Theorem faclim 36726

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 14189	. . 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 7459	. . . . . . 7 ⊢ (𝑎 = 0 → ((1 + (1 / 𝑛))↑𝑎) = ((1 + (1 / 𝑛))↑0))
5		oveq1 7458	. . . . . . . 8 ⊢ (𝑎 = 0 → (𝑎 / 𝑛) = (0 / 𝑛))
6	5	oveq2d 7467	. . . . . . 7 ⊢ (𝑎 = 0 → (1 + (𝑎 / 𝑛)) = (1 + (0 / 𝑛)))
7	4, 6	oveq12d 7469	. . . . . 6 ⊢ (𝑎 = 0 → (((1 + (1 / 𝑛))↑𝑎) / (1 + (𝑎 / 𝑛))) = (((1 + (1 / 𝑛))↑0) / (1 + (0 / 𝑛))))
8	7	mpteq2dv 5205	. . . . 5 ⊢ (𝑎 = 0 → (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑎) / (1 + (𝑎 / 𝑛)))) = (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑0) / (1 + (0 / 𝑛)))))
9	8	seqeq3d 14192	. . . 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 6916	. . . . 5 ⊢ (𝑎 = 0 → (!‘𝑎) = (!‘0))
11		fac0 14461	. . . . 5 ⊢ (!‘0) = 1
12	10, 11	eqtrdi 2814	. . . 4 ⊢ (𝑎 = 0 → (!‘𝑎) = 1)
13	9, 12	breq12d 5121	. . 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 7459	. . . . . . 7 ⊢ (𝑎 = 𝑚 → ((1 + (1 / 𝑛))↑𝑎) = ((1 + (1 / 𝑛))↑𝑚))
15		oveq1 7458	. . . . . . . 8 ⊢ (𝑎 = 𝑚 → (𝑎 / 𝑛) = (𝑚 / 𝑛))
16	15	oveq2d 7467	. . . . . . 7 ⊢ (𝑎 = 𝑚 → (1 + (𝑎 / 𝑛)) = (1 + (𝑚 / 𝑛)))
17	14, 16	oveq12d 7469	. . . . . 6 ⊢ (𝑎 = 𝑚 → (((1 + (1 / 𝑛))↑𝑎) / (1 + (𝑎 / 𝑛))) = (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))
18	17	mpteq2dv 5205	. . . . 5 ⊢ (𝑎 = 𝑚 → (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑎) / (1 + (𝑎 / 𝑛)))) = (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛)))))
19	18	seqeq3d 14192	. . . 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 6916	. . . 4 ⊢ (𝑎 = 𝑚 → (!‘𝑎) = (!‘𝑚))
21	19, 20	breq12d 5121	. . 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 7459	. . . . . . 7 ⊢ (𝑎 = (𝑚 + 1) → ((1 + (1 / 𝑛))↑𝑎) = ((1 + (1 / 𝑛))↑(𝑚 + 1)))
23		oveq1 7458	. . . . . . . 8 ⊢ (𝑎 = (𝑚 + 1) → (𝑎 / 𝑛) = ((𝑚 + 1) / 𝑛))
24	23	oveq2d 7467	. . . . . . 7 ⊢ (𝑎 = (𝑚 + 1) → (1 + (𝑎 / 𝑛)) = (1 + ((𝑚 + 1) / 𝑛)))
25	22, 24	oveq12d 7469	. . . . . 6 ⊢ (𝑎 = (𝑚 + 1) → (((1 + (1 / 𝑛))↑𝑎) / (1 + (𝑎 / 𝑛))) = (((1 + (1 / 𝑛))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑛))))
26	25	mpteq2dv 5205	. . . . 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 14192	. . . 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 6916	. . . 4 ⊢ (𝑎 = (𝑚 + 1) → (!‘𝑎) = (!‘(𝑚 + 1)))
29	27, 28	breq12d 5121	. . 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 7459	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ((1 + (1 / 𝑛))↑𝑎) = ((1 + (1 / 𝑛))↑𝐴))
31		oveq1 7458	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑎 / 𝑛) = (𝐴 / 𝑛))
32	31	oveq2d 7467	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (1 + (𝑎 / 𝑛)) = (1 + (𝐴 / 𝑛)))
33	30, 32	oveq12d 7469	. . . . . 6 ⊢ (𝑎 = 𝐴 → (((1 + (1 / 𝑛))↑𝑎) / (1 + (𝑎 / 𝑛))) = (((1 + (1 / 𝑛))↑𝐴) / (1 + (𝐴 / 𝑛))))
34	33	mpteq2dv 5205	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑎) / (1 + (𝑎 / 𝑛)))) = (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝐴) / (1 + (𝐴 / 𝑛)))))
35	34	seqeq3d 14192	. . . 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 6916	. . . 4 ⊢ (𝑎 = 𝐴 → (!‘𝑎) = (!‘𝐴))
37	35, 36	breq12d 5121	. . 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 11348	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 1 ∈ ℝ)
39		nnrecre 12420	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ)
40	38, 39	readdcld 11377	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (1 + (1 / 𝑛)) ∈ ℝ)
41	40	recnd 11376	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (1 + (1 / 𝑛)) ∈ ℂ)
42	41	exp0d 14323	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ((1 + (1 / 𝑛))↑0) = 1)
43		nncn 12383	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
44		nnne0 12412	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 𝑛 ≠ 0)
45	43, 44	div0d 12132	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (0 / 𝑛) = 0)
46	45	oveq2d 7467	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (1 + (0 / 𝑛)) = (1 + 0))
47		1p0e1 12505	. . . . . . . . . 10 ⊢ (1 + 0) = 1
48	46, 47	eqtrdi 2814	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (1 + (0 / 𝑛)) = 1)
49	42, 48	oveq12d 7469	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (((1 + (1 / 𝑛))↑0) / (1 + (0 / 𝑛))) = (1 / 1))
50		1div1e1 12047	. . . . . . . 8 ⊢ (1 / 1) = 1
51	49, 50	eqtrdi 2814	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (((1 + (1 / 𝑛))↑0) / (1 + (0 / 𝑛))) = 1)
52	51	mpteq2ia 5206	. . . . . 6 ⊢ (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑0) / (1 + (0 / 𝑛)))) = (𝑛 ∈ ℕ ↦ 1)
53		fconstmpt 5720	. . . . . 6 ⊢ (ℕ × {1}) = (𝑛 ∈ ℕ ↦ 1)
54	52, 53	eqtr4i 2789	. . . . 5 ⊢ (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑0) / (1 + (0 / 𝑛)))) = (ℕ × {1})
55		seqeq3 14189	. . . . 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 13044	. . . . . 6 ⊢ ℕ = (ℤ≥‘1)
58		1zzd 12767	. . . . . 6 ⊢ (⊤ → 1 ∈ ℤ)
59	57, 58	climprod1 16176	. . . . 5 ⊢ (⊤ → seq1( · , (ℕ × {1})) ⇝ 1)
60	59	mptru 1577	. . . 4 ⊢ seq1( · , (ℕ × {1})) ⇝ 1
61	56, 60	eqbrtri 5132	. . 3 ⊢ seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑0) / (1 + (0 / 𝑛))))) ⇝ 1
62		1zzd 12767	. . . . . 6 ⊢ ((𝑚 ∈ ℕ0 ∧ seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))) ⇝ (!‘𝑚)) → 1 ∈ ℤ)
63		simpr 490	. . . . . 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 14186	. . . . . . 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 36724	. . . . . . 7 ⊢ (𝑚 ∈ ℕ0 → seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛))))) ⇝ (𝑚 + 1))
67	66	adantr 486	. . . . . 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 13045	. . . . . . . . 9 ⊢ (𝑎 ∈ ℕ ↔ 𝑎 ∈ (ℤ≥‘1))
69	68	bilani 510	. . . . . . . 8 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑎 ∈ ℕ) → 𝑎 ∈ (ℤ≥‘1))
70		1rp 13164	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℝ+
71	70	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → 1 ∈ ℝ+)
72		nnrp 13172	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ+)
73	72	rpreccld 13214	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ+)
74	73	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈ ℝ+)
75	71, 74	rpaddcld 13219	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → (1 + (1 / 𝑛)) ∈ ℝ+)
76		nn0z 12757	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ℕ0 → 𝑚 ∈ ℤ)
77	76	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → 𝑚 ∈ ℤ)
78	75, 77	rpexpcld 14432	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → ((1 + (1 / 𝑛))↑𝑚) ∈ ℝ+)
79		1cnd 11341	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → 1 ∈ ℂ)
80		nn0nndivcl 12718	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → (𝑚 / 𝑛) ∈ ℝ)
81	80	recnd 11376	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → (𝑚 / 𝑛) ∈ ℂ)
82	79, 81	addcomd 11551	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → (1 + (𝑚 / 𝑛)) = ((𝑚 / 𝑛) + 1))
83		nn0ge0div 12808	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → 0 ≤ (𝑚 / 𝑛))
84	80, 83	ge0p1rpd 13234	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → ((𝑚 / 𝑛) + 1) ∈ ℝ+)
85	82, 84	eqeltrd 2864	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → (1 + (𝑚 / 𝑛)) ∈ ℝ+)
86	78, 85	rpdivcld 13221	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))) ∈ ℝ+)
87	86	rpcnd 13206	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))) ∈ ℂ)
88	87	fmpttd 7147	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ0 → (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛)))):ℕ⟶ℂ)
89		elfznn 13727	. . . . . . . . . 10 ⊢ (𝑏 ∈ (1...𝑎) → 𝑏 ∈ ℕ)
90		ffvelcdm 7113	. . . . . . . . . 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)
91	88, 89, 90	syl2an 608	. . . . . . . . 9 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑏 ∈ (1...𝑎)) → ((𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))‘𝑏) ∈ ℂ)
92	91	adantlr 728	. . . . . . . 8 ⊢ (((𝑚 ∈ ℕ0 ∧ 𝑎 ∈ ℕ) ∧ 𝑏 ∈ (1...𝑎)) → ((𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))‘𝑏) ∈ ℂ)
93		mulcl 11323	. . . . . . . . 9 ⊢ ((𝑏 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑏 · 𝑥) ∈ ℂ)
94	93	adantl 487	. . . . . . . 8 ⊢ (((𝑚 ∈ ℕ0 ∧ 𝑎 ∈ ℕ) ∧ (𝑏 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → (𝑏 · 𝑥) ∈ ℂ)
95	69, 92, 94	seqcl 14205	. . . . . . 7 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑎 ∈ ℕ) → (seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛)))))‘𝑎) ∈ ℂ)
96	95	adantlr 728	. . . . . 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)
97	85, 75	rpmulcld 13220	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → ((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) ∈ ℝ+)
98		nn0p1nn 12685	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℕ)
99	98	nnrpd 13202	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℝ+)
100		rpdivcl 13187	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 + 1) ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) → ((𝑚 + 1) / 𝑛) ∈ ℝ+)
101	99, 72, 100	syl2an 608	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → ((𝑚 + 1) / 𝑛) ∈ ℝ+)
102	71, 101	rpaddcld 13219	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → (1 + ((𝑚 + 1) / 𝑛)) ∈ ℝ+)
103	97, 102	rpdivcld 13221	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛))) ∈ ℝ+)
104	103	rpcnd 13206	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛))) ∈ ℂ)
105	104	fmpttd 7147	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ0 → (𝑛 ∈ ℕ ↦ (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛)))):ℕ⟶ℂ)
106		ffvelcdm 7113	. . . . . . . . . 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)
107	105, 89, 106	syl2an 608	. . . . . . . . 9 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑏 ∈ (1...𝑎)) → ((𝑛 ∈ ℕ ↦ (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛))))‘𝑏) ∈ ℂ)
108	107	adantlr 728	. . . . . . . 8 ⊢ (((𝑚 ∈ ℕ0 ∧ 𝑎 ∈ ℕ) ∧ 𝑏 ∈ (1...𝑎)) → ((𝑛 ∈ ℕ ↦ (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛))))‘𝑏) ∈ ℂ)
109	69, 108, 94	seqcl 14205	. . . . . . 7 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑎 ∈ ℕ) → (seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛)))))‘𝑎) ∈ ℂ)
110	109	adantlr 728	. . . . . 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)
111		faclimlem3 36725	. . . . . . . . . . 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)))))
112		oveq2 7459	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑏 → (1 / 𝑛) = (1 / 𝑏))
113	112	oveq2d 7467	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑏 → (1 + (1 / 𝑛)) = (1 + (1 / 𝑏)))
114	113	oveq1d 7466	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑏 → ((1 + (1 / 𝑛))↑(𝑚 + 1)) = ((1 + (1 / 𝑏))↑(𝑚 + 1)))
115		oveq2 7459	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑏 → ((𝑚 + 1) / 𝑛) = ((𝑚 + 1) / 𝑏))
116	115	oveq2d 7467	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑏 → (1 + ((𝑚 + 1) / 𝑛)) = (1 + ((𝑚 + 1) / 𝑏)))
117	114, 116	oveq12d 7469	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑏 → (((1 + (1 / 𝑛))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑛))) = (((1 + (1 / 𝑏))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑏))))
118		eqid 2763	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑛)))) = (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑛))))
119		ovex 7484	. . . . . . . . . . . . 13 ⊢ (((1 + (1 / 𝑏))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑏))) ∈ V
120	117, 118, 119	fvmpt 7024	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑛))))‘𝑏) = (((1 + (1 / 𝑏))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑏))))
121	120	adantl 487	. . . . . . . . . . 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))))
122	113	oveq1d 7466	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑏 → ((1 + (1 / 𝑛))↑𝑚) = ((1 + (1 / 𝑏))↑𝑚))
123		oveq2 7459	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑏 → (𝑚 / 𝑛) = (𝑚 / 𝑏))
124	123	oveq2d 7467	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑏 → (1 + (𝑚 / 𝑛)) = (1 + (𝑚 / 𝑏)))
125	122, 124	oveq12d 7469	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑏 → (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))) = (((1 + (1 / 𝑏))↑𝑚) / (1 + (𝑚 / 𝑏))))
126		eqid 2763	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛)))) = (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))
127		ovex 7484	. . . . . . . . . . . . . 14 ⊢ (((1 + (1 / 𝑏))↑𝑚) / (1 + (𝑚 / 𝑏))) ∈ V
128	125, 126, 127	fvmpt 7024	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))‘𝑏) = (((1 + (1 / 𝑏))↑𝑚) / (1 + (𝑚 / 𝑏))))
129	124, 113	oveq12d 7469	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑏 → ((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) = ((1 + (𝑚 / 𝑏)) · (1 + (1 / 𝑏))))
130	129, 116	oveq12d 7469	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑏 → (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛))) = (((1 + (𝑚 / 𝑏)) · (1 + (1 / 𝑏))) / (1 + ((𝑚 + 1) / 𝑏))))
131		eqid 2763	. . . . . . . . . . . . . 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))))
132		ovex 7484	. . . . . . . . . . . . . 14 ⊢ (((1 + (𝑚 / 𝑏)) · (1 + (1 / 𝑏))) / (1 + ((𝑚 + 1) / 𝑏))) ∈ V
133	130, 131, 132	fvmpt 7024	. . . . . . . . . . . . 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))))
134	128, 133	oveq12d 7469	. . . . . . . . . . . 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)))))
135	134	adantl 487	. . . . . . . . . . 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)))))
136	111, 121, 135	3eqtr4d 2808	. . . . . . . . . 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)))
137	89, 136	sylan2 605	. . . . . . . . 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)))
138	137	adantlr 728	. . . . . . . 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)))
139	69, 92, 108, 138	prodfmul 16103	. . . . . . 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)))
140	139	adantlr 728	. . . . . 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)))
141	57, 62, 63, 65, 67, 96, 110, 140	climmul 15844	. . . . 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)))
142		facp1 14463	. . . . . 6 ⊢ (𝑚 ∈ ℕ0 → (!‘(𝑚 + 1)) = ((!‘𝑚) · (𝑚 + 1)))
143	142	adantr 486	. . . . 5 ⊢ ((𝑚 ∈ ℕ0 ∧ seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))) ⇝ (!‘𝑚)) → (!‘(𝑚 + 1)) = ((!‘𝑚) · (𝑚 + 1)))
144	141, 143	breqtrrd 5139	. . . 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)))
145	144	ex 418	. . 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))))
146	13, 21, 29, 37, 61, 145	nn0ind 12834	. 2 ⊢ (𝐴 ∈ ℕ0 → seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝐴) / (1 + (𝐴 / 𝑛))))) ⇝ (!‘𝐴))
147	3, 146	eqbrtrid 5146	1 ⊢ (𝐴 ∈ ℕ0 → seq1( · , 𝐹) ⇝ (!‘𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ⊤wtru 1571   ∈ wcel 2147  Vcvv 3454  {csn 4588   class class class wbr 5108   ↦ cmpt 5192   × cxp 5656  ⟶wf 6562  ‘cfv 6566  (class class class)co 7451  ℂcc 11237  0cc0 11239  1c1 11240   + caddc 11242   · cmul 11244   / cdiv 12013  ℕcn 12375  ℕ₀cn0 12646  ℤcz 12733  ℤ_≥cuz 13005  ℝ⁺crp 13160  ...cfz 13679  seqcseq 14184  ↑cexp 14244  !cfa 14458   ⇝ cli 15695

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-inf2 9679  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316  ax-pre-sup 11317

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-er 8755  df-pm 8888  df-en 9012  df-dom 9013  df-sdom 9014  df-sup 9471  df-inf 9472  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-div 12014  df-nn 12376  df-2 12445  df-3 12446  df-n0 12647  df-z 12734  df-uz 13006  df-rp 13161  df-fz 13680  df-fzo 13829  df-fl 13972  df-seq 14185  df-exp 14245  df-fac 14459  df-shft 15263  df-cj 15309  df-re 15310  df-im 15311  df-sqrt 15445  df-abs 15446  df-clim 15699  df-rlim 15700

This theorem is used by:  iprodfac  36727