MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  reps Structured version   Unicode version

Theorem reps 12529
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 3087 . . 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 6228 . . 3  |-  ( 0..^ N )  e.  _V
5 mptexg 6059 . . 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 6211 . . . . 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 4480 . . 3  |-  ( ( s  =  S  /\  n  =  N )  ->  ( x  e.  ( 0..^ n )  |->  s )  =  ( x  e.  ( 0..^ N )  |->  S ) )
11 df-reps 12357 . . 3  |- repeatS  =  ( s  e.  _V ,  n  e.  NN0  |->  ( x  e.  ( 0..^ n )  |->  s ) )
1210, 11ovmpt2ga 6333 . 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 1219 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 1370    e. wcel 1758   _Vcvv 3078    |-> cmpt 4461  (class class class)co 6203   0cc0 9396   NN0cn0 10693  ..^cfzo 11668   repeatS creps 12349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-reps 12357
This theorem is referenced by:  repsconst  12531  repsf  12532  repswsymb  12533  repswswrd  12543  repswccat  12544  repswrevw  12545  repsco  12588
  Copyright terms: Public domain W3C validator