Theorem faclim 33691

Description: An infinite product expression relating to factorials. Originally due to Euler. (Contributed by Scott Fenton, 22-Nov-2017.)

Hypothesis

Ref	Expression
faclim.1	⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝐴) / (1 + (𝐴 / 𝑛))))

Assertion

Ref	Expression
faclim	⊢ (𝐴 ∈ ℕ0 → seq1( · , 𝐹) ⇝ (!‘𝐴))

Distinct variable group:   𝐴,𝑛

Allowed substitution hint:   𝐹(𝑛)

Proof of Theorem faclim

Dummy variables 𝑎 𝑏 𝑚 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		faclim.1	. . 3 ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝐴) / (1 + (𝐴 / 𝑛))))
2		seqeq3 13707	. . 3 ⊢ (𝐹 = (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝐴) / (1 + (𝐴 / 𝑛)))) → seq1( · , 𝐹) = seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝐴) / (1 + (𝐴 / 𝑛))))))
3	1, 2	ax-mp 5	. 2 ⊢ seq1( · , 𝐹) = seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝐴) / (1 + (𝐴 / 𝑛)))))
4		oveq2 7276	. . . . . . 7 ⊢ (𝑎 = 0 → ((1 + (1 / 𝑛))↑𝑎) = ((1 + (1 / 𝑛))↑0))
5		oveq1 7275	. . . . . . . 8 ⊢ (𝑎 = 0 → (𝑎 / 𝑛) = (0 / 𝑛))
6	5	oveq2d 7284	. . . . . . 7 ⊢ (𝑎 = 0 → (1 + (𝑎 / 𝑛)) = (1 + (0 / 𝑛)))
7	4, 6	oveq12d 7286	. . . . . 6 ⊢ (𝑎 = 0 → (((1 + (1 / 𝑛))↑𝑎) / (1 + (𝑎 / 𝑛))) = (((1 + (1 / 𝑛))↑0) / (1 + (0 / 𝑛))))
8	7	mpteq2dv 5180	. . . . 5 ⊢ (𝑎 = 0 → (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑎) / (1 + (𝑎 / 𝑛)))) = (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑0) / (1 + (0 / 𝑛)))))
9	8	seqeq3d 13710	. . . 4 ⊢ (𝑎 = 0 → seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑎) / (1 + (𝑎 / 𝑛))))) = seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑0) / (1 + (0 / 𝑛))))))
10		fveq2 6768	. . . . 5 ⊢ (𝑎 = 0 → (!‘𝑎) = (!‘0))
11		fac0 13971	. . . . 5 ⊢ (!‘0) = 1
12	10, 11	eqtrdi 2795	. . . 4 ⊢ (𝑎 = 0 → (!‘𝑎) = 1)
13	9, 12	breq12d 5091	. . 3 ⊢ (𝑎 = 0 → (seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑎) / (1 + (𝑎 / 𝑛))))) ⇝ (!‘𝑎) ↔ seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑0) / (1 + (0 / 𝑛))))) ⇝ 1))
14		oveq2 7276	. . . . . . 7 ⊢ (𝑎 = 𝑚 → ((1 + (1 / 𝑛))↑𝑎) = ((1 + (1 / 𝑛))↑𝑚))
15		oveq1 7275	. . . . . . . 8 ⊢ (𝑎 = 𝑚 → (𝑎 / 𝑛) = (𝑚 / 𝑛))
16	15	oveq2d 7284	. . . . . . 7 ⊢ (𝑎 = 𝑚 → (1 + (𝑎 / 𝑛)) = (1 + (𝑚 / 𝑛)))
17	14, 16	oveq12d 7286	. . . . . 6 ⊢ (𝑎 = 𝑚 → (((1 + (1 / 𝑛))↑𝑎) / (1 + (𝑎 / 𝑛))) = (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))
18	17	mpteq2dv 5180	. . . . 5 ⊢ (𝑎 = 𝑚 → (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑎) / (1 + (𝑎 / 𝑛)))) = (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛)))))
19	18	seqeq3d 13710	. . . 4 ⊢ (𝑎 = 𝑚 → seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑎) / (1 + (𝑎 / 𝑛))))) = seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))))
20		fveq2 6768	. . . 4 ⊢ (𝑎 = 𝑚 → (!‘𝑎) = (!‘𝑚))
21	19, 20	breq12d 5091	. . 3 ⊢ (𝑎 = 𝑚 → (seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑎) / (1 + (𝑎 / 𝑛))))) ⇝ (!‘𝑎) ↔ seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))) ⇝ (!‘𝑚)))
22		oveq2 7276	. . . . . . 7 ⊢ (𝑎 = (𝑚 + 1) → ((1 + (1 / 𝑛))↑𝑎) = ((1 + (1 / 𝑛))↑(𝑚 + 1)))
23		oveq1 7275	. . . . . . . 8 ⊢ (𝑎 = (𝑚 + 1) → (𝑎 / 𝑛) = ((𝑚 + 1) / 𝑛))
24	23	oveq2d 7284	. . . . . . 7 ⊢ (𝑎 = (𝑚 + 1) → (1 + (𝑎 / 𝑛)) = (1 + ((𝑚 + 1) / 𝑛)))
25	22, 24	oveq12d 7286	. . . . . 6 ⊢ (𝑎 = (𝑚 + 1) → (((1 + (1 / 𝑛))↑𝑎) / (1 + (𝑎 / 𝑛))) = (((1 + (1 / 𝑛))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑛))))
26	25	mpteq2dv 5180	. . . . 5 ⊢ (𝑎 = (𝑚 + 1) → (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑎) / (1 + (𝑎 / 𝑛)))) = (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑛)))))
27	26	seqeq3d 13710	. . . 4 ⊢ (𝑎 = (𝑚 + 1) → seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑎) / (1 + (𝑎 / 𝑛))))) = seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑛))))))
28		fveq2 6768	. . . 4 ⊢ (𝑎 = (𝑚 + 1) → (!‘𝑎) = (!‘(𝑚 + 1)))
29	27, 28	breq12d 5091	. . 3 ⊢ (𝑎 = (𝑚 + 1) → (seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑎) / (1 + (𝑎 / 𝑛))))) ⇝ (!‘𝑎) ↔ seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑛))))) ⇝ (!‘(𝑚 + 1))))
30		oveq2 7276	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ((1 + (1 / 𝑛))↑𝑎) = ((1 + (1 / 𝑛))↑𝐴))
31		oveq1 7275	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑎 / 𝑛) = (𝐴 / 𝑛))
32	31	oveq2d 7284	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (1 + (𝑎 / 𝑛)) = (1 + (𝐴 / 𝑛)))
33	30, 32	oveq12d 7286	. . . . . 6 ⊢ (𝑎 = 𝐴 → (((1 + (1 / 𝑛))↑𝑎) / (1 + (𝑎 / 𝑛))) = (((1 + (1 / 𝑛))↑𝐴) / (1 + (𝐴 / 𝑛))))
34	33	mpteq2dv 5180	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑎) / (1 + (𝑎 / 𝑛)))) = (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝐴) / (1 + (𝐴 / 𝑛)))))
35	34	seqeq3d 13710	. . . 4 ⊢ (𝑎 = 𝐴 → seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑎) / (1 + (𝑎 / 𝑛))))) = seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝐴) / (1 + (𝐴 / 𝑛))))))
36		fveq2 6768	. . . 4 ⊢ (𝑎 = 𝐴 → (!‘𝑎) = (!‘𝐴))
37	35, 36	breq12d 5091	. . 3 ⊢ (𝑎 = 𝐴 → (seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑎) / (1 + (𝑎 / 𝑛))))) ⇝ (!‘𝑎) ↔ seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝐴) / (1 + (𝐴 / 𝑛))))) ⇝ (!‘𝐴)))
38		1red 10960	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 1 ∈ ℝ)
39		nnrecre 11998	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ)
40	38, 39	readdcld 10988	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (1 + (1 / 𝑛)) ∈ ℝ)
41	40	recnd 10987	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (1 + (1 / 𝑛)) ∈ ℂ)
42	41	exp0d 13839	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ((1 + (1 / 𝑛))↑0) = 1)
43		nncn 11964	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
44		nnne0 11990	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 𝑛 ≠ 0)
45	43, 44	div0d 11733	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (0 / 𝑛) = 0)
46	45	oveq2d 7284	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (1 + (0 / 𝑛)) = (1 + 0))
47		1p0e1 12080	. . . . . . . . . 10 ⊢ (1 + 0) = 1
48	46, 47	eqtrdi 2795	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (1 + (0 / 𝑛)) = 1)
49	42, 48	oveq12d 7286	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (((1 + (1 / 𝑛))↑0) / (1 + (0 / 𝑛))) = (1 / 1))
50		1div1e1 11648	. . . . . . . 8 ⊢ (1 / 1) = 1
51	49, 50	eqtrdi 2795	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (((1 + (1 / 𝑛))↑0) / (1 + (0 / 𝑛))) = 1)
52	51	mpteq2ia 5181	. . . . . 6 ⊢ (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑0) / (1 + (0 / 𝑛)))) = (𝑛 ∈ ℕ ↦ 1)
53		fconstmpt 5648	. . . . . 6 ⊢ (ℕ × {1}) = (𝑛 ∈ ℕ ↦ 1)
54	52, 53	eqtr4i 2770	. . . . 5 ⊢ (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑0) / (1 + (0 / 𝑛)))) = (ℕ × {1})
55		seqeq3 13707	. . . . 5 ⊢ ((𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑0) / (1 + (0 / 𝑛)))) = (ℕ × {1}) → seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑0) / (1 + (0 / 𝑛))))) = seq1( · , (ℕ × {1})))
56	54, 55	ax-mp 5	. . . 4 ⊢ seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑0) / (1 + (0 / 𝑛))))) = seq1( · , (ℕ × {1}))
57		nnuz 12603	. . . . . 6 ⊢ ℕ = (ℤ≥‘1)
58		1zzd 12334	. . . . . 6 ⊢ (⊤ → 1 ∈ ℤ)
59	57, 58	climprod1 15656	. . . . 5 ⊢ (⊤ → seq1( · , (ℕ × {1})) ⇝ 1)
60	59	mptru 1548	. . . 4 ⊢ seq1( · , (ℕ × {1})) ⇝ 1
61	56, 60	eqbrtri 5099	. . 3 ⊢ seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑0) / (1 + (0 / 𝑛))))) ⇝ 1
62		1zzd 12334	. . . . . 6 ⊢ ((𝑚 ∈ ℕ0 ∧ seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))) ⇝ (!‘𝑚)) → 1 ∈ ℤ)
63		simpr 484	. . . . . 6 ⊢ ((𝑚 ∈ ℕ0 ∧ seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))) ⇝ (!‘𝑚)) → seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))) ⇝ (!‘𝑚))
64		seqex 13704	. . . . . . 7 ⊢ seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑛))))) ∈ V
65	64	a1i 11	. . . . . 6 ⊢ ((𝑚 ∈ ℕ0 ∧ seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))) ⇝ (!‘𝑚)) → seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑛))))) ∈ V)
66		faclimlem2 33689	. . . . . . 7 ⊢ (𝑚 ∈ ℕ0 → seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛))))) ⇝ (𝑚 + 1))
67	66	adantr 480	. . . . . 6 ⊢ ((𝑚 ∈ ℕ0 ∧ seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))) ⇝ (!‘𝑚)) → seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛))))) ⇝ (𝑚 + 1))
68		elnnuz 12604	. . . . . . . . . 10 ⊢ (𝑎 ∈ ℕ ↔ 𝑎 ∈ (ℤ≥‘1))
69	68	biimpi 215	. . . . . . . . 9 ⊢ (𝑎 ∈ ℕ → 𝑎 ∈ (ℤ≥‘1))
70	69	adantl 481	. . . . . . . 8 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑎 ∈ ℕ) → 𝑎 ∈ (ℤ≥‘1))
71		1rp 12716	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℝ+
72	71	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → 1 ∈ ℝ+)
73		nnrp 12723	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ+)
74	73	rpreccld 12764	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ+)
75	74	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈ ℝ+)
76	72, 75	rpaddcld 12769	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → (1 + (1 / 𝑛)) ∈ ℝ+)
77		nn0z 12326	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ℕ0 → 𝑚 ∈ ℤ)
78	77	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → 𝑚 ∈ ℤ)
79	76, 78	rpexpcld 13943	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → ((1 + (1 / 𝑛))↑𝑚) ∈ ℝ+)
80		1cnd 10954	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → 1 ∈ ℂ)
81		nn0nndivcl 12287	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → (𝑚 / 𝑛) ∈ ℝ)
82	81	recnd 10987	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → (𝑚 / 𝑛) ∈ ℂ)
83	80, 82	addcomd 11160	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → (1 + (𝑚 / 𝑛)) = ((𝑚 / 𝑛) + 1))
84		nn0ge0div 12372	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → 0 ≤ (𝑚 / 𝑛))
85	81, 84	ge0p1rpd 12784	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → ((𝑚 / 𝑛) + 1) ∈ ℝ+)
86	83, 85	eqeltrd 2840	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → (1 + (𝑚 / 𝑛)) ∈ ℝ+)
87	79, 86	rpdivcld 12771	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))) ∈ ℝ+)
88	87	rpcnd 12756	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))) ∈ ℂ)
89	88	fmpttd 6983	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ0 → (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛)))):ℕ⟶ℂ)
90		elfznn 13267	. . . . . . . . . 10 ⊢ (𝑏 ∈ (1...𝑎) → 𝑏 ∈ ℕ)
91		ffvelrn 6953	. . . . . . . . . 10 ⊢ (((𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛)))):ℕ⟶ℂ ∧ 𝑏 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))‘𝑏) ∈ ℂ)
92	89, 90, 91	syl2an 595	. . . . . . . . 9 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑏 ∈ (1...𝑎)) → ((𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))‘𝑏) ∈ ℂ)
93	92	adantlr 711	. . . . . . . 8 ⊢ (((𝑚 ∈ ℕ0 ∧ 𝑎 ∈ ℕ) ∧ 𝑏 ∈ (1...𝑎)) → ((𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))‘𝑏) ∈ ℂ)
94		mulcl 10939	. . . . . . . . 9 ⊢ ((𝑏 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑏 · 𝑥) ∈ ℂ)
95	94	adantl 481	. . . . . . . 8 ⊢ (((𝑚 ∈ ℕ0 ∧ 𝑎 ∈ ℕ) ∧ (𝑏 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → (𝑏 · 𝑥) ∈ ℂ)
96	70, 93, 95	seqcl 13724	. . . . . . 7 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑎 ∈ ℕ) → (seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛)))))‘𝑎) ∈ ℂ)
97	96	adantlr 711	. . . . . 6 ⊢ (((𝑚 ∈ ℕ0 ∧ seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))) ⇝ (!‘𝑚)) ∧ 𝑎 ∈ ℕ) → (seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛)))))‘𝑎) ∈ ℂ)
98	86, 76	rpmulcld 12770	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → ((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) ∈ ℝ+)
99		nn0p1nn 12255	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℕ)
100	99	nnrpd 12752	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℝ+)
101		rpdivcl 12737	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 + 1) ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) → ((𝑚 + 1) / 𝑛) ∈ ℝ+)
102	100, 73, 101	syl2an 595	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → ((𝑚 + 1) / 𝑛) ∈ ℝ+)
103	72, 102	rpaddcld 12769	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → (1 + ((𝑚 + 1) / 𝑛)) ∈ ℝ+)
104	98, 103	rpdivcld 12771	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛))) ∈ ℝ+)
105	104	rpcnd 12756	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛))) ∈ ℂ)
106	105	fmpttd 6983	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ0 → (𝑛 ∈ ℕ ↦ (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛)))):ℕ⟶ℂ)
107		ffvelrn 6953	. . . . . . . . . 10 ⊢ (((𝑛 ∈ ℕ ↦ (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛)))):ℕ⟶ℂ ∧ 𝑏 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛))))‘𝑏) ∈ ℂ)
108	106, 90, 107	syl2an 595	. . . . . . . . 9 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑏 ∈ (1...𝑎)) → ((𝑛 ∈ ℕ ↦ (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛))))‘𝑏) ∈ ℂ)
109	108	adantlr 711	. . . . . . . 8 ⊢ (((𝑚 ∈ ℕ0 ∧ 𝑎 ∈ ℕ) ∧ 𝑏 ∈ (1...𝑎)) → ((𝑛 ∈ ℕ ↦ (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛))))‘𝑏) ∈ ℂ)
110	70, 109, 95	seqcl 13724	. . . . . . 7 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑎 ∈ ℕ) → (seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛)))))‘𝑎) ∈ ℂ)
111	110	adantlr 711	. . . . . 6 ⊢ (((𝑚 ∈ ℕ0 ∧ seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))) ⇝ (!‘𝑚)) ∧ 𝑎 ∈ ℕ) → (seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛)))))‘𝑎) ∈ ℂ)
112		faclimlem3 33690	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑏 ∈ ℕ) → (((1 + (1 / 𝑏))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑏))) = ((((1 + (1 / 𝑏))↑𝑚) / (1 + (𝑚 / 𝑏))) · (((1 + (𝑚 / 𝑏)) · (1 + (1 / 𝑏))) / (1 + ((𝑚 + 1) / 𝑏)))))
113		oveq2 7276	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑏 → (1 / 𝑛) = (1 / 𝑏))
114	113	oveq2d 7284	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑏 → (1 + (1 / 𝑛)) = (1 + (1 / 𝑏)))
115	114	oveq1d 7283	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑏 → ((1 + (1 / 𝑛))↑(𝑚 + 1)) = ((1 + (1 / 𝑏))↑(𝑚 + 1)))
116		oveq2 7276	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑏 → ((𝑚 + 1) / 𝑛) = ((𝑚 + 1) / 𝑏))
117	116	oveq2d 7284	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑏 → (1 + ((𝑚 + 1) / 𝑛)) = (1 + ((𝑚 + 1) / 𝑏)))
118	115, 117	oveq12d 7286	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑏 → (((1 + (1 / 𝑛))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑛))) = (((1 + (1 / 𝑏))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑏))))
119		eqid 2739	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑛)))) = (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑛))))
120		ovex 7301	. . . . . . . . . . . . 13 ⊢ (((1 + (1 / 𝑏))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑏))) ∈ V
121	118, 119, 120	fvmpt 6869	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑛))))‘𝑏) = (((1 + (1 / 𝑏))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑏))))
122	121	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑏 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑛))))‘𝑏) = (((1 + (1 / 𝑏))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑏))))
123	114	oveq1d 7283	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑏 → ((1 + (1 / 𝑛))↑𝑚) = ((1 + (1 / 𝑏))↑𝑚))
124		oveq2 7276	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑏 → (𝑚 / 𝑛) = (𝑚 / 𝑏))
125	124	oveq2d 7284	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑏 → (1 + (𝑚 / 𝑛)) = (1 + (𝑚 / 𝑏)))
126	123, 125	oveq12d 7286	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑏 → (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))) = (((1 + (1 / 𝑏))↑𝑚) / (1 + (𝑚 / 𝑏))))
127		eqid 2739	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛)))) = (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))
128		ovex 7301	. . . . . . . . . . . . . 14 ⊢ (((1 + (1 / 𝑏))↑𝑚) / (1 + (𝑚 / 𝑏))) ∈ V
129	126, 127, 128	fvmpt 6869	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))‘𝑏) = (((1 + (1 / 𝑏))↑𝑚) / (1 + (𝑚 / 𝑏))))
130	125, 114	oveq12d 7286	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑏 → ((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) = ((1 + (𝑚 / 𝑏)) · (1 + (1 / 𝑏))))
131	130, 117	oveq12d 7286	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑏 → (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛))) = (((1 + (𝑚 / 𝑏)) · (1 + (1 / 𝑏))) / (1 + ((𝑚 + 1) / 𝑏))))
132		eqid 2739	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ ↦ (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛)))) = (𝑛 ∈ ℕ ↦ (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛))))
133		ovex 7301	. . . . . . . . . . . . . 14 ⊢ (((1 + (𝑚 / 𝑏)) · (1 + (1 / 𝑏))) / (1 + ((𝑚 + 1) / 𝑏))) ∈ V
134	131, 132, 133	fvmpt 6869	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛))))‘𝑏) = (((1 + (𝑚 / 𝑏)) · (1 + (1 / 𝑏))) / (1 + ((𝑚 + 1) / 𝑏))))
135	129, 134	oveq12d 7286	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ ℕ → (((𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))‘𝑏) · ((𝑛 ∈ ℕ ↦ (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛))))‘𝑏)) = ((((1 + (1 / 𝑏))↑𝑚) / (1 + (𝑚 / 𝑏))) · (((1 + (𝑚 / 𝑏)) · (1 + (1 / 𝑏))) / (1 + ((𝑚 + 1) / 𝑏)))))
136	135	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑏 ∈ ℕ) → (((𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))‘𝑏) · ((𝑛 ∈ ℕ ↦ (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛))))‘𝑏)) = ((((1 + (1 / 𝑏))↑𝑚) / (1 + (𝑚 / 𝑏))) · (((1 + (𝑚 / 𝑏)) · (1 + (1 / 𝑏))) / (1 + ((𝑚 + 1) / 𝑏)))))
137	112, 122, 136	3eqtr4d 2789	. . . . . . . . . 10 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑏 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑛))))‘𝑏) = (((𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))‘𝑏) · ((𝑛 ∈ ℕ ↦ (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛))))‘𝑏)))
138	90, 137	sylan2 592	. . . . . . . . 9 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑏 ∈ (1...𝑎)) → ((𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑛))))‘𝑏) = (((𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))‘𝑏) · ((𝑛 ∈ ℕ ↦ (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛))))‘𝑏)))
139	138	adantlr 711	. . . . . . . 8 ⊢ (((𝑚 ∈ ℕ0 ∧ 𝑎 ∈ ℕ) ∧ 𝑏 ∈ (1...𝑎)) → ((𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑛))))‘𝑏) = (((𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))‘𝑏) · ((𝑛 ∈ ℕ ↦ (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛))))‘𝑏)))
140	70, 93, 109, 139	prodfmul 15583	. . . . . . 7 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑎 ∈ ℕ) → (seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑛)))))‘𝑎) = ((seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛)))))‘𝑎) · (seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛)))))‘𝑎)))
141	140	adantlr 711	. . . . . 6 ⊢ (((𝑚 ∈ ℕ0 ∧ seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))) ⇝ (!‘𝑚)) ∧ 𝑎 ∈ ℕ) → (seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑛)))))‘𝑎) = ((seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛)))))‘𝑎) · (seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛)))))‘𝑎)))
142	57, 62, 63, 65, 67, 97, 111, 141	climmul 15323	. . . . 5 ⊢ ((𝑚 ∈ ℕ0 ∧ seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))) ⇝ (!‘𝑚)) → seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑛))))) ⇝ ((!‘𝑚) · (𝑚 + 1)))
143		facp1 13973	. . . . . 6 ⊢ (𝑚 ∈ ℕ0 → (!‘(𝑚 + 1)) = ((!‘𝑚) · (𝑚 + 1)))
144	143	adantr 480	. . . . 5 ⊢ ((𝑚 ∈ ℕ0 ∧ seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))) ⇝ (!‘𝑚)) → (!‘(𝑚 + 1)) = ((!‘𝑚) · (𝑚 + 1)))
145	142, 144	breqtrrd 5106	. . . 4 ⊢ ((𝑚 ∈ ℕ0 ∧ seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))) ⇝ (!‘𝑚)) → seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑛))))) ⇝ (!‘(𝑚 + 1)))
146	145	ex 412	. . 3 ⊢ (𝑚 ∈ ℕ0 → (seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))) ⇝ (!‘𝑚) → seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑛))))) ⇝ (!‘(𝑚 + 1))))
147	13, 21, 29, 37, 61, 146	nn0ind 12398	. 2 ⊢ (𝐴 ∈ ℕ0 → seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝐴) / (1 + (𝐴 / 𝑛))))) ⇝ (!‘𝐴))
148	3, 147	eqbrtrid 5113	1 ⊢ (𝐴 ∈ ℕ0 → seq1( · , 𝐹) ⇝ (!‘𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541  ⊤wtru 1542   ∈ wcel 2109  Vcvv 3430  {csn 4566   class class class wbr 5078   ↦ cmpt 5161   × cxp 5586  ⟶wf 6426  ‘cfv 6430  (class class class)co 7268  ℂcc 10853  0cc0 10855  1c1 10856   + caddc 10858   · cmul 10860   / cdiv 11615  ℕcn 11956  ℕ₀cn0 12216  ℤcz 12302  ℤ_≥cuz 12564  ℝ⁺crp 12712  ...cfz 13221  seqcseq 13702  ↑cexp 13763  !cfa 13968   ⇝ cli 15174

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-rep 5213  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579  ax-inf2 9360  ax-cnex 10911  ax-resscn 10912  ax-1cn 10913  ax-icn 10914  ax-addcl 10915  ax-addrcl 10916  ax-mulcl 10917  ax-mulrcl 10918  ax-mulcom 10919  ax-addass 10920  ax-mulass 10921  ax-distr 10922  ax-i2m1 10923  ax-1ne0 10924  ax-1rid 10925  ax-rnegex 10926  ax-rrecex 10927  ax-cnre 10928  ax-pre-lttri 10929  ax-pre-lttrn 10930  ax-pre-ltadd 10931  ax-pre-mulgt0 10932  ax-pre-sup 10933

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3070  df-rex 3071  df-reu 3072  df-rmo 3073  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-pss 3910  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4845  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-tr 5196  df-id 5488  df-eprel 5494  df-po 5502  df-so 5503  df-fr 5543  df-we 5545  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-pred 6199  df-ord 6266  df-on 6267  df-lim 6268  df-suc 6269  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-riota 7225  df-ov 7271  df-oprab 7272  df-mpo 7273  df-om 7701  df-1st 7817  df-2nd 7818  df-frecs 8081  df-wrecs 8112  df-recs 8186  df-rdg 8225  df-er 8472  df-pm 8592  df-en 8708  df-dom 8709  df-sdom 8710  df-sup 9162  df-inf 9163  df-pnf 10995  df-mnf 10996  df-xr 10997  df-ltxr 10998  df-le 10999  df-sub 11190  df-neg 11191  df-div 11616  df-nn 11957  df-2 12019  df-3 12020  df-n0 12217  df-z 12303  df-uz 12565  df-rp 12713  df-fz 13222  df-fzo 13365  df-fl 13493  df-seq 13703  df-exp 13764  df-fac 13969  df-shft 14759  df-cj 14791  df-re 14792  df-im 14793  df-sqrt 14927  df-abs 14928  df-clim 15178  df-rlim 15179

This theorem is referenced by:  iprodfac  33692