stirlinglem8 - Mathbox for Glauco Siliprandi

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Theorem stirlinglem8 46531

Description: If 𝐴 converges to 𝐶, then 𝐹 converges to C^2 . (Contributed by Glauco Siliprandi, 29-Jun-2017.)

**Hypotheses**
Ref	Expression
stirlinglem8.1	⊢ Ⅎ𝑛𝜑
stirlinglem8.2	⊢ Ⅎ𝑛𝐴
stirlinglem8.3	⊢ Ⅎ𝑛𝐷
stirlinglem8.4	⊢ 𝐷 = (𝑛 ∈ ℕ ↦ (𝐴‘(2 · 𝑛)))
stirlinglem8.5	⊢ (𝜑 → 𝐴:ℕ⟶ℝ⁺)
stirlinglem8.6	⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (((𝐴‘𝑛)↑4) / ((𝐷‘𝑛)↑2)))
stirlinglem8.7	⊢ 𝐿 = (𝑛 ∈ ℕ ↦ ((𝐴‘𝑛)↑4))
stirlinglem8.8	⊢ 𝑀 = (𝑛 ∈ ℕ ↦ ((𝐷‘𝑛)↑2))
stirlinglem8.9	⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) ∈ ℝ⁺)
stirlinglem8.10	⊢ (𝜑 → 𝐶 ∈ ℝ⁺)
stirlinglem8.11	⊢ (𝜑 → 𝐴 ⇝ 𝐶)

**Assertion**
Ref	Expression
stirlinglem8	⊢ (𝜑 → 𝐹 ⇝ (𝐶↑2))

Proof of Theorem stirlinglem8 Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		stirlinglem8.1	. . 3 ⊢ Ⅎ𝑛𝜑
2		stirlinglem8.7	. . . 4 ⊢ 𝐿 = (𝑛 ∈ ℕ ↦ ((𝐴‘𝑛)↑4))
3		nfmpt1 5185	. . . 4 ⊢ Ⅎ𝑛(𝑛 ∈ ℕ ↦ ((𝐴‘𝑛)↑4))
4	2, 3	nfcxfr 2897	. . 3 ⊢ Ⅎ𝑛𝐿
5		stirlinglem8.8	. . . 4 ⊢ 𝑀 = (𝑛 ∈ ℕ ↦ ((𝐷‘𝑛)↑2))
6		nfmpt1 5185	. . . 4 ⊢ Ⅎ𝑛(𝑛 ∈ ℕ ↦ ((𝐷‘𝑛)↑2))
7	5, 6	nfcxfr 2897	. . 3 ⊢ Ⅎ𝑛𝑀
8		stirlinglem8.6	. . . 4 ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (((𝐴‘𝑛)↑4) / ((𝐷‘𝑛)↑2)))
9		nfmpt1 5185	. . . 4 ⊢ Ⅎ𝑛(𝑛 ∈ ℕ ↦ (((𝐴‘𝑛)↑4) / ((𝐷‘𝑛)↑2)))
10	8, 9	nfcxfr 2897	. . 3 ⊢ Ⅎ𝑛𝐹
11		nnuz 12822	. . 3 ⊢ ℕ = (ℤ_≥‘1)
12		1zzd 12553	. . 3 ⊢ (𝜑 → 1 ∈ ℤ)
13		stirlinglem8.2	. . . 4 ⊢ Ⅎ𝑛𝐴
14		stirlinglem8.5	. . . . 5 ⊢ (𝜑 → 𝐴:ℕ⟶ℝ⁺)
15		rrpsscn 46040	. . . . 5 ⊢ ℝ⁺ ⊆ ℂ
16		fss 6680	. . . . 5 ⊢ ((𝐴:ℕ⟶ℝ⁺ ∧ ℝ⁺ ⊆ ℂ) → 𝐴:ℕ⟶ℂ)
17	14, 15, 16	sylancl 587	. . . 4 ⊢ (𝜑 → 𝐴:ℕ⟶ℂ)
18		stirlinglem8.11	. . . 4 ⊢ (𝜑 → 𝐴 ⇝ 𝐶)
19		4nn0 12451	. . . . 5 ⊢ 4 ∈ ℕ₀
20	19	a1i 11	. . . 4 ⊢ (𝜑 → 4 ∈ ℕ₀)
21		nnex 12175	. . . . . . 7 ⊢ ℕ ∈ V
22	21	mptex 7173	. . . . . 6 ⊢ (𝑛 ∈ ℕ ↦ ((𝐴‘𝑛)↑4)) ∈ V
23	2, 22	eqeltri 2833	. . . . 5 ⊢ 𝐿 ∈ V
24	23	a1i 11	. . . 4 ⊢ (𝜑 → 𝐿 ∈ V)
25		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
26	14	ffvelcdmda 7032	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ∈ ℝ⁺)
27	26	rpcnd 12983	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ∈ ℂ)
28	19	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 4 ∈ ℕ₀)
29	27, 28	expcld 14103	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐴‘𝑛)↑4) ∈ ℂ)
30	2	fvmpt2 6955	. . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ ((𝐴‘𝑛)↑4) ∈ ℂ) → (𝐿‘𝑛) = ((𝐴‘𝑛)↑4))
31	25, 29, 30	syl2anc 585	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐿‘𝑛) = ((𝐴‘𝑛)↑4))
32	1, 13, 4, 11, 12, 17, 18, 20, 24, 31	climexp 46057	. . 3 ⊢ (𝜑 → 𝐿 ⇝ (𝐶↑4))
33	21	mptex 7173	. . . . 5 ⊢ (𝑛 ∈ ℕ ↦ (((𝐴‘𝑛)↑4) / ((𝐷‘𝑛)↑2))) ∈ V
34	8, 33	eqeltri 2833	. . . 4 ⊢ 𝐹 ∈ V
35	34	a1i 11	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
36		stirlinglem8.3	. . . 4 ⊢ Ⅎ𝑛𝐷
37	17	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴:ℕ⟶ℂ)
38		2nn 12249	. . . . . . . . 9 ⊢ 2 ∈ ℕ
39	38	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → 2 ∈ ℕ)
40		id 22	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ)
41	39, 40	nnmulcld 12225	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (2 · 𝑛) ∈ ℕ)
42	41	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2 · 𝑛) ∈ ℕ)
43	37, 42	ffvelcdmd 7033	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘(2 · 𝑛)) ∈ ℂ)
44		stirlinglem8.4	. . . . 5 ⊢ 𝐷 = (𝑛 ∈ ℕ ↦ (𝐴‘(2 · 𝑛)))
45	1, 43, 44	fmptdf 7065	. . . 4 ⊢ (𝜑 → 𝐷:ℕ⟶ℂ)
46		nfmpt1 5185	. . . . 5 ⊢ Ⅎ𝑛(𝑛 ∈ ℕ ↦ (2 · 𝑛))
47		fex 7176	. . . . . 6 ⊢ ((𝐴:ℕ⟶ℂ ∧ ℕ ∈ V) → 𝐴 ∈ V)
48	17, 21, 47	sylancl 587	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ V)
49		1nn 12180	. . . . . . 7 ⊢ 1 ∈ ℕ
50		2cnd 12254	. . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℂ)
51		1cnd 11134	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℂ)
52	50, 51	mulcld 11160	. . . . . . 7 ⊢ (𝜑 → (2 · 1) ∈ ℂ)
53		oveq2 7370	. . . . . . . 8 ⊢ (𝑛 = 1 → (2 · 𝑛) = (2 · 1))
54		eqid 2737	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ ↦ (2 · 𝑛)) = (𝑛 ∈ ℕ ↦ (2 · 𝑛))
55	53, 54	fvmptg 6941	. . . . . . 7 ⊢ ((1 ∈ ℕ ∧ (2 · 1) ∈ ℂ) → ((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘1) = (2 · 1))
56	49, 52, 55	sylancr 588	. . . . . 6 ⊢ (𝜑 → ((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘1) = (2 · 1))
57	38	a1i 11	. . . . . . 7 ⊢ (𝜑 → 2 ∈ ℕ)
58	49	a1i 11	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℕ)
59	57, 58	nnmulcld 12225	. . . . . 6 ⊢ (𝜑 → (2 · 1) ∈ ℕ)
60	56, 59	eqeltrd 2837	. . . . 5 ⊢ (𝜑 → ((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘1) ∈ ℕ)
61		1red 11140	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → 1 ∈ ℝ)
62	39	nnred 12184	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → 2 ∈ ℝ)
63	41	nnred 12184	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (2 · 𝑛) ∈ ℝ)
64	39	nnge1d 12220	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → 1 ≤ 2)
65	61, 62, 63, 64	leadd2dd 11760	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → ((2 · 𝑛) + 1) ≤ ((2 · 𝑛) + 2))
66	54	fvmpt2 6955	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ (2 · 𝑛) ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘𝑛) = (2 · 𝑛))
67	41, 66	mpdan 688	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘𝑛) = (2 · 𝑛))
68	67	oveq1d 7377	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘𝑛) + 1) = ((2 · 𝑛) + 1))
69		oveq2 7370	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (2 · 𝑛) = (2 · 𝑘))
70	69	cbvmptv 5190	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ ↦ (2 · 𝑛)) = (𝑘 ∈ ℕ ↦ (2 · 𝑘))
71	70	a1i 11	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (𝑛 ∈ ℕ ↦ (2 · 𝑛)) = (𝑘 ∈ ℕ ↦ (2 · 𝑘)))
72		simpr 484	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 = (𝑛 + 1)) → 𝑘 = (𝑛 + 1))
73	72	oveq2d 7378	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 = (𝑛 + 1)) → (2 · 𝑘) = (2 · (𝑛 + 1)))
74		peano2nn 12181	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
75	39, 74	nnmulcld 12225	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (2 · (𝑛 + 1)) ∈ ℕ)
76	71, 73, 74, 75	fvmptd 6951	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘(𝑛 + 1)) = (2 · (𝑛 + 1)))
77		2cnd 12254	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 2 ∈ ℂ)
78		nncn 12177	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
79		1cnd 11134	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 1 ∈ ℂ)
80	77, 78, 79	adddid 11164	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (2 · (𝑛 + 1)) = ((2 · 𝑛) + (2 · 1)))
81	77	mulridd 11157	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (2 · 1) = 2)
82	81	oveq2d 7378	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ((2 · 𝑛) + (2 · 1)) = ((2 · 𝑛) + 2))
83	76, 80, 82	3eqtrd 2776	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘(𝑛 + 1)) = ((2 · 𝑛) + 2))
84	65, 68, 83	3brtr4d 5118	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘𝑛) + 1) ≤ ((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘(𝑛 + 1)))
85	41	nnzd 12545	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (2 · 𝑛) ∈ ℤ)
86	67, 85	eqeltrd 2837	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘𝑛) ∈ ℤ)
87	86	peano2zd 12631	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘𝑛) + 1) ∈ ℤ)
88	75	nnzd 12545	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (2 · (𝑛 + 1)) ∈ ℤ)
89	76, 88	eqeltrd 2837	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘(𝑛 + 1)) ∈ ℤ)
90		eluz 12797	. . . . . . . 8 ⊢ (((((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘𝑛) + 1) ∈ ℤ ∧ ((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘(𝑛 + 1)) ∈ ℤ) → (((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘(𝑛 + 1)) ∈ (ℤ_≥‘(((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘𝑛) + 1)) ↔ (((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘𝑛) + 1) ≤ ((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘(𝑛 + 1))))
91	87, 89, 90	syl2anc 585	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘(𝑛 + 1)) ∈ (ℤ_≥‘(((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘𝑛) + 1)) ↔ (((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘𝑛) + 1) ≤ ((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘(𝑛 + 1))))
92	84, 91	mpbird 257	. . . . . 6 ⊢ (𝑛 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘(𝑛 + 1)) ∈ (ℤ_≥‘(((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘𝑛) + 1)))
93	92	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘(𝑛 + 1)) ∈ (ℤ_≥‘(((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘𝑛) + 1)))
94	21	mptex 7173	. . . . . . 7 ⊢ (𝑛 ∈ ℕ ↦ (𝐴‘(2 · 𝑛))) ∈ V
95	44, 94	eqeltri 2833	. . . . . 6 ⊢ 𝐷 ∈ V
96	95	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ V)
97	44	fvmpt2 6955	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ ∧ (𝐴‘(2 · 𝑛)) ∈ ℂ) → (𝐷‘𝑛) = (𝐴‘(2 · 𝑛)))
98	25, 43, 97	syl2anc 585	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) = (𝐴‘(2 · 𝑛)))
99	67	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘𝑛) = (2 · 𝑛))
100	99	eqcomd 2743	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2 · 𝑛) = ((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘𝑛))
101	100	fveq2d 6840	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘(2 · 𝑛)) = (𝐴‘((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘𝑛)))
102	98, 101	eqtrd 2772	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) = (𝐴‘((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘𝑛)))
103	1, 13, 36, 46, 11, 12, 48, 27, 18, 60, 93, 96, 102	climsuse 46060	. . . 4 ⊢ (𝜑 → 𝐷 ⇝ 𝐶)
104		2nn0 12449	. . . . 5 ⊢ 2 ∈ ℕ₀
105	104	a1i 11	. . . 4 ⊢ (𝜑 → 2 ∈ ℕ₀)
106	21	mptex 7173	. . . . . 6 ⊢ (𝑛 ∈ ℕ ↦ ((𝐷‘𝑛)↑2)) ∈ V
107	5, 106	eqeltri 2833	. . . . 5 ⊢ 𝑀 ∈ V
108	107	a1i 11	. . . 4 ⊢ (𝜑 → 𝑀 ∈ V)
109		stirlinglem8.9	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) ∈ ℝ⁺)
110	109	rpcnd 12983	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) ∈ ℂ)
111	110	sqcld 14101	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐷‘𝑛)↑2) ∈ ℂ)
112	5	fvmpt2 6955	. . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ ((𝐷‘𝑛)↑2) ∈ ℂ) → (𝑀‘𝑛) = ((𝐷‘𝑛)↑2))
113	25, 111, 112	syl2anc 585	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘𝑛) = ((𝐷‘𝑛)↑2))
114	1, 36, 7, 11, 12, 45, 103, 105, 108, 113	climexp 46057	. . 3 ⊢ (𝜑 → 𝑀 ⇝ (𝐶↑2))
115		stirlinglem8.10	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ⁺)
116	115	rpcnd 12983	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℂ)
117	115	rpne0d 12986	. . . 4 ⊢ (𝜑 → 𝐶 ≠ 0)
118		2z 12554	. . . . 5 ⊢ 2 ∈ ℤ
119	118	a1i 11	. . . 4 ⊢ (𝜑 → 2 ∈ ℤ)
120	116, 117, 119	expne0d 14109	. . 3 ⊢ (𝜑 → (𝐶↑2) ≠ 0)
121	1, 29, 2	fmptdf 7065	. . . 4 ⊢ (𝜑 → 𝐿:ℕ⟶ℂ)
122	121	ffvelcdmda 7032	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐿‘𝑛) ∈ ℂ)
123	113, 111	eqeltrd 2837	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘𝑛) ∈ ℂ)
124	98	oveq1d 7377	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐷‘𝑛)↑2) = ((𝐴‘(2 · 𝑛))↑2))
125	113, 124	eqtrd 2772	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘𝑛) = ((𝐴‘(2 · 𝑛))↑2))
126	98, 109	eqeltrrd 2838	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘(2 · 𝑛)) ∈ ℝ⁺)
127	118	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 2 ∈ ℤ)
128	126, 127	rpexpcld 14204	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐴‘(2 · 𝑛))↑2) ∈ ℝ⁺)
129	125, 128	eqeltrd 2837	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘𝑛) ∈ ℝ⁺)
130	129	rpne0d 12986	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘𝑛) ≠ 0)
131	130	neneqd 2938	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ¬ (𝑀‘𝑛) = 0)
132		0cn 11131	. . . . . 6 ⊢ 0 ∈ ℂ
133		elsn2g 4609	. . . . . 6 ⊢ (0 ∈ ℂ → ((𝑀‘𝑛) ∈ {0} ↔ (𝑀‘𝑛) = 0))
134	132, 133	ax-mp 5	. . . . 5 ⊢ ((𝑀‘𝑛) ∈ {0} ↔ (𝑀‘𝑛) = 0)
135	131, 134	sylnibr 329	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ¬ (𝑀‘𝑛) ∈ {0})
136	123, 135	eldifd 3901	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘𝑛) ∈ (ℂ ∖ {0}))
137	28	nn0zd 12544	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 4 ∈ ℤ)
138	26, 137	rpexpcld 14204	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐴‘𝑛)↑4) ∈ ℝ⁺)
139	109, 127	rpexpcld 14204	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐷‘𝑛)↑2) ∈ ℝ⁺)
140	138, 139	rpdivcld 12998	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((𝐴‘𝑛)↑4) / ((𝐷‘𝑛)↑2)) ∈ ℝ⁺)
141	8	fvmpt2 6955	. . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ (((𝐴‘𝑛)↑4) / ((𝐷‘𝑛)↑2)) ∈ ℝ⁺) → (𝐹‘𝑛) = (((𝐴‘𝑛)↑4) / ((𝐷‘𝑛)↑2)))
142	25, 140, 141	syl2anc 585	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) = (((𝐴‘𝑛)↑4) / ((𝐷‘𝑛)↑2)))
143	2	fvmpt2 6955	. . . . . 6 ⊢ ((𝑛 ∈ ℕ ∧ ((𝐴‘𝑛)↑4) ∈ ℝ⁺) → (𝐿‘𝑛) = ((𝐴‘𝑛)↑4))
144	25, 138, 143	syl2anc 585	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐿‘𝑛) = ((𝐴‘𝑛)↑4))
145	144, 113	oveq12d 7380	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐿‘𝑛) / (𝑀‘𝑛)) = (((𝐴‘𝑛)↑4) / ((𝐷‘𝑛)↑2)))
146	142, 145	eqtr4d 2775	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) = ((𝐿‘𝑛) / (𝑀‘𝑛)))
147	1, 4, 7, 10, 11, 12, 32, 35, 114, 120, 122, 136, 146	climdivf 46064	. 2 ⊢ (𝜑 → 𝐹 ⇝ ((𝐶↑4) / (𝐶↑2)))
148		2cn 12251	. . . . . 6 ⊢ 2 ∈ ℂ
149		2p2e4 12306	. . . . . 6 ⊢ (2 + 2) = 4
150	148, 148, 149	mvlladdi 11407	. . . . 5 ⊢ 2 = (4 − 2)
151	150	a1i 11	. . . 4 ⊢ (𝜑 → 2 = (4 − 2))
152	151	oveq2d 7378	. . 3 ⊢ (𝜑 → (𝐶↑2) = (𝐶↑(4 − 2)))
153	20	nn0zd 12544	. . . 4 ⊢ (𝜑 → 4 ∈ ℤ)
154	116, 117, 119, 153	expsubd 14114	. . 3 ⊢ (𝜑 → (𝐶↑(4 − 2)) = ((𝐶↑4) / (𝐶↑2)))
155	152, 154	eqtrd 2772	. 2 ⊢ (𝜑 → (𝐶↑2) = ((𝐶↑4) / (𝐶↑2)))
156	147, 155	breqtrrd 5114	1 ⊢ (𝜑 → 𝐹 ⇝ (𝐶↑2))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 Ⅎwnf 1785 ∈ wcel 2114 Ⅎwnfc 2884 Vcvv 3430 ⊆ wss 3890 {csn 4568 class class class wbr 5086 ↦ cmpt 5167 ⟶wf 6490 ‘cfv 6494 (class class class)co 7362 ℂcc 11031 0cc0 11033 1c1 11034 + caddc 11036 · cmul 11038 ≤ cle 11175 − cmin 11372 / cdiv 11802 ℕcn 12169 2c2 12231 4c4 12233 ℕ₀cn0 12432 ℤcz 12519 ℤ_≥cuz 12783 ℝ⁺crp 12937 ↑cexp 14018 ⇝ cli 15441

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 5304 ax-pr 5372 ax-un 7684 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110 ax-pre-sup 11111

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-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5521 df-eprel 5526 df-po 5534 df-so 5535 df-fr 5579 df-se 5580 df-we 5581 df-xp 5632 df-rel 5633 df-cnv 5634 df-co 5635 df-dm 5636 df-rn 5637 df-res 5638 df-ima 5639 df-pred 6261 df-ord 6322 df-on 6323 df-lim 6324 df-suc 6325 df-iota 6450 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-isom 6503 df-riota 7319 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7626 df-om 7813 df-1st 7937 df-2nd 7938 df-supp 8106 df-frecs 8226 df-wrecs 8257 df-recs 8306 df-rdg 8344 df-1o 8400 df-2o 8401 df-er 8638 df-map 8770 df-ixp 8841 df-en 8889 df-dom 8890 df-sdom 8891 df-fin 8892 df-fsupp 9270 df-fi 9319 df-sup 9350 df-inf 9351 df-oi 9420 df-card 9858 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-div 11803 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-z 12520 df-dec 12640 df-uz 12784 df-q 12894 df-rp 12938 df-xneg 13058 df-xadd 13059 df-xmul 13060 df-icc 13300 df-fz 13457 df-fzo 13604 df-seq 13959 df-exp 14019 df-hash 14288 df-cj 15056 df-re 15057 df-im 15058 df-sqrt 15192 df-abs 15193 df-clim 15445 df-struct 17112 df-sets 17129 df-slot 17147 df-ndx 17159 df-base 17175 df-ress 17196 df-plusg 17228 df-mulr 17229 df-starv 17230 df-sca 17231 df-vsca 17232 df-ip 17233 df-tset 17234 df-ple 17235 df-ds 17237 df-unif 17238 df-hom 17239 df-cco 17240 df-rest 17380 df-topn 17381 df-0g 17399 df-gsum 17400 df-topgen 17401 df-pt 17402 df-prds 17405 df-xrs 17461 df-qtop 17466 df-imas 17467 df-xps 17469 df-mre 17543 df-mrc 17544 df-acs 17546 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-submnd 18747 df-mulg 19039 df-cntz 19287 df-cmn 19752 df-psmet 21340 df-xmet 21341 df-met 21342 df-bl 21343 df-mopn 21344 df-cnfld 21349 df-top 22873 df-topon 22890 df-topsp 22912 df-bases 22925 df-cn 23206 df-cnp 23207 df-tx 23541 df-hmeo 23734 df-xms 24299 df-ms 24300 df-tms 24301 df-cncf 24859

This theorem is referenced by: stirlinglem15 46538

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