Theorem fprod2d 16046

Description: Write a double product as a product over a two-dimensional region. Compare fsum2d 15833. (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 3962	. 2 ⊢ 𝐴 ⊆ 𝐴
2		fprod2d.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ Fin)
3		sseq1 3965	. . . . . 6 ⊢ (𝑤 = ∅ → (𝑤 ⊆ 𝐴 ↔ ∅ ⊆ 𝐴))
4		prodeq1 15972	. . . . . . 7 ⊢ (𝑤 = ∅ → ∏𝑗 ∈ 𝑤 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑗 ∈ ∅ ∏𝑘 ∈ 𝐵 𝐶)
5		iuneq1 4976	. . . . . . . . 9 ⊢ (𝑤 = ∅ → ∪ 𝑗 ∈ 𝑤 ({𝑗} × 𝐵) = ∪ 𝑗 ∈ ∅ ({𝑗} × 𝐵))
6		0iun 5030	. . . . . . . . 9 ⊢ ∪ 𝑗 ∈ ∅ ({𝑗} × 𝐵) = ∅
7	5, 6	eqtrdi 2817	. . . . . . . 8 ⊢ (𝑤 = ∅ → ∪ 𝑗 ∈ 𝑤 ({𝑗} × 𝐵) = ∅)
8	7	prodeq1d 15985	. . . . . . 7 ⊢ (𝑤 = ∅ → ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑤 ({𝑗} × 𝐵)𝐷 = ∏𝑧 ∈ ∅ 𝐷)
9	4, 8	eqeq12d 2782	. . . . . 6 ⊢ (𝑤 = ∅ → (∏𝑗 ∈ 𝑤 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑤 ({𝑗} × 𝐵)𝐷 ↔ ∏𝑗 ∈ ∅ ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∅ 𝐷))
10	3, 9	imbi12d 347	. . . . 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 343	. . . 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 3965	. . . . . 6 ⊢ (𝑤 = 𝑥 → (𝑤 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐴))
13		prodeq1 15972	. . . . . . 7 ⊢ (𝑤 = 𝑥 → ∏𝑗 ∈ 𝑤 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑗 ∈ 𝑥 ∏𝑘 ∈ 𝐵 𝐶)
14		iuneq1 4976	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → ∪ 𝑗 ∈ 𝑤 ({𝑗} × 𝐵) = ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵))
15	14	prodeq1d 15985	. . . . . . 7 ⊢ (𝑤 = 𝑥 → ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑤 ({𝑗} × 𝐵)𝐷 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷)
16	13, 15	eqeq12d 2782	. . . . . 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 347	. . . . 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 343	. . . 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 3965	. . . . . 6 ⊢ (𝑤 = (𝑥 ∪ {𝑦}) → (𝑤 ⊆ 𝐴 ↔ (𝑥 ∪ {𝑦}) ⊆ 𝐴))
20		prodeq1 15972	. . . . . . 7 ⊢ (𝑤 = (𝑥 ∪ {𝑦}) → ∏𝑗 ∈ 𝑤 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘 ∈ 𝐵 𝐶)
21		iuneq1 4976	. . . . . . . 8 ⊢ (𝑤 = (𝑥 ∪ {𝑦}) → ∪ 𝑗 ∈ 𝑤 ({𝑗} × 𝐵) = ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵))
22	21	prodeq1d 15985	. . . . . . 7 ⊢ (𝑤 = (𝑥 ∪ {𝑦}) → ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑤 ({𝑗} × 𝐵)𝐷 = ∏𝑧 ∈ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷)
23	20, 22	eqeq12d 2782	. . . . . 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 347	. . . . 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 343	. . . 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 3965	. . . . . 6 ⊢ (𝑤 = 𝐴 → (𝑤 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
27		prodeq1 15972	. . . . . . 7 ⊢ (𝑤 = 𝐴 → ∏𝑗 ∈ 𝑤 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑗 ∈ 𝐴 ∏𝑘 ∈ 𝐵 𝐶)
28		iuneq1 4976	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → ∪ 𝑗 ∈ 𝑤 ({𝑗} × 𝐵) = ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
29	28	prodeq1d 15985	. . . . . . 7 ⊢ (𝑤 = 𝐴 → ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑤 ({𝑗} × 𝐵)𝐷 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐷)
30	27, 29	eqeq12d 2782	. . . . . 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 347	. . . . 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 343	. . . 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 16008	. . . . . 6 ⊢ ∏𝑗 ∈ ∅ ∏𝑘 ∈ 𝐵 𝐶 = 1
34		prod0 16008	. . . . . 6 ⊢ ∏𝑧 ∈ ∅ 𝐷 = 1
35	33, 34	eqtr4i 2792	. . . . 5 ⊢ ∏𝑗 ∈ ∅ ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∅ 𝐷
36	35	2a1i 12	. . . 4 ⊢ (𝜑 → (∅ ⊆ 𝐴 → ∏𝑗 ∈ ∅ ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∅ 𝐷))
37		ssun1 4134	. . . . . . . . . 10 ⊢ 𝑥 ⊆ (𝑥 ∪ {𝑦})
38		sstr 3948	. . . . . . . . . 10 ⊢ ((𝑥 ⊆ (𝑥 ∪ {𝑦}) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑥 ⊆ 𝐴)
39	37, 38	mpan 703	. . . . . . . . 9 ⊢ ((𝑥 ∪ {𝑦}) ⊆ 𝐴 → 𝑥 ⊆ 𝐴)
40	39	imim1i 64	. . . . . . . 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 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝐴 ∈ Fin)
43		fprod2d.3	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ Fin)
44	43	ad4ant14 765	. . . . . . . . . . 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 765	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐶 ∈ ℂ)
47		simplr 781	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → ¬ 𝑦 ∈ 𝑥)
48		simpr 490	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ 𝑦 ∈ 𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (𝑥 ∪ {𝑦}) ⊆ 𝐴)
49		biid 264	. . . . . . . . . . 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 16045	. . . . . . . . . 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 425	. . . . . . . . 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 30	. . . . . . . 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 35	. . . . . . 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 419	. . . . . 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 30	. . . . 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 487	. . . 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 9152	. . 3 ⊢ (𝐴 ∈ Fin → (𝜑 → (𝐴 ⊆ 𝐴 → ∏𝑗 ∈ 𝐴 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐷)))
58	2, 57	mpcom 39	. 2 ⊢ (𝜑 → (𝐴 ⊆ 𝐴 → ∏𝑗 ∈ 𝐴 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐷))
59	1, 58	mpi 21	1 ⊢ (𝜑 → ∏𝑗 ∈ 𝐴 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐷)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146   ∪ cun 3906   ⊆ wss 3908  ∅c0 4289  {csn 4592  ⟨cop 4598  ∪ ciun 4959   × cxp 5662  Fincfn 8945  ℂcc 11108  1c1 11111  ∏cprod 15968

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5241  ax-sep 5260  ax-nul 5272  ax-pow 5339  ax-pr 5407  ax-un 7738  ax-inf2 9612  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-nel 3068  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4876  df-int 4916  df-iun 4961  df-br 5113  df-opab 5177  df-mpt 5196  df-tr 5222  df-id 5559  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-se 5618  df-we 5619  df-xp 5670  df-rel 5671  df-cnv 5672  df-co 5673  df-dm 5674  df-rn 5675  df-res 5676  df-ima 5677  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7865  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-er 8696  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-sup 9404  df-oi 9474  df-card 9936  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11453  df-neg 11454  df-div 11882  df-nn 12244  df-2 12313  df-3 12314  df-n0 12515  df-z 12602  df-uz 12873  df-rp 13027  df-fz 13546  df-fzo 13694  df-seq 14049  df-exp 14109  df-hash 14378  df-cj 15161  df-re 15162  df-im 15163  df-sqrt 15297  df-abs 15298  df-clim 15550  df-prod 15969

This theorem is used by:  fprodxp  16047  fprodcom2  16049