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Theorem revs1 11799
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 11759 . . . . 5  |-  <" S ">  e. Word  _V
2 s1len 11760 . . . . . . 7  |-  ( # `  <" S "> )  =  1
3 1nn 10013 . . . . . . 7  |-  1  e.  NN
42, 3eqeltri 2508 . . . . . 6  |-  ( # `  <" S "> )  e.  NN
5 lbfzo0 11172 . . . . . 6  |-  ( 0  e.  ( 0..^ (
# `  <" S "> ) )  <->  ( # `  <" S "> )  e.  NN )
64, 5mpbir 202 . . . . 5  |-  0  e.  ( 0..^ ( # `  <" S "> ) )
7 revfv 11797 . . . . 5  |-  ( (
<" S ">  e. Word  _V  /\  0  e.  ( 0..^ ( # `  <" S "> ) ) )  -> 
( (reverse `  <" S "> ) `  0 )  =  ( <" S "> `  ( (
( # `  <" S "> )  -  1 )  -  0 ) ) )
81, 6, 7mp2an 655 . . . 4  |-  ( (reverse `  <" S "> ) `  0 )  =  ( <" S "> `  ( (
( # `  <" S "> )  -  1 )  -  0 ) )
92oveq1i 6093 . . . . . . . . 9  |-  ( (
# `  <" S "> )  -  1 )  =  ( 1  -  1 )
10 1m1e0 10070 . . . . . . . . 9  |-  ( 1  -  1 )  =  0
119, 10eqtri 2458 . . . . . . . 8  |-  ( (
# `  <" S "> )  -  1 )  =  0
1211oveq1i 6093 . . . . . . 7  |-  ( ( ( # `  <" S "> )  -  1 )  - 
0 )  =  ( 0  -  0 )
13 0cn 9086 . . . . . . . 8  |-  0  e.  CC
1413subidi 9373 . . . . . . 7  |-  ( 0  -  0 )  =  0
1512, 14eqtri 2458 . . . . . 6  |-  ( ( ( # `  <" S "> )  -  1 )  - 
0 )  =  0
1615fveq2i 5733 . . . . 5  |-  ( <" S "> `  ( ( ( # `  <" S "> )  -  1
)  -  0 ) )  =  ( <" S "> `  0 )
17 ids1 11753 . . . . . . 7  |-  <" S ">  =  <" (  _I  `  S ) ">
1817fveq1i 5731 . . . . . 6  |-  ( <" S "> `  0 )  =  (
<" (  _I  `  S ) "> `  0 )
19 fvex 5744 . . . . . . 7  |-  (  _I 
`  S )  e. 
_V
20 s1fv 11762 . . . . . . 7  |-  ( (  _I  `  S )  e.  _V  ->  ( <" (  _I  `  S ) "> `  0 )  =  (  _I  `  S ) )
2119, 20ax-mp 8 . . . . . 6  |-  ( <" (  _I  `  S ) "> `  0 )  =  (  _I  `  S )
2218, 21eqtri 2458 . . . . 5  |-  ( <" S "> `  0 )  =  (  _I  `  S )
2316, 22eqtri 2458 . . . 4  |-  ( <" S "> `  ( ( ( # `  <" S "> )  -  1
)  -  0 ) )  =  (  _I 
`  S )
248, 23eqtri 2458 . . 3  |-  ( (reverse `  <" S "> ) `  0 )  =  (  _I  `  S )
25 s1eq 11755 . . 3  |-  ( ( (reverse `  <" S "> ) `  0
)  =  (  _I 
`  S )  ->  <" ( (reverse `  <" S "> ) `  0 ) ">  =  <" (  _I  `  S ) "> )
2624, 25ax-mp 8 . 2  |-  <" (
(reverse `  <" S "> ) `  0 ) ">  =  <" (  _I  `  S
) ">
27 revcl 11795 . . . 4  |-  ( <" S ">  e. Word  _V  ->  (reverse `  <" S "> )  e. Word  _V )
281, 27ax-mp 8 . . 3  |-  (reverse `  <" S "> )  e. Word  _V
29 revlen 11796 . . . . 5  |-  ( <" S ">  e. Word  _V  ->  ( # `  (reverse ` 
<" S "> ) )  =  (
# `  <" S "> ) )
301, 29ax-mp 8 . . . 4  |-  ( # `  (reverse `  <" S "> ) )  =  ( # `  <" S "> )
3130, 2eqtri 2458 . . 3  |-  ( # `  (reverse `  <" S "> ) )  =  1
32 eqs1 11763 . . 3  |-  ( ( (reverse `  <" S "> )  e. Word  _V  /\  ( # `  (reverse ` 
<" S "> ) )  =  1 )  ->  (reverse `  <" S "> )  =  <" ( (reverse `  <" S "> ) `  0 ) "> )
3328, 31, 32mp2an 655 . 2  |-  (reverse `  <" S "> )  =  <" ( (reverse `  <" S "> ) `  0 ) ">
3426, 33, 173eqtr4i 2468 1  |-  (reverse `  <" S "> )  =  <" S ">
Colors of variables: wff set class
Syntax hints:    = wceq 1653    e. wcel 1726   _Vcvv 2958    _I cid 4495   ` cfv 5456  (class class class)co 6083   0cc0 8992   1c1 8993    - cmin 9293   NNcn 10002  ..^cfzo 11137   #chash 11620  Word cword 11719   <"cs1 11721  reversecreverse 11724
This theorem is referenced by:  gsumwrev  15164  efginvrel2  15361  vrgpinv  15403
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-card 7828  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-n0 10224  df-z 10285  df-uz 10491  df-fz 11046  df-fzo 11138  df-hash 11621  df-word 11725  df-s1 11727  df-reverse 11730
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