Theorem fprod2d 12390

Description: Write a double product as a product over a two-dimensional region. Compare fsum2d 12202. (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 3268	. 2 ⊢ 𝐴 ⊆ 𝐴
2		fprod2d.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ Fin)
3		sseq1 3271	. . . . . 6 ⊢ (𝑤 = ∅ → (𝑤 ⊆ 𝐴 ↔ ∅ ⊆ 𝐴))
4		prodeq1 12320	. . . . . . 7 ⊢ (𝑤 = ∅ → ∏𝑗 ∈ 𝑤 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑗 ∈ ∅ ∏𝑘 ∈ 𝐵 𝐶)
5		iuneq1 4025	. . . . . . . . 9 ⊢ (𝑤 = ∅ → ∪ 𝑗 ∈ 𝑤 ({𝑗} × 𝐵) = ∪ 𝑗 ∈ ∅ ({𝑗} × 𝐵))
6		0iun 4070	. . . . . . . . 9 ⊢ ∪ 𝑗 ∈ ∅ ({𝑗} × 𝐵) = ∅
7	5, 6	eqtrdi 2287	. . . . . . . 8 ⊢ (𝑤 = ∅ → ∪ 𝑗 ∈ 𝑤 ({𝑗} × 𝐵) = ∅)
8	7	prodeq1d 12331	. . . . . . 7 ⊢ (𝑤 = ∅ → ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑤 ({𝑗} × 𝐵)𝐷 = ∏𝑧 ∈ ∅ 𝐷)
9	4, 8	eqeq12d 2253	. . . . . 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 3271	. . . . . 6 ⊢ (𝑤 = 𝑥 → (𝑤 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐴))
13		prodeq1 12320	. . . . . . 7 ⊢ (𝑤 = 𝑥 → ∏𝑗 ∈ 𝑤 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑗 ∈ 𝑥 ∏𝑘 ∈ 𝐵 𝐶)
14		iuneq1 4025	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → ∪ 𝑗 ∈ 𝑤 ({𝑗} × 𝐵) = ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵))
15	14	prodeq1d 12331	. . . . . . 7 ⊢ (𝑤 = 𝑥 → ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑤 ({𝑗} × 𝐵)𝐷 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷)
16	13, 15	eqeq12d 2253	. . . . . 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 3271	. . . . . 6 ⊢ (𝑤 = (𝑥 ∪ {𝑦}) → (𝑤 ⊆ 𝐴 ↔ (𝑥 ∪ {𝑦}) ⊆ 𝐴))
20		prodeq1 12320	. . . . . . 7 ⊢ (𝑤 = (𝑥 ∪ {𝑦}) → ∏𝑗 ∈ 𝑤 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘 ∈ 𝐵 𝐶)
21		iuneq1 4025	. . . . . . . 8 ⊢ (𝑤 = (𝑥 ∪ {𝑦}) → ∪ 𝑗 ∈ 𝑤 ({𝑗} × 𝐵) = ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵))
22	21	prodeq1d 12331	. . . . . . 7 ⊢ (𝑤 = (𝑥 ∪ {𝑦}) → ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑤 ({𝑗} × 𝐵)𝐷 = ∏𝑧 ∈ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷)
23	20, 22	eqeq12d 2253	. . . . . 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 3271	. . . . . 6 ⊢ (𝑤 = 𝐴 → (𝑤 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
27		prodeq1 12320	. . . . . . 7 ⊢ (𝑤 = 𝐴 → ∏𝑗 ∈ 𝑤 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑗 ∈ 𝐴 ∏𝑘 ∈ 𝐵 𝐶)
28		iuneq1 4025	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → ∪ 𝑗 ∈ 𝑤 ({𝑗} × 𝐵) = ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
29	28	prodeq1d 12331	. . . . . . 7 ⊢ (𝑤 = 𝐴 → ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑤 ({𝑗} × 𝐵)𝐷 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐷)
30	27, 29	eqeq12d 2253	. . . . . 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 12352	. . . . . 6 ⊢ ∏𝑗 ∈ ∅ ∏𝑘 ∈ 𝐵 𝐶 = 1
34		prod0 12352	. . . . . 6 ⊢ ∏𝑧 ∈ ∅ 𝐷 = 1
35	33, 34	eqtr4i 2262	. . . . 5 ⊢ ∏𝑗 ∈ ∅ ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∅ 𝐷
36	35	2a1i 27	. . . 4 ⊢ (𝜑 → (∅ ⊆ 𝐴 → ∏𝑗 ∈ ∅ ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∅ 𝐷))
37		ssun1 3392	. . . . . . . . 9 ⊢ 𝑥 ⊆ (𝑥 ∪ {𝑦})
38		sstr 3256	. . . . . . . . 9 ⊢ ((𝑥 ⊆ (𝑥 ∪ {𝑦}) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑥 ⊆ 𝐴)
39	37, 38	mpan 428	. . . . . . . 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 492	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝐴 ∈ Fin)
43		fprod2d.3	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ Fin)
44	43	ad4ant14 518	. . . . . . . . . 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 518	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐶 ∈ ℂ)
47		simplrr 542	. . . . . . . . . 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 541	. . . . . . . . . 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 12389	. . . . . . . . 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 7194	. . 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)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209   ∪ cun 3218   ⊆ wss 3220  ∅c0 3520  {csn 3709  ⟨cop 3712  ∪ ciun 4012   × cxp 4772  Fincfn 7022  ℂcc 8177  1c1 8180  ∏cprod 12317

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-disj 4107  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-oadd 6691  df-er 6807  df-en 7023  df-dom 7024  df-fin 7025  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-n0 9564  df-z 9645  df-uz 9922  df-q 10020  df-rp 10055  df-fz 10412  df-fzo 10550  df-seqfrec 10885  df-exp 10976  df-ihash 11215  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765  df-clim 12045  df-proddc 12318

This theorem is used by:  fprodxp  12391  fprodcom2fi  12393