Theorem fprod2d 16011

Description: Write a double product as a product over a two-dimensional region. Compare fsum2d 15798. (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   𝐷,𝑗,𝑘   𝐵,𝑘,𝑧   𝑧,𝐶   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 3958	. 2 ⊢ 𝐴 ⊆ 𝐴
2		fprod2d.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ Fin)
3		sseq1 3961	. . . . . 6 ⊢ (𝑤 = ∅ → (𝑤 ⊆ 𝐴 ↔ ∅ ⊆ 𝐴))
4		prodeq1 15937	. . . . . . 7 ⊢ (𝑤 = ∅ → ∏𝑗 ∈ 𝑤 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑗 ∈ ∅ ∏𝑘 ∈ 𝐵 𝐶)
5		iuneq1 4966	. . . . . . . . 9 ⊢ (𝑤 = ∅ → ∪ 𝑗 ∈ 𝑤 ({𝑗} × 𝐵) = ∪ 𝑗 ∈ ∅ ({𝑗} × 𝐵))
6		0iun 5020	. . . . . . . . 9 ⊢ ∪ 𝑗 ∈ ∅ ({𝑗} × 𝐵) = ∅
7	5, 6	eqtrdi 2813	. . . . . . . 8 ⊢ (𝑤 = ∅ → ∪ 𝑗 ∈ 𝑤 ({𝑗} × 𝐵) = ∅)
8	7	prodeq1d 15950	. . . . . . 7 ⊢ (𝑤 = ∅ → ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑤 ({𝑗} × 𝐵)𝐷 = ∏𝑧 ∈ ∅ 𝐷)
9	4, 8	eqeq12d 2778	. . . . . 6 ⊢ (𝑤 = ∅ → (∏𝑗 ∈ 𝑤 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑤 ({𝑗} × 𝐵)𝐷 ↔ ∏𝑗 ∈ ∅ ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∅ 𝐷))
10	3, 9	imbi12d 346	. . . . 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 342	. . . 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 3961	. . . . . 6 ⊢ (𝑤 = 𝑥 → (𝑤 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐴))
13		prodeq1 15937	. . . . . . 7 ⊢ (𝑤 = 𝑥 → ∏𝑗 ∈ 𝑤 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑗 ∈ 𝑥 ∏𝑘 ∈ 𝐵 𝐶)
14		iuneq1 4966	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → ∪ 𝑗 ∈ 𝑤 ({𝑗} × 𝐵) = ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵))
15	14	prodeq1d 15950	. . . . . . 7 ⊢ (𝑤 = 𝑥 → ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑤 ({𝑗} × 𝐵)𝐷 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷)
16	13, 15	eqeq12d 2778	. . . . . 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 346	. . . . 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 342	. . . 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 3961	. . . . . 6 ⊢ (𝑤 = (𝑥 ∪ {𝑦}) → (𝑤 ⊆ 𝐴 ↔ (𝑥 ∪ {𝑦}) ⊆ 𝐴))
20		prodeq1 15937	. . . . . . 7 ⊢ (𝑤 = (𝑥 ∪ {𝑦}) → ∏𝑗 ∈ 𝑤 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘 ∈ 𝐵 𝐶)
21		iuneq1 4966	. . . . . . . 8 ⊢ (𝑤 = (𝑥 ∪ {𝑦}) → ∪ 𝑗 ∈ 𝑤 ({𝑗} × 𝐵) = ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵))
22	21	prodeq1d 15950	. . . . . . 7 ⊢ (𝑤 = (𝑥 ∪ {𝑦}) → ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑤 ({𝑗} × 𝐵)𝐷 = ∏𝑧 ∈ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷)
23	20, 22	eqeq12d 2778	. . . . . 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 346	. . . . 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 342	. . . 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 3961	. . . . . 6 ⊢ (𝑤 = 𝐴 → (𝑤 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
27		prodeq1 15937	. . . . . . 7 ⊢ (𝑤 = 𝐴 → ∏𝑗 ∈ 𝑤 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑗 ∈ 𝐴 ∏𝑘 ∈ 𝐵 𝐶)
28		iuneq1 4966	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → ∪ 𝑗 ∈ 𝑤 ({𝑗} × 𝐵) = ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
29	28	prodeq1d 15950	. . . . . . 7 ⊢ (𝑤 = 𝐴 → ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑤 ({𝑗} × 𝐵)𝐷 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐷)
30	27, 29	eqeq12d 2778	. . . . . 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 346	. . . . 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 342	. . . 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 15973	. . . . . 6 ⊢ ∏𝑗 ∈ ∅ ∏𝑘 ∈ 𝐵 𝐶 = 1
34		prod0 15973	. . . . . 6 ⊢ ∏𝑧 ∈ ∅ 𝐷 = 1
35	33, 34	eqtr4i 2788	. . . . 5 ⊢ ∏𝑗 ∈ ∅ ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∅ 𝐷
36	35	2a1i 12	. . . 4 ⊢ (𝜑 → (∅ ⊆ 𝐴 → ∏𝑗 ∈ ∅ ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∅ 𝐷))
37		ssun1 4130	. . . . . . . . . 10 ⊢ 𝑥 ⊆ (𝑥 ∪ {𝑦})
38		sstr 3944	. . . . . . . . . 10 ⊢ ((𝑥 ⊆ (𝑥 ∪ {𝑦}) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑥 ⊆ 𝐴)
39	37, 38	mpan 700	. . . . . . . . 9 ⊢ ((𝑥 ∪ {𝑦}) ⊆ 𝐴 → 𝑥 ⊆ 𝐴)
40	39	imim1i 63	. . . . . . . 8 ⊢ ((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	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → 𝐷 = 𝐶)
42	2	ad2antrr 736	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝐴 ∈ Fin)
43		fprod2d.3	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ Fin)
44	43	ad4ant14 762	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ Fin)
45		fprod2d.4	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐶 ∈ ℂ)
46	45	ad4ant14 762	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐶 ∈ ℂ)
47		simplr 778	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → ¬ 𝑦 ∈ 𝑥)
48		simpr 488	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (𝑥 ∪ {𝑦}) ⊆ 𝐴)
49		biid 263	. . . . . . . . . . 11 ⊢ (∏𝑗 ∈ 𝑥 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷 ↔ ∏𝑗 ∈ 𝑥 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷)
50	41, 42, 44, 46, 47, 48, 49	fprod2dlem 16010	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ¬ 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)
51	50	exp31 423	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 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	a2d 29	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 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)))
53	40, 52	syl5 34	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 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)))
54	53	expcom 417	. . . . . 6 ⊢ (¬ 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))))
55	54	a2d 29	. . . . 5 ⊢ (¬ 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))))
56	55	adantl 485	. . . 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 9134	. . 3 ⊢ (𝐴 ∈ Fin → (𝜑 → (𝐴 ⊆ 𝐴 → ∏𝑗 ∈ 𝐴 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐷)))
58	2, 57	mpcom 38	. 2 ⊢ (𝜑 → (𝐴 ⊆ 𝐴 → ∏𝑗 ∈ 𝐴 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐷))
59	1, 58	mpi 20	1 ⊢ (𝜑 → ∏𝑗 ∈ 𝐴 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐷)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1560   ∈ wcel 2142   ∪ cun 3902   ⊆ wss 3904  ∅c0 4285  {csn 4582  ⟨cop 4588  ∪ ciun 4949   × cxp 5645  Fincfn 8927  ℂcc 11071  1c1 11074  ∏cprod 15933

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718  ax-inf2 9596  ax-cnex 11129  ax-resscn 11130  ax-1cn 11131  ax-icn 11132  ax-addcl 11133  ax-addrcl 11134  ax-mulcl 11135  ax-mulrcl 11136  ax-mulcom 11137  ax-addass 11138  ax-mulass 11139  ax-distr 11140  ax-i2m1 11141  ax-1ne0 11142  ax-1rid 11143  ax-rnegex 11144  ax-rrecex 11145  ax-cnre 11146  ax-pre-lttri 11147  ax-pre-lttrn 11148  ax-pre-ltadd 11149  ax-pre-mulgt0 11150  ax-pre-sup 11151

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4906  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-se 5601  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-isom 6530  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7847  df-1st 7970  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-1o 8437  df-er 8678  df-en 8928  df-dom 8929  df-sdom 8930  df-fin 8931  df-sup 9388  df-oi 9458  df-card 9897  df-pnf 11218  df-mnf 11219  df-xr 11220  df-ltxr 11221  df-le 11222  df-sub 11416  df-neg 11417  df-div 11845  df-nn 12211  df-2 12280  df-3 12281  df-n0 12482  df-z 12569  df-uz 12840  df-rp 12994  df-fz 13513  df-fzo 13660  df-seq 14015  df-exp 14075  df-hash 14344  df-cj 15126  df-re 15127  df-im 15128  df-sqrt 15262  df-abs 15263  df-clim 15515  df-prod 15934

This theorem is referenced by:  fprodxp  16012  fprodcom2  16014