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Theorem efginvrel2 16215
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 5744 . . . 4  |-  (  _I 
` Word  ( I  X.  2o ) )  C_ Word  ( I  X.  2o )
31, 2eqsstri 3381 . . 3  |-  W  C_ Word  ( I  X.  2o )
43sseli 3347 . 2  |-  ( A  e.  W  ->  A  e. Word  ( I  X.  2o ) )
5 id 22 . . . . . 6  |-  ( c  =  (/)  ->  c  =  (/) )
6 fveq2 5686 . . . . . . . . 9  |-  ( c  =  (/)  ->  (reverse `  c
)  =  (reverse `  (/) ) )
7 rev0 12396 . . . . . . . . 9  |-  (reverse `  (/) )  =  (/)
86, 7syl6eq 2486 . . . . . . . 8  |-  ( c  =  (/)  ->  (reverse `  c
)  =  (/) )
98coeq2d 4997 . . . . . . 7  |-  ( c  =  (/)  ->  ( M  o.  (reverse `  c
) )  =  ( M  o.  (/) ) )
10 co02 5346 . . . . . . 7  |-  ( M  o.  (/) )  =  (/)
119, 10syl6eq 2486 . . . . . 6  |-  ( c  =  (/)  ->  ( M  o.  (reverse `  c
) )  =  (/) )
125, 11oveq12d 6104 . . . . 5  |-  ( c  =  (/)  ->  ( c concat 
( M  o.  (reverse `  c ) ) )  =  ( (/) concat  (/) ) )
1312breq1d 4297 . . . 4  |-  ( c  =  (/)  ->  ( ( c concat  ( M  o.  (reverse `  c ) ) )  .~  (/)  <->  ( (/) concat  (/) )  .~  (/) ) )
1413imbi2d 316 . . 3  |-  ( c  =  (/)  ->  ( ( A  e.  W  -> 
( c concat  ( M  o.  (reverse `  c )
) )  .~  (/) )  <->  ( A  e.  W  ->  ( (/) concat  (/) )  .~  (/) ) ) )
15 id 22 . . . . . 6  |-  ( c  =  a  ->  c  =  a )
16 fveq2 5686 . . . . . . 7  |-  ( c  =  a  ->  (reverse `  c )  =  (reverse `  a ) )
1716coeq2d 4997 . . . . . 6  |-  ( c  =  a  ->  ( M  o.  (reverse `  c
) )  =  ( M  o.  (reverse `  a
) ) )
1815, 17oveq12d 6104 . . . . 5  |-  ( c  =  a  ->  (
c concat  ( M  o.  (reverse `  c ) ) )  =  ( a concat  ( M  o.  (reverse `  a
) ) ) )
1918breq1d 4297 . . . 4  |-  ( c  =  a  ->  (
( c concat  ( M  o.  (reverse `  c )
) )  .~  (/)  <->  ( a concat  ( M  o.  (reverse `  a
) ) )  .~  (/) ) )
2019imbi2d 316 . . 3  |-  ( c  =  a  ->  (
( A  e.  W  ->  ( c concat  ( M  o.  (reverse `  c
) ) )  .~  (/) )  <->  ( A  e.  W  ->  ( a concat  ( M  o.  (reverse `  a
) ) )  .~  (/) ) ) )
21 id 22 . . . . . 6  |-  ( c  =  ( a concat  <" b "> )  ->  c  =  ( a concat  <" b "> ) )
22 fveq2 5686 . . . . . . 7  |-  ( c  =  ( a concat  <" b "> )  ->  (reverse `  c )  =  (reverse `  ( a concat  <" b "> ) ) )
2322coeq2d 4997 . . . . . 6  |-  ( c  =  ( a concat  <" b "> )  ->  ( M  o.  (reverse `  c ) )  =  ( M  o.  (reverse `  ( a concat  <" b "> ) ) ) )
2421, 23oveq12d 6104 . . . . 5  |-  ( c  =  ( a concat  <" b "> )  ->  ( c concat  ( M  o.  (reverse `  c
) ) )  =  ( ( a concat  <" b "> ) concat  ( M  o.  (reverse `  (
a concat  <" b "> ) ) ) ) )
2524breq1d 4297 . . . 4  |-  ( c  =  ( a concat  <" b "> )  ->  ( ( c concat  ( M  o.  (reverse `  c
) ) )  .~  (/)  <->  ( ( a concat  <" b "> ) concat  ( M  o.  (reverse `  ( a concat  <" b "> ) ) ) )  .~  (/) ) )
2625imbi2d 316 . . 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 22 . . . . . 6  |-  ( c  =  A  ->  c  =  A )
28 fveq2 5686 . . . . . . 7  |-  ( c  =  A  ->  (reverse `  c )  =  (reverse `  A ) )
2928coeq2d 4997 . . . . . 6  |-  ( c  =  A  ->  ( M  o.  (reverse `  c
) )  =  ( M  o.  (reverse `  A
) ) )
3027, 29oveq12d 6104 . . . . 5  |-  ( c  =  A  ->  (
c concat  ( M  o.  (reverse `  c ) ) )  =  ( A concat  ( M  o.  (reverse `  A
) ) ) )
3130breq1d 4297 . . . 4  |-  ( c  =  A  ->  (
( c concat  ( M  o.  (reverse `  c )
) )  .~  (/)  <->  ( A concat  ( M  o.  (reverse `  A
) ) )  .~  (/) ) )
3231imbi2d 316 . . 3  |-  ( c  =  A  ->  (
( A  e.  W  ->  ( c concat  ( M  o.  (reverse `  c
) ) )  .~  (/) )  <->  ( A  e.  W  ->  ( A concat  ( M  o.  (reverse `  A
) ) )  .~  (/) ) ) )
33 wrd0 12244 . . . . 5  |-  (/)  e. Word  (
I  X.  2o )
34 ccatlid 12276 . . . . 5  |-  ( (/)  e. Word  ( I  X.  2o )  ->  ( (/) concat  (/) )  =  (/) )
3533, 34ax-mp 5 . . . 4  |-  ( (/) concat  (/) )  =  (/)
36 efgval.r . . . . . . 7  |-  .~  =  ( ~FG  `  I )
371, 36efger 16206 . . . . . 6  |-  .~  Er  W
3837a1i 11 . . . . 5  |-  ( A  e.  W  ->  .~  Er  W )
391efgrcl 16203 . . . . . . 7  |-  ( A  e.  W  ->  (
I  e.  _V  /\  W  = Word  ( I  X.  2o ) ) )
4039simprd 463 . . . . . 6  |-  ( A  e.  W  ->  W  = Word  ( I  X.  2o ) )
4133, 40syl5eleqr 2525 . . . . 5  |-  ( A  e.  W  ->  (/)  e.  W
)
4238, 41erref 7113 . . . 4  |-  ( A  e.  W  ->  (/)  .~  (/) )
4335, 42syl5eqbr 4320 . . 3  |-  ( A  e.  W  ->  ( (/) concat  (/) )  .~  (/) )
4437a1i 11 . . . . . . 7  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  .~  Er  W )
45 simprl 755 . . . . . . . . . 10  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  a  e. Word  ( I  X.  2o ) )
46 revcl 12393 . . . . . . . . . . . 12  |-  ( a  e. Word  ( I  X.  2o )  ->  (reverse `  a
)  e. Word  ( I  X.  2o ) )
4746ad2antrl 727 . . . . . . . . . . 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 16201 . . . . . . . . . . 11  |-  M :
( I  X.  2o )
--> ( I  X.  2o )
50 wrdco 12451 . . . . . . . . . . 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 662 . . . . . . . . . 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 12266 . . . . . . . . . 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 661 . . . . . . . . 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 465 . . . . . . . . 9  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  W  = Word  ( I  X.  2o ) )
5553, 54eleqtrrd 2515 . . . . . . . 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 12241 . . . . . . . . . . . . . 14  |-  ( a  e. Word  ( I  X.  2o )  ->  ( # `  a )  e.  NN0 )
5756ad2antrl 727 . . . . . . . . . . . . 13  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 a )  e. 
NN0 )
58 nn0uz 10887 . . . . . . . . . . . . 13  |-  NN0  =  ( ZZ>= `  0 )
5957, 58syl6eleq 2528 . . . . . . . . . . . 12  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 a )  e.  ( ZZ>= `  0 )
)
60 ccatlen 12267 . . . . . . . . . . . . . 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 661 . . . . . . . . . . . . 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 10737 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 a )  e.  ZZ )
63 uzid 10867 . . . . . . . . . . . . . . 15  |-  ( (
# `  a )  e.  ZZ  ->  ( # `  a
)  e.  ( ZZ>= `  ( # `  a ) ) )
6462, 63syl 16 . . . . . . . . . . . . . 14  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 a )  e.  ( ZZ>= `  ( # `  a
) ) )
65 lencl 12241 . . . . . . . . . . . . . . 15  |-  ( ( M  o.  (reverse `  a
) )  e. Word  (
I  X.  2o )  ->  ( # `  ( M  o.  (reverse `  a
) ) )  e. 
NN0 )
6651, 65syl 16 . . . . . . . . . . . . . 14  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 ( M  o.  (reverse `  a ) ) )  e.  NN0 )
67 uzaddcl 10903 . . . . . . . . . . . . . 14  |-  ( ( ( # `  a
)  e.  ( ZZ>= `  ( # `  a ) )  /\  ( # `  ( M  o.  (reverse `  a ) ) )  e.  NN0 )  -> 
( ( # `  a
)  +  ( # `  ( M  o.  (reverse `  a ) ) ) )  e.  ( ZZ>= `  ( # `  a ) ) )
6864, 66, 67syl2anc 661 . . . . . . . . . . . . 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 2512 . . . . . . . . . . . 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 11439 . . . . . . . . . . . 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 664 . . . . . . . . . . 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 756 . . . . . . . . . . 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 16211 . . . . . . . . . . 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 1218 . . . . . . . . . 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 11 . . . . . . . . . . 11  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (/)  e. Word  (
I  X.  2o ) )
7749ffvelrni 5837 . . . . . . . . . . . . 13  |-  ( b  e.  ( I  X.  2o )  ->  ( M `
 b )  e.  ( I  X.  2o ) )
7872, 77syl 16 . . . . . . . . . . . 12  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( M `  b )  e.  ( I  X.  2o ) )
7972, 78s2cld 12488 . . . . . . . . . . 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 12277 . . . . . . . . . . . . . 14  |-  ( a  e. Word  ( I  X.  2o )  ->  ( a concat  (/) )  =  a )
8180ad2antrl 727 . . . . . . . . . . . . 13  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
a concat  (/) )  =  a )
8281eqcomd 2443 . . . . . . . . . . . 12  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  a  =  ( a concat  (/) ) )
8382oveq1d 6101 . . . . . . . . . . 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 2439 . . . . . . . . . . 11  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 a )  =  ( # `  a
) )
85 hash0 12127 . . . . . . . . . . . . 13  |-  ( # `  (/) )  =  0
8685oveq2i 6097 . . . . . . . . . . . 12  |-  ( (
# `  a )  +  ( # `  (/) ) )  =  ( ( # `  a )  +  0 )
8757nn0cnd 10630 . . . . . . . . . . . . 13  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 a )  e.  CC )
8887addid1d 9561 . . . . . . . . . . . 12  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( # `  a )  +  0 )  =  ( # `  a
) )
8986, 88syl5req 2483 . . . . . . . . . . 11  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 a )  =  ( ( # `  a
)  +  ( # `  (/) ) ) )
9045, 76, 51, 79, 83, 84, 89splval2 12391 . . . . . . . . . 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 12286 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  <" b ">  e. Word  ( I  X.  2o ) )
92 revccat 12398 . . . . . . . . . . . . . . . 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 661 . . . . . . . . . . . . . . 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 12397 . . . . . . . . . . . . . . . 16  |-  (reverse `  <" b "> )  =  <" b ">
9594oveq1i 6096 . . . . . . . . . . . . . . 15  |-  ( (reverse `  <" b "> ) concat  (reverse `  a
) )  =  (
<" b "> concat  (reverse `  a ) )
9693, 95syl6eq 2486 . . . . . . . . . . . . . 14  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (reverse `  ( a concat  <" b "> ) )  =  ( <" b "> concat  (reverse `  a )
) )
9796coeq2d 4997 . . . . . . . . . . . . 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 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 12455 . . . . . . . . . . . . . 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 1218 . . . . . . . . . . . . 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 12453 . . . . . . . . . . . . . . 15  |-  ( ( b  e.  ( I  X.  2o )  /\  M : ( I  X.  2o ) --> ( I  X.  2o ) )  ->  ( M  o.  <" b "> )  =  <" ( M `  b
) "> )
10272, 49, 101sylancl 662 . . . . . . . . . . . . . 14  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( M  o.  <" b "> )  =  <" ( M `  b
) "> )
103102oveq1d 6101 . . . . . . . . . . . . 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 2474 . . . . . . . . . . . 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 6102 . . . . . . . . . . 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 12266 . . . . . . . . . . . . 13  |-  ( ( a  e. Word  ( I  X.  2o )  /\  <" b ">  e. Word  ( I  X.  2o ) )  ->  (
a concat  <" b "> )  e. Word  (
I  X.  2o ) )
10745, 91, 106syl2anc 661 . . . . . . . . . . . 12  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
a concat  <" b "> )  e. Word  (
I  X.  2o ) )
10878s1cld 12286 . . . . . . . . . . . 12  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  <" ( M `  b ) ">  e. Word  ( I  X.  2o ) )
109 ccatass 12278 . . . . . . . . . . . 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 1218 . . . . . . . . . . 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 12278 . . . . . . . . . . . . . 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 1218 . . . . . . . . . . . . 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 12467 . . . . . . . . . . . . . 14  |-  <" b
( M `  b
) ">  =  ( <" b "> concat  <" ( M `
 b ) "> )
114113oveq2i 6097 . . . . . . . . . . . . 13  |-  ( a concat  <" b ( M `
 b ) "> )  =  ( a concat  ( <" b "> concat  <" ( M `
 b ) "> ) )
115112, 114syl6eqr 2488 . . . . . . . . . . . 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 6101 . . . . . . . . . . 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 2477 . . . . . . . . . 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 2474 . . . . . . . . 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 16210 . . . . . . . . . . . 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 463 . . . . . . . . . . 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 5554 . . . . . . . . . . 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 20 . . . . . . . . . 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 6233 . . . . . . . . . 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 1218 . . . . . . . . 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 2513 . . . . . . . 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 16213 . . . . . . . 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 661 . . . . . . 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 7105 . . . . . 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 7108 . . . . . 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 675 . . . . 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 435 . . . 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 26 . . 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 12363 . 2  |-  ( A  e. Word  ( I  X.  2o )  ->  ( A  e.  W  ->  ( A concat  ( M  o.  (reverse `  A ) ) )  .~  (/) ) )
1344, 133mpcom 36 1  |-  ( A  e.  W  ->  ( A concat  ( M  o.  (reverse `  A ) ) )  .~  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2967    \ cdif 3320   (/)c0 3632   <.cop 3878   <.cotp 3880   class class class wbr 4287    e. cmpt 4345    _I cid 4626    X. cxp 4833   ran crn 4836    o. ccom 4839    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6086    e. cmpt2 6088   1oc1o 6905   2oc2o 6906    Er wer 7090   0cc0 9274    + caddc 9277   NN0cn0 10571   ZZcz 10638   ZZ>=cuz 10853   ...cfz 11429   #chash 12095  Word cword 12213   concat cconcat 12215   <"cs1 12216   splice csplice 12218  reversecreverse 12219   <"cs2 12460   ~FG cefg 16194
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-ot 3881  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-er 7093  df-ec 7095  df-map 7208  df-pm 7209  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-card 8101  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-n0 10572  df-z 10639  df-uz 10854  df-fz 11430  df-fzo 11541  df-hash 12096  df-word 12221  df-concat 12223  df-s1 12224  df-substr 12225  df-splice 12226  df-reverse 12227  df-s2 12467  df-efg 16197
This theorem is referenced by:  efginvrel1  16216  frgpinv  16252
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