stirlinglem8 - Mathbox for Glauco Siliprandi

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Theorem stirlinglem8 44882

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 5256	. . . 4 ⊢ Ⅎ𝑛(𝑛 ∈ ℕ ↦ ((𝐴‘𝑛)↑4))
4	2, 3	nfcxfr 2901	. . 3 ⊢ Ⅎ𝑛𝐿
5		stirlinglem8.8	. . . 4 ⊢ 𝑀 = (𝑛 ∈ ℕ ↦ ((𝐷‘𝑛)↑2))
6		nfmpt1 5256	. . . 4 ⊢ Ⅎ𝑛(𝑛 ∈ ℕ ↦ ((𝐷‘𝑛)↑2))
7	5, 6	nfcxfr 2901	. . 3 ⊢ Ⅎ𝑛𝑀
8		stirlinglem8.6	. . . 4 ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (((𝐴‘𝑛)↑4) / ((𝐷‘𝑛)↑2)))
9		nfmpt1 5256	. . . 4 ⊢ Ⅎ𝑛(𝑛 ∈ ℕ ↦ (((𝐴‘𝑛)↑4) / ((𝐷‘𝑛)↑2)))
10	8, 9	nfcxfr 2901	. . 3 ⊢ Ⅎ𝑛𝐹
11		nnuz 12867	. . 3 ⊢ ℕ = (ℤ_≥‘1)
12		1zzd 12595	. . 3 ⊢ (𝜑 → 1 ∈ ℤ)
13		stirlinglem8.2	. . . 4 ⊢ Ⅎ𝑛𝐴
14		stirlinglem8.5	. . . . 5 ⊢ (𝜑 → 𝐴:ℕ⟶ℝ⁺)
15		rrpsscn 44389	. . . . 5 ⊢ ℝ⁺ ⊆ ℂ
16		fss 6734	. . . . 5 ⊢ ((𝐴:ℕ⟶ℝ⁺ ∧ ℝ⁺ ⊆ ℂ) → 𝐴:ℕ⟶ℂ)
17	14, 15, 16	sylancl 586	. . . 4 ⊢ (𝜑 → 𝐴:ℕ⟶ℂ)
18		stirlinglem8.11	. . . 4 ⊢ (𝜑 → 𝐴 ⇝ 𝐶)
19		4nn0 12493	. . . . 5 ⊢ 4 ∈ ℕ₀
20	19	a1i 11	. . . 4 ⊢ (𝜑 → 4 ∈ ℕ₀)
21		nnex 12220	. . . . . . 7 ⊢ ℕ ∈ V
22	21	mptex 7227	. . . . . 6 ⊢ (𝑛 ∈ ℕ ↦ ((𝐴‘𝑛)↑4)) ∈ V
23	2, 22	eqeltri 2829	. . . . 5 ⊢ 𝐿 ∈ V
24	23	a1i 11	. . . 4 ⊢ (𝜑 → 𝐿 ∈ V)
25		simpr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
26	14	ffvelcdmda 7086	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ∈ ℝ⁺)
27	26	rpcnd 13020	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ∈ ℂ)
28	19	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 4 ∈ ℕ₀)
29	27, 28	expcld 14113	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐴‘𝑛)↑4) ∈ ℂ)
30	2	fvmpt2 7009	. . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ ((𝐴‘𝑛)↑4) ∈ ℂ) → (𝐿‘𝑛) = ((𝐴‘𝑛)↑4))
31	25, 29, 30	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐿‘𝑛) = ((𝐴‘𝑛)↑4))
32	1, 13, 4, 11, 12, 17, 18, 20, 24, 31	climexp 44406	. . 3 ⊢ (𝜑 → 𝐿 ⇝ (𝐶↑4))
33	21	mptex 7227	. . . . 5 ⊢ (𝑛 ∈ ℕ ↦ (((𝐴‘𝑛)↑4) / ((𝐷‘𝑛)↑2))) ∈ V
34	8, 33	eqeltri 2829	. . . 4 ⊢ 𝐹 ∈ V
35	34	a1i 11	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
36		stirlinglem8.3	. . . 4 ⊢ Ⅎ𝑛𝐷
37	17	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴:ℕ⟶ℂ)
38		2nn 12287	. . . . . . . . 9 ⊢ 2 ∈ ℕ
39	38	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → 2 ∈ ℕ)
40		id 22	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ)
41	39, 40	nnmulcld 12267	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (2 · 𝑛) ∈ ℕ)
42	41	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2 · 𝑛) ∈ ℕ)
43	37, 42	ffvelcdmd 7087	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘(2 · 𝑛)) ∈ ℂ)
44		stirlinglem8.4	. . . . 5 ⊢ 𝐷 = (𝑛 ∈ ℕ ↦ (𝐴‘(2 · 𝑛)))
45	1, 43, 44	fmptdf 7118	. . . 4 ⊢ (𝜑 → 𝐷:ℕ⟶ℂ)
46		nfmpt1 5256	. . . . 5 ⊢ Ⅎ𝑛(𝑛 ∈ ℕ ↦ (2 · 𝑛))
47		fex 7230	. . . . . 6 ⊢ ((𝐴:ℕ⟶ℂ ∧ ℕ ∈ V) → 𝐴 ∈ V)
48	17, 21, 47	sylancl 586	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ V)
49		1nn 12225	. . . . . . 7 ⊢ 1 ∈ ℕ
50		2cnd 12292	. . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℂ)
51		1cnd 11211	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℂ)
52	50, 51	mulcld 11236	. . . . . . 7 ⊢ (𝜑 → (2 · 1) ∈ ℂ)
53		oveq2 7419	. . . . . . . 8 ⊢ (𝑛 = 1 → (2 · 𝑛) = (2 · 1))
54		eqid 2732	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ ↦ (2 · 𝑛)) = (𝑛 ∈ ℕ ↦ (2 · 𝑛))
55	53, 54	fvmptg 6996	. . . . . . 7 ⊢ ((1 ∈ ℕ ∧ (2 · 1) ∈ ℂ) → ((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘1) = (2 · 1))
56	49, 52, 55	sylancr 587	. . . . . 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 12267	. . . . . 6 ⊢ (𝜑 → (2 · 1) ∈ ℕ)
60	56, 59	eqeltrd 2833	. . . . 5 ⊢ (𝜑 → ((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘1) ∈ ℕ)
61		1red 11217	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → 1 ∈ ℝ)
62	39	nnred 12229	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → 2 ∈ ℝ)
63	41	nnred 12229	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (2 · 𝑛) ∈ ℝ)
64	39	nnge1d 12262	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → 1 ≤ 2)
65	61, 62, 63, 64	leadd2dd 11831	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → ((2 · 𝑛) + 1) ≤ ((2 · 𝑛) + 2))
66	54	fvmpt2 7009	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ (2 · 𝑛) ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘𝑛) = (2 · 𝑛))
67	41, 66	mpdan 685	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘𝑛) = (2 · 𝑛))
68	67	oveq1d 7426	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘𝑛) + 1) = ((2 · 𝑛) + 1))
69		oveq2 7419	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (2 · 𝑛) = (2 · 𝑘))
70	69	cbvmptv 5261	. . . . . . . . . . 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 485	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 = (𝑛 + 1)) → 𝑘 = (𝑛 + 1))
73	72	oveq2d 7427	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 = (𝑛 + 1)) → (2 · 𝑘) = (2 · (𝑛 + 1)))
74		peano2nn 12226	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
75	39, 74	nnmulcld 12267	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (2 · (𝑛 + 1)) ∈ ℕ)
76	71, 73, 74, 75	fvmptd 7005	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘(𝑛 + 1)) = (2 · (𝑛 + 1)))
77		2cnd 12292	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 2 ∈ ℂ)
78		nncn 12222	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
79		1cnd 11211	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 1 ∈ ℂ)
80	77, 78, 79	adddid 11240	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (2 · (𝑛 + 1)) = ((2 · 𝑛) + (2 · 1)))
81	77	mulridd 11233	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (2 · 1) = 2)
82	81	oveq2d 7427	. . . . . . . . 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 5180	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘𝑛) + 1) ≤ ((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘(𝑛 + 1)))
85	41	nnzd 12587	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (2 · 𝑛) ∈ ℤ)
86	67, 85	eqeltrd 2833	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘𝑛) ∈ ℤ)
87	86	peano2zd 12671	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘𝑛) + 1) ∈ ℤ)
88	75	nnzd 12587	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (2 · (𝑛 + 1)) ∈ ℤ)
89	76, 88	eqeltrd 2833	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘(𝑛 + 1)) ∈ ℤ)
90		eluz 12838	. . . . . . . 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 584	. . . . . . 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 256	. . . . . 6 ⊢ (𝑛 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘(𝑛 + 1)) ∈ (ℤ_≥‘(((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘𝑛) + 1)))
93	92	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘(𝑛 + 1)) ∈ (ℤ_≥‘(((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘𝑛) + 1)))
94	21	mptex 7227	. . . . . . 7 ⊢ (𝑛 ∈ ℕ ↦ (𝐴‘(2 · 𝑛))) ∈ V
95	44, 94	eqeltri 2829	. . . . . 6 ⊢ 𝐷 ∈ V
96	95	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ V)
97	44	fvmpt2 7009	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ ∧ (𝐴‘(2 · 𝑛)) ∈ ℂ) → (𝐷‘𝑛) = (𝐴‘(2 · 𝑛)))
98	25, 43, 97	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) = (𝐴‘(2 · 𝑛)))
99	67	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘𝑛) = (2 · 𝑛))
100	99	eqcomd 2738	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2 · 𝑛) = ((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘𝑛))
101	100	fveq2d 6895	. . . . . 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 44409	. . . 4 ⊢ (𝜑 → 𝐷 ⇝ 𝐶)
104		2nn0 12491	. . . . 5 ⊢ 2 ∈ ℕ₀
105	104	a1i 11	. . . 4 ⊢ (𝜑 → 2 ∈ ℕ₀)
106	21	mptex 7227	. . . . . 6 ⊢ (𝑛 ∈ ℕ ↦ ((𝐷‘𝑛)↑2)) ∈ V
107	5, 106	eqeltri 2829	. . . . 5 ⊢ 𝑀 ∈ V
108	107	a1i 11	. . . 4 ⊢ (𝜑 → 𝑀 ∈ V)
109		stirlinglem8.9	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) ∈ ℝ⁺)
110	109	rpcnd 13020	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) ∈ ℂ)
111	110	sqcld 14111	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐷‘𝑛)↑2) ∈ ℂ)
112	5	fvmpt2 7009	. . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ ((𝐷‘𝑛)↑2) ∈ ℂ) → (𝑀‘𝑛) = ((𝐷‘𝑛)↑2))
113	25, 111, 112	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘𝑛) = ((𝐷‘𝑛)↑2))
114	1, 36, 7, 11, 12, 45, 103, 105, 108, 113	climexp 44406	. . 3 ⊢ (𝜑 → 𝑀 ⇝ (𝐶↑2))
115		stirlinglem8.10	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ⁺)
116	115	rpcnd 13020	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℂ)
117	115	rpne0d 13023	. . . 4 ⊢ (𝜑 → 𝐶 ≠ 0)
118		2z 12596	. . . . 5 ⊢ 2 ∈ ℤ
119	118	a1i 11	. . . 4 ⊢ (𝜑 → 2 ∈ ℤ)
120	116, 117, 119	expne0d 14119	. . 3 ⊢ (𝜑 → (𝐶↑2) ≠ 0)
121	1, 29, 2	fmptdf 7118	. . . 4 ⊢ (𝜑 → 𝐿:ℕ⟶ℂ)
122	121	ffvelcdmda 7086	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐿‘𝑛) ∈ ℂ)
123	113, 111	eqeltrd 2833	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘𝑛) ∈ ℂ)
124	98	oveq1d 7426	. . . . . . . . 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 2834	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘(2 · 𝑛)) ∈ ℝ⁺)
127	118	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 2 ∈ ℤ)
128	126, 127	rpexpcld 14212	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐴‘(2 · 𝑛))↑2) ∈ ℝ⁺)
129	125, 128	eqeltrd 2833	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘𝑛) ∈ ℝ⁺)
130	129	rpne0d 13023	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘𝑛) ≠ 0)
131	130	neneqd 2945	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ¬ (𝑀‘𝑛) = 0)
132		0cn 11208	. . . . . 6 ⊢ 0 ∈ ℂ
133		elsn2g 4666	. . . . . 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 328	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ¬ (𝑀‘𝑛) ∈ {0})
136	123, 135	eldifd 3959	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘𝑛) ∈ (ℂ ∖ {0}))
137	28	nn0zd 12586	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 4 ∈ ℤ)
138	26, 137	rpexpcld 14212	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐴‘𝑛)↑4) ∈ ℝ⁺)
139	109, 127	rpexpcld 14212	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐷‘𝑛)↑2) ∈ ℝ⁺)
140	138, 139	rpdivcld 13035	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((𝐴‘𝑛)↑4) / ((𝐷‘𝑛)↑2)) ∈ ℝ⁺)
141	8	fvmpt2 7009	. . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ (((𝐴‘𝑛)↑4) / ((𝐷‘𝑛)↑2)) ∈ ℝ⁺) → (𝐹‘𝑛) = (((𝐴‘𝑛)↑4) / ((𝐷‘𝑛)↑2)))
142	25, 140, 141	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) = (((𝐴‘𝑛)↑4) / ((𝐷‘𝑛)↑2)))
143	2	fvmpt2 7009	. . . . . 6 ⊢ ((𝑛 ∈ ℕ ∧ ((𝐴‘𝑛)↑4) ∈ ℝ⁺) → (𝐿‘𝑛) = ((𝐴‘𝑛)↑4))
144	25, 138, 143	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐿‘𝑛) = ((𝐴‘𝑛)↑4))
145	144, 113	oveq12d 7429	. . . 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 44413	. 2 ⊢ (𝜑 → 𝐹 ⇝ ((𝐶↑4) / (𝐶↑2)))
148		2cn 12289	. . . . . 6 ⊢ 2 ∈ ℂ
149		2p2e4 12349	. . . . . 6 ⊢ (2 + 2) = 4
150	148, 148, 149	mvlladdi 11480	. . . . 5 ⊢ 2 = (4 − 2)
151	150	a1i 11	. . . 4 ⊢ (𝜑 → 2 = (4 − 2))
152	151	oveq2d 7427	. . 3 ⊢ (𝜑 → (𝐶↑2) = (𝐶↑(4 − 2)))
153	20	nn0zd 12586	. . . 4 ⊢ (𝜑 → 4 ∈ ℤ)
154	116, 117, 119, 153	expsubd 14124	. . 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 5176	1 ⊢ (𝜑 → 𝐹 ⇝ (𝐶↑2))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 Ⅎwnf 1785 ∈ wcel 2106 Ⅎwnfc 2883 Vcvv 3474 ⊆ wss 3948 {csn 4628 class class class wbr 5148 ↦ cmpt 5231 ⟶wf 6539 ‘cfv 6543 (class class class)co 7411 ℂcc 11110 0cc0 11112 1c1 11113 + caddc 11115 · cmul 11117 ≤ cle 11251 − cmin 11446 / cdiv 11873 ℕcn 12214 2c2 12269 4c4 12271 ℕ₀cn0 12474 ℤcz 12560 ℤ_≥cuz 12824 ℝ⁺crp 12976 ↑cexp 14029 ⇝ cli 15430

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-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 ax-mulf 11192

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-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 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-se 5632 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-isom 6552 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-er 8705 df-map 8824 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-div 11874 df-nn 12215 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12475 df-z 12561 df-dec 12680 df-uz 12825 df-q 12935 df-rp 12977 df-xneg 13094 df-xadd 13095 df-xmul 13096 df-icc 13333 df-fz 13487 df-fzo 13630 df-seq 13969 df-exp 14030 df-hash 14293 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-clim 15434 df-struct 17082 df-sets 17099 df-slot 17117 df-ndx 17129 df-base 17147 df-ress 17176 df-plusg 17212 df-mulr 17213 df-starv 17214 df-sca 17215 df-vsca 17216 df-ip 17217 df-tset 17218 df-ple 17219 df-ds 17221 df-unif 17222 df-hom 17223 df-cco 17224 df-rest 17370 df-topn 17371 df-0g 17389 df-gsum 17390 df-topgen 17391 df-pt 17392 df-prds 17395 df-xrs 17450 df-qtop 17455 df-imas 17456 df-xps 17458 df-mre 17532 df-mrc 17533 df-acs 17535 df-mgm 18563 df-sgrp 18612 df-mnd 18628 df-submnd 18674 df-mulg 18953 df-cntz 19183 df-cmn 19652 df-psmet 20942 df-xmet 20943 df-met 20944 df-bl 20945 df-mopn 20946 df-cnfld 20951 df-top 22403 df-topon 22420 df-topsp 22442 df-bases 22456 df-cn 22738 df-cnp 22739 df-tx 23073 df-hmeo 23266 df-xms 23833 df-ms 23834 df-tms 23835 df-cncf 24401

This theorem is referenced by: stirlinglem15 44889

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