MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  efginvrel2 Structured version   Unicode version

Theorem efginvrel2 16944
Description: The inverse of the reverse of a word composed with the word relates to the identity. (This provides an explicit expression for the representation of the group inverse, given a representative of the free group equivalence class.) (Contributed by Mario Carneiro, 1-Oct-2015.)
Hypotheses
Ref Expression
efgval.w  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
efgval.r  |-  .~  =  ( ~FG  `  I )
efgval2.m  |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
)
efgval2.t  |-  T  =  ( v  e.  W  |->  ( n  e.  ( 0 ... ( # `  v ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. )
) )
Assertion
Ref Expression
efginvrel2  |-  ( A  e.  W  ->  ( A ++  ( M  o.  (reverse `  A ) ) )  .~  (/) )
Distinct variable groups:    y, z    v, n, w, y, z   
n, M, v, w   
n, W, v, w, y, z    y,  .~ , z    n, I, v, w, y, z
Allowed substitution hints:    A( y, z, w, v, n)    .~ ( w, v, n)    T( y, z, w, v, n)    M( y, z)

Proof of Theorem efginvrel2
Dummy variables  a 
b  c  u  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 efgval.w . . . 4  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
2 fviss 5906 . . . 4  |-  (  _I 
` Word  ( I  X.  2o ) )  C_ Word  ( I  X.  2o )
31, 2eqsstri 3519 . . 3  |-  W  C_ Word  ( I  X.  2o )
43sseli 3485 . 2  |-  ( A  e.  W  ->  A  e. Word  ( I  X.  2o ) )
5 id 22 . . . . . 6  |-  ( c  =  (/)  ->  c  =  (/) )
6 fveq2 5848 . . . . . . . . 9  |-  ( c  =  (/)  ->  (reverse `  c
)  =  (reverse `  (/) ) )
7 rev0 12729 . . . . . . . . 9  |-  (reverse `  (/) )  =  (/)
86, 7syl6eq 2511 . . . . . . . 8  |-  ( c  =  (/)  ->  (reverse `  c
)  =  (/) )
98coeq2d 5154 . . . . . . 7  |-  ( c  =  (/)  ->  ( M  o.  (reverse `  c
) )  =  ( M  o.  (/) ) )
10 co02 5504 . . . . . . 7  |-  ( M  o.  (/) )  =  (/)
119, 10syl6eq 2511 . . . . . 6  |-  ( c  =  (/)  ->  ( M  o.  (reverse `  c
) )  =  (/) )
125, 11oveq12d 6288 . . . . 5  |-  ( c  =  (/)  ->  ( c ++  ( M  o.  (reverse `  c ) ) )  =  ( (/) ++  (/) ) )
1312breq1d 4449 . . . 4  |-  ( c  =  (/)  ->  ( ( c ++  ( M  o.  (reverse `  c ) ) )  .~  (/)  <->  ( (/) ++  (/) )  .~  (/) ) )
1413imbi2d 314 . . 3  |-  ( c  =  (/)  ->  ( ( A  e.  W  -> 
( c ++  ( M  o.  (reverse `  c
) ) )  .~  (/) )  <->  ( A  e.  W  ->  ( (/) ++  (/) )  .~  (/) ) ) )
15 id 22 . . . . . 6  |-  ( c  =  a  ->  c  =  a )
16 fveq2 5848 . . . . . . 7  |-  ( c  =  a  ->  (reverse `  c )  =  (reverse `  a ) )
1716coeq2d 5154 . . . . . 6  |-  ( c  =  a  ->  ( M  o.  (reverse `  c
) )  =  ( M  o.  (reverse `  a
) ) )
1815, 17oveq12d 6288 . . . . 5  |-  ( c  =  a  ->  (
c ++  ( M  o.  (reverse `  c ) ) )  =  ( a ++  ( M  o.  (reverse `  a ) ) ) )
1918breq1d 4449 . . . 4  |-  ( c  =  a  ->  (
( c ++  ( M  o.  (reverse `  c
) ) )  .~  (/)  <->  ( a ++  ( M  o.  (reverse `  a ) ) )  .~  (/) ) )
2019imbi2d 314 . . 3  |-  ( c  =  a  ->  (
( A  e.  W  ->  ( c ++  ( M  o.  (reverse `  c
) ) )  .~  (/) )  <->  ( A  e.  W  ->  ( a ++  ( M  o.  (reverse `  a ) ) )  .~  (/) ) ) )
21 id 22 . . . . . 6  |-  ( c  =  ( a ++  <" b "> )  ->  c  =  ( a ++ 
<" b "> ) )
22 fveq2 5848 . . . . . . 7  |-  ( c  =  ( a ++  <" b "> )  ->  (reverse `  c )  =  (reverse `  ( a ++  <" b "> ) ) )
2322coeq2d 5154 . . . . . 6  |-  ( c  =  ( a ++  <" b "> )  ->  ( M  o.  (reverse `  c ) )  =  ( M  o.  (reverse `  ( a ++  <" b "> ) ) ) )
2421, 23oveq12d 6288 . . . . 5  |-  ( c  =  ( a ++  <" b "> )  ->  ( c ++  ( M  o.  (reverse `  c
) ) )  =  ( ( a ++  <" b "> ) ++  ( M  o.  (reverse `  ( a ++  <" b "> ) ) ) ) )
2524breq1d 4449 . . . 4  |-  ( c  =  ( a ++  <" b "> )  ->  ( ( c ++  ( M  o.  (reverse `  c
) ) )  .~  (/)  <->  ( ( a ++  <" b "> ) ++  ( M  o.  (reverse `  (
a ++  <" b "> ) ) ) )  .~  (/) ) )
2625imbi2d 314 . . 3  |-  ( c  =  ( a ++  <" b "> )  ->  ( ( A  e.  W  ->  ( c ++  ( M  o.  (reverse `  c ) ) )  .~  (/) )  <->  ( A  e.  W  ->  ( ( a ++  <" b "> ) ++  ( M  o.  (reverse `  (
a ++  <" b "> ) ) ) )  .~  (/) ) ) )
27 id 22 . . . . . 6  |-  ( c  =  A  ->  c  =  A )
28 fveq2 5848 . . . . . . 7  |-  ( c  =  A  ->  (reverse `  c )  =  (reverse `  A ) )
2928coeq2d 5154 . . . . . 6  |-  ( c  =  A  ->  ( M  o.  (reverse `  c
) )  =  ( M  o.  (reverse `  A
) ) )
3027, 29oveq12d 6288 . . . . 5  |-  ( c  =  A  ->  (
c ++  ( M  o.  (reverse `  c ) ) )  =  ( A ++  ( M  o.  (reverse `  A ) ) ) )
3130breq1d 4449 . . . 4  |-  ( c  =  A  ->  (
( c ++  ( M  o.  (reverse `  c
) ) )  .~  (/)  <->  ( A ++  ( M  o.  (reverse `  A ) ) )  .~  (/) ) )
3231imbi2d 314 . . 3  |-  ( c  =  A  ->  (
( A  e.  W  ->  ( c ++  ( M  o.  (reverse `  c
) ) )  .~  (/) )  <->  ( A  e.  W  ->  ( A ++  ( M  o.  (reverse `  A ) ) )  .~  (/) ) ) )
33 wrd0 12553 . . . . 5  |-  (/)  e. Word  (
I  X.  2o )
34 ccatlid 12592 . . . . 5  |-  ( (/)  e. Word  ( I  X.  2o )  ->  ( (/) ++  (/) )  =  (/) )
3533, 34ax-mp 5 . . . 4  |-  ( (/) ++  (/) )  =  (/)
36 efgval.r . . . . . . 7  |-  .~  =  ( ~FG  `  I )
371, 36efger 16935 . . . . . 6  |-  .~  Er  W
3837a1i 11 . . . . 5  |-  ( A  e.  W  ->  .~  Er  W )
391efgrcl 16932 . . . . . . 7  |-  ( A  e.  W  ->  (
I  e.  _V  /\  W  = Word  ( I  X.  2o ) ) )
4039simprd 461 . . . . . 6  |-  ( A  e.  W  ->  W  = Word  ( I  X.  2o ) )
4133, 40syl5eleqr 2549 . . . . 5  |-  ( A  e.  W  ->  (/)  e.  W
)
4238, 41erref 7323 . . . 4  |-  ( A  e.  W  ->  (/)  .~  (/) )
4335, 42syl5eqbr 4472 . . 3  |-  ( A  e.  W  ->  ( (/) ++  (/) )  .~  (/) )
4437a1i 11 . . . . . . 7  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  .~  Er  W )
45 simprl 754 . . . . . . . . . 10  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  a  e. Word  ( I  X.  2o ) )
46 revcl 12726 . . . . . . . . . . . 12  |-  ( a  e. Word  ( I  X.  2o )  ->  (reverse `  a
)  e. Word  ( I  X.  2o ) )
4746ad2antrl 725 . . . . . . . . . . 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 16930 . . . . . . . . . . 11  |-  M :
( I  X.  2o )
--> ( I  X.  2o )
50 wrdco 12788 . . . . . . . . . . 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 660 . . . . . . . . . 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 12582 . . . . . . . . . 10  |-  ( ( a  e. Word  ( I  X.  2o )  /\  ( M  o.  (reverse `  a ) )  e. Word 
( I  X.  2o ) )  ->  (
a ++  ( M  o.  (reverse `  a ) ) )  e. Word  ( I  X.  2o ) )
5345, 51, 52syl2anc 659 . . . . . . . . 9  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
a ++  ( M  o.  (reverse `  a ) ) )  e. Word  ( I  X.  2o ) )
5440adantr 463 . . . . . . . . 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 ++  ( M  o.  (reverse `  a ) ) )  e.  W )
56 lencl 12549 . . . . . . . . . . . . . 14  |-  ( a  e. Word  ( I  X.  2o )  ->  ( # `  a )  e.  NN0 )
5756ad2antrl 725 . . . . . . . . . . . . 13  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 a )  e. 
NN0 )
58 nn0uz 11116 . . . . . . . . . . . . 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 12583 . . . . . . . . . . . . . 14  |-  ( ( a  e. Word  ( I  X.  2o )  /\  ( M  o.  (reverse `  a ) )  e. Word 
( I  X.  2o ) )  ->  ( # `
 ( a ++  ( M  o.  (reverse `  a
) ) ) )  =  ( ( # `  a )  +  (
# `  ( M  o.  (reverse `  a )
) ) ) )
6145, 51, 60syl2anc 659 . . . . . . . . . . . . 13  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 ( a ++  ( M  o.  (reverse `  a
) ) ) )  =  ( ( # `  a )  +  (
# `  ( M  o.  (reverse `  a )
) ) ) )
6257nn0zd 10963 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 a )  e.  ZZ )
63 uzid 11096 . . . . . . . . . . . . . . 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 12549 . . . . . . . . . . . . . . 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 11138 . . . . . . . . . . . . . 14  |-  ( ( ( # `  a
)  e.  ( ZZ>= `  ( # `  a ) )  /\  ( # `  ( M  o.  (reverse `  a ) ) )  e.  NN0 )  -> 
( ( # `  a
)  +  ( # `  ( M  o.  (reverse `  a ) ) ) )  e.  ( ZZ>= `  ( # `  a ) ) )
6864, 66, 67syl2anc 659 . . . . . . . . . . . . 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 ++  ( M  o.  (reverse `  a
) ) ) )  e.  ( ZZ>= `  ( # `
 a ) ) )
70 elfzuzb 11685 . . . . . . . . . . . 12  |-  ( (
# `  a )  e.  ( 0 ... ( # `
 ( a ++  ( M  o.  (reverse `  a
) ) ) ) )  <->  ( ( # `  a )  e.  (
ZZ>= `  0 )  /\  ( # `  ( a ++  ( M  o.  (reverse `  a ) ) ) )  e.  ( ZZ>= `  ( # `  a ) ) ) )
7159, 69, 70sylanbrc 662 . . . . . . . . . . 11  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 a )  e.  ( 0 ... ( # `
 ( a ++  ( M  o.  (reverse `  a
) ) ) ) ) )
72 simprr 755 . . . . . . . . . . 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 16940 . . . . . . . . . . 11  |-  ( ( ( a ++  ( M  o.  (reverse `  a
) ) )  e.  W  /\  ( # `  a )  e.  ( 0 ... ( # `  ( a ++  ( M  o.  (reverse `  a
) ) ) ) )  /\  b  e.  ( I  X.  2o ) )  ->  (
( # `  a ) ( T `  (
a ++  ( M  o.  (reverse `  a ) ) ) ) b )  =  ( ( a ++  ( M  o.  (reverse `  a ) ) ) splice  <. ( # `  a
) ,  ( # `  a ) ,  <" b ( M `  b ) "> >.
) )
7555, 71, 72, 74syl3anc 1226 . . . . . . . . . 10  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( # `  a ) ( T `  (
a ++  ( M  o.  (reverse `  a ) ) ) ) b )  =  ( ( a ++  ( M  o.  (reverse `  a ) ) ) splice  <. ( # `  a
) ,  ( # `  a ) ,  <" b ( M `  b ) "> >.
) )
7633a1i 11 . . . . . . . . . . 11  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (/)  e. Word  (
I  X.  2o ) )
7749ffvelrni 6006 . . . . . . . . . . . . 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 12825 . . . . . . . . . . 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 12593 . . . . . . . . . . . . . 14  |-  ( a  e. Word  ( I  X.  2o )  ->  ( a ++  (/) )  =  a
)
8180ad2antrl 725 . . . . . . . . . . . . 13  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
a ++  (/) )  =  a )
8281eqcomd 2462 . . . . . . . . . . . 12  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  a  =  ( a ++  (/) ) )
8382oveq1d 6285 . . . . . . . . . . 11  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
a ++  ( M  o.  (reverse `  a ) ) )  =  ( ( a ++  (/) ) ++  ( M  o.  (reverse `  a
) ) ) )
84 eqidd 2455 . . . . . . . . . . 11  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 a )  =  ( # `  a
) )
85 hash0 12420 . . . . . . . . . . . . 13  |-  ( # `  (/) )  =  0
8685oveq2i 6281 . . . . . . . . . . . 12  |-  ( (
# `  a )  +  ( # `  (/) ) )  =  ( ( # `  a )  +  0 )
8757nn0cnd 10850 . . . . . . . . . . . . 13  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 a )  e.  CC )
8887addid1d 9769 . . . . . . . . . . . 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 12724 . . . . . . . . . 10  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( a ++  ( M  o.  (reverse `  a
) ) ) splice  <. (
# `  a ) ,  ( # `  a
) ,  <" b
( M `  b
) "> >. )  =  ( ( a ++ 
<" b ( M `
 b ) "> ) ++  ( M  o.  (reverse `  a
) ) ) )
9172s1cld 12604 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  <" b ">  e. Word  ( I  X.  2o ) )
92 revccat 12731 . . . . . . . . . . . . . . . 16  |-  ( ( a  e. Word  ( I  X.  2o )  /\  <" b ">  e. Word  ( I  X.  2o ) )  ->  (reverse `  ( a ++  <" b "> ) )  =  ( (reverse `  <" b "> ) ++  (reverse `  a ) ) )
9345, 91, 92syl2anc 659 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (reverse `  ( a ++  <" b "> ) )  =  ( (reverse `  <" b "> ) ++  (reverse `  a ) ) )
94 revs1 12730 . . . . . . . . . . . . . . . 16  |-  (reverse `  <" b "> )  =  <" b ">
9594oveq1i 6280 . . . . . . . . . . . . . . 15  |-  ( (reverse `  <" b "> ) ++  (reverse `  a
) )  =  (
<" b "> ++  (reverse `  a ) )
9693, 95syl6eq 2511 . . . . . . . . . . . . . 14  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (reverse `  ( a ++  <" b "> ) )  =  ( <" b "> ++  (reverse `  a )
) )
9796coeq2d 5154 . . . . . . . . . . . . 13  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( M  o.  (reverse `  (
a ++  <" b "> ) ) )  =  ( M  o.  ( <" b "> ++  (reverse `  a )
) ) )
9849a1i 11 . . . . . . . . . . . . . 14  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  M : ( I  X.  2o ) --> ( I  X.  2o ) )
99 ccatco 12792 . . . . . . . . . . . . . 14  |-  ( (
<" b ">  e. Word  ( I  X.  2o )  /\  (reverse `  a
)  e. Word  ( I  X.  2o )  /\  M : ( I  X.  2o ) --> ( I  X.  2o ) )  ->  ( M  o.  ( <" b "> ++  (reverse `  a
) ) )  =  ( ( M  o.  <" b "> ) ++  ( M  o.  (reverse `  a ) ) ) )
10091, 47, 98, 99syl3anc 1226 . . . . . . . . . . . . 13  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( M  o.  ( <" b "> ++  (reverse `  a
) ) )  =  ( ( M  o.  <" b "> ) ++  ( M  o.  (reverse `  a ) ) ) )
101 s1co 12790 . . . . . . . . . . . . . . 15  |-  ( ( b  e.  ( I  X.  2o )  /\  M : ( I  X.  2o ) --> ( I  X.  2o ) )  ->  ( M  o.  <" b "> )  =  <" ( M `  b
) "> )
10272, 49, 101sylancl 660 . . . . . . . . . . . . . 14  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( M  o.  <" b "> )  =  <" ( M `  b
) "> )
103102oveq1d 6285 . . . . . . . . . . . . 13  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( M  o.  <" b "> ) ++  ( M  o.  (reverse `  a ) ) )  =  ( <" ( M `  b ) "> ++  ( M  o.  (reverse `  a ) ) ) )
10497, 100, 1033eqtrd 2499 . . . . . . . . . . . 12  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( M  o.  (reverse `  (
a ++  <" b "> ) ) )  =  ( <" ( M `  b ) "> ++  ( M  o.  (reverse `  a ) ) ) )
105104oveq2d 6286 . . . . . . . . . . 11  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( a ++  <" b "> ) ++  ( M  o.  (reverse `  (
a ++  <" b "> ) ) ) )  =  ( ( a ++  <" b "> ) ++  ( <" ( M `  b ) "> ++  ( M  o.  (reverse `  a
) ) ) ) )
106 ccatcl 12582 . . . . . . . . . . . . 13  |-  ( ( a  e. Word  ( I  X.  2o )  /\  <" b ">  e. Word  ( I  X.  2o ) )  ->  (
a ++  <" b "> )  e. Word  (
I  X.  2o ) )
10745, 91, 106syl2anc 659 . . . . . . . . . . . 12  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
a ++  <" b "> )  e. Word  (
I  X.  2o ) )
10878s1cld 12604 . . . . . . . . . . . 12  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  <" ( M `  b ) ">  e. Word  ( I  X.  2o ) )
109 ccatass 12594 . . . . . . . . . . . 12  |-  ( ( ( a ++  <" b "> )  e. Word  (
I  X.  2o )  /\  <" ( M `
 b ) ">  e. Word  ( I  X.  2o )  /\  ( M  o.  (reverse `  a
) )  e. Word  (
I  X.  2o ) )  ->  ( (
( a ++  <" b "> ) ++  <" ( M `  b ) "> ) ++  ( M  o.  (reverse `  a
) ) )  =  ( ( a ++  <" b "> ) ++  ( <" ( M `
 b ) "> ++  ( M  o.  (reverse `  a ) ) ) ) )
110107, 108, 51, 109syl3anc 1226 . . . . . . . . . . 11  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( ( a ++  <" b "> ) ++  <" ( M `  b ) "> ) ++  ( M  o.  (reverse `  a ) ) )  =  ( ( a ++ 
<" b "> ) ++  ( <" ( M `  b ) "> ++  ( M  o.  (reverse `  a ) ) ) ) )
111 ccatass 12594 . . . . . . . . . . . . . 14  |-  ( ( a  e. Word  ( I  X.  2o )  /\  <" b ">  e. Word  ( I  X.  2o )  /\  <" ( M `
 b ) ">  e. Word  ( I  X.  2o ) )  -> 
( ( a ++  <" b "> ) ++  <" ( M `  b ) "> )  =  ( a ++  ( <" b "> ++  <" ( M `
 b ) "> ) ) )
11245, 91, 108, 111syl3anc 1226 . . . . . . . . . . . . 13  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( a ++  <" b "> ) ++  <" ( M `  b ) "> )  =  ( a ++  ( <" b "> ++  <" ( M `
 b ) "> ) ) )
113 df-s2 12804 . . . . . . . . . . . . . 14  |-  <" b
( M `  b
) ">  =  ( <" b "> ++  <" ( M `
 b ) "> )
114113oveq2i 6281 . . . . . . . . . . . . 13  |-  ( a ++ 
<" b ( M `
 b ) "> )  =  ( a ++  ( <" b "> ++  <" ( M `
 b ) "> ) )
115112, 114syl6eqr 2513 . . . . . . . . . . . 12  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( a ++  <" b "> ) ++  <" ( M `  b ) "> )  =  ( a ++  <" b ( M `  b ) "> ) )
116115oveq1d 6285 . . . . . . . . . . 11  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( ( a ++  <" b "> ) ++  <" ( M `  b ) "> ) ++  ( M  o.  (reverse `  a ) ) )  =  ( ( a ++ 
<" b ( M `
 b ) "> ) ++  ( M  o.  (reverse `  a
) ) ) )
117105, 110, 1163eqtr2rd 2502 . . . . . . . . . 10  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( a ++  <" b
( M `  b
) "> ) ++  ( M  o.  (reverse `  a ) ) )  =  ( ( a ++ 
<" b "> ) ++  ( M  o.  (reverse `  ( a ++  <" b "> ) ) ) ) )
11875, 90, 1173eqtrd 2499 . . . . . . . . 9  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( # `  a ) ( T `  (
a ++  ( M  o.  (reverse `  a ) ) ) ) b )  =  ( ( a ++ 
<" b "> ) ++  ( M  o.  (reverse `  ( a ++  <" b "> ) ) ) ) )
1191, 36, 48, 73efgtf 16939 . . . . . . . . . . . 12  |-  ( ( a ++  ( M  o.  (reverse `  a ) ) )  e.  W  -> 
( ( T `  ( a ++  ( M  o.  (reverse `  a )
) ) )  =  ( m  e.  ( 0 ... ( # `  ( a ++  ( M  o.  (reverse `  a
) ) ) ) ) ,  u  e.  ( I  X.  2o )  |->  ( ( a ++  ( M  o.  (reverse `  a ) ) ) splice  <. m ,  m , 
<" u ( M `
 u ) "> >. ) )  /\  ( T `  ( a ++  ( M  o.  (reverse `  a ) ) ) ) : ( ( 0 ... ( # `  ( a ++  ( M  o.  (reverse `  a
) ) ) ) )  X.  ( I  X.  2o ) ) --> W ) )
120119simprd 461 . . . . . . . . . . 11  |-  ( ( a ++  ( M  o.  (reverse `  a ) ) )  e.  W  -> 
( T `  (
a ++  ( M  o.  (reverse `  a ) ) ) ) : ( ( 0 ... ( # `
 ( a ++  ( M  o.  (reverse `  a
) ) ) ) )  X.  ( I  X.  2o ) ) --> W )
121 ffn 5713 . . . . . . . . . . 11  |-  ( ( T `  ( a ++  ( M  o.  (reverse `  a ) ) ) ) : ( ( 0 ... ( # `  ( a ++  ( M  o.  (reverse `  a
) ) ) ) )  X.  ( I  X.  2o ) ) --> W  ->  ( T `  ( a ++  ( M  o.  (reverse `  a
) ) ) )  Fn  ( ( 0 ... ( # `  (
a ++  ( M  o.  (reverse `  a ) ) ) ) )  X.  ( I  X.  2o ) ) )
12255, 120, 1213syl 20 . . . . . . . . . 10  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( T `  ( a ++  ( M  o.  (reverse `  a ) ) ) )  Fn  ( ( 0 ... ( # `  ( a ++  ( M  o.  (reverse `  a
) ) ) ) )  X.  ( I  X.  2o ) ) )
123 fnovrn 6423 . . . . . . . . . 10  |-  ( ( ( T `  (
a ++  ( M  o.  (reverse `  a ) ) ) )  Fn  (
( 0 ... ( # `
 ( a ++  ( M  o.  (reverse `  a
) ) ) ) )  X.  ( I  X.  2o ) )  /\  ( # `  a
)  e.  ( 0 ... ( # `  (
a ++  ( M  o.  (reverse `  a ) ) ) ) )  /\  b  e.  ( I  X.  2o ) )  -> 
( ( # `  a
) ( T `  ( a ++  ( M  o.  (reverse `  a )
) ) ) b )  e.  ran  ( T `  ( a ++  ( M  o.  (reverse `  a ) ) ) ) )
124122, 71, 72, 123syl3anc 1226 . . . . . . . . 9  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( # `  a ) ( T `  (
a ++  ( M  o.  (reverse `  a ) ) ) ) b )  e.  ran  ( T `
 ( a ++  ( M  o.  (reverse `  a
) ) ) ) )
125118, 124eqeltrrd 2543 . . . . . . . 8  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( a ++  <" b "> ) ++  ( M  o.  (reverse `  (
a ++  <" b "> ) ) ) )  e.  ran  ( T `  ( a ++  ( M  o.  (reverse `  a ) ) ) ) )
1261, 36, 48, 73efgi2 16942 . . . . . . . 8  |-  ( ( ( a ++  ( M  o.  (reverse `  a
) ) )  e.  W  /\  ( ( a ++  <" b "> ) ++  ( M  o.  (reverse `  (
a ++  <" b "> ) ) ) )  e.  ran  ( T `  ( a ++  ( M  o.  (reverse `  a ) ) ) ) )  ->  (
a ++  ( M  o.  (reverse `  a ) ) )  .~  ( ( a ++  <" b "> ) ++  ( M  o.  (reverse `  (
a ++  <" b "> ) ) ) ) )
12755, 125, 126syl2anc 659 . . . . . . 7  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
a ++  ( M  o.  (reverse `  a ) ) )  .~  ( ( a ++  <" b "> ) ++  ( M  o.  (reverse `  (
a ++  <" b "> ) ) ) ) )
12844, 127ersym 7315 . . . . . 6  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( a ++  <" b "> ) ++  ( M  o.  (reverse `  (
a ++  <" b "> ) ) ) )  .~  ( a ++  ( M  o.  (reverse `  a ) ) ) )
12944ertr 7318 . . . . . 6  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( ( ( a ++ 
<" b "> ) ++  ( M  o.  (reverse `  ( a ++  <" b "> ) ) ) )  .~  ( a ++  ( M  o.  (reverse `  a ) ) )  /\  ( a ++  ( M  o.  (reverse `  a
) ) )  .~  (/) )  ->  ( (
a ++  <" b "> ) ++  ( M  o.  (reverse `  (
a ++  <" b "> ) ) ) )  .~  (/) ) )
130128, 129mpand 673 . . . . 5  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( a ++  ( M  o.  (reverse `  a
) ) )  .~  (/) 
->  ( ( a ++  <" b "> ) ++  ( M  o.  (reverse `  ( a ++  <" b "> ) ) ) )  .~  (/) ) )
131130expcom 433 . . . 4  |-  ( ( a  e. Word  ( I  X.  2o )  /\  b  e.  ( I  X.  2o ) )  -> 
( A  e.  W  ->  ( ( a ++  ( M  o.  (reverse `  a
) ) )  .~  (/) 
->  ( ( a ++  <" b "> ) ++  ( M  o.  (reverse `  ( a ++  <" b "> ) ) ) )  .~  (/) ) ) )
132131a2d 26 . . 3  |-  ( ( a  e. Word  ( I  X.  2o )  /\  b  e.  ( I  X.  2o ) )  -> 
( ( A  e.  W  ->  ( a ++  ( M  o.  (reverse `  a ) ) )  .~  (/) )  ->  ( A  e.  W  ->  ( ( a ++  <" b "> ) ++  ( M  o.  (reverse `  (
a ++  <" b "> ) ) ) )  .~  (/) ) ) )
13314, 20, 26, 32, 43, 132wrdind 12693 . 2  |-  ( A  e. Word  ( I  X.  2o )  ->  ( A  e.  W  ->  ( A ++  ( M  o.  (reverse `  A ) ) )  .~  (/) ) )
1344, 133mpcom 36 1  |-  ( A  e.  W  ->  ( A ++  ( M  o.  (reverse `  A ) ) )  .~  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   _Vcvv 3106    \ cdif 3458   (/)c0 3783   <.cop 4022   <.cotp 4024   class class class wbr 4439    |-> cmpt 4497    _I cid 4779    X. cxp 4986   ran crn 4989    o. ccom 4992    Fn wfn 5565   -->wf 5566   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   1oc1o 7115   2oc2o 7116    Er wer 7300   0cc0 9481    + caddc 9484   NN0cn0 10791   ZZcz 10860   ZZ>=cuz 11082   ...cfz 11675   #chash 12387  Word cword 12518   ++ cconcat 12520   <"cs1 12521   splice csplice 12523  reversecreverse 12524   <"cs2 12797   ~FG cefg 16923
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-ot 4025  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-ec 7305  df-map 7414  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-hash 12388  df-word 12526  df-lsw 12527  df-concat 12528  df-s1 12529  df-substr 12530  df-splice 12531  df-reverse 12532  df-s2 12804  df-efg 16926
This theorem is referenced by:  efginvrel1  16945  frgpinv  16981
  Copyright terms: Public domain W3C validator