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Theorem revfv 12403
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 12400 . . 3  |-  ( W  e. Word  A  ->  (reverse `  W )  =  ( x  e.  ( 0..^ ( # `  W
) )  |->  ( W `
 ( ( (
# `  W )  -  1 )  -  x ) ) ) )
21fveq1d 5693 . 2  |-  ( W  e. Word  A  ->  (
(reverse `  W ) `  X )  =  ( ( x  e.  ( 0..^ ( # `  W
) )  |->  ( W `
 ( ( (
# `  W )  -  1 )  -  x ) ) ) `
 X ) )
3 oveq2 6099 . . . 4  |-  ( x  =  X  ->  (
( ( # `  W
)  -  1 )  -  x )  =  ( ( ( # `  W )  -  1 )  -  X ) )
43fveq2d 5695 . . 3  |-  ( x  =  X  ->  ( W `  ( (
( # `  W )  -  1 )  -  x ) )  =  ( W `  (
( ( # `  W
)  -  1 )  -  X ) ) )
5 eqid 2443 . . 3  |-  ( x  e.  ( 0..^ (
# `  W )
)  |->  ( W `  ( ( ( # `  W )  -  1 )  -  x ) ) )  =  ( x  e.  ( 0..^ ( # `  W
) )  |->  ( W `
 ( ( (
# `  W )  -  1 )  -  x ) ) )
6 fvex 5701 . . 3  |-  ( W `
 ( ( (
# `  W )  -  1 )  -  X ) )  e. 
_V
74, 5, 6fvmpt 5774 . 2  |-  ( X  e.  ( 0..^ (
# `  W )
)  ->  ( (
x  e.  ( 0..^ ( # `  W
) )  |->  ( W `
 ( ( (
# `  W )  -  1 )  -  x ) ) ) `
 X )  =  ( W `  (
( ( # `  W
)  -  1 )  -  X ) ) )
82, 7sylan9eq 2495 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 1369    e. wcel 1756    e. cmpt 4350   ` cfv 5418  (class class class)co 6091   0cc0 9282   1c1 9283    - cmin 9595  ..^cfzo 11548   #chash 12103  Word cword 12221  reversecreverse 12227
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-reverse 12235
This theorem is referenced by:  revs1  12405  revccat  12406  revrev  12407  revco  12462
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