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Theorem revrev 11501
Description: Reversion is an involution on words. (Contributed by Mario Carneiro, 1-Oct-2015.)
Assertion
Ref Expression
revrev  |-  ( W  e. Word  A  ->  (reverse `  (reverse `  W )
)  =  W )

Proof of Theorem revrev
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 revcl 11495 . . . . 5  |-  ( W  e. Word  A  ->  (reverse `  W )  e. Word  A
)
2 revcl 11495 . . . . 5  |-  ( (reverse `  W )  e. Word  A  ->  (reverse `  (reverse `  W
) )  e. Word  A
)
31, 2syl 15 . . . 4  |-  ( W  e. Word  A  ->  (reverse `  (reverse `  W )
)  e. Word  A )
4 wrdf 11435 . . . 4  |-  ( (reverse `  (reverse `  W )
)  e. Word  A  ->  (reverse `  (reverse `  W )
) : ( 0..^ ( # `  (reverse `  (reverse `  W )
) ) ) --> A )
5 ffn 5405 . . . 4  |-  ( (reverse `  (reverse `  W )
) : ( 0..^ ( # `  (reverse `  (reverse `  W )
) ) ) --> A  ->  (reverse `  (reverse `  W
) )  Fn  (
0..^ ( # `  (reverse `  (reverse `  W )
) ) ) )
63, 4, 53syl 18 . . 3  |-  ( W  e. Word  A  ->  (reverse `  (reverse `  W )
)  Fn  ( 0..^ ( # `  (reverse `  (reverse `  W )
) ) ) )
7 revlen 11496 . . . . . . 7  |-  ( (reverse `  W )  e. Word  A  ->  ( # `  (reverse `  (reverse `  W )
) )  =  (
# `  (reverse `  W
) ) )
81, 7syl 15 . . . . . 6  |-  ( W  e. Word  A  ->  ( # `
 (reverse `  (reverse `  W ) ) )  =  ( # `  (reverse `  W ) ) )
9 revlen 11496 . . . . . 6  |-  ( W  e. Word  A  ->  ( # `
 (reverse `  W
) )  =  (
# `  W )
)
108, 9eqtrd 2328 . . . . 5  |-  ( W  e. Word  A  ->  ( # `
 (reverse `  (reverse `  W ) ) )  =  ( # `  W
) )
1110oveq2d 5890 . . . 4  |-  ( W  e. Word  A  ->  (
0..^ ( # `  (reverse `  (reverse `  W )
) ) )  =  ( 0..^ ( # `  W ) ) )
1211fneq2d 5352 . . 3  |-  ( W  e. Word  A  ->  (
(reverse `  (reverse `  W
) )  Fn  (
0..^ ( # `  (reverse `  (reverse `  W )
) ) )  <->  (reverse `  (reverse `  W ) )  Fn  ( 0..^ ( # `  W ) ) ) )
136, 12mpbid 201 . 2  |-  ( W  e. Word  A  ->  (reverse `  (reverse `  W )
)  Fn  ( 0..^ ( # `  W
) ) )
14 wrdf 11435 . . 3  |-  ( W  e. Word  A  ->  W : ( 0..^ (
# `  W )
) --> A )
15 ffn 5405 . . 3  |-  ( W : ( 0..^ (
# `  W )
) --> A  ->  W  Fn  ( 0..^ ( # `  W ) ) )
1614, 15syl 15 . 2  |-  ( W  e. Word  A  ->  W  Fn  ( 0..^ ( # `  W ) ) )
171adantr 451 . . . 4  |-  ( ( W  e. Word  A  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
(reverse `  W )  e. Word  A )
18 simpr 447 . . . . 5  |-  ( ( W  e. Word  A  /\  x  e.  ( 0..^ ( # `  W
) ) )  ->  x  e.  ( 0..^ ( # `  W
) ) )
199adantr 451 . . . . . 6  |-  ( ( W  e. Word  A  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( # `  (reverse `  W
) )  =  (
# `  W )
)
2019oveq2d 5890 . . . . 5  |-  ( ( W  e. Word  A  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( 0..^ ( # `  (reverse `  W )
) )  =  ( 0..^ ( # `  W
) ) )
2118, 20eleqtrrd 2373 . . . 4  |-  ( ( W  e. Word  A  /\  x  e.  ( 0..^ ( # `  W
) ) )  ->  x  e.  ( 0..^ ( # `  (reverse `  W ) ) ) )
22 revfv 11497 . . . 4  |-  ( ( (reverse `  W )  e. Word  A  /\  x  e.  ( 0..^ ( # `  (reverse `  W )
) ) )  -> 
( (reverse `  (reverse `  W ) ) `  x )  =  ( (reverse `  W ) `  ( ( ( # `  (reverse `  W )
)  -  1 )  -  x ) ) )
2317, 21, 22syl2anc 642 . . 3  |-  ( ( W  e. Word  A  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( (reverse `  (reverse `  W ) ) `  x )  =  ( (reverse `  W ) `  ( ( ( # `  (reverse `  W )
)  -  1 )  -  x ) ) )
2419oveq1d 5889 . . . . . 6  |-  ( ( W  e. Word  A  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( ( # `  (reverse `  W ) )  - 
1 )  =  ( ( # `  W
)  -  1 ) )
2524oveq1d 5889 . . . . 5  |-  ( ( W  e. Word  A  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( ( ( # `  (reverse `  W )
)  -  1 )  -  x )  =  ( ( ( # `  W )  -  1 )  -  x ) )
2625fveq2d 5545 . . . 4  |-  ( ( W  e. Word  A  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( (reverse `  W
) `  ( (
( # `  (reverse `  W
) )  -  1 )  -  x ) )  =  ( (reverse `  W ) `  (
( ( # `  W
)  -  1 )  -  x ) ) )
27 lencl 11437 . . . . . . . . . . . 12  |-  ( W  e. Word  A  ->  ( # `
 W )  e. 
NN0 )
2827nn0zd 10131 . . . . . . . . . . 11  |-  ( W  e. Word  A  ->  ( # `
 W )  e.  ZZ )
29 fzoval 10892 . . . . . . . . . . 11  |-  ( (
# `  W )  e.  ZZ  ->  ( 0..^ ( # `  W
) )  =  ( 0 ... ( (
# `  W )  -  1 ) ) )
3028, 29syl 15 . . . . . . . . . 10  |-  ( W  e. Word  A  ->  (
0..^ ( # `  W
) )  =  ( 0 ... ( (
# `  W )  -  1 ) ) )
3130eleq2d 2363 . . . . . . . . 9  |-  ( W  e. Word  A  ->  (
x  e.  ( 0..^ ( # `  W
) )  <->  x  e.  ( 0 ... (
( # `  W )  -  1 ) ) ) )
3231biimpa 470 . . . . . . . 8  |-  ( ( W  e. Word  A  /\  x  e.  ( 0..^ ( # `  W
) ) )  ->  x  e.  ( 0 ... ( ( # `  W )  -  1 ) ) )
33 fznn0sub2 10841 . . . . . . . 8  |-  ( x  e.  ( 0 ... ( ( # `  W
)  -  1 ) )  ->  ( (
( # `  W )  -  1 )  -  x )  e.  ( 0 ... ( (
# `  W )  -  1 ) ) )
3432, 33syl 15 . . . . . . 7  |-  ( ( W  e. Word  A  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( ( ( # `  W )  -  1 )  -  x )  e.  ( 0 ... ( ( # `  W
)  -  1 ) ) )
3530adantr 451 . . . . . . 7  |-  ( ( W  e. Word  A  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( 0..^ ( # `  W ) )  =  ( 0 ... (
( # `  W )  -  1 ) ) )
3634, 35eleqtrrd 2373 . . . . . 6  |-  ( ( W  e. Word  A  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( ( ( # `  W )  -  1 )  -  x )  e.  ( 0..^ (
# `  W )
) )
37 revfv 11497 . . . . . 6  |-  ( ( W  e. Word  A  /\  ( ( ( # `  W )  -  1 )  -  x )  e.  ( 0..^ (
# `  W )
) )  ->  (
(reverse `  W ) `  ( ( ( # `  W )  -  1 )  -  x ) )  =  ( W `
 ( ( (
# `  W )  -  1 )  -  ( ( ( # `  W )  -  1 )  -  x ) ) ) )
3836, 37syldan 456 . . . . 5  |-  ( ( W  e. Word  A  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( (reverse `  W
) `  ( (
( # `  W )  -  1 )  -  x ) )  =  ( W `  (
( ( # `  W
)  -  1 )  -  ( ( (
# `  W )  -  1 )  -  x ) ) ) )
39 peano2zm 10078 . . . . . . . . 9  |-  ( (
# `  W )  e.  ZZ  ->  ( ( # `
 W )  - 
1 )  e.  ZZ )
4028, 39syl 15 . . . . . . . 8  |-  ( W  e. Word  A  ->  (
( # `  W )  -  1 )  e.  ZZ )
4140zcnd 10134 . . . . . . 7  |-  ( W  e. Word  A  ->  (
( # `  W )  -  1 )  e.  CC )
42 elfzoelz 10891 . . . . . . . 8  |-  ( x  e.  ( 0..^ (
# `  W )
)  ->  x  e.  ZZ )
4342zcnd 10134 . . . . . . 7  |-  ( x  e.  ( 0..^ (
# `  W )
)  ->  x  e.  CC )
44 nncan 9092 . . . . . . 7  |-  ( ( ( ( # `  W
)  -  1 )  e.  CC  /\  x  e.  CC )  ->  (
( ( # `  W
)  -  1 )  -  ( ( (
# `  W )  -  1 )  -  x ) )  =  x )
4541, 43, 44syl2an 463 . . . . . 6  |-  ( ( W  e. Word  A  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( ( ( # `  W )  -  1 )  -  ( ( ( # `  W
)  -  1 )  -  x ) )  =  x )
4645fveq2d 5545 . . . . 5  |-  ( ( W  e. Word  A  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( W `  (
( ( # `  W
)  -  1 )  -  ( ( (
# `  W )  -  1 )  -  x ) ) )  =  ( W `  x ) )
4738, 46eqtrd 2328 . . . 4  |-  ( ( W  e. Word  A  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( (reverse `  W
) `  ( (
( # `  W )  -  1 )  -  x ) )  =  ( W `  x
) )
4826, 47eqtrd 2328 . . 3  |-  ( ( W  e. Word  A  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( (reverse `  W
) `  ( (
( # `  (reverse `  W
) )  -  1 )  -  x ) )  =  ( W `
 x ) )
4923, 48eqtrd 2328 . 2  |-  ( ( W  e. Word  A  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( (reverse `  (reverse `  W ) ) `  x )  =  ( W `  x ) )
5013, 16, 49eqfnfvd 5641 1  |-  ( W  e. Word  A  ->  (reverse `  (reverse `  W )
)  =  W )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751   0cc0 8753   1c1 8754    - cmin 9053   ZZcz 10040   ...cfz 10798  ..^cfzo 10886   #chash 11353  Word cword 11419  reversecreverse 11424
This theorem is referenced by:  efginvrel1  15053
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-fzo 10887  df-hash 11354  df-word 11425  df-reverse 11430
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