Theorem fprodcom

Description: Interchange product order. (Contributed by Scott Fenton, 2-Feb-2018)

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

Proof

Step	Hyp	Ref	Expression
1		fprodcom.1	⊢ ( 𝜑 → 𝐴 ∈ Fin )
2		fprodcom.2	⊢ ( 𝜑 → 𝐵 ∈ Fin )
3		fprodcom.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	fprodcom2	⊢ ( 𝜑 → ∏ 𝑗 ∈ 𝐴 ∏ 𝑘 ∈ 𝐵 𝐶 = ∏ 𝑘 ∈ 𝐵 ∏ 𝑗 ∈ 𝐴 𝐶 )