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Theorem ofcccat 28135
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 5772 . . . 4  |-  ( K  e.  T  ->  (
( 0..^ ( # `  F ) )  X. 
{ K } ) : ( 0..^ (
# `  F )
) --> T )
5 iswrdi 12512 . . . 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 5772 . . . 4  |-  ( K  e.  T  ->  (
( 0..^ ( # `  G ) )  X. 
{ K } ) : ( 0..^ (
# `  G )
) --> T )
8 iswrdi 12512 . . . 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 12047 . . . . 5  |-  ( 0..^ ( # `  F
) )  e.  Fin
11 snfi 7593 . . . . 5  |-  { K }  e.  Fin
12 hashxp 12452 . . . . 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 12521 . . . . . . . 8  |-  ( F  e. Word  S  ->  F  e.  Fin )
15 hashcl 12390 . . . . . . . 8  |-  ( F  e.  Fin  ->  ( # `
 F )  e. 
NN0 )
161, 14, 153syl 20 . . . . . . 7  |-  ( ph  ->  ( # `  F
)  e.  NN0 )
17 hashfzo0 12447 . . . . . . 7  |-  ( (
# `  F )  e.  NN0  ->  ( # `  (
0..^ ( # `  F
) ) )  =  ( # `  F
) )
1816, 17syl 16 . . . . . 6  |-  ( ph  ->  ( # `  (
0..^ ( # `  F
) ) )  =  ( # `  F
) )
19 hashsng 12400 . . . . . . 7  |-  ( K  e.  T  ->  ( # `
 { K }
)  =  1 )
203, 19syl 16 . . . . . 6  |-  ( ph  ->  ( # `  { K } )  =  1 )
2118, 20oveq12d 6300 . . . . 5  |-  ( ph  ->  ( ( # `  (
0..^ ( # `  F
) ) )  x.  ( # `  { K } ) )  =  ( ( # `  F
)  x.  1 ) )
2216nn0cnd 10850 . . . . . 6  |-  ( ph  ->  ( # `  F
)  e.  CC )
2322mulid1d 9609 . . . . 5  |-  ( ph  ->  ( ( # `  F
)  x.  1 )  =  ( # `  F
) )
2421, 23eqtrd 2508 . . . 4  |-  ( ph  ->  ( ( # `  (
0..^ ( # `  F
) ) )  x.  ( # `  { K } ) )  =  ( # `  F
) )
2513, 24syl5req 2521 . . 3  |-  ( ph  ->  ( # `  F
)  =  ( # `  ( ( 0..^ (
# `  F )
)  X.  { K } ) ) )
26 fzofi 12047 . . . . 5  |-  ( 0..^ ( # `  G
) )  e.  Fin
27 hashxp 12452 . . . . 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 12521 . . . . . . . 8  |-  ( G  e. Word  S  ->  G  e.  Fin )
30 hashcl 12390 . . . . . . . 8  |-  ( G  e.  Fin  ->  ( # `
 G )  e. 
NN0 )
312, 29, 303syl 20 . . . . . . 7  |-  ( ph  ->  ( # `  G
)  e.  NN0 )
32 hashfzo0 12447 . . . . . . 7  |-  ( (
# `  G )  e.  NN0  ->  ( # `  (
0..^ ( # `  G
) ) )  =  ( # `  G
) )
3331, 32syl 16 . . . . . 6  |-  ( ph  ->  ( # `  (
0..^ ( # `  G
) ) )  =  ( # `  G
) )
3433, 20oveq12d 6300 . . . . 5  |-  ( ph  ->  ( ( # `  (
0..^ ( # `  G
) ) )  x.  ( # `  { K } ) )  =  ( ( # `  G
)  x.  1 ) )
3531nn0cnd 10850 . . . . . 6  |-  ( ph  ->  ( # `  G
)  e.  CC )
3635mulid1d 9609 . . . . 5  |-  ( ph  ->  ( ( # `  G
)  x.  1 )  =  ( # `  G
) )
3734, 36eqtrd 2508 . . . 4  |-  ( ph  ->  ( ( # `  (
0..^ ( # `  G
) ) )  x.  ( # `  { K } ) )  =  ( # `  G
) )
3828, 37syl5req 2521 . . 3  |-  ( ph  ->  ( # `  G
)  =  ( # `  ( ( 0..^ (
# `  G )
)  X.  { K } ) ) )
391, 2, 6, 9, 25, 38ofccat 28134 . 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 12552 . . . . . 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 12513 . . . . 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 6307 . . . . 5  |-  ( 0..^ ( # `  ( F concat  G ) ) )  e.  _V
4544a1i 11 . . . 4  |-  ( ph  ->  ( 0..^ ( # `  ( F concat  G ) ) )  e.  _V )
4643, 45, 3ofcof 27743 . . 3  |-  ( ph  ->  ( ( F concat  G
)𝑓/𝑐 R K )  =  ( ( F concat  G )  oF R ( ( 0..^ ( # `  ( F concat  G ) ) )  X.  { K } ) ) )
47 ccatlen 12553 . . . . . . . 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 6298 . . . . . 6  |-  ( ph  ->  ( 0..^ ( # `  ( F concat  G ) ) )  =  ( 0..^ ( ( # `  F )  +  (
# `  G )
) ) )
5049xpeq1d 5022 . . . . 5  |-  ( ph  ->  ( ( 0..^ (
# `  ( F concat  G ) ) )  X. 
{ K } )  =  ( ( 0..^ ( ( # `  F
)  +  ( # `  G ) ) )  X.  { K }
) )
51 eqid 2467 . . . . . 6  |-  ( ( 0..^ ( # `  F
) )  X.  { K } )  =  ( ( 0..^ ( # `  F ) )  X. 
{ K } )
52 eqid 2467 . . . . . 6  |-  ( ( 0..^ ( # `  G
) )  X.  { K } )  =  ( ( 0..^ ( # `  G ) )  X. 
{ K } )
53 eqid 2467 . . . . . 6  |-  ( ( 0..^ ( ( # `  F )  +  (
# `  G )
) )  X.  { K } )  =  ( ( 0..^ ( (
# `  F )  +  ( # `  G
) ) )  X. 
{ K } )
5451, 52, 53, 3, 16, 31ccatmulgnn0dir 28133 . . . . 5  |-  ( ph  ->  ( ( ( 0..^ ( # `  F
) )  X.  { K } ) concat  ( ( 0..^ ( # `  G
) )  X.  { K } ) )  =  ( ( 0..^ ( ( # `  F
)  +  ( # `  G ) ) )  X.  { K }
) )
5550, 54eqtr4d 2511 . . . 4  |-  ( ph  ->  ( ( 0..^ (
# `  ( F concat  G ) ) )  X. 
{ K } )  =  ( ( ( 0..^ ( # `  F
) )  X.  { K } ) concat  ( ( 0..^ ( # `  G
) )  X.  { K } ) ) )
5655oveq2d 6298 . . 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 2508 . 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 12513 . . . . 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 27743 . . 3  |-  ( ph  ->  ( F𝑓/𝑐 R K )  =  ( F  oF R ( ( 0..^ (
# `  F )
)  X.  { K } ) ) )
62 wrdf 12513 . . . . 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 27743 . . 3  |-  ( ph  ->  ( G𝑓/𝑐 R K )  =  ( G  oF R ( ( 0..^ (
# `  G )
)  X.  { K } ) ) )
6661, 65oveq12d 6300 . 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 2518 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 1379    e. wcel 1767   _Vcvv 3113   {csn 4027    X. cxp 4997   -->wf 5582   ` cfv 5586  (class class class)co 6282    oFcof 6520   Fincfn 7513   0cc0 9488   1c1 9489    + caddc 9491    x. cmul 9493   NN0cn0 10791  ..^cfzo 11788   #chash 12367  Word cword 12494   concat cconcat 12496  ∘𝑓/𝑐cofc 27731
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 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
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-rmo 2822  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 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-card 8316  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-fzo 11789  df-hash 12368  df-word 12502  df-concat 12504  df-ofc 27732
This theorem is referenced by:  ofcs2  28139
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