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

Theorem repsundef 12654
Description: A function mapping a half-open range of nonnegative integers with an upper bound not being a nonnegative integer to a constant is the empty set (in the meaning of "undefined"). (Contributed by AV, 5-Nov-2018.)
Assertion
Ref Expression
repsundef  |-  ( N  e/  NN0  ->  ( S repeatS  N )  =  (/) )

Proof of Theorem repsundef
Dummy variables  n  s  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-reps 12453 . . 3  |- repeatS  =  ( s  e.  _V ,  n  e.  NN0  |->  ( x  e.  ( 0..^ n )  |->  s ) )
2 ovex 6224 . . . 4  |-  ( 0..^ n )  e.  _V
32mptex 6044 . . 3  |-  ( x  e.  ( 0..^ n )  |->  s )  e. 
_V
41, 3dmmpt2 6769 . 2  |-  dom repeatS  =  ( _V  X.  NN0 )
5 df-nel 2580 . . . 4  |-  ( N  e/  NN0  <->  -.  N  e.  NN0 )
65biimpi 194 . . 3  |-  ( N  e/  NN0  ->  -.  N  e.  NN0 )
76intnand 914 . 2  |-  ( N  e/  NN0  ->  -.  ( S  e.  _V  /\  N  e.  NN0 ) )
8 ndmovg 6357 . 2  |-  ( ( dom repeatS  =  ( _V  X.  NN0 )  /\  -.  ( S  e.  _V  /\  N  e.  NN0 )
)  ->  ( S repeatS  N )  =  (/) )
94, 7, 8sylancr 661 1  |-  ( N  e/  NN0  ->  ( S repeatS  N )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1826    e/ wnel 2578   _Vcvv 3034   (/)c0 3711    |-> cmpt 4425    X. cxp 4911   dom cdm 4913  (class class class)co 6196   0cc0 9403   NN0cn0 10712  ..^cfzo 11717   repeatS creps 12445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-1st 6699  df-2nd 6700  df-reps 12453
This theorem is referenced by:  repswswrd  12667
  Copyright terms: Public domain W3C validator