Theorem fz0addcom

Description: The addition of two members of a finite set of sequential integers starting at 0 is commutative. (Contributed by Alexander van der Vekens, 22-May-2018) (Revised by Alexander van der Vekens, 9-Jun-2018)

	Ref	Expression
Assertion	fz0addcom	⊢ ( ( 𝐴 ∈ ( 0 ... 𝑁 ) ∧ 𝐵 ∈ ( 0 ... 𝑁 ) ) → ( 𝐴 + 𝐵 ) = ( 𝐵 + 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		elfznn0	⊢ ( 𝐴 ∈ ( 0 ... 𝑁 ) → 𝐴 ∈ ℕ0 )
2	1	nn0cnd	⊢ ( 𝐴 ∈ ( 0 ... 𝑁 ) → 𝐴 ∈ ℂ )
3		elfznn0	⊢ ( 𝐵 ∈ ( 0 ... 𝑁 ) → 𝐵 ∈ ℕ0 )
4	3	nn0cnd	⊢ ( 𝐵 ∈ ( 0 ... 𝑁 ) → 𝐵 ∈ ℂ )
5		addcom	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 + 𝐵 ) = ( 𝐵 + 𝐴 ) )
6	2 4 5	syl2an	⊢ ( ( 𝐴 ∈ ( 0 ... 𝑁 ) ∧ 𝐵 ∈ ( 0 ... 𝑁 ) ) → ( 𝐴 + 𝐵 ) = ( 𝐵 + 𝐴 ) )