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Theorem revfv 12700
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 12697 . . 3  |-  ( W  e. Word  A  ->  (reverse `  W )  =  ( x  e.  ( 0..^ ( # `  W
) )  |->  ( W `
 ( ( (
# `  W )  -  1 )  -  x ) ) ) )
21fveq1d 5807 . 2  |-  ( W  e. Word  A  ->  (
(reverse `  W ) `  X )  =  ( ( x  e.  ( 0..^ ( # `  W
) )  |->  ( W `
 ( ( (
# `  W )  -  1 )  -  x ) ) ) `
 X ) )
3 oveq2 6242 . . . 4  |-  ( x  =  X  ->  (
( ( # `  W
)  -  1 )  -  x )  =  ( ( ( # `  W )  -  1 )  -  X ) )
43fveq2d 5809 . . 3  |-  ( x  =  X  ->  ( W `  ( (
( # `  W )  -  1 )  -  x ) )  =  ( W `  (
( ( # `  W
)  -  1 )  -  X ) ) )
5 eqid 2402 . . 3  |-  ( x  e.  ( 0..^ (
# `  W )
)  |->  ( W `  ( ( ( # `  W )  -  1 )  -  x ) ) )  =  ( x  e.  ( 0..^ ( # `  W
) )  |->  ( W `
 ( ( (
# `  W )  -  1 )  -  x ) ) )
6 fvex 5815 . . 3  |-  ( W `
 ( ( (
# `  W )  -  1 )  -  X ) )  e. 
_V
74, 5, 6fvmpt 5888 . 2  |-  ( X  e.  ( 0..^ (
# `  W )
)  ->  ( (
x  e.  ( 0..^ ( # `  W
) )  |->  ( W `
 ( ( (
# `  W )  -  1 )  -  x ) ) ) `
 X )  =  ( W `  (
( ( # `  W
)  -  1 )  -  X ) ) )
82, 7sylan9eq 2463 1  |-  ( ( W  e. Word  A  /\  X  e.  ( 0..^ ( # `  W
) ) )  -> 
( (reverse `  W
) `  X )  =  ( W `  ( ( ( # `  W )  -  1 )  -  X ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842    |-> cmpt 4452   ` cfv 5525  (class class class)co 6234   0cc0 9442   1c1 9443    - cmin 9761  ..^cfzo 11767   #chash 12359  Word cword 12490  reversecreverse 12496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-ov 6237  df-reverse 12504
This theorem is referenced by:  revs1  12702  revccat  12703  revrev  12704  revco  12763
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