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Theorem repswrevw 12523
Description: The reverse of a "repeated symbol word". (Contributed by AV, 6-Nov-2018.)
Assertion
Ref Expression
repswrevw  |-  ( ( S  e.  V  /\  N  e.  NN0 )  -> 
(reverse `  ( S repeatS  N
) )  =  ( S repeatS  N ) )

Proof of Theorem repswrevw
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 repswlen 12513 . . . . 5  |-  ( ( S  e.  V  /\  N  e.  NN0 )  -> 
( # `  ( S repeatS  N ) )  =  N )
21oveq2d 6203 . . . 4  |-  ( ( S  e.  V  /\  N  e.  NN0 )  -> 
( 0..^ ( # `  ( S repeatS  N )
) )  =  ( 0..^ N ) )
32mpteq1d 4468 . . 3  |-  ( ( S  e.  V  /\  N  e.  NN0 )  -> 
( x  e.  ( 0..^ ( # `  ( S repeatS  N ) ) ) 
|->  ( ( S repeatS  N
) `  ( (
( # `  ( S repeatS  N ) )  - 
1 )  -  x
) ) )  =  ( x  e.  ( 0..^ N )  |->  ( ( S repeatS  N ) `  ( ( ( # `  ( S repeatS  N )
)  -  1 )  -  x ) ) ) )
4 simpll 753 . . . . 5  |-  ( ( ( S  e.  V  /\  N  e.  NN0 )  /\  x  e.  ( 0..^ N ) )  ->  S  e.  V
)
5 simplr 754 . . . . 5  |-  ( ( ( S  e.  V  /\  N  e.  NN0 )  /\  x  e.  ( 0..^ N ) )  ->  N  e.  NN0 )
61adantr 465 . . . . . . . 8  |-  ( ( ( S  e.  V  /\  N  e.  NN0 )  /\  x  e.  ( 0..^ N ) )  ->  ( # `  ( S repeatS  N ) )  =  N )
76oveq1d 6202 . . . . . . 7  |-  ( ( ( S  e.  V  /\  N  e.  NN0 )  /\  x  e.  ( 0..^ N ) )  ->  ( ( # `  ( S repeatS  N )
)  -  1 )  =  ( N  - 
1 ) )
87oveq1d 6202 . . . . . 6  |-  ( ( ( S  e.  V  /\  N  e.  NN0 )  /\  x  e.  ( 0..^ N ) )  ->  ( ( (
# `  ( S repeatS  N ) )  -  1 )  -  x )  =  ( ( N  -  1 )  -  x ) )
9 ubmelm1fzo 11721 . . . . . . . 8  |-  ( x  e.  ( 0..^ N )  ->  ( ( N  -  x )  -  1 )  e.  ( 0..^ N ) )
10 elfzoelz 11651 . . . . . . . . 9  |-  ( x  e.  ( 0..^ N )  ->  x  e.  ZZ )
11 nn0cn 10687 . . . . . . . . . . . . . 14  |-  ( N  e.  NN0  ->  N  e.  CC )
1211ad2antll 728 . . . . . . . . . . . . 13  |-  ( ( x  e.  ZZ  /\  ( S  e.  V  /\  N  e.  NN0 ) )  ->  N  e.  CC )
13 zcn 10749 . . . . . . . . . . . . . 14  |-  ( x  e.  ZZ  ->  x  e.  CC )
1413adantr 465 . . . . . . . . . . . . 13  |-  ( ( x  e.  ZZ  /\  ( S  e.  V  /\  N  e.  NN0 ) )  ->  x  e.  CC )
15 ax-1cn 9438 . . . . . . . . . . . . . 14  |-  1  e.  CC
1615a1i 11 . . . . . . . . . . . . 13  |-  ( ( x  e.  ZZ  /\  ( S  e.  V  /\  N  e.  NN0 ) )  ->  1  e.  CC )
1712, 14, 16sub32d 9849 . . . . . . . . . . . 12  |-  ( ( x  e.  ZZ  /\  ( S  e.  V  /\  N  e.  NN0 ) )  ->  (
( N  -  x
)  -  1 )  =  ( ( N  -  1 )  -  x ) )
1817eleq1d 2519 . . . . . . . . . . 11  |-  ( ( x  e.  ZZ  /\  ( S  e.  V  /\  N  e.  NN0 ) )  ->  (
( ( N  -  x )  -  1 )  e.  ( 0..^ N )  <->  ( ( N  -  1 )  -  x )  e.  ( 0..^ N ) ) )
1918biimpd 207 . . . . . . . . . 10  |-  ( ( x  e.  ZZ  /\  ( S  e.  V  /\  N  e.  NN0 ) )  ->  (
( ( N  -  x )  -  1 )  e.  ( 0..^ N )  ->  (
( N  -  1 )  -  x )  e.  ( 0..^ N ) ) )
2019ex 434 . . . . . . . . 9  |-  ( x  e.  ZZ  ->  (
( S  e.  V  /\  N  e.  NN0 )  ->  ( ( ( N  -  x )  -  1 )  e.  ( 0..^ N )  ->  ( ( N  -  1 )  -  x )  e.  ( 0..^ N ) ) ) )
2110, 20syl 16 . . . . . . . 8  |-  ( x  e.  ( 0..^ N )  ->  ( ( S  e.  V  /\  N  e.  NN0 )  -> 
( ( ( N  -  x )  - 
1 )  e.  ( 0..^ N )  -> 
( ( N  - 
1 )  -  x
)  e.  ( 0..^ N ) ) ) )
229, 21mpid 41 . . . . . . 7  |-  ( x  e.  ( 0..^ N )  ->  ( ( S  e.  V  /\  N  e.  NN0 )  -> 
( ( N  - 
1 )  -  x
)  e.  ( 0..^ N ) ) )
2322impcom 430 . . . . . 6  |-  ( ( ( S  e.  V  /\  N  e.  NN0 )  /\  x  e.  ( 0..^ N ) )  ->  ( ( N  -  1 )  -  x )  e.  ( 0..^ N ) )
248, 23eqeltrd 2537 . . . . 5  |-  ( ( ( S  e.  V  /\  N  e.  NN0 )  /\  x  e.  ( 0..^ N ) )  ->  ( ( (
# `  ( S repeatS  N ) )  -  1 )  -  x )  e.  ( 0..^ N ) )
25 repswsymb 12511 . . . . 5  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  (
( ( # `  ( S repeatS  N ) )  - 
1 )  -  x
)  e.  ( 0..^ N ) )  -> 
( ( S repeatS  N
) `  ( (
( # `  ( S repeatS  N ) )  - 
1 )  -  x
) )  =  S )
264, 5, 24, 25syl3anc 1219 . . . 4  |-  ( ( ( S  e.  V  /\  N  e.  NN0 )  /\  x  e.  ( 0..^ N ) )  ->  ( ( S repeatS  N ) `  (
( ( # `  ( S repeatS  N ) )  - 
1 )  -  x
) )  =  S )
2726mpteq2dva 4473 . . 3  |-  ( ( S  e.  V  /\  N  e.  NN0 )  -> 
( x  e.  ( 0..^ N )  |->  ( ( S repeatS  N ) `  ( ( ( # `  ( S repeatS  N )
)  -  1 )  -  x ) ) )  =  ( x  e.  ( 0..^ N )  |->  S ) )
283, 27eqtrd 2491 . 2  |-  ( ( S  e.  V  /\  N  e.  NN0 )  -> 
( x  e.  ( 0..^ ( # `  ( S repeatS  N ) ) ) 
|->  ( ( S repeatS  N
) `  ( (
( # `  ( S repeatS  N ) )  - 
1 )  -  x
) ) )  =  ( x  e.  ( 0..^ N )  |->  S ) )
29 ovex 6212 . . 3  |-  ( S repeatS  N )  e.  _V
30 revval 12499 . . 3  |-  ( ( S repeatS  N )  e.  _V  ->  (reverse `  ( S repeatS  N ) )  =  ( x  e.  ( 0..^ ( # `  ( S repeatS  N ) ) ) 
|->  ( ( S repeatS  N
) `  ( (
( # `  ( S repeatS  N ) )  - 
1 )  -  x
) ) ) )
3129, 30mp1i 12 . 2  |-  ( ( S  e.  V  /\  N  e.  NN0 )  -> 
(reverse `  ( S repeatS  N
) )  =  ( x  e.  ( 0..^ ( # `  ( S repeatS  N ) ) ) 
|->  ( ( S repeatS  N
) `  ( (
( # `  ( S repeatS  N ) )  - 
1 )  -  x
) ) ) )
32 reps 12507 . 2  |-  ( ( S  e.  V  /\  N  e.  NN0 )  -> 
( S repeatS  N )  =  ( x  e.  ( 0..^ N ) 
|->  S ) )
3328, 31, 323eqtr4d 2501 1  |-  ( ( S  e.  V  /\  N  e.  NN0 )  -> 
(reverse `  ( S repeatS  N
) )  =  ( S repeatS  N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3065    |-> cmpt 4445   ` cfv 5513  (class class class)co 6187   CCcc 9378   0cc0 9380   1c1 9381    - cmin 9693   NN0cn0 10677   ZZcz 10744  ..^cfzo 11646   #chash 12201  reversecreverse 12326   repeatS creps 12327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469  ax-cnex 9436  ax-resscn 9437  ax-1cn 9438  ax-icn 9439  ax-addcl 9440  ax-addrcl 9441  ax-mulcl 9442  ax-mulrcl 9443  ax-mulcom 9444  ax-addass 9445  ax-mulass 9446  ax-distr 9447  ax-i2m1 9448  ax-1ne0 9449  ax-1rid 9450  ax-rnegex 9451  ax-rrecex 9452  ax-cnre 9453  ax-pre-lttri 9454  ax-pre-lttrn 9455  ax-pre-ltadd 9456  ax-pre-mulgt0 9457
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-nel 2645  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-pss 3439  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4187  df-int 4224  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-tr 4481  df-eprel 4727  df-id 4731  df-po 4736  df-so 4737  df-fr 4774  df-we 4776  df-ord 4817  df-on 4818  df-lim 4819  df-suc 4820  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6148  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-om 6574  df-1st 6674  df-2nd 6675  df-recs 6929  df-rdg 6963  df-1o 7017  df-er 7198  df-en 7408  df-dom 7409  df-sdom 7410  df-fin 7411  df-card 8207  df-pnf 9518  df-mnf 9519  df-xr 9520  df-ltxr 9521  df-le 9522  df-sub 9695  df-neg 9696  df-nn 10421  df-n0 10678  df-z 10745  df-uz 10960  df-fz 11536  df-fzo 11647  df-hash 12202  df-reverse 12334  df-reps 12335
This theorem is referenced by: (None)
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