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Theorem revrev 12704
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 12698 . . . 4  |-  ( W  e. Word  A  ->  (reverse `  W )  e. Word  A
)
2 revcl 12698 . . . 4  |-  ( (reverse `  W )  e. Word  A  ->  (reverse `  (reverse `  W
) )  e. Word  A
)
3 wrdf 12519 . . . 4  |-  ( (reverse `  (reverse `  W )
)  e. Word  A  ->  (reverse `  (reverse `  W )
) : ( 0..^ ( # `  (reverse `  (reverse `  W )
) ) ) --> A )
4 ffn 5731 . . . 4  |-  ( (reverse `  (reverse `  W )
) : ( 0..^ ( # `  (reverse `  (reverse `  W )
) ) ) --> A  ->  (reverse `  (reverse `  W
) )  Fn  (
0..^ ( # `  (reverse `  (reverse `  W )
) ) ) )
51, 2, 3, 44syl 21 . . 3  |-  ( W  e. Word  A  ->  (reverse `  (reverse `  W )
)  Fn  ( 0..^ ( # `  (reverse `  (reverse `  W )
) ) ) )
6 revlen 12699 . . . . . . 7  |-  ( (reverse `  W )  e. Word  A  ->  ( # `  (reverse `  (reverse `  W )
) )  =  (
# `  (reverse `  W
) ) )
71, 6syl 16 . . . . . 6  |-  ( W  e. Word  A  ->  ( # `
 (reverse `  (reverse `  W ) ) )  =  ( # `  (reverse `  W ) ) )
8 revlen 12699 . . . . . 6  |-  ( W  e. Word  A  ->  ( # `
 (reverse `  W
) )  =  (
# `  W )
)
97, 8eqtrd 2508 . . . . 5  |-  ( W  e. Word  A  ->  ( # `
 (reverse `  (reverse `  W ) ) )  =  ( # `  W
) )
109oveq2d 6300 . . . 4  |-  ( W  e. Word  A  ->  (
0..^ ( # `  (reverse `  (reverse `  W )
) ) )  =  ( 0..^ ( # `  W ) ) )
1110fneq2d 5672 . . 3  |-  ( W  e. Word  A  ->  (
(reverse `  (reverse `  W
) )  Fn  (
0..^ ( # `  (reverse `  (reverse `  W )
) ) )  <->  (reverse `  (reverse `  W ) )  Fn  ( 0..^ ( # `  W ) ) ) )
125, 11mpbid 210 . 2  |-  ( W  e. Word  A  ->  (reverse `  (reverse `  W )
)  Fn  ( 0..^ ( # `  W
) ) )
13 wrdf 12519 . . 3  |-  ( W  e. Word  A  ->  W : ( 0..^ (
# `  W )
) --> A )
14 ffn 5731 . . 3  |-  ( W : ( 0..^ (
# `  W )
) --> A  ->  W  Fn  ( 0..^ ( # `  W ) ) )
1513, 14syl 16 . 2  |-  ( W  e. Word  A  ->  W  Fn  ( 0..^ ( # `  W ) ) )
161adantr 465 . . . 4  |-  ( ( W  e. Word  A  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
(reverse `  W )  e. Word  A )
17 simpr 461 . . . . 5  |-  ( ( W  e. Word  A  /\  x  e.  ( 0..^ ( # `  W
) ) )  ->  x  e.  ( 0..^ ( # `  W
) ) )
188adantr 465 . . . . . 6  |-  ( ( W  e. Word  A  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( # `  (reverse `  W
) )  =  (
# `  W )
)
1918oveq2d 6300 . . . . 5  |-  ( ( W  e. Word  A  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( 0..^ ( # `  (reverse `  W )
) )  =  ( 0..^ ( # `  W
) ) )
2017, 19eleqtrrd 2558 . . . 4  |-  ( ( W  e. Word  A  /\  x  e.  ( 0..^ ( # `  W
) ) )  ->  x  e.  ( 0..^ ( # `  (reverse `  W ) ) ) )
21 revfv 12700 . . . 4  |-  ( ( (reverse `  W )  e. Word  A  /\  x  e.  ( 0..^ ( # `  (reverse `  W )
) ) )  -> 
( (reverse `  (reverse `  W ) ) `  x )  =  ( (reverse `  W ) `  ( ( ( # `  (reverse `  W )
)  -  1 )  -  x ) ) )
2216, 20, 21syl2anc 661 . . 3  |-  ( ( W  e. Word  A  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( (reverse `  (reverse `  W ) ) `  x )  =  ( (reverse `  W ) `  ( ( ( # `  (reverse `  W )
)  -  1 )  -  x ) ) )
2318oveq1d 6299 . . . . . 6  |-  ( ( W  e. Word  A  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( ( # `  (reverse `  W ) )  - 
1 )  =  ( ( # `  W
)  -  1 ) )
2423oveq1d 6299 . . . . 5  |-  ( ( W  e. Word  A  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( ( ( # `  (reverse `  W )
)  -  1 )  -  x )  =  ( ( ( # `  W )  -  1 )  -  x ) )
2524fveq2d 5870 . . . 4  |-  ( ( W  e. Word  A  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( (reverse `  W
) `  ( (
( # `  (reverse `  W
) )  -  1 )  -  x ) )  =  ( (reverse `  W ) `  (
( ( # `  W
)  -  1 )  -  x ) ) )
26 lencl 12528 . . . . . . . . . . . 12  |-  ( W  e. Word  A  ->  ( # `
 W )  e. 
NN0 )
2726nn0zd 10964 . . . . . . . . . . 11  |-  ( W  e. Word  A  ->  ( # `
 W )  e.  ZZ )
28 fzoval 11798 . . . . . . . . . . 11  |-  ( (
# `  W )  e.  ZZ  ->  ( 0..^ ( # `  W
) )  =  ( 0 ... ( (
# `  W )  -  1 ) ) )
2927, 28syl 16 . . . . . . . . . 10  |-  ( W  e. Word  A  ->  (
0..^ ( # `  W
) )  =  ( 0 ... ( (
# `  W )  -  1 ) ) )
3029eleq2d 2537 . . . . . . . . 9  |-  ( W  e. Word  A  ->  (
x  e.  ( 0..^ ( # `  W
) )  <->  x  e.  ( 0 ... (
( # `  W )  -  1 ) ) ) )
3130biimpa 484 . . . . . . . 8  |-  ( ( W  e. Word  A  /\  x  e.  ( 0..^ ( # `  W
) ) )  ->  x  e.  ( 0 ... ( ( # `  W )  -  1 ) ) )
32 fznn0sub2 11779 . . . . . . . 8  |-  ( x  e.  ( 0 ... ( ( # `  W
)  -  1 ) )  ->  ( (
( # `  W )  -  1 )  -  x )  e.  ( 0 ... ( (
# `  W )  -  1 ) ) )
3331, 32syl 16 . . . . . . 7  |-  ( ( W  e. Word  A  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( ( ( # `  W )  -  1 )  -  x )  e.  ( 0 ... ( ( # `  W
)  -  1 ) ) )
3429adantr 465 . . . . . . 7  |-  ( ( W  e. Word  A  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( 0..^ ( # `  W ) )  =  ( 0 ... (
( # `  W )  -  1 ) ) )
3533, 34eleqtrrd 2558 . . . . . 6  |-  ( ( W  e. Word  A  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( ( ( # `  W )  -  1 )  -  x )  e.  ( 0..^ (
# `  W )
) )
36 revfv 12700 . . . . . 6  |-  ( ( W  e. Word  A  /\  ( ( ( # `  W )  -  1 )  -  x )  e.  ( 0..^ (
# `  W )
) )  ->  (
(reverse `  W ) `  ( ( ( # `  W )  -  1 )  -  x ) )  =  ( W `
 ( ( (
# `  W )  -  1 )  -  ( ( ( # `  W )  -  1 )  -  x ) ) ) )
3735, 36syldan 470 . . . . 5  |-  ( ( W  e. Word  A  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( (reverse `  W
) `  ( (
( # `  W )  -  1 )  -  x ) )  =  ( W `  (
( ( # `  W
)  -  1 )  -  ( ( (
# `  W )  -  1 )  -  x ) ) ) )
38 peano2zm 10906 . . . . . . . . 9  |-  ( (
# `  W )  e.  ZZ  ->  ( ( # `
 W )  - 
1 )  e.  ZZ )
3927, 38syl 16 . . . . . . . 8  |-  ( W  e. Word  A  ->  (
( # `  W )  -  1 )  e.  ZZ )
4039zcnd 10967 . . . . . . 7  |-  ( W  e. Word  A  ->  (
( # `  W )  -  1 )  e.  CC )
41 elfzoelz 11797 . . . . . . . 8  |-  ( x  e.  ( 0..^ (
# `  W )
)  ->  x  e.  ZZ )
4241zcnd 10967 . . . . . . 7  |-  ( x  e.  ( 0..^ (
# `  W )
)  ->  x  e.  CC )
43 nncan 9848 . . . . . . 7  |-  ( ( ( ( # `  W
)  -  1 )  e.  CC  /\  x  e.  CC )  ->  (
( ( # `  W
)  -  1 )  -  ( ( (
# `  W )  -  1 )  -  x ) )  =  x )
4440, 42, 43syl2an 477 . . . . . 6  |-  ( ( W  e. Word  A  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( ( ( # `  W )  -  1 )  -  ( ( ( # `  W
)  -  1 )  -  x ) )  =  x )
4544fveq2d 5870 . . . . 5  |-  ( ( W  e. Word  A  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( W `  (
( ( # `  W
)  -  1 )  -  ( ( (
# `  W )  -  1 )  -  x ) ) )  =  ( W `  x ) )
4637, 45eqtrd 2508 . . . 4  |-  ( ( W  e. Word  A  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( (reverse `  W
) `  ( (
( # `  W )  -  1 )  -  x ) )  =  ( W `  x
) )
4725, 46eqtrd 2508 . . 3  |-  ( ( W  e. Word  A  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( (reverse `  W
) `  ( (
( # `  (reverse `  W
) )  -  1 )  -  x ) )  =  ( W `
 x ) )
4822, 47eqtrd 2508 . 2  |-  ( ( W  e. Word  A  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( (reverse `  (reverse `  W ) ) `  x )  =  ( W `  x ) )
4912, 15, 48eqfnfvd 5978 1  |-  ( W  e. Word  A  ->  (reverse `  (reverse `  W )
)  =  W )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    Fn wfn 5583   -->wf 5584   ` cfv 5588  (class class class)co 6284   CCcc 9490   0cc0 9492   1c1 9493    - cmin 9805   ZZcz 10864   ...cfz 11672  ..^cfzo 11792   #chash 12373  Word cword 12500  reversecreverse 12506
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 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
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 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-card 8320  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-n0 10796  df-z 10865  df-uz 11083  df-fz 11673  df-fzo 11793  df-hash 12374  df-word 12508  df-reverse 12514
This theorem is referenced by:  efginvrel1  16552
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