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Theorem repswrevw 12812
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 12802 . . . . 5  |-  ( ( S  e.  V  /\  N  e.  NN0 )  -> 
( # `  ( S repeatS  N ) )  =  N )
21oveq2d 6293 . . . 4  |-  ( ( S  e.  V  /\  N  e.  NN0 )  -> 
( 0..^ ( # `  ( S repeatS  N )
) )  =  ( 0..^ N ) )
32mpteq1d 4475 . . 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 752 . . . . 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 463 . . . . . . . 8  |-  ( ( ( S  e.  V  /\  N  e.  NN0 )  /\  x  e.  ( 0..^ N ) )  ->  ( # `  ( S repeatS  N ) )  =  N )
76oveq1d 6292 . . . . . . 7  |-  ( ( ( S  e.  V  /\  N  e.  NN0 )  /\  x  e.  ( 0..^ N ) )  ->  ( ( # `  ( S repeatS  N )
)  -  1 )  =  ( N  - 
1 ) )
87oveq1d 6292 . . . . . 6  |-  ( ( ( S  e.  V  /\  N  e.  NN0 )  /\  x  e.  ( 0..^ N ) )  ->  ( ( (
# `  ( S repeatS  N ) )  -  1 )  -  x )  =  ( ( N  -  1 )  -  x ) )
9 ubmelm1fzo 11943 . . . . . . . 8  |-  ( x  e.  ( 0..^ N )  ->  ( ( N  -  x )  -  1 )  e.  ( 0..^ N ) )
10 elfzoelz 11857 . . . . . . . . 9  |-  ( x  e.  ( 0..^ N )  ->  x  e.  ZZ )
11 nn0cn 10845 . . . . . . . . . . . . . 14  |-  ( N  e.  NN0  ->  N  e.  CC )
1211ad2antll 727 . . . . . . . . . . . . 13  |-  ( ( x  e.  ZZ  /\  ( S  e.  V  /\  N  e.  NN0 ) )  ->  N  e.  CC )
13 zcn 10909 . . . . . . . . . . . . . 14  |-  ( x  e.  ZZ  ->  x  e.  CC )
1413adantr 463 . . . . . . . . . . . . 13  |-  ( ( x  e.  ZZ  /\  ( S  e.  V  /\  N  e.  NN0 ) )  ->  x  e.  CC )
15 1cnd 9641 . . . . . . . . . . . . 13  |-  ( ( x  e.  ZZ  /\  ( S  e.  V  /\  N  e.  NN0 ) )  ->  1  e.  CC )
1612, 14, 15sub32d 9998 . . . . . . . . . . . 12  |-  ( ( x  e.  ZZ  /\  ( S  e.  V  /\  N  e.  NN0 ) )  ->  (
( N  -  x
)  -  1 )  =  ( ( N  -  1 )  -  x ) )
1716eleq1d 2471 . . . . . . . . . . 11  |-  ( ( x  e.  ZZ  /\  ( S  e.  V  /\  N  e.  NN0 ) )  ->  (
( ( N  -  x )  -  1 )  e.  ( 0..^ N )  <->  ( ( N  -  1 )  -  x )  e.  ( 0..^ N ) ) )
1817biimpd 207 . . . . . . . . . 10  |-  ( ( x  e.  ZZ  /\  ( S  e.  V  /\  N  e.  NN0 ) )  ->  (
( ( N  -  x )  -  1 )  e.  ( 0..^ N )  ->  (
( N  -  1 )  -  x )  e.  ( 0..^ N ) ) )
1918ex 432 . . . . . . . . 9  |-  ( x  e.  ZZ  ->  (
( S  e.  V  /\  N  e.  NN0 )  ->  ( ( ( N  -  x )  -  1 )  e.  ( 0..^ N )  ->  ( ( N  -  1 )  -  x )  e.  ( 0..^ N ) ) ) )
2010, 19syl 17 . . . . . . . 8  |-  ( x  e.  ( 0..^ N )  ->  ( ( S  e.  V  /\  N  e.  NN0 )  -> 
( ( ( N  -  x )  - 
1 )  e.  ( 0..^ N )  -> 
( ( N  - 
1 )  -  x
)  e.  ( 0..^ N ) ) ) )
219, 20mpid 39 . . . . . . 7  |-  ( x  e.  ( 0..^ N )  ->  ( ( S  e.  V  /\  N  e.  NN0 )  -> 
( ( N  - 
1 )  -  x
)  e.  ( 0..^ N ) ) )
2221impcom 428 . . . . . 6  |-  ( ( ( S  e.  V  /\  N  e.  NN0 )  /\  x  e.  ( 0..^ N ) )  ->  ( ( N  -  1 )  -  x )  e.  ( 0..^ N ) )
238, 22eqeltrd 2490 . . . . 5  |-  ( ( ( S  e.  V  /\  N  e.  NN0 )  /\  x  e.  ( 0..^ N ) )  ->  ( ( (
# `  ( S repeatS  N ) )  -  1 )  -  x )  e.  ( 0..^ N ) )
24 repswsymb 12800 . . . . 5  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  (
( ( # `  ( S repeatS  N ) )  - 
1 )  -  x
)  e.  ( 0..^ N ) )  -> 
( ( S repeatS  N
) `  ( (
( # `  ( S repeatS  N ) )  - 
1 )  -  x
) )  =  S )
254, 5, 23, 24syl3anc 1230 . . . 4  |-  ( ( ( S  e.  V  /\  N  e.  NN0 )  /\  x  e.  ( 0..^ N ) )  ->  ( ( S repeatS  N ) `  (
( ( # `  ( S repeatS  N ) )  - 
1 )  -  x
) )  =  S )
2625mpteq2dva 4480 . . 3  |-  ( ( S  e.  V  /\  N  e.  NN0 )  -> 
( x  e.  ( 0..^ N )  |->  ( ( S repeatS  N ) `  ( ( ( # `  ( S repeatS  N )
)  -  1 )  -  x ) ) )  =  ( x  e.  ( 0..^ N )  |->  S ) )
273, 26eqtrd 2443 . 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 ) )
28 ovex 6305 . . 3  |-  ( S repeatS  N )  e.  _V
29 revval 12788 . . 3  |-  ( ( S repeatS  N )  e.  _V  ->  (reverse `  ( S repeatS  N ) )  =  ( x  e.  ( 0..^ ( # `  ( S repeatS  N ) ) ) 
|->  ( ( S repeatS  N
) `  ( (
( # `  ( S repeatS  N ) )  - 
1 )  -  x
) ) ) )
3028, 29mp1i 13 . 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
) ) ) )
31 reps 12796 . 2  |-  ( ( S  e.  V  /\  N  e.  NN0 )  -> 
( S repeatS  N )  =  ( x  e.  ( 0..^ N ) 
|->  S ) )
3227, 30, 313eqtr4d 2453 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 367    = wceq 1405    e. wcel 1842   _Vcvv 3058    |-> cmpt 4452   ` cfv 5568  (class class class)co 6277   CCcc 9519   0cc0 9521   1c1 9522    - cmin 9840   NN0cn0 10835   ZZcz 10904  ..^cfzo 11852   #chash 12450  reversecreverse 12587   repeatS creps 12588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-er 7347  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-card 8351  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-nn 10576  df-n0 10836  df-z 10905  df-uz 11127  df-fz 11725  df-fzo 11853  df-hash 12451  df-reverse 12595  df-reps 12596
This theorem is referenced by: (None)
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