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Theorem revfv 11497
Description: Reverse of a word at a point. (Contributed by Stefan O'Rear, 26-Aug-2015.)
Assertion
Ref Expression
revfv  |-  ( ( W  e. Word  A  /\  X  e.  ( 0..^ ( # `  W
) ) )  -> 
( (reverse `  W
) `  X )  =  ( W `  ( ( ( # `  W )  -  1 )  -  X ) ) )

Proof of Theorem revfv
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 revval 11494 . . 3  |-  ( W  e. Word  A  ->  (reverse `  W )  =  ( x  e.  ( 0..^ ( # `  W
) )  |->  ( W `
 ( ( (
# `  W )  -  1 )  -  x ) ) ) )
21fveq1d 5543 . 2  |-  ( W  e. Word  A  ->  (
(reverse `  W ) `  X )  =  ( ( x  e.  ( 0..^ ( # `  W
) )  |->  ( W `
 ( ( (
# `  W )  -  1 )  -  x ) ) ) `
 X ) )
3 oveq2 5882 . . . 4  |-  ( x  =  X  ->  (
( ( # `  W
)  -  1 )  -  x )  =  ( ( ( # `  W )  -  1 )  -  X ) )
43fveq2d 5545 . . 3  |-  ( x  =  X  ->  ( W `  ( (
( # `  W )  -  1 )  -  x ) )  =  ( W `  (
( ( # `  W
)  -  1 )  -  X ) ) )
5 eqid 2296 . . 3  |-  ( x  e.  ( 0..^ (
# `  W )
)  |->  ( W `  ( ( ( # `  W )  -  1 )  -  x ) ) )  =  ( x  e.  ( 0..^ ( # `  W
) )  |->  ( W `
 ( ( (
# `  W )  -  1 )  -  x ) ) )
6 fvex 5555 . . 3  |-  ( W `
 ( ( (
# `  W )  -  1 )  -  X ) )  e. 
_V
74, 5, 6fvmpt 5618 . 2  |-  ( X  e.  ( 0..^ (
# `  W )
)  ->  ( (
x  e.  ( 0..^ ( # `  W
) )  |->  ( W `
 ( ( (
# `  W )  -  1 )  -  x ) ) ) `
 X )  =  ( W `  (
( ( # `  W
)  -  1 )  -  X ) ) )
82, 7sylan9eq 2348 1  |-  ( ( W  e. Word  A  /\  X  e.  ( 0..^ ( # `  W
) ) )  -> 
( (reverse `  W
) `  X )  =  ( W `  ( ( ( # `  W )  -  1 )  -  X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   0cc0 8753   1c1 8754    - cmin 9053  ..^cfzo 10886   #chash 11353  Word cword 11419  reversecreverse 11424
This theorem is referenced by:  revs1  11499  revccat  11500  revrev  11501  revco  11505
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-reverse 11430
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