Theorem faclim 35947

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 13962	. . 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 7369	. . . . . . 7 ⊢ (𝑎 = 0 → ((1 + (1 / 𝑛))↑𝑎) = ((1 + (1 / 𝑛))↑0))
5		oveq1 7368	. . . . . . . 8 ⊢ (𝑎 = 0 → (𝑎 / 𝑛) = (0 / 𝑛))
6	5	oveq2d 7377	. . . . . . 7 ⊢ (𝑎 = 0 → (1 + (𝑎 / 𝑛)) = (1 + (0 / 𝑛)))
7	4, 6	oveq12d 7379	. . . . . 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 13965	. . . 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 6835	. . . . 5 ⊢ (𝑎 = 0 → (!‘𝑎) = (!‘0))
11		fac0 14232	. . . . 5 ⊢ (!‘0) = 1
12	10, 11	eqtrdi 2788	. . . 4 ⊢ (𝑎 = 0 → (!‘𝑎) = 1)
13	9, 12	breq12d 5099	. . 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 7369	. . . . . . 7 ⊢ (𝑎 = 𝑚 → ((1 + (1 / 𝑛))↑𝑎) = ((1 + (1 / 𝑛))↑𝑚))
15		oveq1 7368	. . . . . . . 8 ⊢ (𝑎 = 𝑚 → (𝑎 / 𝑛) = (𝑚 / 𝑛))
16	15	oveq2d 7377	. . . . . . 7 ⊢ (𝑎 = 𝑚 → (1 + (𝑎 / 𝑛)) = (1 + (𝑚 / 𝑛)))
17	14, 16	oveq12d 7379	. . . . . 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 13965	. . . 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 6835	. . . 4 ⊢ (𝑎 = 𝑚 → (!‘𝑎) = (!‘𝑚))
21	19, 20	breq12d 5099	. . 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 7369	. . . . . . 7 ⊢ (𝑎 = (𝑚 + 1) → ((1 + (1 / 𝑛))↑𝑎) = ((1 + (1 / 𝑛))↑(𝑚 + 1)))
23		oveq1 7368	. . . . . . . 8 ⊢ (𝑎 = (𝑚 + 1) → (𝑎 / 𝑛) = ((𝑚 + 1) / 𝑛))
24	23	oveq2d 7377	. . . . . . 7 ⊢ (𝑎 = (𝑚 + 1) → (1 + (𝑎 / 𝑛)) = (1 + ((𝑚 + 1) / 𝑛)))
25	22, 24	oveq12d 7379	. . . . . 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 13965	. . . 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 6835	. . . 4 ⊢ (𝑎 = (𝑚 + 1) → (!‘𝑎) = (!‘(𝑚 + 1)))
29	27, 28	breq12d 5099	. . 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 7369	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ((1 + (1 / 𝑛))↑𝑎) = ((1 + (1 / 𝑛))↑𝐴))
31		oveq1 7368	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑎 / 𝑛) = (𝐴 / 𝑛))
32	31	oveq2d 7377	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (1 + (𝑎 / 𝑛)) = (1 + (𝐴 / 𝑛)))
33	30, 32	oveq12d 7379	. . . . . 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 13965	. . . 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 6835	. . . 4 ⊢ (𝑎 = 𝐴 → (!‘𝑎) = (!‘𝐴))
37	35, 36	breq12d 5099	. . 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 11139	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 1 ∈ ℝ)
39		nnrecre 12213	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ)
40	38, 39	readdcld 11168	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (1 + (1 / 𝑛)) ∈ ℝ)
41	40	recnd 11167	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (1 + (1 / 𝑛)) ∈ ℂ)
42	41	exp0d 14096	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ((1 + (1 / 𝑛))↑0) = 1)
43		nncn 12176	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
44		nnne0 12205	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 𝑛 ≠ 0)
45	43, 44	div0d 11924	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (0 / 𝑛) = 0)
46	45	oveq2d 7377	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (1 + (0 / 𝑛)) = (1 + 0))
47		1p0e1 12294	. . . . . . . . . 10 ⊢ (1 + 0) = 1
48	46, 47	eqtrdi 2788	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (1 + (0 / 𝑛)) = 1)
49	42, 48	oveq12d 7379	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (((1 + (1 / 𝑛))↑0) / (1 + (0 / 𝑛))) = (1 / 1))
50		1div1e1 11839	. . . . . . . 8 ⊢ (1 / 1) = 1
51	49, 50	eqtrdi 2788	. . . . . . 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 5687	. . . . . 6 ⊢ (ℕ × {1}) = (𝑛 ∈ ℕ ↦ 1)
54	52, 53	eqtr4i 2763	. . . . 5 ⊢ (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑0) / (1 + (0 / 𝑛)))) = (ℕ × {1})
55		seqeq3 13962	. . . . 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 12821	. . . . . 6 ⊢ ℕ = (ℤ≥‘1)
58		1zzd 12552	. . . . . 6 ⊢ (⊤ → 1 ∈ ℤ)
59	57, 58	climprod1 15924	. . . . 5 ⊢ (⊤ → seq1( · , (ℕ × {1})) ⇝ 1)
60	59	mptru 1549	. . . 4 ⊢ seq1( · , (ℕ × {1})) ⇝ 1
61	56, 60	eqbrtri 5107	. . 3 ⊢ seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑0) / (1 + (0 / 𝑛))))) ⇝ 1
62		1zzd 12552	. . . . . 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 13959	. . . . . . 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 35945	. . . . . . 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 12822	. . . . . . . . . 10 ⊢ (𝑎 ∈ ℕ ↔ 𝑎 ∈ (ℤ≥‘1))
69	68	biimpi 216	. . . . . . . . 9 ⊢ (𝑎 ∈ ℕ → 𝑎 ∈ (ℤ≥‘1))
70	69	adantl 481	. . . . . . . 8 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑎 ∈ ℕ) → 𝑎 ∈ (ℤ≥‘1))
71		1rp 12940	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℝ+
72	71	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → 1 ∈ ℝ+)
73		nnrp 12948	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ+)
74	73	rpreccld 12990	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ+)
75	74	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈ ℝ+)
76	72, 75	rpaddcld 12995	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → (1 + (1 / 𝑛)) ∈ ℝ+)
77		nn0z 12542	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ℕ0 → 𝑚 ∈ ℤ)
78	77	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → 𝑚 ∈ ℤ)
79	76, 78	rpexpcld 14203	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → ((1 + (1 / 𝑛))↑𝑚) ∈ ℝ+)
80		1cnd 11133	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → 1 ∈ ℂ)
81		nn0nndivcl 12503	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → (𝑚 / 𝑛) ∈ ℝ)
82	81	recnd 11167	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → (𝑚 / 𝑛) ∈ ℂ)
83	80, 82	addcomd 11342	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → (1 + (𝑚 / 𝑛)) = ((𝑚 / 𝑛) + 1))
84		nn0ge0div 12592	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → 0 ≤ (𝑚 / 𝑛))
85	81, 84	ge0p1rpd 13010	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → ((𝑚 / 𝑛) + 1) ∈ ℝ+)
86	83, 85	eqeltrd 2837	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → (1 + (𝑚 / 𝑛)) ∈ ℝ+)
87	79, 86	rpdivcld 12997	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))) ∈ ℝ+)
88	87	rpcnd 12982	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))) ∈ ℂ)
89	88	fmpttd 7062	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ0 → (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛)))):ℕ⟶ℂ)
90		elfznn 13501	. . . . . . . . . 10 ⊢ (𝑏 ∈ (1...𝑎) → 𝑏 ∈ ℕ)
91		ffvelcdm 7028	. . . . . . . . . 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 597	. . . . . . . . 9 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑏 ∈ (1...𝑎)) → ((𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))‘𝑏) ∈ ℂ)
93	92	adantlr 716	. . . . . . . 8 ⊢ (((𝑚 ∈ ℕ0 ∧ 𝑎 ∈ ℕ) ∧ 𝑏 ∈ (1...𝑎)) → ((𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))‘𝑏) ∈ ℂ)
94		mulcl 11116	. . . . . . . . 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 13978	. . . . . . 7 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑎 ∈ ℕ) → (seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛)))))‘𝑎) ∈ ℂ)
97	96	adantlr 716	. . . . . 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 12996	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → ((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) ∈ ℝ+)
99		nn0p1nn 12470	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℕ)
100	99	nnrpd 12978	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℝ+)
101		rpdivcl 12963	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 + 1) ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) → ((𝑚 + 1) / 𝑛) ∈ ℝ+)
102	100, 73, 101	syl2an 597	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → ((𝑚 + 1) / 𝑛) ∈ ℝ+)
103	72, 102	rpaddcld 12995	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → (1 + ((𝑚 + 1) / 𝑛)) ∈ ℝ+)
104	98, 103	rpdivcld 12997	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛))) ∈ ℝ+)
105	104	rpcnd 12982	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ) → (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛))) ∈ ℂ)
106	105	fmpttd 7062	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ0 → (𝑛 ∈ ℕ ↦ (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛)))):ℕ⟶ℂ)
107		ffvelcdm 7028	. . . . . . . . . 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 597	. . . . . . . . 9 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑏 ∈ (1...𝑎)) → ((𝑛 ∈ ℕ ↦ (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛))))‘𝑏) ∈ ℂ)
109	108	adantlr 716	. . . . . . . 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 13978	. . . . . . 7 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑎 ∈ ℕ) → (seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛)))))‘𝑎) ∈ ℂ)
111	110	adantlr 716	. . . . . 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 35946	. . . . . . . . . . 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 7369	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑏 → (1 / 𝑛) = (1 / 𝑏))
114	113	oveq2d 7377	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑏 → (1 + (1 / 𝑛)) = (1 + (1 / 𝑏)))
115	114	oveq1d 7376	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑏 → ((1 + (1 / 𝑛))↑(𝑚 + 1)) = ((1 + (1 / 𝑏))↑(𝑚 + 1)))
116		oveq2 7369	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑏 → ((𝑚 + 1) / 𝑛) = ((𝑚 + 1) / 𝑏))
117	116	oveq2d 7377	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑏 → (1 + ((𝑚 + 1) / 𝑛)) = (1 + ((𝑚 + 1) / 𝑏)))
118	115, 117	oveq12d 7379	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑏 → (((1 + (1 / 𝑛))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑛))) = (((1 + (1 / 𝑏))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑏))))
119		eqid 2737	. . . . . . . . . . . . 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 7394	. . . . . . . . . . . . 13 ⊢ (((1 + (1 / 𝑏))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑏))) ∈ V
121	118, 119, 120	fvmpt 6942	. . . . . . . . . . . 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 7376	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑏 → ((1 + (1 / 𝑛))↑𝑚) = ((1 + (1 / 𝑏))↑𝑚))
124		oveq2 7369	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑏 → (𝑚 / 𝑛) = (𝑚 / 𝑏))
125	124	oveq2d 7377	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑏 → (1 + (𝑚 / 𝑛)) = (1 + (𝑚 / 𝑏)))
126	123, 125	oveq12d 7379	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑏 → (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))) = (((1 + (1 / 𝑏))↑𝑚) / (1 + (𝑚 / 𝑏))))
127		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛)))) = (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))
128		ovex 7394	. . . . . . . . . . . . . 14 ⊢ (((1 + (1 / 𝑏))↑𝑚) / (1 + (𝑚 / 𝑏))) ∈ V
129	126, 127, 128	fvmpt 6942	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))‘𝑏) = (((1 + (1 / 𝑏))↑𝑚) / (1 + (𝑚 / 𝑏))))
130	125, 114	oveq12d 7379	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑏 → ((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) = ((1 + (𝑚 / 𝑏)) · (1 + (1 / 𝑏))))
131	130, 117	oveq12d 7379	. . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . 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 7394	. . . . . . . . . . . . . 14 ⊢ (((1 + (𝑚 / 𝑏)) · (1 + (1 / 𝑏))) / (1 + ((𝑚 + 1) / 𝑏))) ∈ V
134	131, 132, 133	fvmpt 6942	. . . . . . . . . . . . 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 7379	. . . . . . . . . . . 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 2782	. . . . . . . . . 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 594	. . . . . . . . 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 716	. . . . . . . 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 15849	. . . . . . 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 716	. . . . . 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 15589	. . . . 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 14234	. . . . . 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 5114	. . . 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 12618	. 2 ⊢ (𝐴 ∈ ℕ0 → seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝐴) / (1 + (𝐴 / 𝑛))))) ⇝ (!‘𝐴))
148	3, 147	eqbrtrid 5121	1 ⊢ (𝐴 ∈ ℕ0 → seq1( · , 𝐹) ⇝ (!‘𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542  ⊤wtru 1543   ∈ wcel 2114  Vcvv 3430  {csn 4568   class class class wbr 5086   ↦ cmpt 5167   × cxp 5623  ⟶wf 6489  ‘cfv 6493  (class class class)co 7361  ℂcc 11030  0cc0 11032  1c1 11033   + caddc 11035   · cmul 11037   / cdiv 11801  ℕcn 12168  ℕ₀cn0 12431  ℤcz 12518  ℤ_≥cuz 12782  ℝ⁺crp 12936  ...cfz 13455  seqcseq 13957  ↑cexp 14017  !cfa 14229   ⇝ cli 15440

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683  ax-inf2 9556  ax-cnex 11088  ax-resscn 11089  ax-1cn 11090  ax-icn 11091  ax-addcl 11092  ax-addrcl 11093  ax-mulcl 11094  ax-mulrcl 11095  ax-mulcom 11096  ax-addass 11097  ax-mulass 11098  ax-distr 11099  ax-i2m1 11100  ax-1ne0 11101  ax-1rid 11102  ax-rnegex 11103  ax-rrecex 11104  ax-cnre 11105  ax-pre-lttri 11106  ax-pre-lttrn 11107  ax-pre-ltadd 11108  ax-pre-mulgt0 11109  ax-pre-sup 11110

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7812  df-1st 7936  df-2nd 7937  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-er 8637  df-pm 8770  df-en 8888  df-dom 8889  df-sdom 8890  df-sup 9349  df-inf 9350  df-pnf 11175  df-mnf 11176  df-xr 11177  df-ltxr 11178  df-le 11179  df-sub 11373  df-neg 11374  df-div 11802  df-nn 12169  df-2 12238  df-3 12239  df-n0 12432  df-z 12519  df-uz 12783  df-rp 12937  df-fz 13456  df-fzo 13603  df-fl 13745  df-seq 13958  df-exp 14018  df-fac 14230  df-shft 15023  df-cj 15055  df-re 15056  df-im 15057  df-sqrt 15191  df-abs 15192  df-clim 15444  df-rlim 15445

This theorem is referenced by:  iprodfac  35948