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Metamath Proof Explorer

Theorem reps

Description: Construct a function mapping a half-open range of nonnegative integers to a constant. (Contributed by AV, 4-Nov-2018)

Ref Expression
Assertion reps 
|- ( ( S e. V /\ N e. NN0 ) -> ( S repeatS N ) = ( x e. ( 0 ..^ N ) |-> S ) )

Proof

Step Hyp Ref Expression
1 elex 
 |-  ( S e. V -> S e. _V )
2 1 adantr 
 |-  ( ( S e. V /\ N e. NN0 ) -> S e. _V )
3 simpr 
 |-  ( ( S e. V /\ N e. NN0 ) -> N e. NN0 )
4 ovex 
 |-  ( 0 ..^ N ) e. _V
5 mptexg 
 |-  ( ( 0 ..^ N ) e. _V -> ( x e. ( 0 ..^ N ) |-> S ) e. _V )
6 4 5 mp1i 
 |-  ( ( S e. V /\ N e. NN0 ) -> ( x e. ( 0 ..^ N ) |-> S ) e. _V )
7 oveq2 
 |-  ( n = N -> ( 0 ..^ n ) = ( 0 ..^ N ) )
8 7 adantl 
 |-  ( ( s = S /\ n = N ) -> ( 0 ..^ n ) = ( 0 ..^ N ) )
9 simpll 
 |-  ( ( ( s = S /\ n = N ) /\ x e. ( 0 ..^ n ) ) -> s = S )
10 8 9 mpteq12dva 
 |-  ( ( s = S /\ n = N ) -> ( x e. ( 0 ..^ n ) |-> s ) = ( x e. ( 0 ..^ N ) |-> S ) )
11 df-reps 
 |-  repeatS = ( s e. _V , n e. NN0 |-> ( x e. ( 0 ..^ n ) |-> s ) )
12 10 11 ovmpoga 
 |-  ( ( 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 
 |-  ( ( S e. V /\ N e. NN0 ) -> ( S repeatS N ) = ( x e. ( 0 ..^ N ) |-> S ) )