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Theorem reps 12754
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 3118 . . 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 6324 . . 3  |-  ( 0..^ N )  e.  _V
5 mptexg 6143 . . 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 6304 . . . . 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 4534 . . 3  |-  ( ( s  =  S  /\  n  =  N )  ->  ( x  e.  ( 0..^ n )  |->  s )  =  ( x  e.  ( 0..^ N )  |->  S ) )
11 df-reps 12553 . . 3  |- repeatS  =  ( s  e.  _V ,  n  e.  NN0  |->  ( x  e.  ( 0..^ n )  |->  s ) )
1210, 11ovmpt2ga 6431 . 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 1395    e. wcel 1819   _Vcvv 3109    |-> cmpt 4515  (class class class)co 6296   0cc0 9509   NN0cn0 10816  ..^cfzo 11821   repeatS creps 12545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  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 6299  df-oprab 6300  df-mpt2 6301  df-reps 12553
This theorem is referenced by:  repsconst  12756  repsf  12757  repswsymb  12758  repswswrd  12768  repswccat  12769  repswrevw  12770  repsco  12817
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