Theorem repsundef 12497

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.

Step	Hyp	Ref	Expression
1		df-reps 12324	. . 3 |- repeatS = ( s e. _V , n e. NN0 |-> ( x e. ( 0..^ n ) |-> s ) )
2		ovex 6201	. . . 4 |- ( 0..^ n ) e. _V
3	2	mptex 6033	. . 3 |- ( x e. ( 0..^ n ) |-> s ) e. _V
4	1, 3	dmmpt2 6730	. 2 |- dom repeatS = ( _V X. NN0 )
5		df-nel 2644	. . . 4 |- ( N e/ NN0 <-> -. N e. NN0 )
6	5	biimpi 194	. . 3 |- ( N e/ NN0 -> -. N e. NN0 )
7	6	intnand 907	. 2 |- ( N e/ NN0 -> -. ( S e. _V /\ N e. NN0 ) )
8		ndmovg 6332	. 2 |- ( ( dom repeatS = ( _V X. NN0 ) /\ -. ( S e. _V /\ N e. NN0 ) ) -> ( S repeatS N ) = (/) )
9	4, 7, 8	sylancr 663	1 |- ( N e/ NN0 -> ( S repeatS N ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   = wceq 1370   e. wcel 1757   e/ wnel 2642   _Vcvv 3054   (/)c0 3721   |-> cmpt 4434   X. cxp 4922   dom cdm 4924  (class class class)co 6176   0cc0 9369   NN0cn0 10666  ..^cfzo 11635   repeatS creps 12316

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 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-rep 4487  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458

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 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-nel 2644  df-ral 2797  df-rex 2798  df-reu 2799  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4176  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-id 4720  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-1st 6663  df-2nd 6664  df-reps 12324

This theorem is referenced by:  repswswrd  12510