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Theorem reps 12873
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 3054 . . 3  |-  ( S  e.  V  ->  S  e.  _V )
21adantr 467 . 2  |-  ( ( S  e.  V  /\  N  e.  NN0 )  ->  S  e.  _V )
3 simpr 463 . 2  |-  ( ( S  e.  V  /\  N  e.  NN0 )  ->  N  e.  NN0 )
4 ovex 6318 . . 3  |-  ( 0..^ N )  e.  _V
5 mptexg 6135 . . 3  |-  ( ( 0..^ N )  e. 
_V  ->  ( x  e.  ( 0..^ N ) 
|->  S )  e.  _V )
64, 5mp1i 13 . 2  |-  ( ( S  e.  V  /\  N  e.  NN0 )  -> 
( x  e.  ( 0..^ N )  |->  S )  e.  _V )
7 oveq2 6298 . . . . 5  |-  ( n  =  N  ->  (
0..^ n )  =  ( 0..^ N ) )
87adantl 468 . . . 4  |-  ( ( s  =  S  /\  n  =  N )  ->  ( 0..^ n )  =  ( 0..^ N ) )
9 simpll 760 . . . 4  |-  ( ( ( s  =  S  /\  n  =  N )  /\  x  e.  ( 0..^ n ) )  ->  s  =  S )
108, 9mpteq12dva 4480 . . 3  |-  ( ( s  =  S  /\  n  =  N )  ->  ( x  e.  ( 0..^ n )  |->  s )  =  ( x  e.  ( 0..^ N )  |->  S ) )
11 df-reps 12671 . . 3  |- repeatS  =  ( s  e.  _V ,  n  e.  NN0  |->  ( x  e.  ( 0..^ n )  |->  s ) )
1210, 11ovmpt2ga 6426 . 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 1268 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 371    = wceq 1444    e. wcel 1887   _Vcvv 3045    |-> cmpt 4461  (class class class)co 6290   0cc0 9539   NN0cn0 10869  ..^cfzo 11915   repeatS creps 12663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-reps 12671
This theorem is referenced by:  repsconst  12875  repsf  12876  repswsymb  12877  repswswrd  12887  repswccat  12888  repswrevw  12889  repsco  12936
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