Theorem fprodrec 12189

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 12113	. . 3 ⊢ (𝑤 = ∅ → ∏𝑘 ∈ 𝑤 (1 / 𝐵) = ∏𝑘 ∈ ∅ (1 / 𝐵))
2		prodeq1 12113	. . . 4 ⊢ (𝑤 = ∅ → ∏𝑘 ∈ 𝑤 𝐵 = ∏𝑘 ∈ ∅ 𝐵)
3	2	oveq2d 6033	. . 3 ⊢ (𝑤 = ∅ → (1 / ∏𝑘 ∈ 𝑤 𝐵) = (1 / ∏𝑘 ∈ ∅ 𝐵))
4	1, 3	eqeq12d 2246	. 2 ⊢ (𝑤 = ∅ → (∏𝑘 ∈ 𝑤 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝑤 𝐵) ↔ ∏𝑘 ∈ ∅ (1 / 𝐵) = (1 / ∏𝑘 ∈ ∅ 𝐵)))
5		prodeq1 12113	. . 3 ⊢ (𝑤 = 𝑦 → ∏𝑘 ∈ 𝑤 (1 / 𝐵) = ∏𝑘 ∈ 𝑦 (1 / 𝐵))
6		prodeq1 12113	. . . 4 ⊢ (𝑤 = 𝑦 → ∏𝑘 ∈ 𝑤 𝐵 = ∏𝑘 ∈ 𝑦 𝐵)
7	6	oveq2d 6033	. . 3 ⊢ (𝑤 = 𝑦 → (1 / ∏𝑘 ∈ 𝑤 𝐵) = (1 / ∏𝑘 ∈ 𝑦 𝐵))
8	5, 7	eqeq12d 2246	. 2 ⊢ (𝑤 = 𝑦 → (∏𝑘 ∈ 𝑤 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝑤 𝐵) ↔ ∏𝑘 ∈ 𝑦 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝑦 𝐵)))
9		prodeq1 12113	. . 3 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ∏𝑘 ∈ 𝑤 (1 / 𝐵) = ∏𝑘 ∈ (𝑦 ∪ {𝑧})(1 / 𝐵))
10		prodeq1 12113	. . . 4 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ∏𝑘 ∈ 𝑤 𝐵 = ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)
11	10	oveq2d 6033	. . 3 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (1 / ∏𝑘 ∈ 𝑤 𝐵) = (1 / ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵))
12	9, 11	eqeq12d 2246	. 2 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (∏𝑘 ∈ 𝑤 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝑤 𝐵) ↔ ∏𝑘 ∈ (𝑦 ∪ {𝑧})(1 / 𝐵) = (1 / ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)))
13		prodeq1 12113	. . 3 ⊢ (𝑤 = 𝐴 → ∏𝑘 ∈ 𝑤 (1 / 𝐵) = ∏𝑘 ∈ 𝐴 (1 / 𝐵))
14		prodeq1 12113	. . . 4 ⊢ (𝑤 = 𝐴 → ∏𝑘 ∈ 𝑤 𝐵 = ∏𝑘 ∈ 𝐴 𝐵)
15	14	oveq2d 6033	. . 3 ⊢ (𝑤 = 𝐴 → (1 / ∏𝑘 ∈ 𝑤 𝐵) = (1 / ∏𝑘 ∈ 𝐴 𝐵))
16	13, 15	eqeq12d 2246	. 2 ⊢ (𝑤 = 𝐴 → (∏𝑘 ∈ 𝑤 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝑤 𝐵) ↔ ∏𝑘 ∈ 𝐴 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝐴 𝐵)))
17		1div1e1 8883	. . . 4 ⊢ (1 / 1) = 1
18		prod0 12145	. . . . 5 ⊢ ∏𝑘 ∈ ∅ 𝐵 = 1
19	18	oveq2i 6028	. . . 4 ⊢ (1 / ∏𝑘 ∈ ∅ 𝐵) = (1 / 1)
20		prod0 12145	. . . 4 ⊢ ∏𝑘 ∈ ∅ (1 / 𝐵) = 1
21	17, 19, 20	3eqtr4ri 2263	. . 3 ⊢ ∏𝑘 ∈ ∅ (1 / 𝐵) = (1 / ∏𝑘 ∈ ∅ 𝐵)
22	21	a1i 9	. 2 ⊢ (𝜑 → ∏𝑘 ∈ ∅ (1 / 𝐵) = (1 / ∏𝑘 ∈ ∅ 𝐵))
23		simpr 110	. . . . . 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 6032	. . . . 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 8194	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∏𝑘 ∈ 𝑦 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝑦 𝐵)) → 1 ∈ ℂ)
26		simplr 529	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑦 ∈ Fin)
27		simplll 535	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝜑)
28		simplrl 537	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝑦 ⊆ 𝐴)
29		simpr 110	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝑘 ∈ 𝑦)
30	28, 29	sseldd 3228	. . . . . . . . . 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 411	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝐵 ∈ ℂ)
33	26, 32	fprodcl 12167	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∏𝑘 ∈ 𝑦 𝐵 ∈ ℂ)
34	33	adantr 276	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∏𝑘 ∈ 𝑦 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝑦 𝐵)) → ∏𝑘 ∈ 𝑦 𝐵 ∈ ℂ)
35		simprr 533	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ (𝐴 ∖ 𝑦))
36	35	eldifad 3211	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ 𝐴)
37	31	ralrimiva 2605	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
38	37	ad2antrr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
39		nfcsb1v 3160	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵
40	39	nfel1 2385	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ
41		csbeq1a 3136	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑘⦌𝐵)
42	41	eleq1d 2300	. . . . . . . . . 10 ⊢ (𝑘 = 𝑧 → (𝐵 ∈ ℂ ↔ ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ))
43	40, 42	rspc 2904	. . . . . . . . 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 276	. . . . . . 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 411	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝐵 # 0)
48	26, 32, 47	fprodap0 12181	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∏𝑘 ∈ 𝑦 𝐵 # 0)
49	48	adantr 276	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∏𝑘 ∈ 𝑦 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝑦 𝐵)) → ∏𝑘 ∈ 𝑦 𝐵 # 0)
50	46	ralrimiva 2605	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 # 0)
51	50	ad2antrr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∀𝑘 ∈ 𝐴 𝐵 # 0)
52		nfcv 2374	. . . . . . . . . . 11 ⊢ Ⅎ𝑘 #
53		nfcv 2374	. . . . . . . . . . 11 ⊢ Ⅎ𝑘0
54	39, 52, 53	nfbr 4135	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵 # 0
55	41	breq1d 4098	. . . . . . . . . 10 ⊢ (𝑘 = 𝑧 → (𝐵 # 0 ↔ ⦋𝑧 / 𝑘⦌𝐵 # 0))
56	54, 55	rspc 2904	. . . . . . . . 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 276	. . . . . . 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 9011	. . . . . 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 9295	. . . . . . 7 ⊢ (1 · 1) = 1
61	60	oveq1i 6027	. . . . . 6 ⊢ ((1 · 1) / (∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵)) = (1 / (∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵))
62	59, 61	eqtrdi 2280	. . . . 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 2264	. . . 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 2374	. . . . . . 7 ⊢ Ⅎ𝑘1
65		nfcv 2374	. . . . . . 7 ⊢ Ⅎ𝑘 /
66	64, 65, 39	nfov 6047	. . . . . 6 ⊢ Ⅎ𝑘(1 / ⦋𝑧 / 𝑘⦌𝐵)
67	35	eldifbd 3212	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ¬ 𝑧 ∈ 𝑦)
68	32, 47	recclapd 8960	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → (1 / 𝐵) ∈ ℂ)
69	44, 57	recclapd 8960	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (1 / ⦋𝑧 / 𝑘⦌𝐵) ∈ ℂ)
70	41	oveq2d 6033	. . . . . 6 ⊢ (𝑘 = 𝑧 → (1 / 𝐵) = (1 / ⦋𝑧 / 𝑘⦌𝐵))
71	66, 26, 35, 67, 68, 69, 70	fprodunsn 12164	. . . . 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 276	. . . 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 12164	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = (∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵))
74	73	oveq2d 6033	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (1 / ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = (1 / (∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵)))
75	74	adantr 276	. . . 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 2274	. . 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 115	. 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 7080	1 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝐴 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1397   ∈ wcel 2202  ∀wral 2510  ⦋csb 3127   ∖ cdif 3197   ∪ cun 3198   ⊆ wss 3200  ∅c0 3494  {csn 3669   class class class wbr 4088  (class class class)co 6017  Fincfn 6908  ℂcc 8029  0cc0 8031  1c1 8032   · cmul 8036   # cap 8760   / cdiv 8851  ∏cprod 12110

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149  ax-arch 8150  ax-caucvg 8151

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-isom 5335  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-irdg 6535  df-frec 6556  df-1o 6581  df-oadd 6585  df-er 6701  df-en 6909  df-dom 6910  df-fin 6911  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-n0 9402  df-z 9479  df-uz 9755  df-q 9853  df-rp 9888  df-fz 10243  df-fzo 10377  df-seqfrec 10709  df-exp 10800  df-ihash 11037  df-cj 11402  df-re 11403  df-im 11404  df-rsqrt 11558  df-abs 11559  df-clim 11839  df-proddc 12111

This theorem is referenced by:  fproddivap  12190