 10278 . . . . . . . . . . . . 13  |-  NN0  =  ( ZZ>= `  0 )
5957, 58syl6eleq 2386 . . . . . . . . . . . 12  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 a )  e.  ( ZZ>= `  0 )
)
60 ccatlen 11446 . . . . . . . . . . . . . 14  |-  ( ( a  e. Word  ( I  X.  2o )  /\  ( M  o.  (reverse `  a ) )  e. Word 
( I  X.  2o ) )  ->  ( # `
 ( a concat  ( M  o.  (reverse `  a
) ) ) )  =  ( ( # `  a )  +  (
# `  ( M  o.  (reverse `  a )
) ) ) )
6145, 51, 60syl2anc 642 . . . . . . . . . . . . 13  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 ( a concat  ( M  o.  (reverse `  a
) ) ) )  =  ( ( # `  a )  +  (
# `  ( M  o.  (reverse `  a )
) ) ) )
6257nn0zd 10131 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 a )  e.  ZZ )
63 uzid 10258 . . . . . . . . . . . . . . 15  |-  ( (
# `  a )  e.  ZZ  ->  ( # `  a
)  e.  ( ZZ>= `  ( # `  a ) ) )
6462, 63syl 15 . . . . . . . . . . . . . 14  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 a )  e.  ( ZZ>= `  ( # `  a
) ) )
65 lencl 11437 . . . . . . . . . . . . . . 15  |-  ( ( M  o.  (reverse `  a
) )  e. Word  (
I  X.  2o )  ->  ( # `  ( M  o.  (reverse `  a
) ) )  e. 
NN0 )
6651, 65syl 15 . . . . . . . . . . . . . 14  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 ( M  o.  (reverse `  a ) ) )  e.  NN0 )
67 uzaddcl 10291 . . . . . . . . . . . . . 14  |-  ( ( ( # `  a
)  e.  ( ZZ>= `  ( # `  a ) )  /\  ( # `  ( M  o.  (reverse `  a ) ) )  e.  NN0 )  -> 
( ( # `  a
)  +  ( # `  ( M  o.  (reverse `  a ) ) ) )  e.  ( ZZ>= `  ( # `  a ) ) )
6864, 66, 67syl2anc 642 . . . . . . . . . . . . 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 2370 . . . . . . . . . . . 12  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 ( a concat  ( M  o.  (reverse `  a
) ) ) )  e.  ( ZZ>= `  ( # `
 a ) ) )
70 elfzuzb 10808 . . . . . . . . . . . 12  |-  ( (
# `  a )  e.  ( 0 ... ( # `
 ( a concat  ( M  o.  (reverse `  a
) ) ) ) )  <->  ( ( # `  a )  e.  (
ZZ>= `  0 )  /\  ( # `  ( a concat 
( M  o.  (reverse `  a ) ) ) )  e.  ( ZZ>= `  ( # `  a ) ) ) )
7159, 69, 70sylanbrc 645 . . . . . . . . . . 11  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 a )  e.  ( 0 ... ( # `
 ( a concat  ( M  o.  (reverse `  a
) ) ) ) ) )
72 simprr 733 . . . . . . . . . . 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 15048 . . . . . . . . . . 11  |-  ( ( ( a concat  ( M  o.  (reverse `  a
) ) )  e.  W  /\  ( # `  a )  e.  ( 0 ... ( # `  ( a concat  ( M  o.  (reverse `  a
) ) ) ) )  /\  b  e.  ( I  X.  2o ) )  ->  (
( # `  a ) ( T `  (
a concat  ( M  o.  (reverse `  a ) ) ) ) b )  =  ( ( a concat  ( M  o.  (reverse `  a
) ) ) splice  <. (
# `  a ) ,  ( # `  a
) ,  <" b
( M `  b
) "> >. )
)
7555, 71, 72, 74syl3anc 1182 . . . . . . . . . 10  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( # `  a ) ( T `  (
a concat  ( M  o.  (reverse `  a ) ) ) ) b )  =  ( ( a concat  ( M  o.  (reverse `  a
) ) ) splice  <. (
# `  a ) ,  ( # `  a
) ,  <" b
( M `  b
) "> >. )
)
7633a1i 10 . . . . . . . . . . 11  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (/)  e. Word  (
I  X.  2o ) )
7749ffvelrni 5680 . . . . . . . . . . . . 13  |-  ( b  e.  ( I  X.  2o )  ->  ( M `
 b )  e.  ( I  X.  2o ) )
7872, 77syl 15 . . . . . . . . . . . 12  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( M `  b )  e.  ( I  X.  2o ) )
7972, 78s2cld 11535 . . . . . . . . . . 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 11451 . . . . . . . . . . . . . 14  |-  ( a  e. Word  ( I  X.  2o )  ->  ( a concat  (/) )  =  a )
8180ad2antrl 708 . . . . . . . . . . . . 13  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
a concat  (/) )  =  a )
8281eqcomd 2301 . . . . . . . . . . . 12  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  a  =  ( a concat  (/) ) )
8382oveq1d 5889 . . . . . . . . . . 11  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
a concat  ( M  o.  (reverse `  a ) ) )  =  ( ( a concat  (/) ) concat  ( M  o.  (reverse `  a ) ) ) )
84 eqidd 2297 . . . . . . . . . . 11  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 a )  =  ( # `  a
) )
85 hash0 11371 . . . . . . . . . . . . 13  |-  ( # `  (/) )  =  0
8685oveq2i 5885 . . . . . . . . . . . 12  |-  ( (
# `  a )  +  ( # `  (/) ) )  =  ( ( # `  a )  +  0 )
8757nn0cnd 10036 . . . . . . . . . . . . 13  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 a )  e.  CC )
8887addid1d 9028 . . . . . . . . . . . 12  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( # `  a )  +  0 )  =  ( # `  a
) )
8986, 88syl5req 2341 . . . . . . . . . . 11  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 a )  =  ( ( # `  a
)  +  ( # `  (/) ) ) )
9045, 76, 51, 79, 83, 84, 89splval2 11488 . . . . . . . . . 10  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( a concat  ( M  o.  (reverse `  a )
) ) splice  <. ( # `  a ) ,  (
# `  a ) ,  <" b ( M `  b ) "> >. )  =  ( ( a concat  <" b ( M `
 b ) "> ) concat  ( M  o.  (reverse `  a )
) ) )
9172s1cld 11458 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  <" b ">  e. Word  ( I  X.  2o ) )
92 revccat 11500 . . . . . . . . . . . . . . . 16  |-  ( ( a  e. Word  ( I  X.  2o )  /\  <" b ">  e. Word  ( I  X.  2o ) )  ->  (reverse `  ( a concat  <" b "> ) )  =  ( (reverse `  <" b "> ) concat  (reverse `  a ) ) )
9345, 91, 92syl2anc 642 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (reverse `  ( a concat  <" b "> ) )  =  ( (reverse `  <" b "> ) concat  (reverse `  a ) ) )
94 revs1 11499 . . . . . . . . . . . . . . . 16  |-  (reverse `  <" b "> )  =  <" b ">
9594oveq1i 5884 . . . . . . . . . . . . . . 15  |-  ( (reverse `  <" b "> ) concat  (reverse `  a
) )  =  (
<" b "> concat  (reverse `  a ) )
9693, 95syl6eq 2344 . . . . . . . . . . . . . 14  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (reverse `  ( a concat  <" b "> ) )  =  ( <" b "> concat  (reverse `  a )
) )
9796coeq2d 4862 . . . . . . . . . . . . 13  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( M  o.  (reverse `  (
a concat  <" b "> ) ) )  =  ( M  o.  ( <" b "> concat  (reverse `  a )
) ) )
9849a1i 10 . . . . . . . . . . . . . 14  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  M : ( I  X.  2o ) --> ( I  X.  2o ) )
99 ccatco 11506 . . . . . . . . . . . . . 14  |-  ( (
<" b ">  e. Word  ( I  X.  2o )  /\  (reverse `  a
)  e. Word  ( I  X.  2o )  /\  M : ( I  X.  2o ) --> ( I  X.  2o ) )  ->  ( M  o.  ( <" b "> concat  (reverse `  a
) ) )  =  ( ( M  o.  <" b "> ) concat  ( M  o.  (reverse `  a ) ) ) )
10091, 47, 98, 99syl3anc 1182 . . . . . . . . . . . . 13  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( M  o.  ( <" b "> concat  (reverse `  a
) ) )  =  ( ( M  o.  <" b "> ) concat  ( M  o.  (reverse `  a ) ) ) )
101 s1co 11504 . . . . . . . . . . . . . . 15  |-  ( ( b  e.  ( I  X.  2o )  /\  M : ( I  X.  2o ) --> ( I  X.  2o ) )  ->  ( M  o.  <" b "> )  =  <" ( M `  b
) "> )
10272, 49, 101sylancl 643 . . . . . . . . . . . . . 14  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( M  o.  <" b "> )  =  <" ( M `  b
) "> )
103102oveq1d 5889 . . . . . . . . . . . . 13  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( M  o.  <" b "> ) concat  ( M  o.  (reverse `  a
) ) )  =  ( <" ( M `  b ) "> concat  ( M  o.  (reverse `  a ) ) ) )
10497, 100, 1033eqtrd 2332 . . . . . . . . . . . 12  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( M  o.  (reverse `  (
a concat  <" b "> ) ) )  =  ( <" ( M `  b ) "> concat  ( M  o.  (reverse `  a ) ) ) )
105104oveq2d 5890 . . . . . . . . . . 11  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( a concat  <" b "> ) concat  ( M  o.  (reverse `  ( a concat  <" b "> ) ) ) )  =  ( ( a concat  <" b "> ) concat  ( <" ( M `  b ) "> concat  ( M  o.  (reverse `  a ) ) ) ) )
106 ccatcl 11445 . . . . . . . . . . . . 13  |-  ( ( a  e. Word  ( I  X.  2o )  /\  <" b ">  e. Word  ( I  X.  2o ) )  ->  (
a concat  <" b "> )  e. Word  (
I  X.  2o ) )
10745, 91, 106syl2anc 642 . . . . . . . . . . . 12  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
a concat  <" b "> )  e. Word  (
I  X.  2o ) )
10878s1cld 11458 . . . . . . . . . . . 12  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  <" ( M `  b ) ">  e. Word  ( I  X.  2o ) )
109 ccatass 11452 . . . . . . . . . . . 12  |-  ( ( ( a concat  <" b "> )  e. Word  (
I  X.  2o )  /\  <" ( M `
 b ) ">  e. Word  ( I  X.  2o )  /\  ( M  o.  (reverse `  a
) )  e. Word  (
I  X.  2o ) )  ->  ( (
( a concat  <" b "> ) concat  <" ( M `  b ) "> ) concat  ( M  o.  (reverse `  a )
) )  =  ( ( a concat  <" b "> ) concat  ( <" ( M `  b
) "> concat  ( M  o.  (reverse `  a
) ) ) ) )
110107, 108, 51, 109syl3anc 1182 . . . . . . . . . . 11  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( ( a concat  <" b "> ) concat  <" ( M `  b ) "> ) concat  ( M  o.  (reverse `  a ) ) )  =  ( ( a concat  <" b "> ) concat  ( <" ( M `  b ) "> concat  ( M  o.  (reverse `  a ) ) ) ) )
111 ccatass 11452 . . . . . . . . . . . . . 14  |-  ( ( a  e. Word  ( I  X.  2o )  /\  <" b ">  e. Word  ( I  X.  2o )  /\  <" ( M `
 b ) ">  e. Word  ( I  X.  2o ) )  -> 
( ( a concat  <" b "> ) concat  <" ( M `  b ) "> )  =  ( a concat  (
<" b "> concat  <" ( M `  b ) "> ) ) )
11245, 91, 108, 111syl3anc 1182 . . . . . . . . . . . . 13  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( a concat  <" b "> ) concat  <" ( M `  b ) "> )  =  ( a concat  ( <" b "> concat  <" ( M `
 b ) "> ) ) )
113 df-s2 11514 . . . . . . . . . . . . . 14  |-  <" b
( M `  b
) ">  =  ( <" b "> concat  <" ( M `
 b ) "> )
114113oveq2i 5885 . . . . . . . . . . . . 13  |-  ( a concat  <" b ( M `
 b ) "> )  =  ( a concat  ( <" b "> concat  <" ( M `
 b ) "> ) )
115112, 114syl6eqr 2346 . . . . . . . . . . . 12  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( a concat  <" b "> ) concat  <" ( M `  b ) "> )  =  ( a concat  <" b ( M `  b ) "> ) )
116115oveq1d 5889 . . . . . . . . . . 11  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( ( a concat  <" b "> ) concat  <" ( M `  b ) "> ) concat  ( M  o.  (reverse `  a ) ) )  =  ( ( a concat  <" b ( M `
 b ) "> ) concat  ( M  o.  (reverse `  a )
) ) )
117105, 110, 1163eqtr2rd 2335 . . . . . . . . . 10  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( a concat  <" b
( M `  b
) "> ) concat  ( M  o.  (reverse `  a
) ) )  =  ( ( a concat  <" b "> ) concat  ( M  o.  (reverse `  (
a concat  <" b "> ) ) ) ) )
11875, 90, 1173eqtrd 2332 . . . . . . . . 9  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( # `  a ) ( T `  (
a concat  ( M  o.  (reverse `  a ) ) ) ) b )  =  ( ( a concat  <" b "> ) concat  ( M  o.  (reverse `  (
a concat  <" b "> ) ) ) ) )
1191, 36, 48, 73efgtf 15047 . . . . . . . . . . . 12  |-  ( ( a concat  ( M  o.  (reverse `  a ) ) )  e.  W  -> 
( ( T `  ( a concat  ( M  o.  (reverse `  a ) ) ) )  =  ( m  e.  ( 0 ... ( # `  (
a concat  ( M  o.  (reverse `  a ) ) ) ) ) ,  u  e.  ( I  X.  2o )  |->  ( ( a concat 
( M  o.  (reverse `  a ) ) ) splice  <. m ,  m , 
<" u ( M `
 u ) "> >. ) )  /\  ( T `  ( a concat 
( M  o.  (reverse `  a ) ) ) ) : ( ( 0 ... ( # `  ( a concat  ( M  o.  (reverse `  a
) ) ) ) )  X.  ( I  X.  2o ) ) --> W ) )
120119simprd 449 . . . . . . . . . . 11  |-  ( ( a concat  ( M  o.  (reverse `  a ) ) )  e.  W  -> 
( T `  (
a concat  ( M  o.  (reverse `  a ) ) ) ) : ( ( 0 ... ( # `  ( a concat  ( M  o.  (reverse `  a
) ) ) ) )  X.  ( I  X.  2o ) ) --> W )
121 ffn 5405 . . . . . . . . . . 11  |-  ( ( T `  ( a concat 
( M  o.  (reverse `  a ) ) ) ) : ( ( 0 ... ( # `  ( a concat  ( M  o.  (reverse `  a
) ) ) ) )  X.  ( I  X.  2o ) ) --> W  ->  ( T `  ( a concat  ( M  o.  (reverse `  a
) ) ) )  Fn  ( ( 0 ... ( # `  (
a concat  ( 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 concat  ( M  o.  (reverse `  a
) ) ) )  Fn  ( ( 0 ... ( # `  (
a concat  ( M  o.  (reverse `  a ) ) ) ) )  X.  (
I  X.  2o ) ) )
123 fnovrn 6011 . . . . . . . . . 10  |-  ( ( ( T `  (
a concat  ( M  o.  (reverse `  a ) ) ) )  Fn  ( ( 0 ... ( # `  ( a concat  ( M  o.  (reverse `  a
) ) ) ) )  X.  ( I  X.  2o ) )  /\  ( # `  a
)  e.  ( 0 ... ( # `  (
a concat  ( M  o.  (reverse `  a ) ) ) ) )  /\  b  e.  ( I  X.  2o ) )  ->  (
( # `  a ) ( T `  (
a concat  ( M  o.  (reverse `  a ) ) ) ) b )  e. 
ran  ( T `  ( a concat  ( M  o.  (reverse `  a ) ) ) ) )
124122, 71, 72, 123syl3anc 1182 . . . . . . . . 9  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( # `  a ) ( T `  (
a concat  ( M  o.  (reverse `  a ) ) ) ) b )  e. 
ran  ( T `  ( a concat  ( M  o.  (reverse `  a ) ) ) ) )
125118, 124eqeltrrd 2371 . . . . . . . 8  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( a concat  <" b "> ) concat  ( M  o.  (reverse `  ( a concat  <" b "> ) ) ) )  e.  ran  ( T `
 ( a concat  ( M  o.  (reverse `  a
) ) ) ) )
1261, 36, 48, 73efgi2 15050 . . . . . . . 8  |-  ( ( ( a concat  ( M  o.  (reverse `  a
) ) )  e.  W  /\  ( ( a concat  <" b "> ) concat  ( M  o.  (reverse `  ( a concat  <" b "> ) ) ) )  e.  ran  ( T `
 ( a concat  ( M  o.  (reverse `  a
) ) ) ) )  ->  ( a concat  ( M  o.  (reverse `  a
) ) )  .~  ( ( a concat  <" b "> ) concat  ( M  o.  (reverse `  (
a concat  <" b "> ) ) ) ) )
12755, 125, 126syl2anc 642 . . . . . . 7  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
a concat  ( M  o.  (reverse `  a ) ) )  .~  ( ( a concat  <" b "> ) concat  ( M  o.  (reverse `  ( a concat  <" b "> ) ) ) ) )
12844, 127ersym 6688 . . . . . 6  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( a concat  <" b "> ) concat  ( M  o.  (reverse `  ( a concat  <" b "> ) ) ) )  .~  ( a concat  ( M  o.  (reverse `  a
) ) ) )
12944ertr 6691 . . . . . 6  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( ( ( a concat  <" b "> ) concat  ( M  o.  (reverse `  ( a concat  <" b "> ) ) ) )  .~  ( a concat 
( M  o.  (reverse `  a ) ) )  /\  ( a concat  ( M  o.  (reverse `  a
) ) )  .~  (/) )  ->  ( (
a concat  <" b "> ) concat  ( M  o.  (reverse `  ( a concat  <" b "> ) ) ) )  .~  (/) ) )
130128, 129mpand 656 . . . . 5  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( a concat  ( M  o.  (reverse `  a )
) )  .~  (/)  ->  (
( a concat  <" b "> ) concat  ( M  o.  (reverse `  ( a concat  <" b "> ) ) ) )  .~  (/) ) )
131130expcom 424 . . . 4  |-  ( ( a  e. Word  ( I  X.  2o )  /\  b  e.  ( I  X.  2o ) )  -> 
( A  e.  W  ->  ( ( a concat  ( M  o.  (reverse `  a
) ) )  .~  (/) 
->  ( ( a concat  <" b "> ) concat  ( M  o.  (reverse `  (
a concat  <" b "> ) ) ) )  .~  (/) ) ) )
132131a2d 23 . . 3  |-  ( ( a  e. Word  ( I  X.  2o )  /\  b  e.  ( I  X.  2o ) )  -> 
( ( A  e.  W  ->  ( a concat  ( M  o.  (reverse `  a
) ) )  .~  (/) )  ->  ( A  e.  W  ->  ( ( a concat  <" b "> ) concat  ( M  o.  (reverse `  ( a concat  <" b "> ) ) ) )  .~  (/) ) ) )
13314, 20, 26, 32, 43, 132wrdind 11493 . 2  |-  ( A  e. Word  ( I  X.  2o )  ->  ( A  e.  W  ->  ( A concat  ( M  o.  (reverse `  A ) ) )  .~  (/) ) )
1344, 133mpcom 32 1  |-  ( A  e.  W  ->  ( A concat  ( M  o.  (reverse `  A ) ) )  .~  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    \ cdif 3162   (/)c0 3468   <.cop 3656   <.cotp 3657   class class class wbr 4039    e. cmpt 4093    _I cid 4320    X. cxp 4703   ran crn 4706    o. ccom 4709    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1oc1o 6488   2oc2o 6489    Er wer 6673   0cc0 8753    + caddc 8756   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246   ...cfz 10798   #chash 11353  Word cword 11419   concat cconcat 11420   <"cs1 11421   splice csplice 11423  reversecreverse 11424   <"cs2 11507   ~FG cefg 15031
This theorem is referenced by:  efginvrel1  15053  frgpinv  15089
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-ot 3663  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-ec 6678  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-fzo 10887  df-hash 11354  df-word 11425  df-concat 11426  df-s1 11427  df-substr 11428  df-splice 11429  df-reverse 11430  df-s2 11514  df-efg 15034
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