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Theorem repswpfx 38689
Description: A prefix of a repeated symbol word is a repeated symbol word. (Contributed by AV, 11-May-2020.)
Assertion
Ref Expression
repswpfx  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  L  e.  ( 0 ... N
) )  ->  (
( S repeatS  N ) prefix  L )  =  ( S repeatS  L ) )

Proof of Theorem repswpfx
Dummy variable  i is distinct from all other variables.
StepHypRef Expression
1 repsw 12869 . . . . 5  |-  ( ( S  e.  V  /\  N  e.  NN0 )  -> 
( S repeatS  N )  e. Word  V )
213adant3 1025 . . . 4  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  L  e.  ( 0 ... N
) )  ->  ( S repeatS  N )  e. Word  V
)
3 repswlen 12870 . . . . . . . 8  |-  ( ( S  e.  V  /\  N  e.  NN0 )  -> 
( # `  ( S repeatS  N ) )  =  N )
43eqcomd 2430 . . . . . . 7  |-  ( ( S  e.  V  /\  N  e.  NN0 )  ->  N  =  ( # `  ( S repeatS  N ) ) )
54oveq2d 6318 . . . . . 6  |-  ( ( S  e.  V  /\  N  e.  NN0 )  -> 
( 0 ... N
)  =  ( 0 ... ( # `  ( S repeatS  N ) ) ) )
65eleq2d 2492 . . . . 5  |-  ( ( S  e.  V  /\  N  e.  NN0 )  -> 
( L  e.  ( 0 ... N )  <-> 
L  e.  ( 0 ... ( # `  ( S repeatS  N ) ) ) ) )
76biimp3a 1364 . . . 4  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  L  e.  ( 0 ... N
) )  ->  L  e.  ( 0 ... ( # `
 ( S repeatS  N
) ) ) )
8 pfxlen 38644 . . . 4  |-  ( ( ( S repeatS  N )  e. Word  V  /\  L  e.  ( 0 ... ( # `
 ( S repeatS  N
) ) ) )  ->  ( # `  (
( S repeatS  N ) prefix  L ) )  =  L )
92, 7, 8syl2anc 665 . . 3  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  L  e.  ( 0 ... N
) )  ->  ( # `
 ( ( S repeatS  N ) prefix  L ) )  =  L )
10 elfznn0 11888 . . . . . 6  |-  ( L  e.  ( 0 ... N )  ->  L  e.  NN0 )
1110anim2i 571 . . . . 5  |-  ( ( S  e.  V  /\  L  e.  ( 0 ... N ) )  ->  ( S  e.  V  /\  L  e. 
NN0 ) )
12113adant2 1024 . . . 4  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  L  e.  ( 0 ... N
) )  ->  ( S  e.  V  /\  L  e.  NN0 ) )
13 repswlen 12870 . . . 4  |-  ( ( S  e.  V  /\  L  e.  NN0 )  -> 
( # `  ( S repeatS  L ) )  =  L )
1412, 13syl 17 . . 3  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  L  e.  ( 0 ... N
) )  ->  ( # `
 ( S repeatS  L
) )  =  L )
159, 14eqtr4d 2466 . 2  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  L  e.  ( 0 ... N
) )  ->  ( # `
 ( ( S repeatS  N ) prefix  L ) )  =  ( # `  ( S repeatS  L ) ) )
16 simpl1 1008 . . . . 5  |-  ( ( ( S  e.  V  /\  N  e.  NN0  /\  L  e.  ( 0 ... N ) )  /\  i  e.  ( 0..^ ( # `  (
( S repeatS  N ) prefix  L ) ) ) )  ->  S  e.  V
)
17 simpl2 1009 . . . . 5  |-  ( ( ( S  e.  V  /\  N  e.  NN0  /\  L  e.  ( 0 ... N ) )  /\  i  e.  ( 0..^ ( # `  (
( S repeatS  N ) prefix  L ) ) ) )  ->  N  e.  NN0 )
18 elfzuz3 11798 . . . . . . . . 9  |-  ( L  e.  ( 0 ... N )  ->  N  e.  ( ZZ>= `  L )
)
19183ad2ant3 1028 . . . . . . . 8  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  L  e.  ( 0 ... N
) )  ->  N  e.  ( ZZ>= `  L )
)
209fveq2d 5882 . . . . . . . 8  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  L  e.  ( 0 ... N
) )  ->  ( ZZ>=
`  ( # `  (
( S repeatS  N ) prefix  L ) ) )  =  ( ZZ>= `  L )
)
2119, 20eleqtrrd 2513 . . . . . . 7  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  L  e.  ( 0 ... N
) )  ->  N  e.  ( ZZ>= `  ( # `  (
( S repeatS  N ) prefix  L ) ) ) )
22 fzoss2 11947 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  ( # `
 ( ( S repeatS  N ) prefix  L ) ) )  ->  ( 0..^ ( # `  (
( S repeatS  N ) prefix  L ) ) )  C_  ( 0..^ N ) )
2321, 22syl 17 . . . . . 6  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  L  e.  ( 0 ... N
) )  ->  (
0..^ ( # `  (
( S repeatS  N ) prefix  L ) ) )  C_  ( 0..^ N ) )
2423sselda 3464 . . . . 5  |-  ( ( ( S  e.  V  /\  N  e.  NN0  /\  L  e.  ( 0 ... N ) )  /\  i  e.  ( 0..^ ( # `  (
( S repeatS  N ) prefix  L ) ) ) )  ->  i  e.  ( 0..^ N ) )
25 repswsymb 12868 . . . . 5  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  i  e.  ( 0..^ N ) )  ->  ( ( S repeatS  N ) `  i
)  =  S )
2616, 17, 24, 25syl3anc 1264 . . . 4  |-  ( ( ( S  e.  V  /\  N  e.  NN0  /\  L  e.  ( 0 ... N ) )  /\  i  e.  ( 0..^ ( # `  (
( S repeatS  N ) prefix  L ) ) ) )  ->  ( ( S repeatS  N ) `  i
)  =  S )
272adantr 466 . . . . 5  |-  ( ( ( S  e.  V  /\  N  e.  NN0  /\  L  e.  ( 0 ... N ) )  /\  i  e.  ( 0..^ ( # `  (
( S repeatS  N ) prefix  L ) ) ) )  ->  ( S repeatS  N
)  e. Word  V )
287adantr 466 . . . . 5  |-  ( ( ( S  e.  V  /\  N  e.  NN0  /\  L  e.  ( 0 ... N ) )  /\  i  e.  ( 0..^ ( # `  (
( S repeatS  N ) prefix  L ) ) ) )  ->  L  e.  ( 0 ... ( # `  ( S repeatS  N )
) ) )
299oveq2d 6318 . . . . . . 7  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  L  e.  ( 0 ... N
) )  ->  (
0..^ ( # `  (
( S repeatS  N ) prefix  L ) ) )  =  ( 0..^ L ) )
3029eleq2d 2492 . . . . . 6  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  L  e.  ( 0 ... N
) )  ->  (
i  e.  ( 0..^ ( # `  (
( S repeatS  N ) prefix  L ) ) )  <->  i  e.  ( 0..^ L ) ) )
3130biimpa 486 . . . . 5  |-  ( ( ( S  e.  V  /\  N  e.  NN0  /\  L  e.  ( 0 ... N ) )  /\  i  e.  ( 0..^ ( # `  (
( S repeatS  N ) prefix  L ) ) ) )  ->  i  e.  ( 0..^ L ) )
32 pfxfv 38652 . . . . 5  |-  ( ( ( S repeatS  N )  e. Word  V  /\  L  e.  ( 0 ... ( # `
 ( S repeatS  N
) ) )  /\  i  e.  ( 0..^ L ) )  -> 
( ( ( S repeatS  N ) prefix  L ) `  i )  =  ( ( S repeatS  N ) `  i ) )
3327, 28, 31, 32syl3anc 1264 . . . 4  |-  ( ( ( S  e.  V  /\  N  e.  NN0  /\  L  e.  ( 0 ... N ) )  /\  i  e.  ( 0..^ ( # `  (
( S repeatS  N ) prefix  L ) ) ) )  ->  ( ( ( S repeatS  N ) prefix  L ) `
 i )  =  ( ( S repeatS  N
) `  i )
)
34103ad2ant3 1028 . . . . . 6  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  L  e.  ( 0 ... N
) )  ->  L  e.  NN0 )
3534adantr 466 . . . . 5  |-  ( ( ( S  e.  V  /\  N  e.  NN0  /\  L  e.  ( 0 ... N ) )  /\  i  e.  ( 0..^ ( # `  (
( S repeatS  N ) prefix  L ) ) ) )  ->  L  e.  NN0 )
36 repswsymb 12868 . . . . 5  |-  ( ( S  e.  V  /\  L  e.  NN0  /\  i  e.  ( 0..^ L ) )  ->  ( ( S repeatS  L ) `  i
)  =  S )
3716, 35, 31, 36syl3anc 1264 . . . 4  |-  ( ( ( S  e.  V  /\  N  e.  NN0  /\  L  e.  ( 0 ... N ) )  /\  i  e.  ( 0..^ ( # `  (
( S repeatS  N ) prefix  L ) ) ) )  ->  ( ( S repeatS  L ) `  i
)  =  S )
3826, 33, 373eqtr4d 2473 . . 3  |-  ( ( ( S  e.  V  /\  N  e.  NN0  /\  L  e.  ( 0 ... N ) )  /\  i  e.  ( 0..^ ( # `  (
( S repeatS  N ) prefix  L ) ) ) )  ->  ( ( ( S repeatS  N ) prefix  L ) `
 i )  =  ( ( S repeatS  L
) `  i )
)
3938ralrimiva 2839 . 2  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  L  e.  ( 0 ... N
) )  ->  A. i  e.  ( 0..^ ( # `  ( ( S repeatS  N
) prefix  L ) ) ) ( ( ( S repeatS  N ) prefix  L ) `  i )  =  ( ( S repeatS  L ) `  i ) )
40 pfxcl 38639 . . . 4  |-  ( ( S repeatS  N )  e. Word  V  ->  ( ( S repeatS  N
) prefix  L )  e. Word  V
)
412, 40syl 17 . . 3  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  L  e.  ( 0 ... N
) )  ->  (
( S repeatS  N ) prefix  L )  e. Word  V )
42 repsw 12869 . . . 4  |-  ( ( S  e.  V  /\  L  e.  NN0 )  -> 
( S repeatS  L )  e. Word  V )
4312, 42syl 17 . . 3  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  L  e.  ( 0 ... N
) )  ->  ( S repeatS  L )  e. Word  V
)
44 eqwrd 12701 . . 3  |-  ( ( ( ( S repeatS  N
) prefix  L )  e. Word  V  /\  ( S repeatS  L )  e. Word  V )  ->  (
( ( S repeatS  N
) prefix  L )  =  ( S repeatS  L )  <->  ( ( # `
 ( ( S repeatS  N ) prefix  L ) )  =  ( # `  ( S repeatS  L ) )  /\  A. i  e.  ( 0..^ ( # `  (
( S repeatS  N ) prefix  L ) ) ) ( ( ( S repeatS  N
) prefix  L ) `  i
)  =  ( ( S repeatS  L ) `  i
) ) ) )
4541, 43, 44syl2anc 665 . 2  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  L  e.  ( 0 ... N
) )  ->  (
( ( S repeatS  N
) prefix  L )  =  ( S repeatS  L )  <->  ( ( # `
 ( ( S repeatS  N ) prefix  L ) )  =  ( # `  ( S repeatS  L ) )  /\  A. i  e.  ( 0..^ ( # `  (
( S repeatS  N ) prefix  L ) ) ) ( ( ( S repeatS  N
) prefix  L ) `  i
)  =  ( ( S repeatS  L ) `  i
) ) ) )
4615, 39, 45mpbir2and 930 1  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  L  e.  ( 0 ... N
) )  ->  (
( S repeatS  N ) prefix  L )  =  ( S repeatS  L ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868   A.wral 2775    C_ wss 3436   ` cfv 5598  (class class class)co 6302   0cc0 9540   NN0cn0 10870   ZZ>=cuz 11160   ...cfz 11785  ..^cfzo 11916   #chash 12515  Word cword 12649   repeatS creps 12656   prefix cpfx 38634
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 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-cnex 9596  ax-resscn 9597  ax-1cn 9598  ax-icn 9599  ax-addcl 9600  ax-addrcl 9601  ax-mulcl 9602  ax-mulrcl 9603  ax-mulcom 9604  ax-addass 9605  ax-mulass 9606  ax-distr 9607  ax-i2m1 9608  ax-1ne0 9609  ax-1rid 9610  ax-rnegex 9611  ax-rrecex 9612  ax-cnre 9613  ax-pre-lttri 9614  ax-pre-lttrn 9615  ax-pre-ltadd 9616  ax-pre-mulgt0 9617
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 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-riota 6264  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-om 6704  df-1st 6804  df-2nd 6805  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-card 8375  df-cda 8599  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-sub 9863  df-neg 9864  df-nn 10611  df-2 10669  df-n0 10871  df-z 10939  df-uz 11161  df-fz 11786  df-fzo 11917  df-hash 12516  df-word 12657  df-substr 12661  df-reps 12664  df-pfx 38635
This theorem is referenced by: (None)
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