faclim - Mathbox for Scott Fenton

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Theorem faclim 35008

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 13975	. . 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 7419	. . . . . . 7 ⊢ (𝑎 = 0 → ((1 + (1 / 𝑛))↑𝑎) = ((1 + (1 / 𝑛))↑0))
5		oveq1 7418	. . . . . . . 8 ⊢ (𝑎 = 0 → (𝑎 / 𝑛) = (0 / 𝑛))
6	5	oveq2d 7427	. . . . . . 7 ⊢ (𝑎 = 0 → (1 + (𝑎 / 𝑛)) = (1 + (0 / 𝑛)))
7	4, 6	oveq12d 7429	. . . . . 6 ⊢ (𝑎 = 0 → (((1 + (1 / 𝑛))↑𝑎) / (1 + (𝑎 / 𝑛))) = (((1 + (1 / 𝑛))↑0) / (1 + (0 / 𝑛))))
8	7	mpteq2dv 5250	. . . . 5 ⊢ (𝑎 = 0 → (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑎) / (1 + (𝑎 / 𝑛)))) = (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑0) / (1 + (0 / 𝑛)))))
9	8	seqeq3d 13978	. . . 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 6891	. . . . 5 ⊢ (𝑎 = 0 → (!‘𝑎) = (!‘0))
11		fac0 14240	. . . . 5 ⊢ (!‘0) = 1
12	10, 11	eqtrdi 2788	. . . 4 ⊢ (𝑎 = 0 → (!‘𝑎) = 1)
13	9, 12	breq12d 5161	. . 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 7419	. . . . . . 7 ⊢ (𝑎 = 𝑚 → ((1 + (1 / 𝑛))↑𝑎) = ((1 + (1 / 𝑛))↑𝑚))
15		oveq1 7418	. . . . . . . 8 ⊢ (𝑎 = 𝑚 → (𝑎 / 𝑛) = (𝑚 / 𝑛))
16	15	oveq2d 7427	. . . . . . 7 ⊢ (𝑎 = 𝑚 → (1 + (𝑎 / 𝑛)) = (1 + (𝑚 / 𝑛)))
17	14, 16	oveq12d 7429	. . . . . 6 ⊢ (𝑎 = 𝑚 → (((1 + (1 / 𝑛))↑𝑎) / (1 + (𝑎 / 𝑛))) = (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))
18	17	mpteq2dv 5250	. . . . 5 ⊢ (𝑎 = 𝑚 → (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑎) / (1 + (𝑎 / 𝑛)))) = (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛)))))
19	18	seqeq3d 13978	. . . 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 6891	. . . 4 ⊢ (𝑎 = 𝑚 → (!‘𝑎) = (!‘𝑚))
21	19, 20	breq12d 5161	. . 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 7419	. . . . . . 7 ⊢ (𝑎 = (𝑚 + 1) → ((1 + (1 / 𝑛))↑𝑎) = ((1 + (1 / 𝑛))↑(𝑚 + 1)))
23		oveq1 7418	. . . . . . . 8 ⊢ (𝑎 = (𝑚 + 1) → (𝑎 / 𝑛) = ((𝑚 + 1) / 𝑛))
24	23	oveq2d 7427	. . . . . . 7 ⊢ (𝑎 = (𝑚 + 1) → (1 + (𝑎 / 𝑛)) = (1 + ((𝑚 + 1) / 𝑛)))
25	22, 24	oveq12d 7429	. . . . . 6 ⊢ (𝑎 = (𝑚 + 1) → (((1 + (1 / 𝑛))↑𝑎) / (1 + (𝑎 / 𝑛))) = (((1 + (1 / 𝑛))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑛))))
26	25	mpteq2dv 5250	. . . . 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 13978	. . . 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 6891	. . . 4 ⊢ (𝑎 = (𝑚 + 1) → (!‘𝑎) = (!‘(𝑚 + 1)))
29	27, 28	breq12d 5161	. . 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 7419	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ((1 + (1 / 𝑛))↑𝑎) = ((1 + (1 / 𝑛))↑𝐴))
31		oveq1 7418	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑎 / 𝑛) = (𝐴 / 𝑛))
32	31	oveq2d 7427	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (1 + (𝑎 / 𝑛)) = (1 + (𝐴 / 𝑛)))
33	30, 32	oveq12d 7429	. . . . . 6 ⊢ (𝑎 = 𝐴 → (((1 + (1 / 𝑛))↑𝑎) / (1 + (𝑎 / 𝑛))) = (((1 + (1 / 𝑛))↑𝐴) / (1 + (𝐴 / 𝑛))))
34	33	mpteq2dv 5250	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑎) / (1 + (𝑎 / 𝑛)))) = (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝐴) / (1 + (𝐴 / 𝑛)))))
35	34	seqeq3d 13978	. . . 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 6891	. . . 4 ⊢ (𝑎 = 𝐴 → (!‘𝑎) = (!‘𝐴))
37	35, 36	breq12d 5161	. . 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 11219	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 1 ∈ ℝ)
39		nnrecre 12258	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ)
40	38, 39	readdcld 11247	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (1 + (1 / 𝑛)) ∈ ℝ)
41	40	recnd 11246	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (1 + (1 / 𝑛)) ∈ ℂ)
42	41	exp0d 14109	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ((1 + (1 / 𝑛))↑0) = 1)
43		nncn 12224	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
44		nnne0 12250	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 𝑛 ≠ 0)
45	43, 44	div0d 11993	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (0 / 𝑛) = 0)
46	45	oveq2d 7427	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (1 + (0 / 𝑛)) = (1 + 0))
47		1p0e1 12340	. . . . . . . . . 10 ⊢ (1 + 0) = 1
48	46, 47	eqtrdi 2788	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (1 + (0 / 𝑛)) = 1)
49	42, 48	oveq12d 7429	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (((1 + (1 / 𝑛))↑0) / (1 + (0 / 𝑛))) = (1 / 1))
50		1div1e1 11908	. . . . . . . 8 ⊢ (1 / 1) = 1
51	49, 50	eqtrdi 2788	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (((1 + (1 / 𝑛))↑0) / (1 + (0 / 𝑛))) = 1)
52	51	mpteq2ia 5251	. . . . . 6 ⊢ (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑0) / (1 + (0 / 𝑛)))) = (𝑛 ∈ ℕ ↦ 1)
53		fconstmpt 5738	. . . . . 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 13975	. . . . 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 12869	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
58		1zzd 12597	. . . . . 6 ⊢ (⊤ → 1 ∈ ℤ)
59	57, 58	climprod1 15913	. . . . 5 ⊢ (⊤ → seq1( · , (ℕ × {1})) ⇝ 1)
60	59	mptru 1548	. . . 4 ⊢ seq1( · , (ℕ × {1})) ⇝ 1
61	56, 60	eqbrtri 5169	. . 3 ⊢ seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑0) / (1 + (0 / 𝑛))))) ⇝ 1
62		1zzd 12597	. . . . . 6 ⊢ ((𝑚 ∈ ℕ₀ ∧ seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))) ⇝ (!‘𝑚)) → 1 ∈ ℤ)
63		simpr 485	. . . . . 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 13972	. . . . . . 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 35006	. . . . . . 7 ⊢ (𝑚 ∈ ℕ₀ → seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛))))) ⇝ (𝑚 + 1))
67	66	adantr 481	. . . . . 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 12870	. . . . . . . . . 10 ⊢ (𝑎 ∈ ℕ ↔ 𝑎 ∈ (ℤ_≥‘1))
69	68	biimpi 215	. . . . . . . . 9 ⊢ (𝑎 ∈ ℕ → 𝑎 ∈ (ℤ_≥‘1))
70	69	adantl 482	. . . . . . . 8 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝑎 ∈ ℕ) → 𝑎 ∈ (ℤ_≥‘1))
71		1rp 12982	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℝ⁺
72	71	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝑛 ∈ ℕ) → 1 ∈ ℝ⁺)
73		nnrp 12989	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ⁺)
74	73	rpreccld 13030	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ⁺)
75	74	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈ ℝ⁺)
76	72, 75	rpaddcld 13035	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝑛 ∈ ℕ) → (1 + (1 / 𝑛)) ∈ ℝ⁺)
77		nn0z 12587	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ℕ₀ → 𝑚 ∈ ℤ)
78	77	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝑛 ∈ ℕ) → 𝑚 ∈ ℤ)
79	76, 78	rpexpcld 14214	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝑛 ∈ ℕ) → ((1 + (1 / 𝑛))↑𝑚) ∈ ℝ⁺)
80		1cnd 11213	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝑛 ∈ ℕ) → 1 ∈ ℂ)
81		nn0nndivcl 12547	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝑛 ∈ ℕ) → (𝑚 / 𝑛) ∈ ℝ)
82	81	recnd 11246	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝑛 ∈ ℕ) → (𝑚 / 𝑛) ∈ ℂ)
83	80, 82	addcomd 11420	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝑛 ∈ ℕ) → (1 + (𝑚 / 𝑛)) = ((𝑚 / 𝑛) + 1))
84		nn0ge0div 12635	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝑛 ∈ ℕ) → 0 ≤ (𝑚 / 𝑛))
85	81, 84	ge0p1rpd 13050	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝑛 ∈ ℕ) → ((𝑚 / 𝑛) + 1) ∈ ℝ⁺)
86	83, 85	eqeltrd 2833	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝑛 ∈ ℕ) → (1 + (𝑚 / 𝑛)) ∈ ℝ⁺)
87	79, 86	rpdivcld 13037	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝑛 ∈ ℕ) → (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))) ∈ ℝ⁺)
88	87	rpcnd 13022	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝑛 ∈ ℕ) → (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))) ∈ ℂ)
89	88	fmpttd 7116	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ₀ → (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛)))):ℕ⟶ℂ)
90		elfznn 13534	. . . . . . . . . 10 ⊢ (𝑏 ∈ (1...𝑎) → 𝑏 ∈ ℕ)
91		ffvelcdm 7083	. . . . . . . . . 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 596	. . . . . . . . 9 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝑏 ∈ (1...𝑎)) → ((𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))‘𝑏) ∈ ℂ)
93	92	adantlr 713	. . . . . . . 8 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝑎 ∈ ℕ) ∧ 𝑏 ∈ (1...𝑎)) → ((𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))‘𝑏) ∈ ℂ)
94		mulcl 11196	. . . . . . . . 9 ⊢ ((𝑏 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑏 · 𝑥) ∈ ℂ)
95	94	adantl 482	. . . . . . . 8 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝑎 ∈ ℕ) ∧ (𝑏 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → (𝑏 · 𝑥) ∈ ℂ)
96	70, 93, 95	seqcl 13992	. . . . . . 7 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝑎 ∈ ℕ) → (seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛)))))‘𝑎) ∈ ℂ)
97	96	adantlr 713	. . . . . 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 13036	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝑛 ∈ ℕ) → ((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) ∈ ℝ⁺)
99		nn0p1nn 12515	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ ℕ₀ → (𝑚 + 1) ∈ ℕ)
100	99	nnrpd 13018	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ℕ₀ → (𝑚 + 1) ∈ ℝ⁺)
101		rpdivcl 13003	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 + 1) ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → ((𝑚 + 1) / 𝑛) ∈ ℝ⁺)
102	100, 73, 101	syl2an 596	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝑛 ∈ ℕ) → ((𝑚 + 1) / 𝑛) ∈ ℝ⁺)
103	72, 102	rpaddcld 13035	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝑛 ∈ ℕ) → (1 + ((𝑚 + 1) / 𝑛)) ∈ ℝ⁺)
104	98, 103	rpdivcld 13037	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝑛 ∈ ℕ) → (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛))) ∈ ℝ⁺)
105	104	rpcnd 13022	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝑛 ∈ ℕ) → (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛))) ∈ ℂ)
106	105	fmpttd 7116	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ₀ → (𝑛 ∈ ℕ ↦ (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛)))):ℕ⟶ℂ)
107		ffvelcdm 7083	. . . . . . . . . 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 596	. . . . . . . . 9 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝑏 ∈ (1...𝑎)) → ((𝑛 ∈ ℕ ↦ (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛))))‘𝑏) ∈ ℂ)
109	108	adantlr 713	. . . . . . . 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 13992	. . . . . . 7 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝑎 ∈ ℕ) → (seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛)))))‘𝑎) ∈ ℂ)
111	110	adantlr 713	. . . . . 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 35007	. . . . . . . . . . 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 7419	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑏 → (1 / 𝑛) = (1 / 𝑏))
114	113	oveq2d 7427	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑏 → (1 + (1 / 𝑛)) = (1 + (1 / 𝑏)))
115	114	oveq1d 7426	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑏 → ((1 + (1 / 𝑛))↑(𝑚 + 1)) = ((1 + (1 / 𝑏))↑(𝑚 + 1)))
116		oveq2 7419	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑏 → ((𝑚 + 1) / 𝑛) = ((𝑚 + 1) / 𝑏))
117	116	oveq2d 7427	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑏 → (1 + ((𝑚 + 1) / 𝑛)) = (1 + ((𝑚 + 1) / 𝑏)))
118	115, 117	oveq12d 7429	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑏 → (((1 + (1 / 𝑛))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑛))) = (((1 + (1 / 𝑏))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑏))))
119		eqid 2732	. . . . . . . . . . . . 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 7444	. . . . . . . . . . . . 13 ⊢ (((1 + (1 / 𝑏))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑏))) ∈ V
121	118, 119, 120	fvmpt 6998	. . . . . . . . . . . 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 482	. . . . . . . . . . 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 7426	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑏 → ((1 + (1 / 𝑛))↑𝑚) = ((1 + (1 / 𝑏))↑𝑚))
124		oveq2 7419	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑏 → (𝑚 / 𝑛) = (𝑚 / 𝑏))
125	124	oveq2d 7427	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑏 → (1 + (𝑚 / 𝑛)) = (1 + (𝑚 / 𝑏)))
126	123, 125	oveq12d 7429	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑏 → (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))) = (((1 + (1 / 𝑏))↑𝑚) / (1 + (𝑚 / 𝑏))))
127		eqid 2732	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛)))) = (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))
128		ovex 7444	. . . . . . . . . . . . . 14 ⊢ (((1 + (1 / 𝑏))↑𝑚) / (1 + (𝑚 / 𝑏))) ∈ V
129	126, 127, 128	fvmpt 6998	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))‘𝑏) = (((1 + (1 / 𝑏))↑𝑚) / (1 + (𝑚 / 𝑏))))
130	125, 114	oveq12d 7429	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑏 → ((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) = ((1 + (𝑚 / 𝑏)) · (1 + (1 / 𝑏))))
131	130, 117	oveq12d 7429	. . . . . . . . . . . . . 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 2732	. . . . . . . . . . . . . 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 7444	. . . . . . . . . . . . . 14 ⊢ (((1 + (𝑚 / 𝑏)) · (1 + (1 / 𝑏))) / (1 + ((𝑚 + 1) / 𝑏))) ∈ V
134	131, 132, 133	fvmpt 6998	. . . . . . . . . . . . 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 7429	. . . . . . . . . . . 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 482	. . . . . . . . . . 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 593	. . . . . . . . 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 713	. . . . . . . 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 15840	. . . . . . 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 713	. . . . . 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 15581	. . . . 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 14242	. . . . . 6 ⊢ (𝑚 ∈ ℕ₀ → (!‘(𝑚 + 1)) = ((!‘𝑚) · (𝑚 + 1)))
144	143	adantr 481	. . . . 5 ⊢ ((𝑚 ∈ ℕ₀ ∧ seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))) ⇝ (!‘𝑚)) → (!‘(𝑚 + 1)) = ((!‘𝑚) · (𝑚 + 1)))
145	142, 144	breqtrrd 5176	. . . 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 413	. . 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 12661	. 2 ⊢ (𝐴 ∈ ℕ₀ → seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝐴) / (1 + (𝐴 / 𝑛))))) ⇝ (!‘𝐴))
148	3, 147	eqbrtrid 5183	1 ⊢ (𝐴 ∈ ℕ₀ → seq1( · , 𝐹) ⇝ (!‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ⊤wtru 1542 ∈ wcel 2106 Vcvv 3474 {csn 4628 class class class wbr 5148 ↦ cmpt 5231 × cxp 5674 ⟶wf 6539 ‘cfv 6543 (class class class)co 7411 ℂcc 11110 0cc0 11112 1c1 11113 + caddc 11115 · cmul 11117 / cdiv 11875 ℕcn 12216 ℕ₀cn0 12476 ℤcz 12562 ℤ_≥cuz 12826 ℝ⁺crp 12978 ...cfz 13488 seqcseq 13970 ↑cexp 14031 !cfa 14237 ⇝ cli 15432

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

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-op 4635 df-uni 4909 df-iun 4999 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-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-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-er 8705 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-sup 9439 df-inf 9440 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12979 df-fz 13489 df-fzo 13632 df-fl 13761 df-seq 13971 df-exp 14032 df-fac 14238 df-shft 15018 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-clim 15436 df-rlim 15437

This theorem is referenced by: iprodfac 35009

W3C validator