Theorem reps 13517

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.

Step	Hyp	Ref	Expression
1		elex 3212	. . 3 |- ( S e. V -> S e. _V )
2	1	adantr 481	. 2 |- ( ( S e. V /\ N e. NN0 ) -> S e. _V )
3		simpr 477	. 2 |- ( ( S e. V /\ N e. NN0 ) -> N e. NN0 )
4		ovex 6678	. . 3 |- ( 0..^ N ) e. _V
5		mptexg 6484	. . 3 |- ( ( 0..^ N ) e. _V -> ( x e. ( 0..^ N ) |-> S ) e. _V )
6	4, 5	mp1i 13	. 2 |- ( ( S e. V /\ N e. NN0 ) -> ( x e. ( 0..^ N ) |-> S ) e. _V )
7		oveq2 6658	. . . . 5 |- ( n = N -> ( 0..^ n ) = ( 0..^ N ) )
8	7	adantl 482	. . . 4 |- ( ( s = S /\ n = N ) -> ( 0..^ n ) = ( 0..^ N ) )
9		simpll 790	. . . 4 |- ( ( ( s = S /\ n = N ) /\ x e. ( 0..^ n ) ) -> s = S )
10	8, 9	mpteq12dva 4732	. . 3 |- ( ( s = S /\ n = N ) -> ( x e. ( 0..^ n ) |-> s ) = ( x e. ( 0..^ N ) |-> S ) )
11		df-reps 13306	. . 3 |- repeatS = ( s e. _V , n e. NN0 |-> ( x e. ( 0..^ n ) |-> s ) )
12	10, 11	ovmpt2ga 6790	. 2 |- ( ( S e. _V /\ N e. NN0 /\ ( x e. ( 0..^ N ) |-> S ) e. _V ) -> ( S repeatS N ) = ( x e. ( 0..^ N ) |-> S ) )
13	2, 3, 6, 12	syl3anc 1326	1 |- ( ( S e. V /\ N e. NN0 ) -> ( S repeatS N ) = ( x e. ( 0..^ N ) |-> S ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   |-> cmpt 4729  (class class class)co 6650   0cc0 9936   NN0cn0 11292  ..^cfzo 12465   repeatS creps 13298

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-reps 13306

This theorem is referenced by:  repsconst  13519  repsf  13520  repswsymb  13521  repswswrd  13531  repswccat  13532  repswrevw  13533  repsco  13585