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Theorem repswsymballbi 12715
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 5866 . . . . 5  |-  ( W  =  (/)  ->  ( # `  W )  =  (
# `  (/) ) )
2 hash0 12405 . . . . 5  |-  ( # `  (/) )  =  0
31, 2syl6eq 2524 . . . 4  |-  ( W  =  (/)  ->  ( # `  W )  =  0 )
4 fvex 5876 . . . . . . . 8  |-  ( W `
 0 )  e. 
_V
5 repsw0 12712 . . . . . . . 8  |-  ( ( W `  0 )  e.  _V  ->  (
( W `  0
) repeatS  0 )  =  (/) )
64, 5ax-mp 5 . . . . . . 7  |-  ( ( W `  0 ) repeatS 
0 )  =  (/)
76eqcomi 2480 . . . . . 6  |-  (/)  =  ( ( W `  0
) repeatS  0 )
8 simpr 461 . . . . . 6  |-  ( ( ( # `  W
)  =  0  /\  W  =  (/) )  ->  W  =  (/) )
9 oveq2 6292 . . . . . . 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 2534 . . . . 5  |-  ( ( ( # `  W
)  =  0  /\  W  =  (/) )  ->  W  =  ( ( W `  0 ) repeatS  (
# `  W )
) )
12 ral0 3932 . . . . . . 7  |-  A. i  e.  (/)  ( W `  i )  =  ( W `  0 )
13 oveq2 6292 . . . . . . . . 9  |-  ( (
# `  W )  =  0  ->  (
0..^ ( # `  W
) )  =  ( 0..^ 0 ) )
14 fzo0 11817 . . . . . . . . 9  |-  ( 0..^ 0 )  =  (/)
1513, 14syl6eq 2524 . . . . . . . 8  |-  ( (
# `  W )  =  0  ->  (
0..^ ( # `  W
) )  =  (/) )
1615raleqdv 3064 . . . . . . 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 975 . . . . 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 12545 . . . . . 6  |-  ( ( W  e. Word  V  /\  W  =/=  (/) )  ->  ( W `  0 )  e.  V )
2524ancoms 453 . . . . 5  |-  ( ( W  =/=  (/)  /\  W  e. Word  V )  ->  ( W `  0 )  e.  V )
26 lencl 12528 . . . . . 6  |-  ( W  e. Word  V  ->  ( # `
 W )  e. 
NN0 )
2726adantl 466 . . . . 5  |-  ( ( W  =/=  (/)  /\  W  e. Word  V )  ->  ( # `
 W )  e. 
NN0 )
28 repsdf2 12713 . . . . 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 2468 . . . . . 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 2780 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   _Vcvv 3113   (/)c0 3785   ` cfv 5588  (class class class)co 6284   0cc0 9492   NN0cn0 10795  ..^cfzo 11792   #chash 12373  Word cword 12500   repeatS creps 12507
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-reps 12515
This theorem is referenced by:  cshw1repsw  12754  cshwsidrepsw  14436  cshwshashlem1  14438  cshwshash  14447
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