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Theorem repsco 12910
Description: Mapping of words commutes with the "repeated symbol" operation. (Contributed by AV, 11-Nov-2018.)
Assertion
Ref Expression
repsco  |-  ( ( S  e.  A  /\  N  e.  NN0  /\  F : A --> B )  -> 
( F  o.  ( S repeatS  N ) )  =  ( ( F `  S ) repeatS  N ) )

Proof of Theorem repsco
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simpl1 1008 . . . . 5  |-  ( ( ( S  e.  A  /\  N  e.  NN0  /\  F : A --> B )  /\  x  e.  ( 0..^ N ) )  ->  S  e.  A
)
2 simpl2 1009 . . . . 5  |-  ( ( ( S  e.  A  /\  N  e.  NN0  /\  F : A --> B )  /\  x  e.  ( 0..^ N ) )  ->  N  e.  NN0 )
3 simpr 462 . . . . 5  |-  ( ( ( S  e.  A  /\  N  e.  NN0  /\  F : A --> B )  /\  x  e.  ( 0..^ N ) )  ->  x  e.  ( 0..^ N ) )
4 repswsymb 12851 . . . . 5  |-  ( ( S  e.  A  /\  N  e.  NN0  /\  x  e.  ( 0..^ N ) )  ->  ( ( S repeatS  N ) `  x
)  =  S )
51, 2, 3, 4syl3anc 1264 . . . 4  |-  ( ( ( S  e.  A  /\  N  e.  NN0  /\  F : A --> B )  /\  x  e.  ( 0..^ N ) )  ->  ( ( S repeatS  N ) `  x
)  =  S )
65fveq2d 5876 . . 3  |-  ( ( ( S  e.  A  /\  N  e.  NN0  /\  F : A --> B )  /\  x  e.  ( 0..^ N ) )  ->  ( F `  ( ( S repeatS  N
) `  x )
)  =  ( F `
 S ) )
76mpteq2dva 4503 . 2  |-  ( ( S  e.  A  /\  N  e.  NN0  /\  F : A --> B )  -> 
( x  e.  ( 0..^ N )  |->  ( F `  ( ( S repeatS  N ) `  x
) ) )  =  ( x  e.  ( 0..^ N )  |->  ( F `  S ) ) )
8 simp3 1007 . . 3  |-  ( ( S  e.  A  /\  N  e.  NN0  /\  F : A --> B )  ->  F : A --> B )
9 repsf 12850 . . . 4  |-  ( ( S  e.  A  /\  N  e.  NN0 )  -> 
( S repeatS  N ) : ( 0..^ N ) --> A )
1093adant3 1025 . . 3  |-  ( ( S  e.  A  /\  N  e.  NN0  /\  F : A --> B )  -> 
( S repeatS  N ) : ( 0..^ N ) --> A )
11 fcompt 6065 . . 3  |-  ( ( F : A --> B  /\  ( S repeatS  N ) : ( 0..^ N ) --> A )  ->  ( F  o.  ( S repeatS  N ) )  =  ( x  e.  ( 0..^ N )  |->  ( F `
 ( ( S repeatS  N ) `  x
) ) ) )
128, 10, 11syl2anc 665 . 2  |-  ( ( S  e.  A  /\  N  e.  NN0  /\  F : A --> B )  -> 
( F  o.  ( S repeatS  N ) )  =  ( x  e.  ( 0..^ N )  |->  ( F `  ( ( S repeatS  N ) `  x
) ) ) )
13 fvex 5882 . . . . . 6  |-  ( F `
 S )  e. 
_V
1413a1i 11 . . . . 5  |-  ( S  e.  A  ->  ( F `  S )  e.  _V )
1514anim1i 570 . . . 4  |-  ( ( S  e.  A  /\  N  e.  NN0 )  -> 
( ( F `  S )  e.  _V  /\  N  e.  NN0 )
)
16153adant3 1025 . . 3  |-  ( ( S  e.  A  /\  N  e.  NN0  /\  F : A --> B )  -> 
( ( F `  S )  e.  _V  /\  N  e.  NN0 )
)
17 reps 12847 . . 3  |-  ( ( ( F `  S
)  e.  _V  /\  N  e.  NN0 )  -> 
( ( F `  S ) repeatS  N )  =  ( x  e.  ( 0..^ N )  |->  ( F `  S ) ) )
1816, 17syl 17 . 2  |-  ( ( S  e.  A  /\  N  e.  NN0  /\  F : A --> B )  -> 
( ( F `  S ) repeatS  N )  =  ( x  e.  ( 0..^ N )  |->  ( F `  S ) ) )
197, 12, 183eqtr4d 2471 1  |-  ( ( S  e.  A  /\  N  e.  NN0  /\  F : A --> B )  -> 
( F  o.  ( S repeatS  N ) )  =  ( ( F `  S ) repeatS  N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867   _Vcvv 3078    |-> cmpt 4475    o. ccom 4849   -->wf 5588   ` cfv 5592  (class class class)co 6296   0cc0 9528   NN0cn0 10858  ..^cfzo 11902   repeatS creps 12639
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 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-reps 12647
This theorem is referenced by: (None)
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