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Theorem revs1 12702
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 12579 . . . . 5  |-  <" S ">  e. Word  _V
2 s1len 12580 . . . . . . 7  |-  ( # `  <" S "> )  =  1
3 1nn 10547 . . . . . . 7  |-  1  e.  NN
42, 3eqeltri 2551 . . . . . 6  |-  ( # `  <" S "> )  e.  NN
5 lbfzo0 11830 . . . . . 6  |-  ( 0  e.  ( 0..^ (
# `  <" S "> ) )  <->  ( # `  <" S "> )  e.  NN )
64, 5mpbir 209 . . . . 5  |-  0  e.  ( 0..^ ( # `  <" S "> ) )
7 revfv 12700 . . . . 5  |-  ( (
<" S ">  e. Word  _V  /\  0  e.  ( 0..^ ( # `  <" S "> ) ) )  -> 
( (reverse `  <" S "> ) `  0 )  =  ( <" S "> `  ( (
( # `  <" S "> )  -  1 )  -  0 ) ) )
81, 6, 7mp2an 672 . . . 4  |-  ( (reverse `  <" S "> ) `  0 )  =  ( <" S "> `  ( (
( # `  <" S "> )  -  1 )  -  0 ) )
92oveq1i 6294 . . . . . . . . 9  |-  ( (
# `  <" S "> )  -  1 )  =  ( 1  -  1 )
10 1m1e0 10604 . . . . . . . . 9  |-  ( 1  -  1 )  =  0
119, 10eqtri 2496 . . . . . . . 8  |-  ( (
# `  <" S "> )  -  1 )  =  0
1211oveq1i 6294 . . . . . . 7  |-  ( ( ( # `  <" S "> )  -  1 )  - 
0 )  =  ( 0  -  0 )
13 0m0e0 10645 . . . . . . 7  |-  ( 0  -  0 )  =  0
1412, 13eqtri 2496 . . . . . 6  |-  ( ( ( # `  <" S "> )  -  1 )  - 
0 )  =  0
1514fveq2i 5869 . . . . 5  |-  ( <" S "> `  ( ( ( # `  <" S "> )  -  1
)  -  0 ) )  =  ( <" S "> `  0 )
16 ids1 12573 . . . . . . 7  |-  <" S ">  =  <" (  _I  `  S ) ">
1716fveq1i 5867 . . . . . 6  |-  ( <" S "> `  0 )  =  (
<" (  _I  `  S ) "> `  0 )
18 fvex 5876 . . . . . . 7  |-  (  _I 
`  S )  e. 
_V
19 s1fv 12582 . . . . . . 7  |-  ( (  _I  `  S )  e.  _V  ->  ( <" (  _I  `  S ) "> `  0 )  =  (  _I  `  S ) )
2018, 19ax-mp 5 . . . . . 6  |-  ( <" (  _I  `  S ) "> `  0 )  =  (  _I  `  S )
2117, 20eqtri 2496 . . . . 5  |-  ( <" S "> `  0 )  =  (  _I  `  S )
2215, 21eqtri 2496 . . . 4  |-  ( <" S "> `  ( ( ( # `  <" S "> )  -  1
)  -  0 ) )  =  (  _I 
`  S )
238, 22eqtri 2496 . . 3  |-  ( (reverse `  <" S "> ) `  0 )  =  (  _I  `  S )
24 s1eq 12575 . . 3  |-  ( ( (reverse `  <" S "> ) `  0
)  =  (  _I 
`  S )  ->  <" ( (reverse `  <" S "> ) `  0 ) ">  =  <" (  _I  `  S ) "> )
2523, 24ax-mp 5 . 2  |-  <" (
(reverse `  <" S "> ) `  0 ) ">  =  <" (  _I  `  S
) ">
26 revcl 12698 . . . 4  |-  ( <" S ">  e. Word  _V  ->  (reverse `  <" S "> )  e. Word  _V )
271, 26ax-mp 5 . . 3  |-  (reverse `  <" S "> )  e. Word  _V
28 revlen 12699 . . . . 5  |-  ( <" S ">  e. Word  _V  ->  ( # `  (reverse ` 
<" S "> ) )  =  (
# `  <" S "> ) )
291, 28ax-mp 5 . . . 4  |-  ( # `  (reverse `  <" S "> ) )  =  ( # `  <" S "> )
3029, 2eqtri 2496 . . 3  |-  ( # `  (reverse `  <" S "> ) )  =  1
31 eqs1 12584 . . 3  |-  ( ( (reverse `  <" S "> )  e. Word  _V  /\  ( # `  (reverse ` 
<" S "> ) )  =  1 )  ->  (reverse `  <" S "> )  =  <" ( (reverse `  <" S "> ) `  0 ) "> )
3227, 30, 31mp2an 672 . 2  |-  (reverse `  <" S "> )  =  <" ( (reverse `  <" S "> ) `  0 ) ">
3325, 32, 163eqtr4i 2506 1  |-  (reverse `  <" S "> )  =  <" S ">
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    e. wcel 1767   _Vcvv 3113    _I cid 4790   ` cfv 5588  (class class class)co 6284   0cc0 9492   1c1 9493    - cmin 9805   NNcn 10536  ..^cfzo 11792   #chash 12373  Word cword 12500   <"cs1 12503  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-s1 12511  df-reverse 12514
This theorem is referenced by:  gsumwrev  16206  efginvrel2  16551  vrgpinv  16593
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