Theorem repsundef 12722

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 12529	. . 3 |- repeatS = ( s e. _V , n e. NN0 |-> ( x e. ( 0..^ n ) |-> s ) )
2		ovex 6320	. . . 4 |- ( 0..^ n ) e. _V
3	2	mptex 6142	. . 3 |- ( x e. ( 0..^ n ) |-> s ) e. _V
4	1, 3	dmmpt2 6865	. 2 |- dom repeatS = ( _V X. NN0 )
5		df-nel 2665	. . . 4 |- ( N e/ NN0 <-> -. N e. NN0 )
6	5	biimpi 194	. . 3 |- ( N e/ NN0 -> -. N e. NN0 )
7	6	intnand 914	. 2 |- ( N e/ NN0 -> -. ( S e. _V /\ N e. NN0 ) )
8		ndmovg 6453	. 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 1379   e. wcel 1767   e/ wnel 2663   _Vcvv 3118   (/)c0 3790   |-> cmpt 4511   X. cxp 5003   dom cdm 5005  (class class class)co 6295   0cc0 9504   NN0cn0 10807  ..^cfzo 11804   repeatS creps 12521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-reps 12529

This theorem is referenced by:  repswswrd  12735