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Theorem revfv 12694
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 12691 . . 3  |-  ( W  e. Word  A  ->  (reverse `  W )  =  ( x  e.  ( 0..^ ( # `  W
) )  |->  ( W `
 ( ( (
# `  W )  -  1 )  -  x ) ) ) )
21fveq1d 5866 . 2  |-  ( W  e. Word  A  ->  (
(reverse `  W ) `  X )  =  ( ( x  e.  ( 0..^ ( # `  W
) )  |->  ( W `
 ( ( (
# `  W )  -  1 )  -  x ) ) ) `
 X ) )
3 oveq2 6290 . . . 4  |-  ( x  =  X  ->  (
( ( # `  W
)  -  1 )  -  x )  =  ( ( ( # `  W )  -  1 )  -  X ) )
43fveq2d 5868 . . 3  |-  ( x  =  X  ->  ( W `  ( (
( # `  W )  -  1 )  -  x ) )  =  ( W `  (
( ( # `  W
)  -  1 )  -  X ) ) )
5 eqid 2467 . . 3  |-  ( x  e.  ( 0..^ (
# `  W )
)  |->  ( W `  ( ( ( # `  W )  -  1 )  -  x ) ) )  =  ( x  e.  ( 0..^ ( # `  W
) )  |->  ( W `
 ( ( (
# `  W )  -  1 )  -  x ) ) )
6 fvex 5874 . . 3  |-  ( W `
 ( ( (
# `  W )  -  1 )  -  X ) )  e. 
_V
74, 5, 6fvmpt 5948 . 2  |-  ( X  e.  ( 0..^ (
# `  W )
)  ->  ( (
x  e.  ( 0..^ ( # `  W
) )  |->  ( W `
 ( ( (
# `  W )  -  1 )  -  x ) ) ) `
 X )  =  ( W `  (
( ( # `  W
)  -  1 )  -  X ) ) )
82, 7sylan9eq 2528 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 369    = wceq 1379    e. wcel 1767    |-> cmpt 4505   ` cfv 5586  (class class class)co 6282   0cc0 9488   1c1 9489    - cmin 9801  ..^cfzo 11788   #chash 12367  Word cword 12494  reversecreverse 12500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-reverse 12508
This theorem is referenced by:  revs1  12696  revccat  12697  revrev  12698  revco  12757
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