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Theorem repswsymballbi 12868
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 5881 . . . . 5  |-  ( W  =  (/)  ->  ( # `  W )  =  (
# `  (/) ) )
2 hash0 12545 . . . . 5  |-  ( # `  (/) )  =  0
31, 2syl6eq 2486 . . . 4  |-  ( W  =  (/)  ->  ( # `  W )  =  0 )
4 fvex 5891 . . . . . . . 8  |-  ( W `
 0 )  e. 
_V
5 repsw0 12865 . . . . . . . 8  |-  ( ( W `  0 )  e.  _V  ->  (
( W `  0
) repeatS  0 )  =  (/) )
64, 5ax-mp 5 . . . . . . 7  |-  ( ( W `  0 ) repeatS 
0 )  =  (/)
76eqcomi 2442 . . . . . 6  |-  (/)  =  ( ( W `  0
) repeatS  0 )
8 simpr 462 . . . . . 6  |-  ( ( ( # `  W
)  =  0  /\  W  =  (/) )  ->  W  =  (/) )
9 oveq2 6313 . . . . . . 7  |-  ( (
# `  W )  =  0  ->  (
( W `  0
) repeatS  ( # `  W
) )  =  ( ( W `  0
) repeatS  0 ) )
109adantr 466 . . . . . 6  |-  ( ( ( # `  W
)  =  0  /\  W  =  (/) )  -> 
( ( W ` 
0 ) repeatS  ( # `  W
) )  =  ( ( W `  0
) repeatS  0 ) )
117, 8, 103eqtr4a 2496 . . . . 5  |-  ( ( ( # `  W
)  =  0  /\  W  =  (/) )  ->  W  =  ( ( W `  0 ) repeatS  (
# `  W )
) )
12 ral0 3908 . . . . . . 7  |-  A. i  e.  (/)  ( W `  i )  =  ( W `  0 )
13 oveq2 6313 . . . . . . . . 9  |-  ( (
# `  W )  =  0  ->  (
0..^ ( # `  W
) )  =  ( 0..^ 0 ) )
14 fzo0 11940 . . . . . . . . 9  |-  ( 0..^ 0 )  =  (/)
1513, 14syl6eq 2486 . . . . . . . 8  |-  ( (
# `  W )  =  0  ->  (
0..^ ( # `  W
) )  =  (/) )
1615raleqdv 3038 . . . . . . 7  |-  ( (
# `  W )  =  0  ->  ( A. i  e.  (
0..^ ( # `  W
) ) ( W `
 i )  =  ( W `  0
)  <->  A. i  e.  (/)  ( W `  i )  =  ( W ` 
0 ) ) )
1712, 16mpbiri 236 . . . . . 6  |-  ( (
# `  W )  =  0  ->  A. i  e.  ( 0..^ ( # `  W ) ) ( W `  i )  =  ( W ` 
0 ) )
1817adantr 466 . . . . 5  |-  ( ( ( # `  W
)  =  0  /\  W  =  (/) )  ->  A. i  e.  (
0..^ ( # `  W
) ) ( W `
 i )  =  ( W `  0
) )
1911, 182thd 243 . . . 4  |-  ( ( ( # `  W
)  =  0  /\  W  =  (/) )  -> 
( W  =  ( ( W `  0
) repeatS  ( # `  W
) )  <->  A. i  e.  ( 0..^ ( # `  W ) ) ( W `  i )  =  ( W ` 
0 ) ) )
203, 19mpancom 673 . . 3  |-  ( W  =  (/)  ->  ( W  =  ( ( W `
 0 ) repeatS  ( # `
 W ) )  <->  A. i  e.  (
0..^ ( # `  W
) ) ( W `
 i )  =  ( W `  0
) ) )
2120a1d 26 . 2  |-  ( W  =  (/)  ->  ( W  e. Word  V  ->  ( W  =  ( ( W `  0 ) repeatS  (
# `  W )
)  <->  A. i  e.  ( 0..^ ( # `  W
) ) ( W `
 i )  =  ( W `  0
) ) ) )
22 df-3an 984 . . . . 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 12693 . . . . . 6  |-  ( ( W  e. Word  V  /\  W  =/=  (/) )  ->  ( W `  0 )  e.  V )
2524ancoms 454 . . . . 5  |-  ( ( W  =/=  (/)  /\  W  e. Word  V )  ->  ( W `  0 )  e.  V )
26 lencl 12674 . . . . . 6  |-  ( W  e. Word  V  ->  ( # `
 W )  e. 
NN0 )
2726adantl 467 . . . . 5  |-  ( ( W  =/=  (/)  /\  W  e. Word  V )  ->  ( # `
 W )  e. 
NN0 )
28 repsdf2 12866 . . . . 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 665 . . . 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 462 . . . . . 6  |-  ( ( W  =/=  (/)  /\  W  e. Word  V )  ->  W  e. Word  V )
31 eqidd 2430 . . . . . 6  |-  ( ( W  =/=  (/)  /\  W  e. Word  V )  ->  ( # `
 W )  =  ( # `  W
) )
3230, 31jca 534 . . . . 5  |-  ( ( W  =/=  (/)  /\  W  e. Word  V )  ->  ( W  e. Word  V  /\  ( # `
 W )  =  ( # `  W
) ) )
3332biantrurd 510 . . . 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 288 . . 3  |-  ( ( W  =/=  (/)  /\  W  e. Word  V )  ->  ( W  =  ( ( W `  0 ) repeatS  (
# `  W )
)  <->  A. i  e.  ( 0..^ ( # `  W
) ) ( W `
 i )  =  ( W `  0
) ) )
3534ex 435 . 2  |-  ( W  =/=  (/)  ->  ( W  e. Word  V  ->  ( W  =  ( ( W `
 0 ) repeatS  ( # `
 W ) )  <->  A. i  e.  (
0..^ ( # `  W
) ) ( W `
 i )  =  ( W `  0
) ) ) )
3621, 35pm2.61ine 2744 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   A.wral 2782   _Vcvv 3087   (/)c0 3767   ` cfv 5601  (class class class)co 6305   0cc0 9538   NN0cn0 10869  ..^cfzo 11913   #chash 12512  Word cword 12643   repeatS creps 12650
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11783  df-fzo 11914  df-hash 12513  df-word 12651  df-reps 12658
This theorem is referenced by:  cshw1repsw  12907  cshwsidrepsw  15027  cshwshashlem1  15029  cshwshash  15038
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