Theorem fsumcom

Description: Interchange order of summation. (Contributed by NM, 15-Nov-2005) (Revised by Mario Carneiro, 23-Apr-2014)

	Ref	Expression
Hypotheses	fsumcom.1	⊢ ( 𝜑 → 𝐴 ∈ Fin )
	fsumcom.2	⊢ ( 𝜑 → 𝐵 ∈ Fin )
	fsumcom.3	⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ) ) → 𝐶 ∈ ℂ )
Assertion	fsumcom	⊢ ( 𝜑 → Σ 𝑗 ∈ 𝐴 Σ 𝑘 ∈ 𝐵 𝐶 = Σ 𝑘 ∈ 𝐵 Σ 𝑗 ∈ 𝐴 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		fsumcom.1	⊢ ( 𝜑 → 𝐴 ∈ Fin )
2		fsumcom.2	⊢ ( 𝜑 → 𝐵 ∈ Fin )
3		fsumcom.3	⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ) ) → 𝐶 ∈ ℂ )
4	2	adantr	⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝐴 ) → 𝐵 ∈ Fin )
5		ancom	⊢ ( ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ) ↔ ( 𝑘 ∈ 𝐵 ∧ 𝑗 ∈ 𝐴 ) )
6	5	a1i	⊢ ( 𝜑 → ( ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ) ↔ ( 𝑘 ∈ 𝐵 ∧ 𝑗 ∈ 𝐴 ) ) )
7	1 2 4 6 3	fsumcom2	⊢ ( 𝜑 → Σ 𝑗 ∈ 𝐴 Σ 𝑘 ∈ 𝐵 𝐶 = Σ 𝑘 ∈ 𝐵 Σ 𝑗 ∈ 𝐴 𝐶 )