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Theorem revfv 12868
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 12865 . . 3  |-  ( W  e. Word  A  ->  (reverse `  W )  =  ( x  e.  ( 0..^ ( # `  W
) )  |->  ( W `
 ( ( (
# `  W )  -  1 )  -  x ) ) ) )
21fveq1d 5867 . 2  |-  ( W  e. Word  A  ->  (
(reverse `  W ) `  X )  =  ( ( x  e.  ( 0..^ ( # `  W
) )  |->  ( W `
 ( ( (
# `  W )  -  1 )  -  x ) ) ) `
 X ) )
3 oveq2 6298 . . . 4  |-  ( x  =  X  ->  (
( ( # `  W
)  -  1 )  -  x )  =  ( ( ( # `  W )  -  1 )  -  X ) )
43fveq2d 5869 . . 3  |-  ( x  =  X  ->  ( W `  ( (
( # `  W )  -  1 )  -  x ) )  =  ( W `  (
( ( # `  W
)  -  1 )  -  X ) ) )
5 eqid 2451 . . 3  |-  ( x  e.  ( 0..^ (
# `  W )
)  |->  ( W `  ( ( ( # `  W )  -  1 )  -  x ) ) )  =  ( x  e.  ( 0..^ ( # `  W
) )  |->  ( W `
 ( ( (
# `  W )  -  1 )  -  x ) ) )
6 fvex 5875 . . 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 2505 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 371    = wceq 1444    e. wcel 1887    |-> cmpt 4461   ` cfv 5582  (class class class)co 6290   0cc0 9539   1c1 9540    - cmin 9860  ..^cfzo 11915   #chash 12515  Word cword 12656  reversecreverse 12662
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-ov 6293  df-reverse 12670
This theorem is referenced by:  revs1  12870  revccat  12871  revrev  12872  revco  12931
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