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Theorem revco 11758
Description: Mapping of words commutes with reversal. (Contributed by Stefan O'Rear, 27-Aug-2015.)
Assertion
Ref Expression
revco  |-  ( ( W  e. Word  A  /\  F : A --> B )  ->  ( F  o.  (reverse `  W ) )  =  (reverse `  ( F  o.  W )
) )

Proof of Theorem revco
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 wrdf 11688 . . . . . . . 8  |-  ( W  e. Word  A  ->  W : ( 0..^ (
# `  W )
) --> A )
2 ffn 5550 . . . . . . . 8  |-  ( W : ( 0..^ (
# `  W )
) --> A  ->  W  Fn  ( 0..^ ( # `  W ) ) )
31, 2syl 16 . . . . . . 7  |-  ( W  e. Word  A  ->  W  Fn  ( 0..^ ( # `  W ) ) )
43ad2antrr 707 . . . . . 6  |-  ( ( ( W  e. Word  A  /\  F : A --> B )  /\  x  e.  ( 0..^ ( # `  W
) ) )  ->  W  Fn  ( 0..^ ( # `  W
) ) )
5 lencl 11690 . . . . . . . . . . . . 13  |-  ( W  e. Word  A  ->  ( # `
 W )  e. 
NN0 )
65nn0zd 10329 . . . . . . . . . . . 12  |-  ( W  e. Word  A  ->  ( # `
 W )  e.  ZZ )
7 fzoval 11096 . . . . . . . . . . . 12  |-  ( (
# `  W )  e.  ZZ  ->  ( 0..^ ( # `  W
) )  =  ( 0 ... ( (
# `  W )  -  1 ) ) )
86, 7syl 16 . . . . . . . . . . 11  |-  ( W  e. Word  A  ->  (
0..^ ( # `  W
) )  =  ( 0 ... ( (
# `  W )  -  1 ) ) )
98adantr 452 . . . . . . . . . 10  |-  ( ( W  e. Word  A  /\  F : A --> B )  ->  ( 0..^ (
# `  W )
)  =  ( 0 ... ( ( # `  W )  -  1 ) ) )
109eleq2d 2471 . . . . . . . . 9  |-  ( ( W  e. Word  A  /\  F : A --> B )  ->  ( x  e.  ( 0..^ ( # `  W ) )  <->  x  e.  ( 0 ... (
( # `  W )  -  1 ) ) ) )
1110biimpa 471 . . . . . . . 8  |-  ( ( ( W  e. Word  A  /\  F : A --> B )  /\  x  e.  ( 0..^ ( # `  W
) ) )  ->  x  e.  ( 0 ... ( ( # `  W )  -  1 ) ) )
12 fznn0sub2 11042 . . . . . . . 8  |-  ( x  e.  ( 0 ... ( ( # `  W
)  -  1 ) )  ->  ( (
( # `  W )  -  1 )  -  x )  e.  ( 0 ... ( (
# `  W )  -  1 ) ) )
1311, 12syl 16 . . . . . . 7  |-  ( ( ( W  e. Word  A  /\  F : A --> B )  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( ( ( # `  W )  -  1 )  -  x )  e.  ( 0 ... ( ( # `  W
)  -  1 ) ) )
149adantr 452 . . . . . . 7  |-  ( ( ( W  e. Word  A  /\  F : A --> B )  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( 0..^ ( # `  W ) )  =  ( 0 ... (
( # `  W )  -  1 ) ) )
1513, 14eleqtrrd 2481 . . . . . 6  |-  ( ( ( W  e. Word  A  /\  F : A --> B )  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( ( ( # `  W )  -  1 )  -  x )  e.  ( 0..^ (
# `  W )
) )
16 fvco2 5757 . . . . . 6  |-  ( ( W  Fn  ( 0..^ ( # `  W
) )  /\  (
( ( # `  W
)  -  1 )  -  x )  e.  ( 0..^ ( # `  W ) ) )  ->  ( ( F  o.  W ) `  ( ( ( # `  W )  -  1 )  -  x ) )  =  ( F `
 ( W `  ( ( ( # `  W )  -  1 )  -  x ) ) ) )
174, 15, 16syl2anc 643 . . . . 5  |-  ( ( ( W  e. Word  A  /\  F : A --> B )  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( ( F  o.  W ) `  (
( ( # `  W
)  -  1 )  -  x ) )  =  ( F `  ( W `  ( ( ( # `  W
)  -  1 )  -  x ) ) ) )
18 lenco 11756 . . . . . . . . 9  |-  ( ( W  e. Word  A  /\  F : A --> B )  ->  ( # `  ( F  o.  W )
)  =  ( # `  W ) )
1918oveq1d 6055 . . . . . . . 8  |-  ( ( W  e. Word  A  /\  F : A --> B )  ->  ( ( # `  ( F  o.  W
) )  -  1 )  =  ( (
# `  W )  -  1 ) )
2019oveq1d 6055 . . . . . . 7  |-  ( ( W  e. Word  A  /\  F : A --> B )  ->  ( ( (
# `  ( F  o.  W ) )  - 
1 )  -  x
)  =  ( ( ( # `  W
)  -  1 )  -  x ) )
2120adantr 452 . . . . . 6  |-  ( ( ( W  e. Word  A  /\  F : A --> B )  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( ( ( # `  ( F  o.  W
) )  -  1 )  -  x )  =  ( ( (
# `  W )  -  1 )  -  x ) )
2221fveq2d 5691 . . . . 5  |-  ( ( ( W  e. Word  A  /\  F : A --> B )  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( ( F  o.  W ) `  (
( ( # `  ( F  o.  W )
)  -  1 )  -  x ) )  =  ( ( F  o.  W ) `  ( ( ( # `  W )  -  1 )  -  x ) ) )
23 revfv 11750 . . . . . . 7  |-  ( ( W  e. Word  A  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( (reverse `  W
) `  x )  =  ( W `  ( ( ( # `  W )  -  1 )  -  x ) ) )
2423adantlr 696 . . . . . 6  |-  ( ( ( W  e. Word  A  /\  F : A --> B )  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( (reverse `  W
) `  x )  =  ( W `  ( ( ( # `  W )  -  1 )  -  x ) ) )
2524fveq2d 5691 . . . . 5  |-  ( ( ( W  e. Word  A  /\  F : A --> B )  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( F `  (
(reverse `  W ) `  x ) )  =  ( F `  ( W `  ( (
( # `  W )  -  1 )  -  x ) ) ) )
2617, 22, 253eqtr4d 2446 . . . 4  |-  ( ( ( W  e. Word  A  /\  F : A --> B )  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( ( F  o.  W ) `  (
( ( # `  ( F  o.  W )
)  -  1 )  -  x ) )  =  ( F `  ( (reverse `  W ) `  x ) ) )
2726mpteq2dva 4255 . . 3  |-  ( ( W  e. Word  A  /\  F : A --> B )  ->  ( x  e.  ( 0..^ ( # `  W ) )  |->  ( ( F  o.  W
) `  ( (
( # `  ( F  o.  W ) )  -  1 )  -  x ) ) )  =  ( x  e.  ( 0..^ ( # `  W ) )  |->  ( F `  ( (reverse `  W ) `  x
) ) ) )
2818oveq2d 6056 . . . 4  |-  ( ( W  e. Word  A  /\  F : A --> B )  ->  ( 0..^ (
# `  ( F  o.  W ) ) )  =  ( 0..^ (
# `  W )
) )
2928mpteq1d 4250 . . 3  |-  ( ( W  e. Word  A  /\  F : A --> B )  ->  ( x  e.  ( 0..^ ( # `  ( F  o.  W
) ) )  |->  ( ( F  o.  W
) `  ( (
( # `  ( F  o.  W ) )  -  1 )  -  x ) ) )  =  ( x  e.  ( 0..^ ( # `  W ) )  |->  ( ( F  o.  W
) `  ( (
( # `  ( F  o.  W ) )  -  1 )  -  x ) ) ) )
30 revlen 11749 . . . . . 6  |-  ( W  e. Word  A  ->  ( # `
 (reverse `  W
) )  =  (
# `  W )
)
3130adantr 452 . . . . 5  |-  ( ( W  e. Word  A  /\  F : A --> B )  ->  ( # `  (reverse `  W ) )  =  ( # `  W
) )
3231oveq2d 6056 . . . 4  |-  ( ( W  e. Word  A  /\  F : A --> B )  ->  ( 0..^ (
# `  (reverse `  W
) ) )  =  ( 0..^ ( # `  W ) ) )
3332mpteq1d 4250 . . 3  |-  ( ( W  e. Word  A  /\  F : A --> B )  ->  ( x  e.  ( 0..^ ( # `  (reverse `  W )
) )  |->  ( F `
 ( (reverse `  W
) `  x )
) )  =  ( x  e.  ( 0..^ ( # `  W
) )  |->  ( F `
 ( (reverse `  W
) `  x )
) ) )
3427, 29, 333eqtr4rd 2447 . 2  |-  ( ( W  e. Word  A  /\  F : A --> B )  ->  ( x  e.  ( 0..^ ( # `  (reverse `  W )
) )  |->  ( F `
 ( (reverse `  W
) `  x )
) )  =  ( x  e.  ( 0..^ ( # `  ( F  o.  W )
) )  |->  ( ( F  o.  W ) `
 ( ( (
# `  ( F  o.  W ) )  - 
1 )  -  x
) ) ) )
35 simpr 448 . . 3  |-  ( ( W  e. Word  A  /\  F : A --> B )  ->  F : A --> B )
36 revcl 11748 . . . . 5  |-  ( W  e. Word  A  ->  (reverse `  W )  e. Word  A
)
37 wrdf 11688 . . . . 5  |-  ( (reverse `  W )  e. Word  A  ->  (reverse `  W ) : ( 0..^ (
# `  (reverse `  W
) ) ) --> A )
3836, 37syl 16 . . . 4  |-  ( W  e. Word  A  ->  (reverse `  W ) : ( 0..^ ( # `  (reverse `  W ) ) ) --> A )
3938adantr 452 . . 3  |-  ( ( W  e. Word  A  /\  F : A --> B )  ->  (reverse `  W ) : ( 0..^ (
# `  (reverse `  W
) ) ) --> A )
40 fcompt 5863 . . 3  |-  ( ( F : A --> B  /\  (reverse `  W ) : ( 0..^ ( # `  (reverse `  W )
) ) --> A )  ->  ( F  o.  (reverse `  W ) )  =  ( x  e.  ( 0..^ ( # `  (reverse `  W )
) )  |->  ( F `
 ( (reverse `  W
) `  x )
) ) )
4135, 39, 40syl2anc 643 . 2  |-  ( ( W  e. Word  A  /\  F : A --> B )  ->  ( F  o.  (reverse `  W ) )  =  ( x  e.  ( 0..^ ( # `  (reverse `  W )
) )  |->  ( F `
 ( (reverse `  W
) `  x )
) ) )
42 ffun 5552 . . . . 5  |-  ( F : A --> B  ->  Fun  F )
4342adantl 453 . . . 4  |-  ( ( W  e. Word  A  /\  F : A --> B )  ->  Fun  F )
44 simpl 444 . . . 4  |-  ( ( W  e. Word  A  /\  F : A --> B )  ->  W  e. Word  A
)
45 cofunexg 5918 . . . 4  |-  ( ( Fun  F  /\  W  e. Word  A )  ->  ( F  o.  W )  e.  _V )
4643, 44, 45syl2anc 643 . . 3  |-  ( ( W  e. Word  A  /\  F : A --> B )  ->  ( F  o.  W )  e.  _V )
47 revval 11747 . . 3  |-  ( ( F  o.  W )  e.  _V  ->  (reverse `  ( F  o.  W
) )  =  ( x  e.  ( 0..^ ( # `  ( F  o.  W )
) )  |->  ( ( F  o.  W ) `
 ( ( (
# `  ( F  o.  W ) )  - 
1 )  -  x
) ) ) )
4846, 47syl 16 . 2  |-  ( ( W  e. Word  A  /\  F : A --> B )  ->  (reverse `  ( F  o.  W ) )  =  ( x  e.  ( 0..^ ( # `  ( F  o.  W )
) )  |->  ( ( F  o.  W ) `
 ( ( (
# `  ( F  o.  W ) )  - 
1 )  -  x
) ) ) )
4934, 41, 483eqtr4d 2446 1  |-  ( ( W  e. Word  A  /\  F : A --> B )  ->  ( F  o.  (reverse `  W ) )  =  (reverse `  ( F  o.  W )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2916    e. cmpt 4226    o. ccom 4841   Fun wfun 5407    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040   0cc0 8946   1c1 8947    - cmin 9247   ZZcz 10238   ...cfz 10999  ..^cfzo 11090   #chash 11573  Word cword 11672  reversecreverse 11677
This theorem is referenced by:  efginvrel1  15315
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-fzo 11091  df-hash 11574  df-word 11678  df-reverse 11683
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