Theorem fsumxp

Description: Combine two sums into a single sum over the cartesian product. (Contributed by Mario Carneiro, 23-Apr-2014)

	Ref	Expression
Hypotheses	fsumxp.1	⊢ ( 𝑧 = ⟨ 𝑗 , 𝑘 ⟩ → 𝐷 = 𝐶 )
	fsumxp.2	⊢ ( 𝜑 → 𝐴 ∈ Fin )
	fsumxp.3	⊢ ( 𝜑 → 𝐵 ∈ Fin )
	fsumxp.4	⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ) ) → 𝐶 ∈ ℂ )
Assertion	fsumxp	⊢ ( 𝜑 → Σ 𝑗 ∈ 𝐴 Σ 𝑘 ∈ 𝐵 𝐶 = Σ 𝑧 ∈ ( 𝐴 × 𝐵 ) 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		fsumxp.1	⊢ ( 𝑧 = ⟨ 𝑗 , 𝑘 ⟩ → 𝐷 = 𝐶 )
2		fsumxp.2	⊢ ( 𝜑 → 𝐴 ∈ Fin )
3		fsumxp.3	⊢ ( 𝜑 → 𝐵 ∈ Fin )
4		fsumxp.4	⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ) ) → 𝐶 ∈ ℂ )
5	3	adantr	⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝐴 ) → 𝐵 ∈ Fin )
6	1 2 5 4	fsum2d	⊢ ( 𝜑 → Σ 𝑗 ∈ 𝐴 Σ 𝑘 ∈ 𝐵 𝐶 = Σ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐷 )
7		iunxpconst	⊢ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) = ( 𝐴 × 𝐵 )
8	7	sumeq1i	⊢ Σ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐷 = Σ 𝑧 ∈ ( 𝐴 × 𝐵 ) 𝐷
9	6 8	syl6eq	⊢ ( 𝜑 → Σ 𝑗 ∈ 𝐴 Σ 𝑘 ∈ 𝐵 𝐶 = Σ 𝑧 ∈ ( 𝐴 × 𝐵 ) 𝐷 )