Theorem fprodrec 11570

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 11494	. . 3 ⊢ (𝑤 = ∅ → ∏𝑘 ∈ 𝑤 (1 / 𝐵) = ∏𝑘 ∈ ∅ (1 / 𝐵))
2		prodeq1 11494	. . . 4 ⊢ (𝑤 = ∅ → ∏𝑘 ∈ 𝑤 𝐵 = ∏𝑘 ∈ ∅ 𝐵)
3	2	oveq2d 5858	. . 3 ⊢ (𝑤 = ∅ → (1 / ∏𝑘 ∈ 𝑤 𝐵) = (1 / ∏𝑘 ∈ ∅ 𝐵))
4	1, 3	eqeq12d 2180	. 2 ⊢ (𝑤 = ∅ → (∏𝑘 ∈ 𝑤 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝑤 𝐵) ↔ ∏𝑘 ∈ ∅ (1 / 𝐵) = (1 / ∏𝑘 ∈ ∅ 𝐵)))
5		prodeq1 11494	. . 3 ⊢ (𝑤 = 𝑦 → ∏𝑘 ∈ 𝑤 (1 / 𝐵) = ∏𝑘 ∈ 𝑦 (1 / 𝐵))
6		prodeq1 11494	. . . 4 ⊢ (𝑤 = 𝑦 → ∏𝑘 ∈ 𝑤 𝐵 = ∏𝑘 ∈ 𝑦 𝐵)
7	6	oveq2d 5858	. . 3 ⊢ (𝑤 = 𝑦 → (1 / ∏𝑘 ∈ 𝑤 𝐵) = (1 / ∏𝑘 ∈ 𝑦 𝐵))
8	5, 7	eqeq12d 2180	. 2 ⊢ (𝑤 = 𝑦 → (∏𝑘 ∈ 𝑤 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝑤 𝐵) ↔ ∏𝑘 ∈ 𝑦 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝑦 𝐵)))
9		prodeq1 11494	. . 3 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ∏𝑘 ∈ 𝑤 (1 / 𝐵) = ∏𝑘 ∈ (𝑦 ∪ {𝑧})(1 / 𝐵))
10		prodeq1 11494	. . . 4 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ∏𝑘 ∈ 𝑤 𝐵 = ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)
11	10	oveq2d 5858	. . 3 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (1 / ∏𝑘 ∈ 𝑤 𝐵) = (1 / ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵))
12	9, 11	eqeq12d 2180	. 2 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (∏𝑘 ∈ 𝑤 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝑤 𝐵) ↔ ∏𝑘 ∈ (𝑦 ∪ {𝑧})(1 / 𝐵) = (1 / ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)))
13		prodeq1 11494	. . 3 ⊢ (𝑤 = 𝐴 → ∏𝑘 ∈ 𝑤 (1 / 𝐵) = ∏𝑘 ∈ 𝐴 (1 / 𝐵))
14		prodeq1 11494	. . . 4 ⊢ (𝑤 = 𝐴 → ∏𝑘 ∈ 𝑤 𝐵 = ∏𝑘 ∈ 𝐴 𝐵)
15	14	oveq2d 5858	. . 3 ⊢ (𝑤 = 𝐴 → (1 / ∏𝑘 ∈ 𝑤 𝐵) = (1 / ∏𝑘 ∈ 𝐴 𝐵))
16	13, 15	eqeq12d 2180	. 2 ⊢ (𝑤 = 𝐴 → (∏𝑘 ∈ 𝑤 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝑤 𝐵) ↔ ∏𝑘 ∈ 𝐴 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝐴 𝐵)))
17		1div1e1 8600	. . . 4 ⊢ (1 / 1) = 1
18		prod0 11526	. . . . 5 ⊢ ∏𝑘 ∈ ∅ 𝐵 = 1
19	18	oveq2i 5853	. . . 4 ⊢ (1 / ∏𝑘 ∈ ∅ 𝐵) = (1 / 1)
20		prod0 11526	. . . 4 ⊢ ∏𝑘 ∈ ∅ (1 / 𝐵) = 1
21	17, 19, 20	3eqtr4ri 2197	. . 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 5857	. . . . 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 7915	. . . . . . 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 3143	. . . . . . . . . 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 11548	. . . . . . . 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 3127	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ 𝐴)
37	31	ralrimiva 2539	. . . . . . . . . 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 3078	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵
40	39	nfel1 2319	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ
41		csbeq1a 3054	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑘⦌𝐵)
42	41	eleq1d 2235	. . . . . . . . . 10 ⊢ (𝑘 = 𝑧 → (𝐵 ∈ ℂ ↔ ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ))
43	40, 42	rspc 2824	. . . . . . . . 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 11562	. . . . . . . 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 2539	. . . . . . . . . 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 2308	. . . . . . . . . . 11 ⊢ Ⅎ𝑘 #
53		nfcv 2308	. . . . . . . . . . 11 ⊢ Ⅎ𝑘0
54	39, 52, 53	nfbr 4028	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵 # 0
55	41	breq1d 3992	. . . . . . . . . 10 ⊢ (𝑘 = 𝑧 → (𝐵 # 0 ↔ ⦋𝑧 / 𝑘⦌𝐵 # 0))
56	54, 55	rspc 2824	. . . . . . . . 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 8728	. . . . . 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 9009	. . . . . . 7 ⊢ (1 · 1) = 1
61	60	oveq1i 5852	. . . . . 6 ⊢ ((1 · 1) / (∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵)) = (1 / (∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵))
62	59, 61	eqtrdi 2215	. . . . 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 2198	. . . 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 2308	. . . . . . 7 ⊢ Ⅎ𝑘1
65		nfcv 2308	. . . . . . 7 ⊢ Ⅎ𝑘 /
66	64, 65, 39	nfov 5872	. . . . . 6 ⊢ Ⅎ𝑘(1 / ⦋𝑧 / 𝑘⦌𝐵)
67	35	eldifbd 3128	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ¬ 𝑧 ∈ 𝑦)
68	32, 47	recclapd 8677	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → (1 / 𝐵) ∈ ℂ)
69	44, 57	recclapd 8677	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (1 / ⦋𝑧 / 𝑘⦌𝐵) ∈ ℂ)
70	41	oveq2d 5858	. . . . . 6 ⊢ (𝑘 = 𝑧 → (1 / 𝐵) = (1 / ⦋𝑧 / 𝑘⦌𝐵))
71	66, 26, 35, 67, 68, 69, 70	fprodunsn 11545	. . . . 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 11545	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = (∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵))
74	73	oveq2d 5858	. . . . 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 2208	. . 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 6858	1 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝐴 𝐵))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1343   ∈ wcel 2136  ∀wral 2444  ⦋csb 3045   ∖ cdif 3113   ∪ cun 3114   ⊆ wss 3116  ∅c0 3409  {csn 3576   class class class wbr 3982  (class class class)co 5842  Fincfn 6706  ℂcc 7751  0cc0 7753  1c1 7754   · cmul 7758   # cap 8479   / cdiv 8568  ∏cprod 11491

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-isom 5197  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-frec 6359  df-1o 6384  df-oadd 6388  df-er 6501  df-en 6707  df-dom 6708  df-fin 6709  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-fz 9945  df-fzo 10078  df-seqfrec 10381  df-exp 10455  df-ihash 10689  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-clim 11220  df-proddc 11492

This theorem is referenced by:  fproddivap  11571