Theorem fprodrec 11639

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 11563	. . 3 ⊢ (𝑤 = ∅ → ∏𝑘 ∈ 𝑤 (1 / 𝐵) = ∏𝑘 ∈ ∅ (1 / 𝐵))
2		prodeq1 11563	. . . 4 ⊢ (𝑤 = ∅ → ∏𝑘 ∈ 𝑤 𝐵 = ∏𝑘 ∈ ∅ 𝐵)
3	2	oveq2d 5893	. . 3 ⊢ (𝑤 = ∅ → (1 / ∏𝑘 ∈ 𝑤 𝐵) = (1 / ∏𝑘 ∈ ∅ 𝐵))
4	1, 3	eqeq12d 2192	. 2 ⊢ (𝑤 = ∅ → (∏𝑘 ∈ 𝑤 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝑤 𝐵) ↔ ∏𝑘 ∈ ∅ (1 / 𝐵) = (1 / ∏𝑘 ∈ ∅ 𝐵)))
5		prodeq1 11563	. . 3 ⊢ (𝑤 = 𝑦 → ∏𝑘 ∈ 𝑤 (1 / 𝐵) = ∏𝑘 ∈ 𝑦 (1 / 𝐵))
6		prodeq1 11563	. . . 4 ⊢ (𝑤 = 𝑦 → ∏𝑘 ∈ 𝑤 𝐵 = ∏𝑘 ∈ 𝑦 𝐵)
7	6	oveq2d 5893	. . 3 ⊢ (𝑤 = 𝑦 → (1 / ∏𝑘 ∈ 𝑤 𝐵) = (1 / ∏𝑘 ∈ 𝑦 𝐵))
8	5, 7	eqeq12d 2192	. 2 ⊢ (𝑤 = 𝑦 → (∏𝑘 ∈ 𝑤 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝑤 𝐵) ↔ ∏𝑘 ∈ 𝑦 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝑦 𝐵)))
9		prodeq1 11563	. . 3 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ∏𝑘 ∈ 𝑤 (1 / 𝐵) = ∏𝑘 ∈ (𝑦 ∪ {𝑧})(1 / 𝐵))
10		prodeq1 11563	. . . 4 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ∏𝑘 ∈ 𝑤 𝐵 = ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)
11	10	oveq2d 5893	. . 3 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (1 / ∏𝑘 ∈ 𝑤 𝐵) = (1 / ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵))
12	9, 11	eqeq12d 2192	. 2 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (∏𝑘 ∈ 𝑤 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝑤 𝐵) ↔ ∏𝑘 ∈ (𝑦 ∪ {𝑧})(1 / 𝐵) = (1 / ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)))
13		prodeq1 11563	. . 3 ⊢ (𝑤 = 𝐴 → ∏𝑘 ∈ 𝑤 (1 / 𝐵) = ∏𝑘 ∈ 𝐴 (1 / 𝐵))
14		prodeq1 11563	. . . 4 ⊢ (𝑤 = 𝐴 → ∏𝑘 ∈ 𝑤 𝐵 = ∏𝑘 ∈ 𝐴 𝐵)
15	14	oveq2d 5893	. . 3 ⊢ (𝑤 = 𝐴 → (1 / ∏𝑘 ∈ 𝑤 𝐵) = (1 / ∏𝑘 ∈ 𝐴 𝐵))
16	13, 15	eqeq12d 2192	. 2 ⊢ (𝑤 = 𝐴 → (∏𝑘 ∈ 𝑤 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝑤 𝐵) ↔ ∏𝑘 ∈ 𝐴 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝐴 𝐵)))
17		1div1e1 8663	. . . 4 ⊢ (1 / 1) = 1
18		prod0 11595	. . . . 5 ⊢ ∏𝑘 ∈ ∅ 𝐵 = 1
19	18	oveq2i 5888	. . . 4 ⊢ (1 / ∏𝑘 ∈ ∅ 𝐵) = (1 / 1)
20		prod0 11595	. . . 4 ⊢ ∏𝑘 ∈ ∅ (1 / 𝐵) = 1
21	17, 19, 20	3eqtr4ri 2209	. . 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 5892	. . . . 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 7975	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∏𝑘 ∈ 𝑦 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝑦 𝐵)) → 1 ∈ ℂ)
26		simplr 528	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑦 ∈ Fin)
27		simplll 533	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝜑)
28		simplrl 535	. . . . . . . . . . 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 3158	. . . . . . . . . 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 11617	. . . . . . . 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 531	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ (𝐴 ∖ 𝑦))
36	35	eldifad 3142	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ 𝐴)
37	31	ralrimiva 2550	. . . . . . . . . 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 3092	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵
40	39	nfel1 2330	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ
41		csbeq1a 3068	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑘⦌𝐵)
42	41	eleq1d 2246	. . . . . . . . . 10 ⊢ (𝑘 = 𝑧 → (𝐵 ∈ ℂ ↔ ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ))
43	40, 42	rspc 2837	. . . . . . . . 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 11631	. . . . . . . 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 2550	. . . . . . . . . 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 2319	. . . . . . . . . . 11 ⊢ Ⅎ𝑘 #
53		nfcv 2319	. . . . . . . . . . 11 ⊢ Ⅎ𝑘0
54	39, 52, 53	nfbr 4051	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵 # 0
55	41	breq1d 4015	. . . . . . . . . 10 ⊢ (𝑘 = 𝑧 → (𝐵 # 0 ↔ ⦋𝑧 / 𝑘⦌𝐵 # 0))
56	54, 55	rspc 2837	. . . . . . . . 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 8791	. . . . . 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 9073	. . . . . . 7 ⊢ (1 · 1) = 1
61	60	oveq1i 5887	. . . . . 6 ⊢ ((1 · 1) / (∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵)) = (1 / (∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵))
62	59, 61	eqtrdi 2226	. . . . 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 2210	. . . 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 2319	. . . . . . 7 ⊢ Ⅎ𝑘1
65		nfcv 2319	. . . . . . 7 ⊢ Ⅎ𝑘 /
66	64, 65, 39	nfov 5907	. . . . . 6 ⊢ Ⅎ𝑘(1 / ⦋𝑧 / 𝑘⦌𝐵)
67	35	eldifbd 3143	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ¬ 𝑧 ∈ 𝑦)
68	32, 47	recclapd 8740	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → (1 / 𝐵) ∈ ℂ)
69	44, 57	recclapd 8740	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (1 / ⦋𝑧 / 𝑘⦌𝐵) ∈ ℂ)
70	41	oveq2d 5893	. . . . . 6 ⊢ (𝑘 = 𝑧 → (1 / 𝐵) = (1 / ⦋𝑧 / 𝑘⦌𝐵))
71	66, 26, 35, 67, 68, 69, 70	fprodunsn 11614	. . . . 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 11614	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = (∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵))
74	73	oveq2d 5893	. . . . 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 2220	. . 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 6894	1 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝐴 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1353   ∈ wcel 2148  ∀wral 2455  ⦋csb 3059   ∖ cdif 3128   ∪ cun 3129   ⊆ wss 3131  ∅c0 3424  {csn 3594   class class class wbr 4005  (class class class)co 5877  Fincfn 6742  ℂcc 7811  0cc0 7813  1c1 7814   · cmul 7818   # cap 8540   / cdiv 8631  ∏cprod 11560

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931  ax-arch 7932  ax-caucvg 7933

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-isom 5227  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-frec 6394  df-1o 6419  df-oadd 6423  df-er 6537  df-en 6743  df-dom 6744  df-fin 6745  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-inn 8922  df-2 8980  df-3 8981  df-4 8982  df-n0 9179  df-z 9256  df-uz 9531  df-q 9622  df-rp 9656  df-fz 10011  df-fzo 10145  df-seqfrec 10448  df-exp 10522  df-ihash 10758  df-cj 10853  df-re 10854  df-im 10855  df-rsqrt 11009  df-abs 11010  df-clim 11289  df-proddc 11561

This theorem is referenced by:  fproddivap  11640