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Theorem efginvrel2 15036
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 5580 . . . 4  |-  (  _I 
` Word  ( I  X.  2o ) )  C_ Word  ( I  X.  2o )
31, 2eqsstri 3208 . . 3  |-  W  C_ Word  ( I  X.  2o )
43sseli 3176 . 2  |-  ( A  e.  W  ->  A  e. Word  ( I  X.  2o ) )
5 id 19 . . . . . 6  |-  ( c  =  (/)  ->  c  =  (/) )
6 fveq2 5525 . . . . . . . . 9  |-  ( c  =  (/)  ->  (reverse `  c
)  =  (reverse `  (/) ) )
7 rev0 11482 . . . . . . . . 9  |-  (reverse `  (/) )  =  (/)
86, 7syl6eq 2331 . . . . . . . 8  |-  ( c  =  (/)  ->  (reverse `  c
)  =  (/) )
98coeq2d 4846 . . . . . . 7  |-  ( c  =  (/)  ->  ( M  o.  (reverse `  c
) )  =  ( M  o.  (/) ) )
10 co02 5186 . . . . . . 7  |-  ( M  o.  (/) )  =  (/)
119, 10syl6eq 2331 . . . . . 6  |-  ( c  =  (/)  ->  ( M  o.  (reverse `  c
) )  =  (/) )
125, 11oveq12d 5876 . . . . 5  |-  ( c  =  (/)  ->  ( c concat 
( M  o.  (reverse `  c ) ) )  =  ( (/) concat  (/) ) )
1312breq1d 4033 . . . 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 5525 . . . . . . 7  |-  ( c  =  a  ->  (reverse `  c )  =  (reverse `  a ) )
1716coeq2d 4846 . . . . . 6  |-  ( c  =  a  ->  ( M  o.  (reverse `  c
) )  =  ( M  o.  (reverse `  a
) ) )
1815, 17oveq12d 5876 . . . . 5  |-  ( c  =  a  ->  (
c concat  ( M  o.  (reverse `  c ) ) )  =  ( a concat  ( M  o.  (reverse `  a
) ) ) )
1918breq1d 4033 . . . 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 5525 . . . . . . 7  |-  ( c  =  ( a concat  <" b "> )  ->  (reverse `  c )  =  (reverse `  ( a concat  <" b "> ) ) )
2322coeq2d 4846 . . . . . 6  |-  ( c  =  ( a concat  <" b "> )  ->  ( M  o.  (reverse `  c ) )  =  ( M  o.  (reverse `  ( a concat  <" b "> ) ) ) )
2421, 23oveq12d 5876 . . . . 5  |-  ( c  =  ( a concat  <" b "> )  ->  ( c concat  ( M  o.  (reverse `  c
) ) )  =  ( ( a concat  <" b "> ) concat  ( M  o.  (reverse `  (
a concat  <" b "> ) ) ) ) )
2524breq1d 4033 . . . 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 5525 . . . . . . 7  |-  ( c  =  A  ->  (reverse `  c )  =  (reverse `  A ) )
2928coeq2d 4846 . . . . . 6  |-  ( c  =  A  ->  ( M  o.  (reverse `  c
) )  =  ( M  o.  (reverse `  A
) ) )
3027, 29oveq12d 5876 . . . . 5  |-  ( c  =  A  ->  (
c concat  ( M  o.  (reverse `  c ) ) )  =  ( A concat  ( M  o.  (reverse `  A
) ) ) )
3130breq1d 4033 . . . 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 11418 . . . . 5  |-  (/)  e. Word  (
I  X.  2o )
34 ccatlid 11434 . . . . 5  |-  ( (/)  e. Word  ( I  X.  2o )  ->  ( (/) concat  (/) )  =  (/) )
3533, 34ax-mp 8 . . . 4  |-  ( (/) concat  (/) )  =  (/)
36 efgval.r . . . . . . 7  |-  .~  =  ( ~FG  `  I )
371, 36efger 15027 . . . . . 6  |-  .~  Er  W
3837a1i 10 . . . . 5  |-  ( A  e.  W  ->  .~  Er  W )
391efgrcl 15024 . . . . . . 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 2370 . . . . 5  |-  ( A  e.  W  ->  (/)  e.  W
)
4238, 41erref 6680 . . . 4  |-  ( A  e.  W  ->  (/)  .~  (/) )
4335, 42syl5eqbr 4056 . . 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 11479 . . . . . . . . . . . 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 15022 . . . . . . . . . . 11  |-  M :
( I  X.  2o )
--> ( I  X.  2o )
50 wrdco 11486 . . . . . . . . . . 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 11429 . . . . . . . . . 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 2360 . . . . . . . 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 11421 . . . . . . . . . . . . . 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 10262 . . . . . . . . . . . . 13  |-  NN0  =  ( ZZ>= `  0 )
5957, 58syl6eleq 2373 . . . . . . . . . . . 12  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 a )  e.  ( ZZ>= `  0 )
)
60 ccatlen 11430 . . . . . . . . . . . . . 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 10115 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 a )  e.  ZZ )
63 uzid 10242 . . . . . . . . . . . . . . 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 11421 . . . . . . . . . . . . . . 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 10275 . . . . . . . . . . . . . 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 2357 . . . . . . . . . . . 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 10792 . . . . . . . . . . . 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 15032 . . . . . . . . . . 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 5664 . . . . . . . . . . . . 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 11519 . . . . . . . . . . 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 11435 . . . . . . . . . . . . . 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 2288 . . . . . . . . . . . 12  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  a  =  ( a concat  (/) ) )
8382oveq1d 5873 . . . . . . . . . . 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 2284 . . . . . . . . . . 11  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 a )  =  ( # `  a
) )
85 hash0 11355 . . . . . . . . . . . . 13  |-  ( # `  (/) )  =  0
8685oveq2i 5869 . . . . . . . . . . . 12  |-  ( (
# `  a )  +  ( # `  (/) ) )  =  ( ( # `  a )  +  0 )
8757nn0cnd 10020 . . . . . . . . . . . . 13  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 a )  e.  CC )
8887addid1d 9012 . . . . . . . . . . . 12  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( # `  a )  +  0 )  =  ( # `  a
) )
8986, 88syl5req 2328 . . . . . . . . . . 11  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 a )  =  ( ( # `  a
)  +  ( # `  (/) ) ) )
9045, 76, 51, 79, 83, 84, 89splval2 11472 . . . . . . . . . 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 11442 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  <" b ">  e. Word  ( I  X.  2o ) )
92 revccat 11484 . . . . . . . . . . . . . . . 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 11483 . . . . . . . . . . . . . . . 16  |-  (reverse `  <" b "> )  =  <" b ">
9594oveq1i 5868 . . . . . . . . . . . . . . 15  |-  ( (reverse `  <" b "> ) concat  (reverse `  a
) )  =  (
<" b "> concat  (reverse `  a ) )
9693, 95syl6eq 2331 . . . . . . . . . . . . . 14  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (reverse `  ( a concat  <" b "> ) )  =  ( <" b "> concat  (reverse `  a )
) )
9796coeq2d 4846 . . . . . . . . . . . . 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 11490 . . . . . . . . . . . . . 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 11488 . . . . . . . . . . . . . . 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 5873 . . . . . . . . . . . . 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 2319 . . . . . . . . . . . 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 5874 . . . . . . . . . . 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 11429 . . . . . . . . . . . . 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 11442 . . . . . . . . . . . 12  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  <" ( M `  b ) ">  e. Word  ( I  X.  2o ) )
109 ccatass 11436 . . . . . . . . . . . 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 11436 . . . . . . . . . . . . . 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 11498 . . . . . . . . . . . . . 14  |-  <" b
( M `  b
) ">  =  ( <" b "> concat  <" ( M `
 b ) "> )
114113oveq2i 5869 . . . . . . . . . . . . 13  |-  ( a concat  <" b ( M `
 b ) "> )  =  ( a concat  ( <" b "> concat  <" ( M `
 b ) "> ) )
115112, 114syl6eqr 2333 . . . . . . . . . . . 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 5873 . . . . . . . . . . 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 2322 . . . . . . . . . 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 2319 . . . . . . . . 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 15031 . . . . . . . . . . . 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 5389 . . . . . . . . . . 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 5995 . . . . . . . . . 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 2358 . . . . . . . 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 15034 . . . . . . . 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 6672 . . . . . 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 6675 . . . . . 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 11477 . 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 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149   (/)c0 3455   <.cop 3643   <.cotp 3644   class class class wbr 4023    e. cmpt 4077    _I cid 4304    X. cxp 4687   ran crn 4690    o. ccom 4693    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1oc1o 6472   2oc2o 6473    Er wer 6657   0cc0 8737    + caddc 8740   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230   ...cfz 10782   #chash 11337  Word cword 11403   concat cconcat 11404   <"cs1 11405   splice csplice 11407  reversecreverse 11408   <"cs2 11491   ~FG cefg 15015
This theorem is referenced by:  efginvrel1  15037  frgpinv  15073
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-ot 3650  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-ec 6662  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-fzo 10871  df-hash 11338  df-word 11409  df-concat 11410  df-s1 11411  df-substr 11412  df-splice 11413  df-reverse 11414  df-s2 11498  df-efg 15018
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