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Theorem repswsymballbi 12535
Description: A word is a "repeated symbol word" iff each of its symbols equals the first symbol of the word. (Contributed by AV, 10-Nov-2018.)
Assertion
Ref Expression
repswsymballbi  |-  ( W  e. Word  V  ->  ( W  =  ( ( W `  0 ) repeatS  (
# `  W )
)  <->  A. i  e.  ( 0..^ ( # `  W
) ) ( W `
 i )  =  ( W `  0
) ) )
Distinct variable group:    i, W
Allowed substitution hint:    V( i)

Proof of Theorem repswsymballbi
StepHypRef Expression
1 fveq2 5798 . . . . 5  |-  ( W  =  (/)  ->  ( # `  W )  =  (
# `  (/) ) )
2 hash0 12251 . . . . 5  |-  ( # `  (/) )  =  0
31, 2syl6eq 2511 . . . 4  |-  ( W  =  (/)  ->  ( # `  W )  =  0 )
4 fvex 5808 . . . . . . . 8  |-  ( W `
 0 )  e. 
_V
5 repsw0 12532 . . . . . . . 8  |-  ( ( W `  0 )  e.  _V  ->  (
( W `  0
) repeatS  0 )  =  (/) )
64, 5ax-mp 5 . . . . . . 7  |-  ( ( W `  0 ) repeatS 
0 )  =  (/)
76eqcomi 2467 . . . . . 6  |-  (/)  =  ( ( W `  0
) repeatS  0 )
8 simpr 461 . . . . . 6  |-  ( ( ( # `  W
)  =  0  /\  W  =  (/) )  ->  W  =  (/) )
9 oveq2 6207 . . . . . . 7  |-  ( (
# `  W )  =  0  ->  (
( W `  0
) repeatS  ( # `  W
) )  =  ( ( W `  0
) repeatS  0 ) )
109adantr 465 . . . . . 6  |-  ( ( ( # `  W
)  =  0  /\  W  =  (/) )  -> 
( ( W ` 
0 ) repeatS  ( # `  W
) )  =  ( ( W `  0
) repeatS  0 ) )
117, 8, 103eqtr4a 2521 . . . . 5  |-  ( ( ( # `  W
)  =  0  /\  W  =  (/) )  ->  W  =  ( ( W `  0 ) repeatS  (
# `  W )
) )
12 ral0 3891 . . . . . . 7  |-  A. i  e.  (/)  ( W `  i )  =  ( W `  0 )
13 oveq2 6207 . . . . . . . . 9  |-  ( (
# `  W )  =  0  ->  (
0..^ ( # `  W
) )  =  ( 0..^ 0 ) )
14 fzo0 11689 . . . . . . . . 9  |-  ( 0..^ 0 )  =  (/)
1513, 14syl6eq 2511 . . . . . . . 8  |-  ( (
# `  W )  =  0  ->  (
0..^ ( # `  W
) )  =  (/) )
1615raleqdv 3027 . . . . . . 7  |-  ( (
# `  W )  =  0  ->  ( A. i  e.  (
0..^ ( # `  W
) ) ( W `
 i )  =  ( W `  0
)  <->  A. i  e.  (/)  ( W `  i )  =  ( W ` 
0 ) ) )
1712, 16mpbiri 233 . . . . . 6  |-  ( (
# `  W )  =  0  ->  A. i  e.  ( 0..^ ( # `  W ) ) ( W `  i )  =  ( W ` 
0 ) )
1817adantr 465 . . . . 5  |-  ( ( ( # `  W
)  =  0  /\  W  =  (/) )  ->  A. i  e.  (
0..^ ( # `  W
) ) ( W `
 i )  =  ( W `  0
) )
1911, 182thd 240 . . . 4  |-  ( ( ( # `  W
)  =  0  /\  W  =  (/) )  -> 
( W  =  ( ( W `  0
) repeatS  ( # `  W
) )  <->  A. i  e.  ( 0..^ ( # `  W ) ) ( W `  i )  =  ( W ` 
0 ) ) )
203, 19mpancom 669 . . 3  |-  ( W  =  (/)  ->  ( W  =  ( ( W `
 0 ) repeatS  ( # `
 W ) )  <->  A. i  e.  (
0..^ ( # `  W
) ) ( W `
 i )  =  ( W `  0
) ) )
2120a1d 25 . 2  |-  ( W  =  (/)  ->  ( W  e. Word  V  ->  ( W  =  ( ( W `  0 ) repeatS  (
# `  W )
)  <->  A. i  e.  ( 0..^ ( # `  W
) ) ( W `
 i )  =  ( W `  0
) ) ) )
22 df-3an 967 . . . . 5  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  ( # `  W
)  /\  A. i  e.  ( 0..^ ( # `  W ) ) ( W `  i )  =  ( W ` 
0 ) )  <->  ( ( W  e. Word  V  /\  ( # `
 W )  =  ( # `  W
) )  /\  A. i  e.  ( 0..^ ( # `  W
) ) ( W `
 i )  =  ( W `  0
) ) )
2322a1i 11 . . . 4  |-  ( ( W  =/=  (/)  /\  W  e. Word  V )  ->  (
( W  e. Word  V  /\  ( # `  W
)  =  ( # `  W )  /\  A. i  e.  ( 0..^ ( # `  W
) ) ( W `
 i )  =  ( W `  0
) )  <->  ( ( W  e. Word  V  /\  ( # `
 W )  =  ( # `  W
) )  /\  A. i  e.  ( 0..^ ( # `  W
) ) ( W `
 i )  =  ( W `  0
) ) ) )
24 fstwrdne 12380 . . . . . 6  |-  ( ( W  e. Word  V  /\  W  =/=  (/) )  ->  ( W `  0 )  e.  V )
2524ancoms 453 . . . . 5  |-  ( ( W  =/=  (/)  /\  W  e. Word  V )  ->  ( W `  0 )  e.  V )
26 lencl 12366 . . . . . 6  |-  ( W  e. Word  V  ->  ( # `
 W )  e. 
NN0 )
2726adantl 466 . . . . 5  |-  ( ( W  =/=  (/)  /\  W  e. Word  V )  ->  ( # `
 W )  e. 
NN0 )
28 repsdf2 12533 . . . . 5  |-  ( ( ( W `  0
)  e.  V  /\  ( # `  W )  e.  NN0 )  -> 
( W  =  ( ( W `  0
) repeatS  ( # `  W
) )  <->  ( W  e. Word  V  /\  ( # `  W )  =  (
# `  W )  /\  A. i  e.  ( 0..^ ( # `  W
) ) ( W `
 i )  =  ( W `  0
) ) ) )
2925, 27, 28syl2anc 661 . . . 4  |-  ( ( W  =/=  (/)  /\  W  e. Word  V )  ->  ( W  =  ( ( W `  0 ) repeatS  (
# `  W )
)  <->  ( W  e. Word  V  /\  ( # `  W
)  =  ( # `  W )  /\  A. i  e.  ( 0..^ ( # `  W
) ) ( W `
 i )  =  ( W `  0
) ) ) )
30 simpr 461 . . . . . 6  |-  ( ( W  =/=  (/)  /\  W  e. Word  V )  ->  W  e. Word  V )
31 eqidd 2455 . . . . . 6  |-  ( ( W  =/=  (/)  /\  W  e. Word  V )  ->  ( # `
 W )  =  ( # `  W
) )
3230, 31jca 532 . . . . 5  |-  ( ( W  =/=  (/)  /\  W  e. Word  V )  ->  ( W  e. Word  V  /\  ( # `
 W )  =  ( # `  W
) ) )
3332biantrurd 508 . . . 4  |-  ( ( W  =/=  (/)  /\  W  e. Word  V )  ->  ( A. i  e.  (
0..^ ( # `  W
) ) ( W `
 i )  =  ( W `  0
)  <->  ( ( W  e. Word  V  /\  ( # `
 W )  =  ( # `  W
) )  /\  A. i  e.  ( 0..^ ( # `  W
) ) ( W `
 i )  =  ( W `  0
) ) ) )
3423, 29, 333bitr4d 285 . . 3  |-  ( ( W  =/=  (/)  /\  W  e. Word  V )  ->  ( W  =  ( ( W `  0 ) repeatS  (
# `  W )
)  <->  A. i  e.  ( 0..^ ( # `  W
) ) ( W `
 i )  =  ( W `  0
) ) )
3534ex 434 . 2  |-  ( W  =/=  (/)  ->  ( W  e. Word  V  ->  ( W  =  ( ( W `
 0 ) repeatS  ( # `
 W ) )  <->  A. i  e.  (
0..^ ( # `  W
) ) ( W `
 i )  =  ( W `  0
) ) ) )
3621, 35pm2.61ine 2764 1  |-  ( W  e. Word  V  ->  ( W  =  ( ( W `  0 ) repeatS  (
# `  W )
)  <->  A. i  e.  ( 0..^ ( # `  W
) ) ( W `
 i )  =  ( W `  0
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2647   A.wral 2798   _Vcvv 3076   (/)c0 3744   ` cfv 5525  (class class class)co 6199   0cc0 9392   NN0cn0 10689  ..^cfzo 11664   #chash 12219  Word cword 12338   repeatS creps 12345
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 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-om 6586  df-1st 6686  df-2nd 6687  df-recs 6941  df-rdg 6975  df-1o 7029  df-oadd 7033  df-er 7210  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-card 8219  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-nn 10433  df-n0 10690  df-z 10757  df-uz 10972  df-fz 11554  df-fzo 11665  df-hash 12220  df-word 12346  df-reps 12353
This theorem is referenced by:  cshw1repsw  12574  cshwsidrepsw  14237  cshwshashlem1  14239  cshwshash  14248
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