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Theorem efginvrel2 16344
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 5857 . . . 4  |-  (  _I 
` Word  ( I  X.  2o ) )  C_ Word  ( I  X.  2o )
31, 2eqsstri 3493 . . 3  |-  W  C_ Word  ( I  X.  2o )
43sseli 3459 . 2  |-  ( A  e.  W  ->  A  e. Word  ( I  X.  2o ) )
5 id 22 . . . . . 6  |-  ( c  =  (/)  ->  c  =  (/) )
6 fveq2 5798 . . . . . . . . 9  |-  ( c  =  (/)  ->  (reverse `  c
)  =  (reverse `  (/) ) )
7 rev0 12521 . . . . . . . . 9  |-  (reverse `  (/) )  =  (/)
86, 7syl6eq 2511 . . . . . . . 8  |-  ( c  =  (/)  ->  (reverse `  c
)  =  (/) )
98coeq2d 5109 . . . . . . 7  |-  ( c  =  (/)  ->  ( M  o.  (reverse `  c
) )  =  ( M  o.  (/) ) )
10 co02 5458 . . . . . . 7  |-  ( M  o.  (/) )  =  (/)
119, 10syl6eq 2511 . . . . . 6  |-  ( c  =  (/)  ->  ( M  o.  (reverse `  c
) )  =  (/) )
125, 11oveq12d 6217 . . . . 5  |-  ( c  =  (/)  ->  ( c concat 
( M  o.  (reverse `  c ) ) )  =  ( (/) concat  (/) ) )
1312breq1d 4409 . . . 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 5798 . . . . . . 7  |-  ( c  =  a  ->  (reverse `  c )  =  (reverse `  a ) )
1716coeq2d 5109 . . . . . 6  |-  ( c  =  a  ->  ( M  o.  (reverse `  c
) )  =  ( M  o.  (reverse `  a
) ) )
1815, 17oveq12d 6217 . . . . 5  |-  ( c  =  a  ->  (
c concat  ( M  o.  (reverse `  c ) ) )  =  ( a concat  ( M  o.  (reverse `  a
) ) ) )
1918breq1d 4409 . . . 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 5798 . . . . . . 7  |-  ( c  =  ( a concat  <" b "> )  ->  (reverse `  c )  =  (reverse `  ( a concat  <" b "> ) ) )
2322coeq2d 5109 . . . . . 6  |-  ( c  =  ( a concat  <" b "> )  ->  ( M  o.  (reverse `  c ) )  =  ( M  o.  (reverse `  ( a concat  <" b "> ) ) ) )
2421, 23oveq12d 6217 . . . . 5  |-  ( c  =  ( a concat  <" b "> )  ->  ( c concat  ( M  o.  (reverse `  c
) ) )  =  ( ( a concat  <" b "> ) concat  ( M  o.  (reverse `  (
a concat  <" b "> ) ) ) ) )
2524breq1d 4409 . . . 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 5798 . . . . . . 7  |-  ( c  =  A  ->  (reverse `  c )  =  (reverse `  A ) )
2928coeq2d 5109 . . . . . 6  |-  ( c  =  A  ->  ( M  o.  (reverse `  c
) )  =  ( M  o.  (reverse `  A
) ) )
3027, 29oveq12d 6217 . . . . 5  |-  ( c  =  A  ->  (
c concat  ( M  o.  (reverse `  c ) ) )  =  ( A concat  ( M  o.  (reverse `  A
) ) ) )
3130breq1d 4409 . . . 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 12369 . . . . 5  |-  (/)  e. Word  (
I  X.  2o )
34 ccatlid 12401 . . . . 5  |-  ( (/)  e. Word  ( I  X.  2o )  ->  ( (/) concat  (/) )  =  (/) )
3533, 34ax-mp 5 . . . 4  |-  ( (/) concat  (/) )  =  (/)
36 efgval.r . . . . . . 7  |-  .~  =  ( ~FG  `  I )
371, 36efger 16335 . . . . . 6  |-  .~  Er  W
3837a1i 11 . . . . 5  |-  ( A  e.  W  ->  .~  Er  W )
391efgrcl 16332 . . . . . . 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 2549 . . . . 5  |-  ( A  e.  W  ->  (/)  e.  W
)
4238, 41erref 7230 . . . 4  |-  ( A  e.  W  ->  (/)  .~  (/) )
4335, 42syl5eqbr 4432 . . 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 12518 . . . . . . . . . . . 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 16330 . . . . . . . . . . 11  |-  M :
( I  X.  2o )
--> ( I  X.  2o )
50 wrdco 12576 . . . . . . . . . . 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 12391 . . . . . . . . . 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 2545 . . . . . . . 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 12366 . . . . . . . . . . . . . 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 11005 . . . . . . . . . . . . 13  |-  NN0  =  ( ZZ>= `  0 )
5957, 58syl6eleq 2552 . . . . . . . . . . . 12  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 a )  e.  ( ZZ>= `  0 )
)
60 ccatlen 12392 . . . . . . . . . . . . . 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 10855 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 a )  e.  ZZ )
63 uzid 10985 . . . . . . . . . . . . . . 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 12366 . . . . . . . . . . . . . . 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 11021 . . . . . . . . . . . . . 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 2542 . . . . . . . . . . . 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 11563 . . . . . . . . . . . 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 16340 . . . . . . . . . . 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 1219 . . . . . . . . . 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 5950 . . . . . . . . . . . . 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 12613 . . . . . . . . . . 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 12402 . . . . . . . . . . . . . 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 2462 . . . . . . . . . . . 12  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  a  =  ( a concat  (/) ) )
8382oveq1d 6214 . . . . . . . . . . 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 2455 . . . . . . . . . . 11  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 a )  =  ( # `  a
) )
85 hash0 12251 . . . . . . . . . . . . 13  |-  ( # `  (/) )  =  0
8685oveq2i 6210 . . . . . . . . . . . 12  |-  ( (
# `  a )  +  ( # `  (/) ) )  =  ( ( # `  a )  +  0 )
8757nn0cnd 10748 . . . . . . . . . . . . 13  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 a )  e.  CC )
8887addid1d 9679 . . . . . . . . . . . 12  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( # `  a )  +  0 )  =  ( # `  a
) )
8986, 88syl5req 2508 . . . . . . . . . . 11  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 a )  =  ( ( # `  a
)  +  ( # `  (/) ) ) )
9045, 76, 51, 79, 83, 84, 89splval2 12516 . . . . . . . . . 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 12411 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  <" b ">  e. Word  ( I  X.  2o ) )
92 revccat 12523 . . . . . . . . . . . . . . . 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 12522 . . . . . . . . . . . . . . . 16  |-  (reverse `  <" b "> )  =  <" b ">
9594oveq1i 6209 . . . . . . . . . . . . . . 15  |-  ( (reverse `  <" b "> ) concat  (reverse `  a
) )  =  (
<" b "> concat  (reverse `  a ) )
9693, 95syl6eq 2511 . . . . . . . . . . . . . 14  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (reverse `  ( a concat  <" b "> ) )  =  ( <" b "> concat  (reverse `  a )
) )
9796coeq2d 5109 . . . . . . . . . . . . 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 12580 . . . . . . . . . . . . . 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 1219 . . . . . . . . . . . . 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 12578 . . . . . . . . . . . . . . 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 6214 . . . . . . . . . . . . 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 2499 . . . . . . . . . . . 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 6215 . . . . . . . . . . 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 12391 . . . . . . . . . . . . 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 12411 . . . . . . . . . . . 12  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  <" ( M `  b ) ">  e. Word  ( I  X.  2o ) )
109 ccatass 12403 . . . . . . . . . . . 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 1219 . . . . . . . . . . 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 12403 . . . . . . . . . . . . . 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 1219 . . . . . . . . . . . . 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 12592 . . . . . . . . . . . . . 14  |-  <" b
( M `  b
) ">  =  ( <" b "> concat  <" ( M `
 b ) "> )
114113oveq2i 6210 . . . . . . . . . . . . 13  |-  ( a concat  <" b ( M `
 b ) "> )  =  ( a concat  ( <" b "> concat  <" ( M `
 b ) "> ) )
115112, 114syl6eqr 2513 . . . . . . . . . . . 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 6214 . . . . . . . . . . 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 2502 . . . . . . . . . 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 2499 . . . . . . . . 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 16339 . . . . . . . . . . . 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 5666 . . . . . . . . . . 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 6347 . . . . . . . . . 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 1219 . . . . . . . . 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 2543 . . . . . . . 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 16342 . . . . . . . 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 7222 . . . . . 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 7225 . . . . . 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 12488 . 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 1370    e. wcel 1758   _Vcvv 3076    \ cdif 3432   (/)c0 3744   <.cop 3990   <.cotp 3992   class class class wbr 4399    |-> cmpt 4457    _I cid 4738    X. cxp 4945   ran crn 4948    o. ccom 4951    Fn wfn 5520   -->wf 5521   ` cfv 5525  (class class class)co 6199    |-> cmpt2 6201   1oc1o 7022   2oc2o 7023    Er wer 7207   0cc0 9392    + caddc 9395   NN0cn0 10689   ZZcz 10756   ZZ>=cuz 10971   ...cfz 11553   #chash 12219  Word cword 12338   concat cconcat 12340   <"cs1 12341   splice csplice 12343  reversecreverse 12344   <"cs2 12585   ~FG cefg 16323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-ot 3993  df-uni 4199  df-int 4236  df-iun 4280  df-iin 4281  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-om 6586  df-1st 6686  df-2nd 6687  df-recs 6941  df-rdg 6975  df-1o 7029  df-2o 7030  df-oadd 7033  df-er 7210  df-ec 7212  df-map 7325  df-pm 7326  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-card 8219  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-nn 10433  df-n0 10690  df-z 10757  df-uz 10972  df-fz 11554  df-fzo 11665  df-hash 12220  df-word 12346  df-concat 12348  df-s1 12349  df-substr 12350  df-splice 12351  df-reverse 12352  df-s2 12592  df-efg 16326
This theorem is referenced by:  efginvrel1  16345  frgpinv  16381
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