Theorem hashrepr

Description: Develop the number of representations of an integer M as a sum of nonnegative integers in set A . (Contributed by Thierry Arnoux, 14-Dec-2021)

	Ref	Expression
Hypotheses	hashrepr.a	⊢ ( 𝜑 → 𝐴 ⊆ ℕ )
	hashrepr.m	⊢ ( 𝜑 → 𝑀 ∈ ℕ0 )
	hashrepr.s	⊢ ( 𝜑 → 𝑆 ∈ ℕ0 )
Assertion	hashrepr	⊢ ( 𝜑 → ( ♯ ‘ ( 𝐴 ( repr ‘ 𝑆 ) 𝑀 ) ) = Σ 𝑐 ∈ ( ℕ ( repr ‘ 𝑆 ) 𝑀 ) ∏ 𝑎 ∈ ( 0 ..^ 𝑆 ) ( ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) ‘ ( 𝑐 ‘ 𝑎 ) ) )

Proof

Step	Hyp	Ref	Expression
1		hashrepr.a	⊢ ( 𝜑 → 𝐴 ⊆ ℕ )
2		hashrepr.m	⊢ ( 𝜑 → 𝑀 ∈ ℕ0 )
3		hashrepr.s	⊢ ( 𝜑 → 𝑆 ∈ ℕ0 )
4	2	nn0zd	⊢ ( 𝜑 → 𝑀 ∈ ℤ )
5		fzfid	⊢ ( 𝜑 → ( 1 ... 𝑀 ) ∈ Fin )
6		fz1ssnn	⊢ ( 1 ... 𝑀 ) ⊆ ℕ
7	6	a1i	⊢ ( 𝜑 → ( 1 ... 𝑀 ) ⊆ ℕ )
8	1 4 3 5 7	hashreprin	⊢ ( 𝜑 → ( ♯ ‘ ( ( 𝐴 ∩ ( 1 ... 𝑀 ) ) ( repr ‘ 𝑆 ) 𝑀 ) ) = Σ 𝑐 ∈ ( ( 1 ... 𝑀 ) ( repr ‘ 𝑆 ) 𝑀 ) ∏ 𝑎 ∈ ( 0 ..^ 𝑆 ) ( ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) ‘ ( 𝑐 ‘ 𝑎 ) ) )
9	2 3 1	reprinfz1	⊢ ( 𝜑 → ( 𝐴 ( repr ‘ 𝑆 ) 𝑀 ) = ( ( 𝐴 ∩ ( 1 ... 𝑀 ) ) ( repr ‘ 𝑆 ) 𝑀 ) )
10	9	fveq2d	⊢ ( 𝜑 → ( ♯ ‘ ( 𝐴 ( repr ‘ 𝑆 ) 𝑀 ) ) = ( ♯ ‘ ( ( 𝐴 ∩ ( 1 ... 𝑀 ) ) ( repr ‘ 𝑆 ) 𝑀 ) ) )
11	2 3	reprfz1	⊢ ( 𝜑 → ( ℕ ( repr ‘ 𝑆 ) 𝑀 ) = ( ( 1 ... 𝑀 ) ( repr ‘ 𝑆 ) 𝑀 ) )
12	11	sumeq1d	⊢ ( 𝜑 → Σ 𝑐 ∈ ( ℕ ( repr ‘ 𝑆 ) 𝑀 ) ∏ 𝑎 ∈ ( 0 ..^ 𝑆 ) ( ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) ‘ ( 𝑐 ‘ 𝑎 ) ) = Σ 𝑐 ∈ ( ( 1 ... 𝑀 ) ( repr ‘ 𝑆 ) 𝑀 ) ∏ 𝑎 ∈ ( 0 ..^ 𝑆 ) ( ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) ‘ ( 𝑐 ‘ 𝑎 ) ) )
13	8 10 12	3eqtr4d	⊢ ( 𝜑 → ( ♯ ‘ ( 𝐴 ( repr ‘ 𝑆 ) 𝑀 ) ) = Σ 𝑐 ∈ ( ℕ ( repr ‘ 𝑆 ) 𝑀 ) ∏ 𝑎 ∈ ( 0 ..^ 𝑆 ) ( ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) ‘ ( 𝑐 ‘ 𝑎 ) ) )