Theorem fprodrec 11526

Description: The finite product of reciprocals is the reciprocal of the product. (Contributed by Jim Kingdon, 28-Aug-2024.)

Hypotheses

Ref	Expression
fprodrec.a	⊢ (𝜑 → 𝐴 ∈ Fin)
fprodrec.ccl	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
fprodrec.cap	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 # 0)

Assertion

Ref	Expression
fprodrec	⊢ (𝜑 → ∏𝑘 ∈ 𝐴 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝐴 𝐵))

Distinct variable groups:   𝐴,𝑘   𝜑,𝑘

Allowed substitution hint:   𝐵(𝑘)

Proof of Theorem fprodrec

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

Step	Hyp	Ref	Expression
1		prodeq1 11450	. . 3 ⊢ (𝑤 = ∅ → ∏𝑘 ∈ 𝑤 (1 / 𝐵) = ∏𝑘 ∈ ∅ (1 / 𝐵))
2		prodeq1 11450	. . . 4 ⊢ (𝑤 = ∅ → ∏𝑘 ∈ 𝑤 𝐵 = ∏𝑘 ∈ ∅ 𝐵)
3	2	oveq2d 5840	. . 3 ⊢ (𝑤 = ∅ → (1 / ∏𝑘 ∈ 𝑤 𝐵) = (1 / ∏𝑘 ∈ ∅ 𝐵))
4	1, 3	eqeq12d 2172	. 2 ⊢ (𝑤 = ∅ → (∏𝑘 ∈ 𝑤 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝑤 𝐵) ↔ ∏𝑘 ∈ ∅ (1 / 𝐵) = (1 / ∏𝑘 ∈ ∅ 𝐵)))
5		prodeq1 11450	. . 3 ⊢ (𝑤 = 𝑦 → ∏𝑘 ∈ 𝑤 (1 / 𝐵) = ∏𝑘 ∈ 𝑦 (1 / 𝐵))
6		prodeq1 11450	. . . 4 ⊢ (𝑤 = 𝑦 → ∏𝑘 ∈ 𝑤 𝐵 = ∏𝑘 ∈ 𝑦 𝐵)
7	6	oveq2d 5840	. . 3 ⊢ (𝑤 = 𝑦 → (1 / ∏𝑘 ∈ 𝑤 𝐵) = (1 / ∏𝑘 ∈ 𝑦 𝐵))
8	5, 7	eqeq12d 2172	. 2 ⊢ (𝑤 = 𝑦 → (∏𝑘 ∈ 𝑤 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝑤 𝐵) ↔ ∏𝑘 ∈ 𝑦 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝑦 𝐵)))
9		prodeq1 11450	. . 3 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ∏𝑘 ∈ 𝑤 (1 / 𝐵) = ∏𝑘 ∈ (𝑦 ∪ {𝑧})(1 / 𝐵))
10		prodeq1 11450	. . . 4 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ∏𝑘 ∈ 𝑤 𝐵 = ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)
11	10	oveq2d 5840	. . 3 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (1 / ∏𝑘 ∈ 𝑤 𝐵) = (1 / ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵))
12	9, 11	eqeq12d 2172	. 2 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (∏𝑘 ∈ 𝑤 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝑤 𝐵) ↔ ∏𝑘 ∈ (𝑦 ∪ {𝑧})(1 / 𝐵) = (1 / ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)))
13		prodeq1 11450	. . 3 ⊢ (𝑤 = 𝐴 → ∏𝑘 ∈ 𝑤 (1 / 𝐵) = ∏𝑘 ∈ 𝐴 (1 / 𝐵))
14		prodeq1 11450	. . . 4 ⊢ (𝑤 = 𝐴 → ∏𝑘 ∈ 𝑤 𝐵 = ∏𝑘 ∈ 𝐴 𝐵)
15	14	oveq2d 5840	. . 3 ⊢ (𝑤 = 𝐴 → (1 / ∏𝑘 ∈ 𝑤 𝐵) = (1 / ∏𝑘 ∈ 𝐴 𝐵))
16	13, 15	eqeq12d 2172	. 2 ⊢ (𝑤 = 𝐴 → (∏𝑘 ∈ 𝑤 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝑤 𝐵) ↔ ∏𝑘 ∈ 𝐴 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝐴 𝐵)))
17		1div1e1 8577	. . . 4 ⊢ (1 / 1) = 1
18		prod0 11482	. . . . 5 ⊢ ∏𝑘 ∈ ∅ 𝐵 = 1
19	18	oveq2i 5835	. . . 4 ⊢ (1 / ∏𝑘 ∈ ∅ 𝐵) = (1 / 1)
20		prod0 11482	. . . 4 ⊢ ∏𝑘 ∈ ∅ (1 / 𝐵) = 1
21	17, 19, 20	3eqtr4ri 2189	. . 3 ⊢ ∏𝑘 ∈ ∅ (1 / 𝐵) = (1 / ∏𝑘 ∈ ∅ 𝐵)
22	21	a1i 9	. 2 ⊢ (𝜑 → ∏𝑘 ∈ ∅ (1 / 𝐵) = (1 / ∏𝑘 ∈ ∅ 𝐵))
23		simpr 109	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∏𝑘 ∈ 𝑦 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝑦 𝐵)) → ∏𝑘 ∈ 𝑦 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝑦 𝐵))
24	23	oveq1d 5839	. . . . 5 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∏𝑘 ∈ 𝑦 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝑦 𝐵)) → (∏𝑘 ∈ 𝑦 (1 / 𝐵) · (1 / ⦋𝑧 / 𝑘⦌𝐵)) = ((1 / ∏𝑘 ∈ 𝑦 𝐵) · (1 / ⦋𝑧 / 𝑘⦌𝐵)))
25		1cnd 7894	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∏𝑘 ∈ 𝑦 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝑦 𝐵)) → 1 ∈ ℂ)
26		simplr 520	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑦 ∈ Fin)
27		simplll 523	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝜑)
28		simplrl 525	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝑦 ⊆ 𝐴)
29		simpr 109	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝑘 ∈ 𝑦)
30	28, 29	sseldd 3129	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝑘 ∈ 𝐴)
31		fprodrec.ccl	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
32	27, 30, 31	syl2anc 409	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝐵 ∈ ℂ)
33	26, 32	fprodcl 11504	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∏𝑘 ∈ 𝑦 𝐵 ∈ ℂ)
34	33	adantr 274	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∏𝑘 ∈ 𝑦 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝑦 𝐵)) → ∏𝑘 ∈ 𝑦 𝐵 ∈ ℂ)
35		simprr 522	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ (𝐴 ∖ 𝑦))
36	35	eldifad 3113	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ 𝐴)
37	31	ralrimiva 2530	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
38	37	ad2antrr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
39		nfcsb1v 3064	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵
40	39	nfel1 2310	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ
41		csbeq1a 3040	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑘⦌𝐵)
42	41	eleq1d 2226	. . . . . . . . . 10 ⊢ (𝑘 = 𝑧 → (𝐵 ∈ ℂ ↔ ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ))
43	40, 42	rspc 2810	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐴 → (∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ))
44	36, 38, 43	sylc 62	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ)
45	44	adantr 274	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∏𝑘 ∈ 𝑦 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝑦 𝐵)) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ)
46		fprodrec.cap	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 # 0)
47	27, 30, 46	syl2anc 409	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝐵 # 0)
48	26, 32, 47	fprodap0 11518	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∏𝑘 ∈ 𝑦 𝐵 # 0)
49	48	adantr 274	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∏𝑘 ∈ 𝑦 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝑦 𝐵)) → ∏𝑘 ∈ 𝑦 𝐵 # 0)
50	46	ralrimiva 2530	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 # 0)
51	50	ad2antrr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∀𝑘 ∈ 𝐴 𝐵 # 0)
52		nfcv 2299	. . . . . . . . . . 11 ⊢ Ⅎ𝑘 #
53		nfcv 2299	. . . . . . . . . . 11 ⊢ Ⅎ𝑘0
54	39, 52, 53	nfbr 4010	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵 # 0
55	41	breq1d 3975	. . . . . . . . . 10 ⊢ (𝑘 = 𝑧 → (𝐵 # 0 ↔ ⦋𝑧 / 𝑘⦌𝐵 # 0))
56	54, 55	rspc 2810	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐴 → (∀𝑘 ∈ 𝐴 𝐵 # 0 → ⦋𝑧 / 𝑘⦌𝐵 # 0))
57	36, 51, 56	sylc 62	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ⦋𝑧 / 𝑘⦌𝐵 # 0)
58	57	adantr 274	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∏𝑘 ∈ 𝑦 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝑦 𝐵)) → ⦋𝑧 / 𝑘⦌𝐵 # 0)
59	25, 34, 25, 45, 49, 58	divmuldivapd 8705	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∏𝑘 ∈ 𝑦 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝑦 𝐵)) → ((1 / ∏𝑘 ∈ 𝑦 𝐵) · (1 / ⦋𝑧 / 𝑘⦌𝐵)) = ((1 · 1) / (∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵)))
60		1t1e1 8985	. . . . . . 7 ⊢ (1 · 1) = 1
61	60	oveq1i 5834	. . . . . 6 ⊢ ((1 · 1) / (∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵)) = (1 / (∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵))
62	59, 61	eqtrdi 2206	. . . . 5 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∏𝑘 ∈ 𝑦 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝑦 𝐵)) → ((1 / ∏𝑘 ∈ 𝑦 𝐵) · (1 / ⦋𝑧 / 𝑘⦌𝐵)) = (1 / (∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵)))
63	24, 62	eqtrd 2190	. . . 4 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∏𝑘 ∈ 𝑦 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝑦 𝐵)) → (∏𝑘 ∈ 𝑦 (1 / 𝐵) · (1 / ⦋𝑧 / 𝑘⦌𝐵)) = (1 / (∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵)))
64		nfcv 2299	. . . . . . 7 ⊢ Ⅎ𝑘1
65		nfcv 2299	. . . . . . 7 ⊢ Ⅎ𝑘 /
66	64, 65, 39	nfov 5851	. . . . . 6 ⊢ Ⅎ𝑘(1 / ⦋𝑧 / 𝑘⦌𝐵)
67	35	eldifbd 3114	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ¬ 𝑧 ∈ 𝑦)
68	32, 47	recclapd 8654	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → (1 / 𝐵) ∈ ℂ)
69	44, 57	recclapd 8654	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (1 / ⦋𝑧 / 𝑘⦌𝐵) ∈ ℂ)
70	41	oveq2d 5840	. . . . . 6 ⊢ (𝑘 = 𝑧 → (1 / 𝐵) = (1 / ⦋𝑧 / 𝑘⦌𝐵))
71	66, 26, 35, 67, 68, 69, 70	fprodunsn 11501	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})(1 / 𝐵) = (∏𝑘 ∈ 𝑦 (1 / 𝐵) · (1 / ⦋𝑧 / 𝑘⦌𝐵)))
72	71	adantr 274	. . . 4 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∏𝑘 ∈ 𝑦 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝑦 𝐵)) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})(1 / 𝐵) = (∏𝑘 ∈ 𝑦 (1 / 𝐵) · (1 / ⦋𝑧 / 𝑘⦌𝐵)))
73	39, 26, 35, 67, 32, 44, 41	fprodunsn 11501	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = (∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵))
74	73	oveq2d 5840	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (1 / ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = (1 / (∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵)))
75	74	adantr 274	. . . 4 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∏𝑘 ∈ 𝑦 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝑦 𝐵)) → (1 / ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = (1 / (∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵)))
76	63, 72, 75	3eqtr4d 2200	. . 3 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∏𝑘 ∈ 𝑦 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝑦 𝐵)) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})(1 / 𝐵) = (1 / ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵))
77	76	ex 114	. 2 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (∏𝑘 ∈ 𝑦 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝑦 𝐵) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})(1 / 𝐵) = (1 / ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)))
78		fprodrec.a	. 2 ⊢ (𝜑 → 𝐴 ∈ Fin)
79	4, 8, 12, 16, 22, 77, 78	findcard2sd 6837	1 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝐴 𝐵))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1335   ∈ wcel 2128  ∀wral 2435  ⦋csb 3031   ∖ cdif 3099   ∪ cun 3100   ⊆ wss 3102  ∅c0 3394  {csn 3560   class class class wbr 3965  (class class class)co 5824  Fincfn 6685  ℂcc 7730  0cc0 7732  1c1 7733   · cmul 7737   # cap 8456   / cdiv 8545  ∏cprod 11447

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547  ax-cnex 7823  ax-resscn 7824  ax-1cn 7825  ax-1re 7826  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-mulrcl 7831  ax-addcom 7832  ax-mulcom 7833  ax-addass 7834  ax-mulass 7835  ax-distr 7836  ax-i2m1 7837  ax-0lt1 7838  ax-1rid 7839  ax-0id 7840  ax-rnegex 7841  ax-precex 7842  ax-cnre 7843  ax-pre-ltirr 7844  ax-pre-ltwlin 7845  ax-pre-lttrn 7846  ax-pre-apti 7847  ax-pre-ltadd 7848  ax-pre-mulgt0 7849  ax-pre-mulext 7850  ax-arch 7851  ax-caucvg 7852

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-ilim 4329  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-isom 5179  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-recs 6252  df-irdg 6317  df-frec 6338  df-1o 6363  df-oadd 6367  df-er 6480  df-en 6686  df-dom 6687  df-fin 6688  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917  df-le 7918  df-sub 8048  df-neg 8049  df-reap 8450  df-ap 8457  df-div 8546  df-inn 8834  df-2 8892  df-3 8893  df-4 8894  df-n0 9091  df-z 9168  df-uz 9440  df-q 9529  df-rp 9561  df-fz 9913  df-fzo 10042  df-seqfrec 10345  df-exp 10419  df-ihash 10650  df-cj 10742  df-re 10743  df-im 10744  df-rsqrt 10898  df-abs 10899  df-clim 11176  df-proddc 11448

This theorem is referenced by:  fproddivap  11527