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Theorem ofcccat 27106
Description: Letterwise operations on word concatenations. (Contributed by Thierry Arnoux, 5-Oct-2018.)
Hypotheses
Ref Expression
ofcccat.1  |-  ( ph  ->  F  e. Word  S )
ofcccat.2  |-  ( ph  ->  G  e. Word  S )
ofcccat.3  |-  ( ph  ->  K  e.  T )
Assertion
Ref Expression
ofcccat  |-  ( ph  ->  ( ( F concat  G
)𝑓/𝑐 R K )  =  ( ( F𝑓/𝑐 R K ) concat  ( G𝑓/𝑐 R K ) ) )

Proof of Theorem ofcccat
StepHypRef Expression
1 ofcccat.1 . . 3  |-  ( ph  ->  F  e. Word  S )
2 ofcccat.2 . . 3  |-  ( ph  ->  G  e. Word  S )
3 ofcccat.3 . . . 4  |-  ( ph  ->  K  e.  T )
4 fconst6g 5710 . . . 4  |-  ( K  e.  T  ->  (
( 0..^ ( # `  F ) )  X. 
{ K } ) : ( 0..^ (
# `  F )
) --> T )
5 iswrdi 12360 . . . 4  |-  ( ( ( 0..^ ( # `  F ) )  X. 
{ K } ) : ( 0..^ (
# `  F )
) --> T  ->  (
( 0..^ ( # `  F ) )  X. 
{ K } )  e. Word  T )
63, 4, 53syl 20 . . 3  |-  ( ph  ->  ( ( 0..^ (
# `  F )
)  X.  { K } )  e. Word  T
)
7 fconst6g 5710 . . . 4  |-  ( K  e.  T  ->  (
( 0..^ ( # `  G ) )  X. 
{ K } ) : ( 0..^ (
# `  G )
) --> T )
8 iswrdi 12360 . . . 4  |-  ( ( ( 0..^ ( # `  G ) )  X. 
{ K } ) : ( 0..^ (
# `  G )
) --> T  ->  (
( 0..^ ( # `  G ) )  X. 
{ K } )  e. Word  T )
93, 7, 83syl 20 . . 3  |-  ( ph  ->  ( ( 0..^ (
# `  G )
)  X.  { K } )  e. Word  T
)
10 fzofi 11916 . . . . 5  |-  ( 0..^ ( # `  F
) )  e.  Fin
11 snfi 7503 . . . . 5  |-  { K }  e.  Fin
12 hashxp 12317 . . . . 5  |-  ( ( ( 0..^ ( # `  F ) )  e. 
Fin  /\  { K }  e.  Fin )  ->  ( # `  (
( 0..^ ( # `  F ) )  X. 
{ K } ) )  =  ( (
# `  ( 0..^ ( # `  F ) ) )  x.  ( # `
 { K }
) ) )
1310, 11, 12mp2an 672 . . . 4  |-  ( # `  ( ( 0..^ (
# `  F )
)  X.  { K } ) )  =  ( ( # `  (
0..^ ( # `  F
) ) )  x.  ( # `  { K } ) )
14 wrdfin 12369 . . . . . . . 8  |-  ( F  e. Word  S  ->  F  e.  Fin )
15 hashcl 12246 . . . . . . . 8  |-  ( F  e.  Fin  ->  ( # `
 F )  e. 
NN0 )
161, 14, 153syl 20 . . . . . . 7  |-  ( ph  ->  ( # `  F
)  e.  NN0 )
17 hashfzo0 12312 . . . . . . 7  |-  ( (
# `  F )  e.  NN0  ->  ( # `  (
0..^ ( # `  F
) ) )  =  ( # `  F
) )
1816, 17syl 16 . . . . . 6  |-  ( ph  ->  ( # `  (
0..^ ( # `  F
) ) )  =  ( # `  F
) )
19 hashsng 12256 . . . . . . 7  |-  ( K  e.  T  ->  ( # `
 { K }
)  =  1 )
203, 19syl 16 . . . . . 6  |-  ( ph  ->  ( # `  { K } )  =  1 )
2118, 20oveq12d 6221 . . . . 5  |-  ( ph  ->  ( ( # `  (
0..^ ( # `  F
) ) )  x.  ( # `  { K } ) )  =  ( ( # `  F
)  x.  1 ) )
2216nn0cnd 10752 . . . . . 6  |-  ( ph  ->  ( # `  F
)  e.  CC )
2322mulid1d 9517 . . . . 5  |-  ( ph  ->  ( ( # `  F
)  x.  1 )  =  ( # `  F
) )
2421, 23eqtrd 2495 . . . 4  |-  ( ph  ->  ( ( # `  (
0..^ ( # `  F
) ) )  x.  ( # `  { K } ) )  =  ( # `  F
) )
2513, 24syl5req 2508 . . 3  |-  ( ph  ->  ( # `  F
)  =  ( # `  ( ( 0..^ (
# `  F )
)  X.  { K } ) ) )
26 fzofi 11916 . . . . 5  |-  ( 0..^ ( # `  G
) )  e.  Fin
27 hashxp 12317 . . . . 5  |-  ( ( ( 0..^ ( # `  G ) )  e. 
Fin  /\  { K }  e.  Fin )  ->  ( # `  (
( 0..^ ( # `  G ) )  X. 
{ K } ) )  =  ( (
# `  ( 0..^ ( # `  G ) ) )  x.  ( # `
 { K }
) ) )
2826, 11, 27mp2an 672 . . . 4  |-  ( # `  ( ( 0..^ (
# `  G )
)  X.  { K } ) )  =  ( ( # `  (
0..^ ( # `  G
) ) )  x.  ( # `  { K } ) )
29 wrdfin 12369 . . . . . . . 8  |-  ( G  e. Word  S  ->  G  e.  Fin )
30 hashcl 12246 . . . . . . . 8  |-  ( G  e.  Fin  ->  ( # `
 G )  e. 
NN0 )
312, 29, 303syl 20 . . . . . . 7  |-  ( ph  ->  ( # `  G
)  e.  NN0 )
32 hashfzo0 12312 . . . . . . 7  |-  ( (
# `  G )  e.  NN0  ->  ( # `  (
0..^ ( # `  G
) ) )  =  ( # `  G
) )
3331, 32syl 16 . . . . . 6  |-  ( ph  ->  ( # `  (
0..^ ( # `  G
) ) )  =  ( # `  G
) )
3433, 20oveq12d 6221 . . . . 5  |-  ( ph  ->  ( ( # `  (
0..^ ( # `  G
) ) )  x.  ( # `  { K } ) )  =  ( ( # `  G
)  x.  1 ) )
3531nn0cnd 10752 . . . . . 6  |-  ( ph  ->  ( # `  G
)  e.  CC )
3635mulid1d 9517 . . . . 5  |-  ( ph  ->  ( ( # `  G
)  x.  1 )  =  ( # `  G
) )
3734, 36eqtrd 2495 . . . 4  |-  ( ph  ->  ( ( # `  (
0..^ ( # `  G
) ) )  x.  ( # `  { K } ) )  =  ( # `  G
) )
3828, 37syl5req 2508 . . 3  |-  ( ph  ->  ( # `  G
)  =  ( # `  ( ( 0..^ (
# `  G )
)  X.  { K } ) ) )
391, 2, 6, 9, 25, 38ofccat 27105 . 2  |-  ( ph  ->  ( ( F concat  G
)  oF R ( ( ( 0..^ ( # `  F
) )  X.  { K } ) concat  ( ( 0..^ ( # `  G
) )  X.  { K } ) ) )  =  ( ( F  oF R ( ( 0..^ ( # `  F ) )  X. 
{ K } ) ) concat  ( G  oF R ( ( 0..^ ( # `  G
) )  X.  { K } ) ) ) )
40 ccatcl 12395 . . . . . 6  |-  ( ( F  e. Word  S  /\  G  e. Word  S )  ->  ( F concat  G )  e. Word  S )
411, 2, 40syl2anc 661 . . . . 5  |-  ( ph  ->  ( F concat  G )  e. Word  S )
42 wrdf 12361 . . . . 5  |-  ( ( F concat  G )  e. Word  S  ->  ( F concat  G
) : ( 0..^ ( # `  ( F concat  G ) ) ) --> S )
4341, 42syl 16 . . . 4  |-  ( ph  ->  ( F concat  G ) : ( 0..^ (
# `  ( F concat  G ) ) ) --> S )
44 ovex 6228 . . . . 5  |-  ( 0..^ ( # `  ( F concat  G ) ) )  e.  _V
4544a1i 11 . . . 4  |-  ( ph  ->  ( 0..^ ( # `  ( F concat  G ) ) )  e.  _V )
4643, 45, 3ofcof 26714 . . 3  |-  ( ph  ->  ( ( F concat  G
)𝑓/𝑐 R K )  =  ( ( F concat  G )  oF R ( ( 0..^ ( # `  ( F concat  G ) ) )  X.  { K } ) ) )
47 ccatlen 12396 . . . . . . . 8  |-  ( ( F  e. Word  S  /\  G  e. Word  S )  ->  ( # `  ( F concat  G ) )  =  ( ( # `  F
)  +  ( # `  G ) ) )
481, 2, 47syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( # `  ( F concat  G ) )  =  ( ( # `  F
)  +  ( # `  G ) ) )
4948oveq2d 6219 . . . . . 6  |-  ( ph  ->  ( 0..^ ( # `  ( F concat  G ) ) )  =  ( 0..^ ( ( # `  F )  +  (
# `  G )
) ) )
5049xpeq1d 4974 . . . . 5  |-  ( ph  ->  ( ( 0..^ (
# `  ( F concat  G ) ) )  X. 
{ K } )  =  ( ( 0..^ ( ( # `  F
)  +  ( # `  G ) ) )  X.  { K }
) )
51 eqid 2454 . . . . . 6  |-  ( ( 0..^ ( # `  F
) )  X.  { K } )  =  ( ( 0..^ ( # `  F ) )  X. 
{ K } )
52 eqid 2454 . . . . . 6  |-  ( ( 0..^ ( # `  G
) )  X.  { K } )  =  ( ( 0..^ ( # `  G ) )  X. 
{ K } )
53 eqid 2454 . . . . . 6  |-  ( ( 0..^ ( ( # `  F )  +  (
# `  G )
) )  X.  { K } )  =  ( ( 0..^ ( (
# `  F )  +  ( # `  G
) ) )  X. 
{ K } )
5451, 52, 53, 3, 16, 31ccatmulgnn0dir 27104 . . . . 5  |-  ( ph  ->  ( ( ( 0..^ ( # `  F
) )  X.  { K } ) concat  ( ( 0..^ ( # `  G
) )  X.  { K } ) )  =  ( ( 0..^ ( ( # `  F
)  +  ( # `  G ) ) )  X.  { K }
) )
5550, 54eqtr4d 2498 . . . 4  |-  ( ph  ->  ( ( 0..^ (
# `  ( F concat  G ) ) )  X. 
{ K } )  =  ( ( ( 0..^ ( # `  F
) )  X.  { K } ) concat  ( ( 0..^ ( # `  G
) )  X.  { K } ) ) )
5655oveq2d 6219 . . 3  |-  ( ph  ->  ( ( F concat  G
)  oF R ( ( 0..^ (
# `  ( F concat  G ) ) )  X. 
{ K } ) )  =  ( ( F concat  G )  oF R ( ( ( 0..^ ( # `  F ) )  X. 
{ K } ) concat 
( ( 0..^ (
# `  G )
)  X.  { K } ) ) ) )
5746, 56eqtrd 2495 . 2  |-  ( ph  ->  ( ( F concat  G
)𝑓/𝑐 R K )  =  ( ( F concat  G )  oF R ( ( ( 0..^ (
# `  F )
)  X.  { K } ) concat  ( (
0..^ ( # `  G
) )  X.  { K } ) ) ) )
58 wrdf 12361 . . . . 5  |-  ( F  e. Word  S  ->  F : ( 0..^ (
# `  F )
) --> S )
591, 58syl 16 . . . 4  |-  ( ph  ->  F : ( 0..^ ( # `  F
) ) --> S )
6010a1i 11 . . . 4  |-  ( ph  ->  ( 0..^ ( # `  F ) )  e. 
Fin )
6159, 60, 3ofcof 26714 . . 3  |-  ( ph  ->  ( F𝑓/𝑐 R K )  =  ( F  oF R ( ( 0..^ (
# `  F )
)  X.  { K } ) ) )
62 wrdf 12361 . . . . 5  |-  ( G  e. Word  S  ->  G : ( 0..^ (
# `  G )
) --> S )
632, 62syl 16 . . . 4  |-  ( ph  ->  G : ( 0..^ ( # `  G
) ) --> S )
6426a1i 11 . . . 4  |-  ( ph  ->  ( 0..^ ( # `  G ) )  e. 
Fin )
6563, 64, 3ofcof 26714 . . 3  |-  ( ph  ->  ( G𝑓/𝑐 R K )  =  ( G  oF R ( ( 0..^ (
# `  G )
)  X.  { K } ) ) )
6661, 65oveq12d 6221 . 2  |-  ( ph  ->  ( ( F𝑓/𝑐 R K ) concat  ( G𝑓/𝑐 R K ) )  =  ( ( F  oF R ( ( 0..^ ( # `  F
) )  X.  { K } ) ) concat  ( G  oF R ( ( 0..^ ( # `  G ) )  X. 
{ K } ) ) ) )
6739, 57, 663eqtr4d 2505 1  |-  ( ph  ->  ( ( F concat  G
)𝑓/𝑐 R K )  =  ( ( F𝑓/𝑐 R K ) concat  ( G𝑓/𝑐 R K ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   _Vcvv 3078   {csn 3988    X. cxp 4949   -->wf 5525   ` cfv 5529  (class class class)co 6203    oFcof 6431   Fincfn 7423   0cc0 9396   1c1 9397    + caddc 9399    x. cmul 9401   NN0cn0 10693  ..^cfzo 11668   #chash 12223  Word cword 12342   concat cconcat 12344  ∘𝑓/𝑐cofc 26702
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 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473
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 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-of 6433  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-card 8223  df-cda 8451  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-nn 10437  df-n0 10694  df-z 10761  df-uz 10976  df-fz 11558  df-fzo 11669  df-hash 12224  df-word 12350  df-concat 12352  df-ofc 26703
This theorem is referenced by:  ofcs2  27110
  Copyright terms: Public domain W3C validator