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Theorem revco 12759
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 12515 . . . . . . . 8  |-  ( W  e. Word  A  ->  W : ( 0..^ (
# `  W )
) --> A )
2 ffn 5729 . . . . . . . 8  |-  ( W : ( 0..^ (
# `  W )
) --> A  ->  W  Fn  ( 0..^ ( # `  W ) ) )
31, 2syl 16 . . . . . . 7  |-  ( W  e. Word  A  ->  W  Fn  ( 0..^ ( # `  W ) ) )
43ad2antrr 725 . . . . . 6  |-  ( ( ( W  e. Word  A  /\  F : A --> B )  /\  x  e.  ( 0..^ ( # `  W
) ) )  ->  W  Fn  ( 0..^ ( # `  W
) ) )
5 lencl 12524 . . . . . . . . . . . . 13  |-  ( W  e. Word  A  ->  ( # `
 W )  e. 
NN0 )
65nn0zd 10960 . . . . . . . . . . . 12  |-  ( W  e. Word  A  ->  ( # `
 W )  e.  ZZ )
7 fzoval 11794 . . . . . . . . . . . 12  |-  ( (
# `  W )  e.  ZZ  ->  ( 0..^ ( # `  W
) )  =  ( 0 ... ( (
# `  W )  -  1 ) ) )
86, 7syl 16 . . . . . . . . . . 11  |-  ( W  e. Word  A  ->  (
0..^ ( # `  W
) )  =  ( 0 ... ( (
# `  W )  -  1 ) ) )
98adantr 465 . . . . . . . . . 10  |-  ( ( W  e. Word  A  /\  F : A --> B )  ->  ( 0..^ (
# `  W )
)  =  ( 0 ... ( ( # `  W )  -  1 ) ) )
109eleq2d 2537 . . . . . . . . 9  |-  ( ( W  e. Word  A  /\  F : A --> B )  ->  ( x  e.  ( 0..^ ( # `  W ) )  <->  x  e.  ( 0 ... (
( # `  W )  -  1 ) ) ) )
1110biimpa 484 . . . . . . . 8  |-  ( ( ( W  e. Word  A  /\  F : A --> B )  /\  x  e.  ( 0..^ ( # `  W
) ) )  ->  x  e.  ( 0 ... ( ( # `  W )  -  1 ) ) )
12 fznn0sub2 11775 . . . . . . . 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 465 . . . . . . 7  |-  ( ( ( W  e. Word  A  /\  F : A --> B )  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( 0..^ ( # `  W ) )  =  ( 0 ... (
( # `  W )  -  1 ) ) )
1513, 14eleqtrrd 2558 . . . . . 6  |-  ( ( ( W  e. Word  A  /\  F : A --> B )  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( ( ( # `  W )  -  1 )  -  x )  e.  ( 0..^ (
# `  W )
) )
16 fvco2 5940 . . . . . 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 661 . . . . 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 12757 . . . . . . . . 9  |-  ( ( W  e. Word  A  /\  F : A --> B )  ->  ( # `  ( F  o.  W )
)  =  ( # `  W ) )
1918oveq1d 6297 . . . . . . . 8  |-  ( ( W  e. Word  A  /\  F : A --> B )  ->  ( ( # `  ( F  o.  W
) )  -  1 )  =  ( (
# `  W )  -  1 ) )
2019oveq1d 6297 . . . . . . 7  |-  ( ( W  e. Word  A  /\  F : A --> B )  ->  ( ( (
# `  ( F  o.  W ) )  - 
1 )  -  x
)  =  ( ( ( # `  W
)  -  1 )  -  x ) )
2120adantr 465 . . . . . 6  |-  ( ( ( W  e. Word  A  /\  F : A --> B )  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( ( ( # `  ( F  o.  W
) )  -  1 )  -  x )  =  ( ( (
# `  W )  -  1 )  -  x ) )
2221fveq2d 5868 . . . . 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 12696 . . . . . . 7  |-  ( ( W  e. Word  A  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( (reverse `  W
) `  x )  =  ( W `  ( ( ( # `  W )  -  1 )  -  x ) ) )
2423adantlr 714 . . . . . 6  |-  ( ( ( W  e. Word  A  /\  F : A --> B )  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( (reverse `  W
) `  x )  =  ( W `  ( ( ( # `  W )  -  1 )  -  x ) ) )
2524fveq2d 5868 . . . . 5  |-  ( ( ( W  e. Word  A  /\  F : A --> B )  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( F `  (
(reverse `  W ) `  x ) )  =  ( F `  ( W `  ( (
( # `  W )  -  1 )  -  x ) ) ) )
2617, 22, 253eqtr4d 2518 . . . 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 4533 . . 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 6298 . . . 4  |-  ( ( W  e. Word  A  /\  F : A --> B )  ->  ( 0..^ (
# `  ( F  o.  W ) ) )  =  ( 0..^ (
# `  W )
) )
2928mpteq1d 4528 . . 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 12695 . . . . . 6  |-  ( W  e. Word  A  ->  ( # `
 (reverse `  W
) )  =  (
# `  W )
)
3130adantr 465 . . . . 5  |-  ( ( W  e. Word  A  /\  F : A --> B )  ->  ( # `  (reverse `  W ) )  =  ( # `  W
) )
3231oveq2d 6298 . . . 4  |-  ( ( W  e. Word  A  /\  F : A --> B )  ->  ( 0..^ (
# `  (reverse `  W
) ) )  =  ( 0..^ ( # `  W ) ) )
3332mpteq1d 4528 . . 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 2519 . 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 461 . . 3  |-  ( ( W  e. Word  A  /\  F : A --> B )  ->  F : A --> B )
36 revcl 12694 . . . . 5  |-  ( W  e. Word  A  ->  (reverse `  W )  e. Word  A
)
37 wrdf 12515 . . . . 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 465 . . 3  |-  ( ( W  e. Word  A  /\  F : A --> B )  ->  (reverse `  W ) : ( 0..^ (
# `  (reverse `  W
) ) ) --> A )
40 fcompt 6055 . . 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 661 . 2  |-  ( ( W  e. Word  A  /\  F : A --> B )  ->  ( F  o.  (reverse `  W ) )  =  ( x  e.  ( 0..^ ( # `  (reverse `  W )
) )  |->  ( F `
 ( (reverse `  W
) `  x )
) ) )
42 ffun 5731 . . . . 5  |-  ( F : A --> B  ->  Fun  F )
4342adantl 466 . . . 4  |-  ( ( W  e. Word  A  /\  F : A --> B )  ->  Fun  F )
44 simpl 457 . . . 4  |-  ( ( W  e. Word  A  /\  F : A --> B )  ->  W  e. Word  A
)
45 cofunexg 6745 . . . 4  |-  ( ( Fun  F  /\  W  e. Word  A )  ->  ( F  o.  W )  e.  _V )
4643, 44, 45syl2anc 661 . . 3  |-  ( ( W  e. Word  A  /\  F : A --> B )  ->  ( F  o.  W )  e.  _V )
47 revval 12693 . . 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 2518 1  |-  ( ( W  e. Word  A  /\  F : A --> B )  ->  ( F  o.  (reverse `  W ) )  =  (reverse `  ( F  o.  W )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113    |-> cmpt 4505    o. ccom 5003   Fun wfun 5580    Fn wfn 5581   -->wf 5582   ` cfv 5586  (class class class)co 6282   0cc0 9488   1c1 9489    - cmin 9801   ZZcz 10860   ...cfz 11668  ..^cfzo 11788   #chash 12369  Word cword 12496  reversecreverse 12502
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-8 1769  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-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  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-nel 2665  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-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  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-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-fzo 11789  df-hash 12370  df-word 12504  df-reverse 12510
This theorem is referenced by:  efginvrel1  16542
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