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Theorem revs1 11483
Description: Singleton words are their own reverses. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
Assertion
Ref Expression
revs1  |-  (reverse `  <" S "> )  =  <" S ">

Proof of Theorem revs1
StepHypRef Expression
1 s1cli 11443 . . . . 5  |-  <" S ">  e. Word  _V
2 s1len 11444 . . . . . . 7  |-  ( # `  <" S "> )  =  1
3 1nn 9757 . . . . . . 7  |-  1  e.  NN
42, 3eqeltri 2353 . . . . . 6  |-  ( # `  <" S "> )  e.  NN
5 lbfzo0 10903 . . . . . 6  |-  ( 0  e.  ( 0..^ (
# `  <" S "> ) )  <->  ( # `  <" S "> )  e.  NN )
64, 5mpbir 200 . . . . 5  |-  0  e.  ( 0..^ ( # `  <" S "> ) )
7 revfv 11481 . . . . 5  |-  ( (
<" S ">  e. Word  _V  /\  0  e.  ( 0..^ ( # `  <" S "> ) ) )  -> 
( (reverse `  <" S "> ) `  0 )  =  ( <" S "> `  ( (
( # `  <" S "> )  -  1 )  -  0 ) ) )
81, 6, 7mp2an 653 . . . 4  |-  ( (reverse `  <" S "> ) `  0 )  =  ( <" S "> `  ( (
( # `  <" S "> )  -  1 )  -  0 ) )
92oveq1i 5868 . . . . . . . . 9  |-  ( (
# `  <" S "> )  -  1 )  =  ( 1  -  1 )
10 1m1e0 9814 . . . . . . . . 9  |-  ( 1  -  1 )  =  0
119, 10eqtri 2303 . . . . . . . 8  |-  ( (
# `  <" S "> )  -  1 )  =  0
1211oveq1i 5868 . . . . . . 7  |-  ( ( ( # `  <" S "> )  -  1 )  - 
0 )  =  ( 0  -  0 )
13 0cn 8831 . . . . . . . 8  |-  0  e.  CC
1413subidi 9117 . . . . . . 7  |-  ( 0  -  0 )  =  0
1512, 14eqtri 2303 . . . . . 6  |-  ( ( ( # `  <" S "> )  -  1 )  - 
0 )  =  0
1615fveq2i 5528 . . . . 5  |-  ( <" S "> `  ( ( ( # `  <" S "> )  -  1
)  -  0 ) )  =  ( <" S "> `  0 )
17 ids1 11437 . . . . . . 7  |-  <" S ">  =  <" (  _I  `  S ) ">
1817fveq1i 5526 . . . . . 6  |-  ( <" S "> `  0 )  =  (
<" (  _I  `  S ) "> `  0 )
19 fvex 5539 . . . . . . 7  |-  (  _I 
`  S )  e. 
_V
20 s1fv 11446 . . . . . . 7  |-  ( (  _I  `  S )  e.  _V  ->  ( <" (  _I  `  S ) "> `  0 )  =  (  _I  `  S ) )
2119, 20ax-mp 8 . . . . . 6  |-  ( <" (  _I  `  S ) "> `  0 )  =  (  _I  `  S )
2218, 21eqtri 2303 . . . . 5  |-  ( <" S "> `  0 )  =  (  _I  `  S )
2316, 22eqtri 2303 . . . 4  |-  ( <" S "> `  ( ( ( # `  <" S "> )  -  1
)  -  0 ) )  =  (  _I 
`  S )
248, 23eqtri 2303 . . 3  |-  ( (reverse `  <" S "> ) `  0 )  =  (  _I  `  S )
25 s1eq 11439 . . 3  |-  ( ( (reverse `  <" S "> ) `  0
)  =  (  _I 
`  S )  ->  <" ( (reverse `  <" S "> ) `  0 ) ">  =  <" (  _I  `  S ) "> )
2624, 25ax-mp 8 . 2  |-  <" (
(reverse `  <" S "> ) `  0 ) ">  =  <" (  _I  `  S
) ">
27 revcl 11479 . . . 4  |-  ( <" S ">  e. Word  _V  ->  (reverse `  <" S "> )  e. Word  _V )
281, 27ax-mp 8 . . 3  |-  (reverse `  <" S "> )  e. Word  _V
29 revlen 11480 . . . . 5  |-  ( <" S ">  e. Word  _V  ->  ( # `  (reverse ` 
<" S "> ) )  =  (
# `  <" S "> ) )
301, 29ax-mp 8 . . . 4  |-  ( # `  (reverse `  <" S "> ) )  =  ( # `  <" S "> )
3130, 2eqtri 2303 . . 3  |-  ( # `  (reverse `  <" S "> ) )  =  1
32 eqs1 11447 . . 3  |-  ( ( (reverse `  <" S "> )  e. Word  _V  /\  ( # `  (reverse ` 
<" S "> ) )  =  1 )  ->  (reverse `  <" S "> )  =  <" ( (reverse `  <" S "> ) `  0 ) "> )
3328, 31, 32mp2an 653 . 2  |-  (reverse `  <" S "> )  =  <" ( (reverse `  <" S "> ) `  0 ) ">
3426, 33, 173eqtr4i 2313 1  |-  (reverse `  <" S "> )  =  <" S ">
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684   _Vcvv 2788    _I cid 4304   ` cfv 5255  (class class class)co 5858   0cc0 8737   1c1 8738    - cmin 9037   NNcn 9746  ..^cfzo 10870   #chash 11337  Word cword 11403   <"cs1 11405  reversecreverse 11408
This theorem is referenced by:  gsumwrev  14839  efginvrel2  15036  vrgpinv  15078
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-fzo 10871  df-hash 11338  df-word 11409  df-s1 11411  df-reverse 11414
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