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Theorem iprodgam 34781

Description: An infinite product version of Euler's gamma function. (Contributed by Scott Fenton, 12-Feb-2018.)

**Hypothesis**
Ref	Expression
iprodgam.1	⊢ (𝜑 → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))

**Assertion**
Ref	Expression
iprodgam	⊢ (𝜑 → (Γ‘𝐴) = (∏𝑘 ∈ ℕ (((1 + (1 / 𝑘))↑_𝑐𝐴) / (1 + (𝐴 / 𝑘))) / 𝐴))

Distinct variable groups: 𝐴,𝑘 𝜑,𝑘

Proof of Theorem iprodgam Dummy variables 𝑗 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iprodgam.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))
2		eflgam 26556	. . 3 ⊢ (𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → (exp‘(log Γ‘𝐴)) = (Γ‘𝐴))
3	1, 2	syl 17	. 2 ⊢ (𝜑 → (exp‘(log Γ‘𝐴)) = (Γ‘𝐴))
4		oveq1 7418	. . . . . . . . 9 ⊢ (𝑧 = 𝐴 → (𝑧 · (log‘((𝑘 + 1) / 𝑘))) = (𝐴 · (log‘((𝑘 + 1) / 𝑘))))
5		oveq1 7418	. . . . . . . . . 10 ⊢ (𝑧 = 𝐴 → (𝑧 / 𝑘) = (𝐴 / 𝑘))
6	5	fvoveq1d 7433	. . . . . . . . 9 ⊢ (𝑧 = 𝐴 → (log‘((𝑧 / 𝑘) + 1)) = (log‘((𝐴 / 𝑘) + 1)))
7	4, 6	oveq12d 7429	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → ((𝑧 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝑧 / 𝑘) + 1))) = ((𝐴 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1))))
8	7	sumeq2sdv 15652	. . . . . . 7 ⊢ (𝑧 = 𝐴 → Σ𝑘 ∈ ℕ ((𝑧 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝑧 / 𝑘) + 1))) = Σ𝑘 ∈ ℕ ((𝐴 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1))))
9		fveq2 6891	. . . . . . 7 ⊢ (𝑧 = 𝐴 → (log‘𝑧) = (log‘𝐴))
10	8, 9	oveq12d 7429	. . . . . 6 ⊢ (𝑧 = 𝐴 → (Σ𝑘 ∈ ℕ ((𝑧 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝑧 / 𝑘) + 1))) − (log‘𝑧)) = (Σ𝑘 ∈ ℕ ((𝐴 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1))) − (log‘𝐴)))
11		df-lgam 26530	. . . . . 6 ⊢ log Γ = (𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↦ (Σ𝑘 ∈ ℕ ((𝑧 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝑧 / 𝑘) + 1))) − (log‘𝑧)))
12		ovex 7444	. . . . . 6 ⊢ (Σ𝑘 ∈ ℕ ((𝐴 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1))) − (log‘𝐴)) ∈ V
13	10, 11, 12	fvmpt 6998	. . . . 5 ⊢ (𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → (log Γ‘𝐴) = (Σ𝑘 ∈ ℕ ((𝐴 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1))) − (log‘𝐴)))
14	1, 13	syl 17	. . . 4 ⊢ (𝜑 → (log Γ‘𝐴) = (Σ𝑘 ∈ ℕ ((𝐴 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1))) − (log‘𝐴)))
15	14	fveq2d 6895	. . 3 ⊢ (𝜑 → (exp‘(log Γ‘𝐴)) = (exp‘(Σ𝑘 ∈ ℕ ((𝐴 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1))) − (log‘𝐴))))
16		nnuz 12867	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
17		1zzd 12595	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℤ)
18		oveq1 7418	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑘 → (𝑗 + 1) = (𝑘 + 1))
19		id 22	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑘 → 𝑗 = 𝑘)
20	18, 19	oveq12d 7429	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → ((𝑗 + 1) / 𝑗) = ((𝑘 + 1) / 𝑘))
21	20	fveq2d 6895	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → (log‘((𝑗 + 1) / 𝑗)) = (log‘((𝑘 + 1) / 𝑘)))
22	21	oveq2d 7427	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → (𝐴 · (log‘((𝑗 + 1) / 𝑗))) = (𝐴 · (log‘((𝑘 + 1) / 𝑘))))
23		oveq2 7419	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → (𝐴 / 𝑗) = (𝐴 / 𝑘))
24	23	fvoveq1d 7433	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → (log‘((𝐴 / 𝑗) + 1)) = (log‘((𝐴 / 𝑘) + 1)))
25	22, 24	oveq12d 7429	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → ((𝐴 · (log‘((𝑗 + 1) / 𝑗))) − (log‘((𝐴 / 𝑗) + 1))) = ((𝐴 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1))))
26		eqid 2732	. . . . . . . 8 ⊢ (𝑗 ∈ ℕ ↦ ((𝐴 · (log‘((𝑗 + 1) / 𝑗))) − (log‘((𝐴 / 𝑗) + 1)))) = (𝑗 ∈ ℕ ↦ ((𝐴 · (log‘((𝑗 + 1) / 𝑗))) − (log‘((𝐴 / 𝑗) + 1))))
27		ovex 7444	. . . . . . . 8 ⊢ ((𝐴 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1))) ∈ V
28	25, 26, 27	fvmpt 6998	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → ((𝑗 ∈ ℕ ↦ ((𝐴 · (log‘((𝑗 + 1) / 𝑗))) − (log‘((𝐴 / 𝑗) + 1))))‘𝑘) = ((𝐴 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1))))
29	28	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑗 ∈ ℕ ↦ ((𝐴 · (log‘((𝑗 + 1) / 𝑗))) − (log‘((𝐴 / 𝑗) + 1))))‘𝑘) = ((𝐴 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1))))
30	1	eldifad 3960	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ)
31	30	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ℂ)
32		peano2nn 12226	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
33	32	adantl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℕ)
34	33	nncnd 12230	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℂ)
35		nncn 12222	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
36	35	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℂ)
37		nnne0 12248	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → 𝑘 ≠ 0)
38	37	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ≠ 0)
39	34, 36, 38	divcld 11992	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑘 + 1) / 𝑘) ∈ ℂ)
40	33	nnne0d 12264	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ≠ 0)
41	34, 36, 40, 38	divne0d 12008	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑘 + 1) / 𝑘) ≠ 0)
42	39, 41	logcld 26086	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (log‘((𝑘 + 1) / 𝑘)) ∈ ℂ)
43	31, 42	mulcld 11236	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐴 · (log‘((𝑘 + 1) / 𝑘))) ∈ ℂ)
44	31, 36, 38	divcld 11992	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐴 / 𝑘) ∈ ℂ)
45		1cnd 11211	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 1 ∈ ℂ)
46	44, 45	addcld 11235	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐴 / 𝑘) + 1) ∈ ℂ)
47	1	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))
48		simpr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
49	47, 48	dmgmdivn0 26539	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐴 / 𝑘) + 1) ≠ 0)
50	46, 49	logcld 26086	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (log‘((𝐴 / 𝑘) + 1)) ∈ ℂ)
51	43, 50	subcld 11573	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐴 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1))) ∈ ℂ)
52	26, 1	lgamcvg 26565	. . . . . . 7 ⊢ (𝜑 → seq1( + , (𝑗 ∈ ℕ ↦ ((𝐴 · (log‘((𝑗 + 1) / 𝑗))) − (log‘((𝐴 / 𝑗) + 1))))) ⇝ ((log Γ‘𝐴) + (log‘𝐴)))
53		seqex 13970	. . . . . . . 8 ⊢ seq1( + , (𝑗 ∈ ℕ ↦ ((𝐴 · (log‘((𝑗 + 1) / 𝑗))) − (log‘((𝐴 / 𝑗) + 1))))) ∈ V
54		ovex 7444	. . . . . . . 8 ⊢ ((log Γ‘𝐴) + (log‘𝐴)) ∈ V
55	53, 54	breldm 5908	. . . . . . 7 ⊢ (seq1( + , (𝑗 ∈ ℕ ↦ ((𝐴 · (log‘((𝑗 + 1) / 𝑗))) − (log‘((𝐴 / 𝑗) + 1))))) ⇝ ((log Γ‘𝐴) + (log‘𝐴)) → seq1( + , (𝑗 ∈ ℕ ↦ ((𝐴 · (log‘((𝑗 + 1) / 𝑗))) − (log‘((𝐴 / 𝑗) + 1))))) ∈ dom ⇝ )
56	52, 55	syl 17	. . . . . 6 ⊢ (𝜑 → seq1( + , (𝑗 ∈ ℕ ↦ ((𝐴 · (log‘((𝑗 + 1) / 𝑗))) − (log‘((𝐴 / 𝑗) + 1))))) ∈ dom ⇝ )
57	16, 17, 29, 51, 56	isumcl 15709	. . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ ℕ ((𝐴 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1))) ∈ ℂ)
58	1	dmgmn0 26537	. . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 0)
59	30, 58	logcld 26086	. . . . 5 ⊢ (𝜑 → (log‘𝐴) ∈ ℂ)
60		efsub 16045	. . . . 5 ⊢ ((Σ𝑘 ∈ ℕ ((𝐴 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1))) ∈ ℂ ∧ (log‘𝐴) ∈ ℂ) → (exp‘(Σ𝑘 ∈ ℕ ((𝐴 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1))) − (log‘𝐴))) = ((exp‘Σ𝑘 ∈ ℕ ((𝐴 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1)))) / (exp‘(log‘𝐴))))
61	57, 59, 60	syl2anc 584	. . . 4 ⊢ (𝜑 → (exp‘(Σ𝑘 ∈ ℕ ((𝐴 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1))) − (log‘𝐴))) = ((exp‘Σ𝑘 ∈ ℕ ((𝐴 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1)))) / (exp‘(log‘𝐴))))
62	16, 17, 29, 51, 56	iprodefisum 34780	. . . . . 6 ⊢ (𝜑 → ∏𝑘 ∈ ℕ (exp‘((𝐴 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1)))) = (exp‘Σ𝑘 ∈ ℕ ((𝐴 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1)))))
63		efsub 16045	. . . . . . . . 9 ⊢ (((𝐴 · (log‘((𝑘 + 1) / 𝑘))) ∈ ℂ ∧ (log‘((𝐴 / 𝑘) + 1)) ∈ ℂ) → (exp‘((𝐴 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1)))) = ((exp‘(𝐴 · (log‘((𝑘 + 1) / 𝑘)))) / (exp‘(log‘((𝐴 / 𝑘) + 1)))))
64	43, 50, 63	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (exp‘((𝐴 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1)))) = ((exp‘(𝐴 · (log‘((𝑘 + 1) / 𝑘)))) / (exp‘(log‘((𝐴 / 𝑘) + 1)))))
65	36, 45, 36, 38	divdird 12030	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑘 + 1) / 𝑘) = ((𝑘 / 𝑘) + (1 / 𝑘)))
66	36, 38	dividd 11990	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘 / 𝑘) = 1)
67	66	oveq1d 7426	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑘 / 𝑘) + (1 / 𝑘)) = (1 + (1 / 𝑘)))
68	65, 67	eqtrd 2772	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑘 + 1) / 𝑘) = (1 + (1 / 𝑘)))
69	68	fveq2d 6895	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (log‘((𝑘 + 1) / 𝑘)) = (log‘(1 + (1 / 𝑘))))
70	69	oveq2d 7427	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐴 · (log‘((𝑘 + 1) / 𝑘))) = (𝐴 · (log‘(1 + (1 / 𝑘)))))
71	70	fveq2d 6895	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (exp‘(𝐴 · (log‘((𝑘 + 1) / 𝑘)))) = (exp‘(𝐴 · (log‘(1 + (1 / 𝑘))))))
72		1rp 12980	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℝ⁺
73	72	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 1 ∈ ℝ⁺)
74	48	nnrpd 13016	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℝ⁺)
75	74	rpreccld 13028	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1 / 𝑘) ∈ ℝ⁺)
76	73, 75	rpaddcld 13033	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1 + (1 / 𝑘)) ∈ ℝ⁺)
77	76	rpcnd 13020	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1 + (1 / 𝑘)) ∈ ℂ)
78	76	rpne0d 13023	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1 + (1 / 𝑘)) ≠ 0)
79	77, 78, 31	cxpefd 26227	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((1 + (1 / 𝑘))↑_𝑐𝐴) = (exp‘(𝐴 · (log‘(1 + (1 / 𝑘))))))
80	71, 79	eqtr4d 2775	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (exp‘(𝐴 · (log‘((𝑘 + 1) / 𝑘)))) = ((1 + (1 / 𝑘))↑_𝑐𝐴))
81		eflog 26092	. . . . . . . . . . 11 ⊢ ((((𝐴 / 𝑘) + 1) ∈ ℂ ∧ ((𝐴 / 𝑘) + 1) ≠ 0) → (exp‘(log‘((𝐴 / 𝑘) + 1))) = ((𝐴 / 𝑘) + 1))
82	46, 49, 81	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (exp‘(log‘((𝐴 / 𝑘) + 1))) = ((𝐴 / 𝑘) + 1))
83	44, 45	addcomd 11418	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐴 / 𝑘) + 1) = (1 + (𝐴 / 𝑘)))
84	82, 83	eqtrd 2772	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (exp‘(log‘((𝐴 / 𝑘) + 1))) = (1 + (𝐴 / 𝑘)))
85	80, 84	oveq12d 7429	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((exp‘(𝐴 · (log‘((𝑘 + 1) / 𝑘)))) / (exp‘(log‘((𝐴 / 𝑘) + 1)))) = (((1 + (1 / 𝑘))↑_𝑐𝐴) / (1 + (𝐴 / 𝑘))))
86	64, 85	eqtrd 2772	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (exp‘((𝐴 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1)))) = (((1 + (1 / 𝑘))↑_𝑐𝐴) / (1 + (𝐴 / 𝑘))))
87	86	prodeq2dv 15869	. . . . . 6 ⊢ (𝜑 → ∏𝑘 ∈ ℕ (exp‘((𝐴 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1)))) = ∏𝑘 ∈ ℕ (((1 + (1 / 𝑘))↑_𝑐𝐴) / (1 + (𝐴 / 𝑘))))
88	62, 87	eqtr3d 2774	. . . . 5 ⊢ (𝜑 → (exp‘Σ𝑘 ∈ ℕ ((𝐴 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1)))) = ∏𝑘 ∈ ℕ (((1 + (1 / 𝑘))↑_𝑐𝐴) / (1 + (𝐴 / 𝑘))))
89		eflog 26092	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (exp‘(log‘𝐴)) = 𝐴)
90	30, 58, 89	syl2anc 584	. . . . 5 ⊢ (𝜑 → (exp‘(log‘𝐴)) = 𝐴)
91	88, 90	oveq12d 7429	. . . 4 ⊢ (𝜑 → ((exp‘Σ𝑘 ∈ ℕ ((𝐴 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1)))) / (exp‘(log‘𝐴))) = (∏𝑘 ∈ ℕ (((1 + (1 / 𝑘))↑_𝑐𝐴) / (1 + (𝐴 / 𝑘))) / 𝐴))
92	61, 91	eqtrd 2772	. . 3 ⊢ (𝜑 → (exp‘(Σ𝑘 ∈ ℕ ((𝐴 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1))) − (log‘𝐴))) = (∏𝑘 ∈ ℕ (((1 + (1 / 𝑘))↑_𝑐𝐴) / (1 + (𝐴 / 𝑘))) / 𝐴))
93	15, 92	eqtrd 2772	. 2 ⊢ (𝜑 → (exp‘(log Γ‘𝐴)) = (∏𝑘 ∈ ℕ (((1 + (1 / 𝑘))↑_𝑐𝐴) / (1 + (𝐴 / 𝑘))) / 𝐴))
94	3, 93	eqtr3d 2774	1 ⊢ (𝜑 → (Γ‘𝐴) = (∏𝑘 ∈ ℕ (((1 + (1 / 𝑘))↑_𝑐𝐴) / (1 + (𝐴 / 𝑘))) / 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∖ cdif 3945 class class class wbr 5148 ↦ cmpt 5231 dom cdm 5676 ‘cfv 6543 (class class class)co 7411 ℂcc 11110 0cc0 11112 1c1 11113 + caddc 11115 · cmul 11117 − cmin 11446 / cdiv 11873 ℕcn 12214 ℤcz 12560 ℝ⁺crp 12976 seqcseq 13968 ⇝ cli 15430 Σcsu 15634 ∏cprod 15851 expce 16007 logclog 26070 ↑_𝑐ccxp 26071 log Γclgam 26527 Γcgam 26528

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-oadd 8472 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-dju 9898 df-card 9936 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-div 11874 df-nn 12215 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12475 df-z 12561 df-dec 12680 df-uz 12825 df-q 12935 df-rp 12977 df-xneg 13094 df-xadd 13095 df-xmul 13096 df-ioo 13330 df-ioc 13331 df-ico 13332 df-icc 13333 df-fz 13487 df-fzo 13630 df-fl 13759 df-mod 13837 df-seq 13969 df-exp 14030 df-fac 14236 df-bc 14265 df-hash 14293 df-shft 15016 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-limsup 15417 df-clim 15434 df-rlim 15435 df-sum 15635 df-prod 15852 df-ef 16013 df-sin 16015 df-cos 16016 df-tan 16017 df-pi 16018 df-struct 17082 df-sets 17099 df-slot 17117 df-ndx 17129 df-base 17147 df-ress 17176 df-plusg 17212 df-mulr 17213 df-starv 17214 df-sca 17215 df-vsca 17216 df-ip 17217 df-tset 17218 df-ple 17219 df-ds 17221 df-unif 17222 df-hom 17223 df-cco 17224 df-rest 17370 df-topn 17371 df-0g 17389 df-gsum 17390 df-topgen 17391 df-pt 17392 df-prds 17395 df-xrs 17450 df-qtop 17455 df-imas 17456 df-xps 17458 df-mre 17532 df-mrc 17533 df-acs 17535 df-mgm 18563 df-sgrp 18612 df-mnd 18628 df-submnd 18674 df-mulg 18953 df-cntz 19183 df-cmn 19652 df-psmet 20942 df-xmet 20943 df-met 20944 df-bl 20945 df-mopn 20946 df-fbas 20947 df-fg 20948 df-cnfld 20951 df-top 22403 df-topon 22420 df-topsp 22442 df-bases 22456 df-cld 22530 df-ntr 22531 df-cls 22532 df-nei 22609 df-lp 22647 df-perf 22648 df-cn 22738 df-cnp 22739 df-haus 22826 df-cmp 22898 df-tx 23073 df-hmeo 23266 df-fil 23357 df-fm 23449 df-flim 23450 df-flf 23451 df-xms 23833 df-ms 23834 df-tms 23835 df-cncf 24401 df-limc 25390 df-dv 25391 df-ulm 25896 df-log 26072 df-cxp 26073 df-lgam 26530 df-gam 26531

This theorem is referenced by: (None)

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