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Metamath Proof Explorer

Theorem repsconst

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

		Ref	Expression
	Assertion	repsconst	⊢ ( ( 𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀ ) → ( 𝑆 repeatS 𝑁 ) = ( ( 0 ..^ 𝑁 ) × { 𝑆 } ) )

Proof

Step	Hyp	Ref	Expression
1		reps	⊢ ( ( 𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀ ) → ( 𝑆 repeatS 𝑁 ) = ( 𝑥 ∈ ( 0 ..^ 𝑁 ) ↦ 𝑆 ) )
2		fconstmpt	⊢ ( ( 0 ..^ 𝑁 ) × { 𝑆 } ) = ( 𝑥 ∈ ( 0 ..^ 𝑁 ) ↦ 𝑆 )
3	1 2	eqtr4di	⊢ ( ( 𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀ ) → ( 𝑆 repeatS 𝑁 ) = ( ( 0 ..^ 𝑁 ) × { 𝑆 } ) )