Theorem fprodrec 11794

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 11718	. . 3 ⊢ (𝑤 = ∅ → ∏𝑘 ∈ 𝑤 (1 / 𝐵) = ∏𝑘 ∈ ∅ (1 / 𝐵))
2		prodeq1 11718	. . . 4 ⊢ (𝑤 = ∅ → ∏𝑘 ∈ 𝑤 𝐵 = ∏𝑘 ∈ ∅ 𝐵)
3	2	oveq2d 5938	. . 3 ⊢ (𝑤 = ∅ → (1 / ∏𝑘 ∈ 𝑤 𝐵) = (1 / ∏𝑘 ∈ ∅ 𝐵))
4	1, 3	eqeq12d 2211	. 2 ⊢ (𝑤 = ∅ → (∏𝑘 ∈ 𝑤 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝑤 𝐵) ↔ ∏𝑘 ∈ ∅ (1 / 𝐵) = (1 / ∏𝑘 ∈ ∅ 𝐵)))
5		prodeq1 11718	. . 3 ⊢ (𝑤 = 𝑦 → ∏𝑘 ∈ 𝑤 (1 / 𝐵) = ∏𝑘 ∈ 𝑦 (1 / 𝐵))
6		prodeq1 11718	. . . 4 ⊢ (𝑤 = 𝑦 → ∏𝑘 ∈ 𝑤 𝐵 = ∏𝑘 ∈ 𝑦 𝐵)
7	6	oveq2d 5938	. . 3 ⊢ (𝑤 = 𝑦 → (1 / ∏𝑘 ∈ 𝑤 𝐵) = (1 / ∏𝑘 ∈ 𝑦 𝐵))
8	5, 7	eqeq12d 2211	. 2 ⊢ (𝑤 = 𝑦 → (∏𝑘 ∈ 𝑤 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝑤 𝐵) ↔ ∏𝑘 ∈ 𝑦 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝑦 𝐵)))
9		prodeq1 11718	. . 3 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ∏𝑘 ∈ 𝑤 (1 / 𝐵) = ∏𝑘 ∈ (𝑦 ∪ {𝑧})(1 / 𝐵))
10		prodeq1 11718	. . . 4 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ∏𝑘 ∈ 𝑤 𝐵 = ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)
11	10	oveq2d 5938	. . 3 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (1 / ∏𝑘 ∈ 𝑤 𝐵) = (1 / ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵))
12	9, 11	eqeq12d 2211	. 2 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (∏𝑘 ∈ 𝑤 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝑤 𝐵) ↔ ∏𝑘 ∈ (𝑦 ∪ {𝑧})(1 / 𝐵) = (1 / ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)))
13		prodeq1 11718	. . 3 ⊢ (𝑤 = 𝐴 → ∏𝑘 ∈ 𝑤 (1 / 𝐵) = ∏𝑘 ∈ 𝐴 (1 / 𝐵))
14		prodeq1 11718	. . . 4 ⊢ (𝑤 = 𝐴 → ∏𝑘 ∈ 𝑤 𝐵 = ∏𝑘 ∈ 𝐴 𝐵)
15	14	oveq2d 5938	. . 3 ⊢ (𝑤 = 𝐴 → (1 / ∏𝑘 ∈ 𝑤 𝐵) = (1 / ∏𝑘 ∈ 𝐴 𝐵))
16	13, 15	eqeq12d 2211	. 2 ⊢ (𝑤 = 𝐴 → (∏𝑘 ∈ 𝑤 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝑤 𝐵) ↔ ∏𝑘 ∈ 𝐴 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝐴 𝐵)))
17		1div1e1 8731	. . . 4 ⊢ (1 / 1) = 1
18		prod0 11750	. . . . 5 ⊢ ∏𝑘 ∈ ∅ 𝐵 = 1
19	18	oveq2i 5933	. . . 4 ⊢ (1 / ∏𝑘 ∈ ∅ 𝐵) = (1 / 1)
20		prod0 11750	. . . 4 ⊢ ∏𝑘 ∈ ∅ (1 / 𝐵) = 1
21	17, 19, 20	3eqtr4ri 2228	. . 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 5937	. . . . 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 8042	. . . . . . 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 3184	. . . . . . . . . 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 11772	. . . . . . . 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 3168	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ 𝐴)
37	31	ralrimiva 2570	. . . . . . . . . 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 3117	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵
40	39	nfel1 2350	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ
41		csbeq1a 3093	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑘⦌𝐵)
42	41	eleq1d 2265	. . . . . . . . . 10 ⊢ (𝑘 = 𝑧 → (𝐵 ∈ ℂ ↔ ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ))
43	40, 42	rspc 2862	. . . . . . . . 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 11786	. . . . . . . 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 2570	. . . . . . . . . 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 2339	. . . . . . . . . . 11 ⊢ Ⅎ𝑘 #
53		nfcv 2339	. . . . . . . . . . 11 ⊢ Ⅎ𝑘0
54	39, 52, 53	nfbr 4079	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵 # 0
55	41	breq1d 4043	. . . . . . . . . 10 ⊢ (𝑘 = 𝑧 → (𝐵 # 0 ↔ ⦋𝑧 / 𝑘⦌𝐵 # 0))
56	54, 55	rspc 2862	. . . . . . . . 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 8859	. . . . . 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 9143	. . . . . . 7 ⊢ (1 · 1) = 1
61	60	oveq1i 5932	. . . . . 6 ⊢ ((1 · 1) / (∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵)) = (1 / (∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵))
62	59, 61	eqtrdi 2245	. . . . 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 2229	. . . 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 2339	. . . . . . 7 ⊢ Ⅎ𝑘1
65		nfcv 2339	. . . . . . 7 ⊢ Ⅎ𝑘 /
66	64, 65, 39	nfov 5952	. . . . . 6 ⊢ Ⅎ𝑘(1 / ⦋𝑧 / 𝑘⦌𝐵)
67	35	eldifbd 3169	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ¬ 𝑧 ∈ 𝑦)
68	32, 47	recclapd 8808	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → (1 / 𝐵) ∈ ℂ)
69	44, 57	recclapd 8808	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (1 / ⦋𝑧 / 𝑘⦌𝐵) ∈ ℂ)
70	41	oveq2d 5938	. . . . . 6 ⊢ (𝑘 = 𝑧 → (1 / 𝐵) = (1 / ⦋𝑧 / 𝑘⦌𝐵))
71	66, 26, 35, 67, 68, 69, 70	fprodunsn 11769	. . . . 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 11769	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = (∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵))
74	73	oveq2d 5938	. . . . 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 2239	. . 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 6953	1 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝐴 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2167  ∀wral 2475  ⦋csb 3084   ∖ cdif 3154   ∪ cun 3155   ⊆ wss 3157  ∅c0 3450  {csn 3622   class class class wbr 4033  (class class class)co 5922  Fincfn 6799  ℂcc 7877  0cc0 7879  1c1 7880   · cmul 7884   # cap 8608   / cdiv 8699  ∏cprod 11715

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998  ax-caucvg 7999

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-isom 5267  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-frec 6449  df-1o 6474  df-oadd 6478  df-er 6592  df-en 6800  df-dom 6801  df-fin 6802  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-n0 9250  df-z 9327  df-uz 9602  df-q 9694  df-rp 9729  df-fz 10084  df-fzo 10218  df-seqfrec 10540  df-exp 10631  df-ihash 10868  df-cj 11007  df-re 11008  df-im 11009  df-rsqrt 11163  df-abs 11164  df-clim 11444  df-proddc 11716

This theorem is referenced by:  fproddivap  11795