Theorem fprod2d 11788

Description: Write a double product as a product over a two-dimensional region. Compare fsum2d 11600. (Contributed by Scott Fenton, 30-Jan-2018.)

Hypotheses

Ref	Expression
fprod2d.1	⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → 𝐷 = 𝐶)
fprod2d.2	⊢ (𝜑 → 𝐴 ∈ Fin)
fprod2d.3	⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ Fin)
fprod2d.4	⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐶 ∈ ℂ)

Assertion

Ref	Expression
fprod2d	⊢ (𝜑 → ∏𝑗 ∈ 𝐴 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐷)

Distinct variable groups:   𝐴,𝑗,𝑘,𝑧   𝐵,𝑘,𝑧   𝑧,𝐶   𝐷,𝑗,𝑘   𝜑,𝑗,𝑘,𝑧

Allowed substitution hints:   𝐵(𝑗)   𝐶(𝑗,𝑘)   𝐷(𝑧)

Proof of Theorem fprod2d

Dummy variables 𝑤 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssid 3203	. 2 ⊢ 𝐴 ⊆ 𝐴
2		fprod2d.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ Fin)
3		sseq1 3206	. . . . . 6 ⊢ (𝑤 = ∅ → (𝑤 ⊆ 𝐴 ↔ ∅ ⊆ 𝐴))
4		prodeq1 11718	. . . . . . 7 ⊢ (𝑤 = ∅ → ∏𝑗 ∈ 𝑤 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑗 ∈ ∅ ∏𝑘 ∈ 𝐵 𝐶)
5		iuneq1 3929	. . . . . . . . 9 ⊢ (𝑤 = ∅ → ∪ 𝑗 ∈ 𝑤 ({𝑗} × 𝐵) = ∪ 𝑗 ∈ ∅ ({𝑗} × 𝐵))
6		0iun 3974	. . . . . . . . 9 ⊢ ∪ 𝑗 ∈ ∅ ({𝑗} × 𝐵) = ∅
7	5, 6	eqtrdi 2245	. . . . . . . 8 ⊢ (𝑤 = ∅ → ∪ 𝑗 ∈ 𝑤 ({𝑗} × 𝐵) = ∅)
8	7	prodeq1d 11729	. . . . . . 7 ⊢ (𝑤 = ∅ → ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑤 ({𝑗} × 𝐵)𝐷 = ∏𝑧 ∈ ∅ 𝐷)
9	4, 8	eqeq12d 2211	. . . . . 6 ⊢ (𝑤 = ∅ → (∏𝑗 ∈ 𝑤 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑤 ({𝑗} × 𝐵)𝐷 ↔ ∏𝑗 ∈ ∅ ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∅ 𝐷))
10	3, 9	imbi12d 234	. . . . 5 ⊢ (𝑤 = ∅ → ((𝑤 ⊆ 𝐴 → ∏𝑗 ∈ 𝑤 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑤 ({𝑗} × 𝐵)𝐷) ↔ (∅ ⊆ 𝐴 → ∏𝑗 ∈ ∅ ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∅ 𝐷)))
11	10	imbi2d 230	. . . 4 ⊢ (𝑤 = ∅ → ((𝜑 → (𝑤 ⊆ 𝐴 → ∏𝑗 ∈ 𝑤 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑤 ({𝑗} × 𝐵)𝐷)) ↔ (𝜑 → (∅ ⊆ 𝐴 → ∏𝑗 ∈ ∅ ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∅ 𝐷))))
12		sseq1 3206	. . . . . 6 ⊢ (𝑤 = 𝑥 → (𝑤 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐴))
13		prodeq1 11718	. . . . . . 7 ⊢ (𝑤 = 𝑥 → ∏𝑗 ∈ 𝑤 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑗 ∈ 𝑥 ∏𝑘 ∈ 𝐵 𝐶)
14		iuneq1 3929	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → ∪ 𝑗 ∈ 𝑤 ({𝑗} × 𝐵) = ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵))
15	14	prodeq1d 11729	. . . . . . 7 ⊢ (𝑤 = 𝑥 → ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑤 ({𝑗} × 𝐵)𝐷 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷)
16	13, 15	eqeq12d 2211	. . . . . 6 ⊢ (𝑤 = 𝑥 → (∏𝑗 ∈ 𝑤 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑤 ({𝑗} × 𝐵)𝐷 ↔ ∏𝑗 ∈ 𝑥 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷))
17	12, 16	imbi12d 234	. . . . 5 ⊢ (𝑤 = 𝑥 → ((𝑤 ⊆ 𝐴 → ∏𝑗 ∈ 𝑤 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑤 ({𝑗} × 𝐵)𝐷) ↔ (𝑥 ⊆ 𝐴 → ∏𝑗 ∈ 𝑥 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷)))
18	17	imbi2d 230	. . . 4 ⊢ (𝑤 = 𝑥 → ((𝜑 → (𝑤 ⊆ 𝐴 → ∏𝑗 ∈ 𝑤 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑤 ({𝑗} × 𝐵)𝐷)) ↔ (𝜑 → (𝑥 ⊆ 𝐴 → ∏𝑗 ∈ 𝑥 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷))))
19		sseq1 3206	. . . . . 6 ⊢ (𝑤 = (𝑥 ∪ {𝑦}) → (𝑤 ⊆ 𝐴 ↔ (𝑥 ∪ {𝑦}) ⊆ 𝐴))
20		prodeq1 11718	. . . . . . 7 ⊢ (𝑤 = (𝑥 ∪ {𝑦}) → ∏𝑗 ∈ 𝑤 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘 ∈ 𝐵 𝐶)
21		iuneq1 3929	. . . . . . . 8 ⊢ (𝑤 = (𝑥 ∪ {𝑦}) → ∪ 𝑗 ∈ 𝑤 ({𝑗} × 𝐵) = ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵))
22	21	prodeq1d 11729	. . . . . . 7 ⊢ (𝑤 = (𝑥 ∪ {𝑦}) → ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑤 ({𝑗} × 𝐵)𝐷 = ∏𝑧 ∈ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷)
23	20, 22	eqeq12d 2211	. . . . . 6 ⊢ (𝑤 = (𝑥 ∪ {𝑦}) → (∏𝑗 ∈ 𝑤 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑤 ({𝑗} × 𝐵)𝐷 ↔ ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷))
24	19, 23	imbi12d 234	. . . . 5 ⊢ (𝑤 = (𝑥 ∪ {𝑦}) → ((𝑤 ⊆ 𝐴 → ∏𝑗 ∈ 𝑤 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑤 ({𝑗} × 𝐵)𝐷) ↔ ((𝑥 ∪ {𝑦}) ⊆ 𝐴 → ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷)))
25	24	imbi2d 230	. . . 4 ⊢ (𝑤 = (𝑥 ∪ {𝑦}) → ((𝜑 → (𝑤 ⊆ 𝐴 → ∏𝑗 ∈ 𝑤 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑤 ({𝑗} × 𝐵)𝐷)) ↔ (𝜑 → ((𝑥 ∪ {𝑦}) ⊆ 𝐴 → ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷))))
26		sseq1 3206	. . . . . 6 ⊢ (𝑤 = 𝐴 → (𝑤 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
27		prodeq1 11718	. . . . . . 7 ⊢ (𝑤 = 𝐴 → ∏𝑗 ∈ 𝑤 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑗 ∈ 𝐴 ∏𝑘 ∈ 𝐵 𝐶)
28		iuneq1 3929	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → ∪ 𝑗 ∈ 𝑤 ({𝑗} × 𝐵) = ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
29	28	prodeq1d 11729	. . . . . . 7 ⊢ (𝑤 = 𝐴 → ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑤 ({𝑗} × 𝐵)𝐷 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐷)
30	27, 29	eqeq12d 2211	. . . . . 6 ⊢ (𝑤 = 𝐴 → (∏𝑗 ∈ 𝑤 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑤 ({𝑗} × 𝐵)𝐷 ↔ ∏𝑗 ∈ 𝐴 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐷))
31	26, 30	imbi12d 234	. . . . 5 ⊢ (𝑤 = 𝐴 → ((𝑤 ⊆ 𝐴 → ∏𝑗 ∈ 𝑤 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑤 ({𝑗} × 𝐵)𝐷) ↔ (𝐴 ⊆ 𝐴 → ∏𝑗 ∈ 𝐴 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐷)))
32	31	imbi2d 230	. . . 4 ⊢ (𝑤 = 𝐴 → ((𝜑 → (𝑤 ⊆ 𝐴 → ∏𝑗 ∈ 𝑤 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑤 ({𝑗} × 𝐵)𝐷)) ↔ (𝜑 → (𝐴 ⊆ 𝐴 → ∏𝑗 ∈ 𝐴 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐷))))
33		prod0 11750	. . . . . 6 ⊢ ∏𝑗 ∈ ∅ ∏𝑘 ∈ 𝐵 𝐶 = 1
34		prod0 11750	. . . . . 6 ⊢ ∏𝑧 ∈ ∅ 𝐷 = 1
35	33, 34	eqtr4i 2220	. . . . 5 ⊢ ∏𝑗 ∈ ∅ ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∅ 𝐷
36	35	2a1i 27	. . . 4 ⊢ (𝜑 → (∅ ⊆ 𝐴 → ∏𝑗 ∈ ∅ ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∅ 𝐷))
37		ssun1 3326	. . . . . . . . 9 ⊢ 𝑥 ⊆ (𝑥 ∪ {𝑦})
38		sstr 3191	. . . . . . . . 9 ⊢ ((𝑥 ⊆ (𝑥 ∪ {𝑦}) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑥 ⊆ 𝐴)
39	37, 38	mpan 424	. . . . . . . 8 ⊢ ((𝑥 ∪ {𝑦}) ⊆ 𝐴 → 𝑥 ⊆ 𝐴)
40	39	imim1i 60	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝐴 → ∏𝑗 ∈ 𝑥 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷) → ((𝑥 ∪ {𝑦}) ⊆ 𝐴 → ∏𝑗 ∈ 𝑥 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷))
41		fprod2d.1	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → 𝐷 = 𝐶)
42	2	ad2antrr 488	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝐴 ∈ Fin)
43		fprod2d.3	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ Fin)
44	43	ad4ant14 514	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ Fin)
45		fprod2d.4	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐶 ∈ ℂ)
46	45	ad4ant14 514	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐶 ∈ ℂ)
47		simplrr 536	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → ¬ 𝑦 ∈ 𝑥)
48		simpr 110	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (𝑥 ∪ {𝑦}) ⊆ 𝐴)
49		simplrl 535	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑥 ∈ Fin)
50		biid 171	. . . . . . . . . 10 ⊢ (∏𝑗 ∈ 𝑥 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷 ↔ ∏𝑗 ∈ 𝑥 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷)
51	41, 42, 44, 46, 47, 48, 49, 50	fprod2dlemstep 11787	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ ∏𝑗 ∈ 𝑥 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷) → ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷)
52	51	exp31 364	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥)) → ((𝑥 ∪ {𝑦}) ⊆ 𝐴 → (∏𝑗 ∈ 𝑥 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷 → ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷)))
53	52	a2d 26	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥)) → (((𝑥 ∪ {𝑦}) ⊆ 𝐴 → ∏𝑗 ∈ 𝑥 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷) → ((𝑥 ∪ {𝑦}) ⊆ 𝐴 → ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷)))
54	40, 53	syl5 32	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥)) → ((𝑥 ⊆ 𝐴 → ∏𝑗 ∈ 𝑥 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷) → ((𝑥 ∪ {𝑦}) ⊆ 𝐴 → ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷)))
55	54	expcom 116	. . . . 5 ⊢ ((𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥) → (𝜑 → ((𝑥 ⊆ 𝐴 → ∏𝑗 ∈ 𝑥 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷) → ((𝑥 ∪ {𝑦}) ⊆ 𝐴 → ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷))))
56	55	a2d 26	. . . 4 ⊢ ((𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥) → ((𝜑 → (𝑥 ⊆ 𝐴 → ∏𝑗 ∈ 𝑥 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷)) → (𝜑 → ((𝑥 ∪ {𝑦}) ⊆ 𝐴 → ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷))))
57	11, 18, 25, 32, 36, 56	findcard2s 6951	. . 3 ⊢ (𝐴 ∈ Fin → (𝜑 → (𝐴 ⊆ 𝐴 → ∏𝑗 ∈ 𝐴 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐷)))
58	2, 57	mpcom 36	. 2 ⊢ (𝜑 → (𝐴 ⊆ 𝐴 → ∏𝑗 ∈ 𝐴 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐷))
59	1, 58	mpi 15	1 ⊢ (𝜑 → ∏𝑗 ∈ 𝐴 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐷)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2167   ∪ cun 3155   ⊆ wss 3157  ∅c0 3450  {csn 3622  ⟨cop 3625  ∪ ciun 3916   × cxp 4661  Fincfn 6799  ℂcc 7877  1c1 7880  ∏cprod 11715

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998  ax-caucvg 7999

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-disj 4011  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-isom 5267  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-frec 6449  df-1o 6474  df-oadd 6478  df-er 6592  df-en 6800  df-dom 6801  df-fin 6802  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-n0 9250  df-z 9327  df-uz 9602  df-q 9694  df-rp 9729  df-fz 10084  df-fzo 10218  df-seqfrec 10540  df-exp 10631  df-ihash 10868  df-cj 11007  df-re 11008  df-im 11009  df-rsqrt 11163  df-abs 11164  df-clim 11444  df-proddc 11716

This theorem is referenced by:  fprodxp  11789  fprodcom2fi  11791