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Theorem efginvrel2 16204
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 5737 . . . 4  |-  (  _I 
` Word  ( I  X.  2o ) )  C_ Word  ( I  X.  2o )
31, 2eqsstri 3374 . . 3  |-  W  C_ Word  ( I  X.  2o )
43sseli 3340 . 2  |-  ( A  e.  W  ->  A  e. Word  ( I  X.  2o ) )
5 id 22 . . . . . 6  |-  ( c  =  (/)  ->  c  =  (/) )
6 fveq2 5679 . . . . . . . . 9  |-  ( c  =  (/)  ->  (reverse `  c
)  =  (reverse `  (/) ) )
7 rev0 12388 . . . . . . . . 9  |-  (reverse `  (/) )  =  (/)
86, 7syl6eq 2481 . . . . . . . 8  |-  ( c  =  (/)  ->  (reverse `  c
)  =  (/) )
98coeq2d 4989 . . . . . . 7  |-  ( c  =  (/)  ->  ( M  o.  (reverse `  c
) )  =  ( M  o.  (/) ) )
10 co02 5339 . . . . . . 7  |-  ( M  o.  (/) )  =  (/)
119, 10syl6eq 2481 . . . . . 6  |-  ( c  =  (/)  ->  ( M  o.  (reverse `  c
) )  =  (/) )
125, 11oveq12d 6098 . . . . 5  |-  ( c  =  (/)  ->  ( c concat 
( M  o.  (reverse `  c ) ) )  =  ( (/) concat  (/) ) )
1312breq1d 4290 . . . 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 5679 . . . . . . 7  |-  ( c  =  a  ->  (reverse `  c )  =  (reverse `  a ) )
1716coeq2d 4989 . . . . . 6  |-  ( c  =  a  ->  ( M  o.  (reverse `  c
) )  =  ( M  o.  (reverse `  a
) ) )
1815, 17oveq12d 6098 . . . . 5  |-  ( c  =  a  ->  (
c concat  ( M  o.  (reverse `  c ) ) )  =  ( a concat  ( M  o.  (reverse `  a
) ) ) )
1918breq1d 4290 . . . 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 5679 . . . . . . 7  |-  ( c  =  ( a concat  <" b "> )  ->  (reverse `  c )  =  (reverse `  ( a concat  <" b "> ) ) )
2322coeq2d 4989 . . . . . 6  |-  ( c  =  ( a concat  <" b "> )  ->  ( M  o.  (reverse `  c ) )  =  ( M  o.  (reverse `  ( a concat  <" b "> ) ) ) )
2421, 23oveq12d 6098 . . . . 5  |-  ( c  =  ( a concat  <" b "> )  ->  ( c concat  ( M  o.  (reverse `  c
) ) )  =  ( ( a concat  <" b "> ) concat  ( M  o.  (reverse `  (
a concat  <" b "> ) ) ) ) )
2524breq1d 4290 . . . 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 5679 . . . . . . 7  |-  ( c  =  A  ->  (reverse `  c )  =  (reverse `  A ) )
2928coeq2d 4989 . . . . . 6  |-  ( c  =  A  ->  ( M  o.  (reverse `  c
) )  =  ( M  o.  (reverse `  A
) ) )
3027, 29oveq12d 6098 . . . . 5  |-  ( c  =  A  ->  (
c concat  ( M  o.  (reverse `  c ) ) )  =  ( A concat  ( M  o.  (reverse `  A
) ) ) )
3130breq1d 4290 . . . 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 12236 . . . . 5  |-  (/)  e. Word  (
I  X.  2o )
34 ccatlid 12268 . . . . 5  |-  ( (/)  e. Word  ( I  X.  2o )  ->  ( (/) concat  (/) )  =  (/) )
3533, 34ax-mp 5 . . . 4  |-  ( (/) concat  (/) )  =  (/)
36 efgval.r . . . . . . 7  |-  .~  =  ( ~FG  `  I )
371, 36efger 16195 . . . . . 6  |-  .~  Er  W
3837a1i 11 . . . . 5  |-  ( A  e.  W  ->  .~  Er  W )
391efgrcl 16192 . . . . . . 7  |-  ( A  e.  W  ->  (
I  e.  _V  /\  W  = Word  ( I  X.  2o ) ) )
4039simprd 460 . . . . . 6  |-  ( A  e.  W  ->  W  = Word  ( I  X.  2o ) )
4133, 40syl5eleqr 2520 . . . . 5  |-  ( A  e.  W  ->  (/)  e.  W
)
4238, 41erref 7109 . . . 4  |-  ( A  e.  W  ->  (/)  .~  (/) )
4335, 42syl5eqbr 4313 . . 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 748 . . . . . . . . . 10  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  a  e. Word  ( I  X.  2o ) )
46 revcl 12385 . . . . . . . . . . . 12  |-  ( a  e. Word  ( I  X.  2o )  ->  (reverse `  a
)  e. Word  ( I  X.  2o ) )
4746ad2antrl 720 . . . . . . . . . . 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 16190 . . . . . . . . . . 11  |-  M :
( I  X.  2o )
--> ( I  X.  2o )
50 wrdco 12443 . . . . . . . . . . 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 655 . . . . . . . . . 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 12258 . . . . . . . . . 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 654 . . . . . . . . 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 462 . . . . . . . . 9  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  W  = Word  ( I  X.  2o ) )
5553, 54eleqtrrd 2510 . . . . . . . 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 12233 . . . . . . . . . . . . . 14  |-  ( a  e. Word  ( I  X.  2o )  ->  ( # `  a )  e.  NN0 )
5756ad2antrl 720 . . . . . . . . . . . . 13  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 a )  e. 
NN0 )
58 nn0uz 10883 . . . . . . . . . . . . 13  |-  NN0  =  ( ZZ>= `  0 )
5957, 58syl6eleq 2523 . . . . . . . . . . . 12  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 a )  e.  ( ZZ>= `  0 )
)
60 ccatlen 12259 . . . . . . . . . . . . . 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 654 . . . . . . . . . . . . 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 10733 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 a )  e.  ZZ )
63 uzid 10863 . . . . . . . . . . . . . . 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 12233 . . . . . . . . . . . . . . 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 10899 . . . . . . . . . . . . . 14  |-  ( ( ( # `  a
)  e.  ( ZZ>= `  ( # `  a ) )  /\  ( # `  ( M  o.  (reverse `  a ) ) )  e.  NN0 )  -> 
( ( # `  a
)  +  ( # `  ( M  o.  (reverse `  a ) ) ) )  e.  ( ZZ>= `  ( # `  a ) ) )
6864, 66, 67syl2anc 654 . . . . . . . . . . . . 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 2507 . . . . . . . . . . . 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 11434 . . . . . . . . . . . 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 657 . . . . . . . . . . 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 749 . . . . . . . . . . 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 16200 . . . . . . . . . . 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 1211 . . . . . . . . . 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 5830 . . . . . . . . . . . . 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 12480 . . . . . . . . . . 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 12269 . . . . . . . . . . . . . 14  |-  ( a  e. Word  ( I  X.  2o )  ->  ( a concat  (/) )  =  a )
8180ad2antrl 720 . . . . . . . . . . . . 13  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
a concat  (/) )  =  a )
8281eqcomd 2438 . . . . . . . . . . . 12  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  a  =  ( a concat  (/) ) )
8382oveq1d 6095 . . . . . . . . . . 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 2434 . . . . . . . . . . 11  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 a )  =  ( # `  a
) )
85 hash0 12119 . . . . . . . . . . . . 13  |-  ( # `  (/) )  =  0
8685oveq2i 6091 . . . . . . . . . . . 12  |-  ( (
# `  a )  +  ( # `  (/) ) )  =  ( ( # `  a )  +  0 )
8757nn0cnd 10626 . . . . . . . . . . . . 13  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 a )  e.  CC )
8887addid1d 9557 . . . . . . . . . . . 12  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
( # `  a )  +  0 )  =  ( # `  a
) )
8986, 88syl5req 2478 . . . . . . . . . . 11  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `
 a )  =  ( ( # `  a
)  +  ( # `  (/) ) ) )
9045, 76, 51, 79, 83, 84, 89splval2 12383 . . . . . . . . . 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 12278 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  <" b ">  e. Word  ( I  X.  2o ) )
92 revccat 12390 . . . . . . . . . . . . . . . 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 654 . . . . . . . . . . . . . . 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 12389 . . . . . . . . . . . . . . . 16  |-  (reverse `  <" b "> )  =  <" b ">
9594oveq1i 6090 . . . . . . . . . . . . . . 15  |-  ( (reverse `  <" b "> ) concat  (reverse `  a
) )  =  (
<" b "> concat  (reverse `  a ) )
9693, 95syl6eq 2481 . . . . . . . . . . . . . 14  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (reverse `  ( a concat  <" b "> ) )  =  ( <" b "> concat  (reverse `  a )
) )
9796coeq2d 4989 . . . . . . . . . . . . 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 12447 . . . . . . . . . . . . . 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 1211 . . . . . . . . . . . . 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 12445 . . . . . . . . . . . . . . 15  |-  ( ( b  e.  ( I  X.  2o )  /\  M : ( I  X.  2o ) --> ( I  X.  2o ) )  ->  ( M  o.  <" b "> )  =  <" ( M `  b
) "> )
10272, 49, 101sylancl 655 . . . . . . . . . . . . . 14  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( M  o.  <" b "> )  =  <" ( M `  b
) "> )
103102oveq1d 6095 . . . . . . . . . . . . 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 2469 . . . . . . . . . . . 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 6096 . . . . . . . . . . 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 12258 . . . . . . . . . . . . 13  |-  ( ( a  e. Word  ( I  X.  2o )  /\  <" b ">  e. Word  ( I  X.  2o ) )  ->  (
a concat  <" b "> )  e. Word  (
I  X.  2o ) )
10745, 91, 106syl2anc 654 . . . . . . . . . . . 12  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  (
a concat  <" b "> )  e. Word  (
I  X.  2o ) )
10878s1cld 12278 . . . . . . . . . . . 12  |-  ( ( A  e.  W  /\  ( a  e. Word  (
I  X.  2o )  /\  b  e.  ( I  X.  2o ) ) )  ->  <" ( M `  b ) ">  e. Word  ( I  X.  2o ) )
109 ccatass 12270 . . . . . . . . . . . 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 1211 . . . . . . . . . . 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 12270 . . . . . . . . . . . . . 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 1211 . . . . . . . . . . . . 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 12459 . . . . . . . . . . . . . 14  |-  <" b
( M `  b
) ">  =  ( <" b "> concat  <" ( M `
 b ) "> )
114113oveq2i 6091 . . . . . . . . . . . . 13  |-  ( a concat  <" b ( M `
 b ) "> )  =  ( a concat  ( <" b "> concat  <" ( M `
 b ) "> ) )
115112, 114syl6eqr 2483 . . . . . . . . . . . 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 6095 . . . . . . . . . . 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 2472 . . . . . . . . . 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 2469 . . . . . . . . 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 16199 . . . . . . . . . . . 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 460 . . . . . . . . . . 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 5547 . . . . . . . . . . 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 6227 . . . . . . . . . 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 1211 . . . . . . . . 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 2508 . . . . . . . 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 16202 . . . . . . . 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 654 . . . . . . 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 7101 . . . . . 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 7104 . . . . . 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 668 . . . . 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 12355 . 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 1362    e. wcel 1755   _Vcvv 2962    \ cdif 3313   (/)c0 3625   <.cop 3871   <.cotp 3873   class class class wbr 4280    e. cmpt 4338    _I cid 4618    X. cxp 4825   ran crn 4828    o. ccom 4831    Fn wfn 5401   -->wf 5402   ` cfv 5406  (class class class)co 6080    e. cmpt2 6082   1oc1o 6901   2oc2o 6902    Er wer 7086   0cc0 9270    + caddc 9273   NN0cn0 10567   ZZcz 10634   ZZ>=cuz 10849   ...cfz 11424   #chash 12087  Word cword 12205   concat cconcat 12207   <"cs1 12208   splice csplice 12210  reversecreverse 12211   <"cs2 12452   ~FG cefg 16183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-ot 3874  df-uni 4080  df-int 4117  df-iun 4161  df-iin 4162  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-1o 6908  df-2o 6909  df-oadd 6912  df-er 7089  df-ec 7091  df-map 7204  df-pm 7205  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-card 8097  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-nn 10311  df-n0 10568  df-z 10635  df-uz 10850  df-fz 11425  df-fzo 11533  df-hash 12088  df-word 12213  df-concat 12215  df-s1 12216  df-substr 12217  df-splice 12218  df-reverse 12219  df-s2 12459  df-efg 16186
This theorem is referenced by:  efginvrel1  16205  frgpinv  16241
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