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Theorem reps 12722
Description: Construct a function mapping a half-open range of nonnegative integers to a constant. (Contributed by AV, 4-Nov-2018.)
Assertion
Ref Expression
reps  |-  ( ( S  e.  V  /\  N  e.  NN0 )  -> 
( S repeatS  N )  =  ( x  e.  ( 0..^ N ) 
|->  S ) )
Distinct variable groups:    x, N    x, S
Allowed substitution hint:    V( x)

Proof of Theorem reps
Dummy variables  n  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3127 . . 3  |-  ( S  e.  V  ->  S  e.  _V )
21adantr 465 . 2  |-  ( ( S  e.  V  /\  N  e.  NN0 )  ->  S  e.  _V )
3 simpr 461 . 2  |-  ( ( S  e.  V  /\  N  e.  NN0 )  ->  N  e.  NN0 )
4 ovex 6320 . . 3  |-  ( 0..^ N )  e.  _V
5 mptexg 6141 . . 3  |-  ( ( 0..^ N )  e. 
_V  ->  ( x  e.  ( 0..^ N ) 
|->  S )  e.  _V )
64, 5mp1i 12 . 2  |-  ( ( S  e.  V  /\  N  e.  NN0 )  -> 
( x  e.  ( 0..^ N )  |->  S )  e.  _V )
7 oveq2 6303 . . . . 5  |-  ( n  =  N  ->  (
0..^ n )  =  ( 0..^ N ) )
87adantl 466 . . . 4  |-  ( ( s  =  S  /\  n  =  N )  ->  ( 0..^ n )  =  ( 0..^ N ) )
9 simpll 753 . . . 4  |-  ( ( ( s  =  S  /\  n  =  N )  /\  x  e.  ( 0..^ n ) )  ->  s  =  S )
108, 9mpteq12dva 4530 . . 3  |-  ( ( s  =  S  /\  n  =  N )  ->  ( x  e.  ( 0..^ n )  |->  s )  =  ( x  e.  ( 0..^ N )  |->  S ) )
11 df-reps 12530 . . 3  |- repeatS  =  ( s  e.  _V ,  n  e.  NN0  |->  ( x  e.  ( 0..^ n )  |->  s ) )
1210, 11ovmpt2ga 6427 . 2  |-  ( ( S  e.  _V  /\  N  e.  NN0  /\  (
x  e.  ( 0..^ N )  |->  S )  e.  _V )  -> 
( S repeatS  N )  =  ( x  e.  ( 0..^ N ) 
|->  S ) )
132, 3, 6, 12syl3anc 1228 1  |-  ( ( S  e.  V  /\  N  e.  NN0 )  -> 
( S repeatS  N )  =  ( x  e.  ( 0..^ N ) 
|->  S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3118    |-> cmpt 4511  (class class class)co 6295   0cc0 9504   NN0cn0 10807  ..^cfzo 11804   repeatS creps 12522
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-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-reps 12530
This theorem is referenced by:  repsconst  12724  repsf  12725  repswsymb  12726  repswswrd  12736  repswccat  12737  repswrevw  12738  repsco  12785
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