stirlinglem8 - Mathbox for Glauco Siliprandi

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Theorem stirlinglem8 46660

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 5201	. . . 4 ⊢ Ⅎ𝑛(𝑛 ∈ ℕ ↦ ((𝐴‘𝑛)↑4))
4	2, 3	nfcxfr 2924	. . 3 ⊢ Ⅎ𝑛𝐿
5		stirlinglem8.8	. . . 4 ⊢ 𝑀 = (𝑛 ∈ ℕ ↦ ((𝐷‘𝑛)↑2))
6		nfmpt1 5201	. . . 4 ⊢ Ⅎ𝑛(𝑛 ∈ ℕ ↦ ((𝐷‘𝑛)↑2))
7	5, 6	nfcxfr 2924	. . 3 ⊢ Ⅎ𝑛𝑀
8		stirlinglem8.6	. . . 4 ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (((𝐴‘𝑛)↑4) / ((𝐷‘𝑛)↑2)))
9		nfmpt1 5201	. . . 4 ⊢ Ⅎ𝑛(𝑛 ∈ ℕ ↦ (((𝐴‘𝑛)↑4) / ((𝐷‘𝑛)↑2)))
10	8, 9	nfcxfr 2924	. . 3 ⊢ Ⅎ𝑛𝐹
11		nnuz 12880	. . 3 ⊢ ℕ = (ℤ_≥‘1)
12		1zzd 12604	. . 3 ⊢ (𝜑 → 1 ∈ ℤ)
13		stirlinglem8.2	. . . 4 ⊢ Ⅎ𝑛𝐴
14		stirlinglem8.5	. . . . 5 ⊢ (𝜑 → 𝐴:ℕ⟶ℝ⁺)
15		rrpsscn 46169	. . . . 5 ⊢ ℝ⁺ ⊆ ℂ
16		fss 6710	. . . . 5 ⊢ ((𝐴:ℕ⟶ℝ⁺ ∧ ℝ⁺ ⊆ ℂ) → 𝐴:ℕ⟶ℂ)
17	14, 15, 16	sylancl 595	. . . 4 ⊢ (𝜑 → 𝐴:ℕ⟶ℂ)
18		stirlinglem8.11	. . . 4 ⊢ (𝜑 → 𝐴 ⇝ 𝐶)
19		4nn0 12502	. . . . 5 ⊢ 4 ∈ ℕ₀
20	19	a1i 11	. . . 4 ⊢ (𝜑 → 4 ∈ ℕ₀)
21		nnex 12218	. . . . . . 7 ⊢ ℕ ∈ V
22	21	mptex 7209	. . . . . 6 ⊢ (𝑛 ∈ ℕ ↦ ((𝐴‘𝑛)↑4)) ∈ V
23	2, 22	eqeltri 2860	. . . . 5 ⊢ 𝐿 ∈ V
24	23	a1i 11	. . . 4 ⊢ (𝜑 → 𝐿 ∈ V)
25		simpr 488	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
26	14	ffvelcdmda 7067	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ∈ ℝ⁺)
27	26	rpcnd 13041	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ∈ ℂ)
28	19	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 4 ∈ ℕ₀)
29	27, 28	expcld 14161	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐴‘𝑛)↑4) ∈ ℂ)
30	2	fvmpt2 6989	. . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ ((𝐴‘𝑛)↑4) ∈ ℂ) → (𝐿‘𝑛) = ((𝐴‘𝑛)↑4))
31	25, 29, 30	syl2anc 593	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐿‘𝑛) = ((𝐴‘𝑛)↑4))
32	1, 13, 4, 11, 12, 17, 18, 20, 24, 31	climexp 46186	. . 3 ⊢ (𝜑 → 𝐿 ⇝ (𝐶↑4))
33	21	mptex 7209	. . . . 5 ⊢ (𝑛 ∈ ℕ ↦ (((𝐴‘𝑛)↑4) / ((𝐷‘𝑛)↑2))) ∈ V
34	8, 33	eqeltri 2860	. . . 4 ⊢ 𝐹 ∈ V
35	34	a1i 11	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
36		stirlinglem8.3	. . . 4 ⊢ Ⅎ𝑛𝐷
37	17	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴:ℕ⟶ℂ)
38		2nn 12293	. . . . . . . . 9 ⊢ 2 ∈ ℕ
39	38	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → 2 ∈ ℕ)
40		id 22	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ)
41	39, 40	nnmulcld 12268	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (2 · 𝑛) ∈ ℕ)
42	41	adantl 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2 · 𝑛) ∈ ℕ)
43	37, 42	ffvelcdmd 7068	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘(2 · 𝑛)) ∈ ℂ)
44		stirlinglem8.4	. . . . 5 ⊢ 𝐷 = (𝑛 ∈ ℕ ↦ (𝐴‘(2 · 𝑛)))
45	1, 43, 44	fmptdf 7100	. . . 4 ⊢ (𝜑 → 𝐷:ℕ⟶ℂ)
46		nfmpt1 5201	. . . . 5 ⊢ Ⅎ𝑛(𝑛 ∈ ℕ ↦ (2 · 𝑛))
47		fex 7212	. . . . . 6 ⊢ ((𝐴:ℕ⟶ℂ ∧ ℕ ∈ V) → 𝐴 ∈ V)
48	17, 21, 47	sylancl 595	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ V)
49		1nn 12223	. . . . . . 7 ⊢ 1 ∈ ℕ
50		2cnd 12298	. . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℂ)
51		1cnd 11177	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℂ)
52	50, 51	mulcld 11204	. . . . . . 7 ⊢ (𝜑 → (2 · 1) ∈ ℂ)
53		oveq2 7406	. . . . . . . 8 ⊢ (𝑛 = 1 → (2 · 𝑛) = (2 · 1))
54		eqid 2764	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ ↦ (2 · 𝑛)) = (𝑛 ∈ ℕ ↦ (2 · 𝑛))
55	53, 54	fvmptg 6975	. . . . . . 7 ⊢ ((1 ∈ ℕ ∧ (2 · 1) ∈ ℂ) → ((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘1) = (2 · 1))
56	49, 52, 55	sylancr 596	. . . . . 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 12268	. . . . . 6 ⊢ (𝜑 → (2 · 1) ∈ ℕ)
60	56, 59	eqeltrd 2864	. . . . 5 ⊢ (𝜑 → ((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘1) ∈ ℕ)
61		1red 11184	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → 1 ∈ ℝ)
62	39	nnred 12227	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → 2 ∈ ℝ)
63	41	nnred 12227	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (2 · 𝑛) ∈ ℝ)
64	39	nnge1d 12263	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → 1 ≤ 2)
65	61, 62, 63, 64	leadd2dd 11804	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → ((2 · 𝑛) + 1) ≤ ((2 · 𝑛) + 2))
66	54	fvmpt2 6989	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ (2 · 𝑛) ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘𝑛) = (2 · 𝑛))
67	41, 66	mpdan 697	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘𝑛) = (2 · 𝑛))
68	67	oveq1d 7413	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘𝑛) + 1) = ((2 · 𝑛) + 1))
69		oveq2 7406	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (2 · 𝑛) = (2 · 𝑘))
70	69	cbvmptv 5206	. . . . . . . . . . 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 488	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 = (𝑛 + 1)) → 𝑘 = (𝑛 + 1))
73	72	oveq2d 7414	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 = (𝑛 + 1)) → (2 · 𝑘) = (2 · (𝑛 + 1)))
74		peano2nn 12224	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
75	39, 74	nnmulcld 12268	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (2 · (𝑛 + 1)) ∈ ℕ)
76	71, 73, 74, 75	fvmptd 6985	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘(𝑛 + 1)) = (2 · (𝑛 + 1)))
77		2cnd 12298	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 2 ∈ ℂ)
78		nncn 12220	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
79		1cnd 11177	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 1 ∈ ℂ)
80	77, 78, 79	adddid 11208	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (2 · (𝑛 + 1)) = ((2 · 𝑛) + (2 · 1)))
81	77	mulridd 11201	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (2 · 1) = 2)
82	81	oveq2d 7414	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ((2 · 𝑛) + (2 · 1)) = ((2 · 𝑛) + 2))
83	76, 80, 82	3eqtrd 2803	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘(𝑛 + 1)) = ((2 · 𝑛) + 2))
84	65, 68, 83	3brtr4d 5134	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘𝑛) + 1) ≤ ((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘(𝑛 + 1)))
85	41	nnzd 12596	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (2 · 𝑛) ∈ ℤ)
86	67, 85	eqeltrd 2864	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘𝑛) ∈ ℤ)
87	86	peano2zd 12682	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘𝑛) + 1) ∈ ℤ)
88	75	nnzd 12596	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (2 · (𝑛 + 1)) ∈ ℤ)
89	76, 88	eqeltrd 2864	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘(𝑛 + 1)) ∈ ℤ)
90		eluz 12855	. . . . . . . 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 593	. . . . . . 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 259	. . . . . 6 ⊢ (𝑛 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘(𝑛 + 1)) ∈ (ℤ_≥‘(((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘𝑛) + 1)))
93	92	adantl 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘(𝑛 + 1)) ∈ (ℤ_≥‘(((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘𝑛) + 1)))
94	21	mptex 7209	. . . . . . 7 ⊢ (𝑛 ∈ ℕ ↦ (𝐴‘(2 · 𝑛))) ∈ V
95	44, 94	eqeltri 2860	. . . . . 6 ⊢ 𝐷 ∈ V
96	95	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ V)
97	44	fvmpt2 6989	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ ∧ (𝐴‘(2 · 𝑛)) ∈ ℂ) → (𝐷‘𝑛) = (𝐴‘(2 · 𝑛)))
98	25, 43, 97	syl2anc 593	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) = (𝐴‘(2 · 𝑛)))
99	67	adantl 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘𝑛) = (2 · 𝑛))
100	99	eqcomd 2770	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2 · 𝑛) = ((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘𝑛))
101	100	fveq2d 6873	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘(2 · 𝑛)) = (𝐴‘((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘𝑛)))
102	98, 101	eqtrd 2799	. . . . 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 46189	. . . 4 ⊢ (𝜑 → 𝐷 ⇝ 𝐶)
104		2nn0 12500	. . . . 5 ⊢ 2 ∈ ℕ₀
105	104	a1i 11	. . . 4 ⊢ (𝜑 → 2 ∈ ℕ₀)
106	21	mptex 7209	. . . . . 6 ⊢ (𝑛 ∈ ℕ ↦ ((𝐷‘𝑛)↑2)) ∈ V
107	5, 106	eqeltri 2860	. . . . 5 ⊢ 𝑀 ∈ V
108	107	a1i 11	. . . 4 ⊢ (𝜑 → 𝑀 ∈ V)
109		stirlinglem8.9	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) ∈ ℝ⁺)
110	109	rpcnd 13041	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) ∈ ℂ)
111	110	sqcld 14159	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐷‘𝑛)↑2) ∈ ℂ)
112	5	fvmpt2 6989	. . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ ((𝐷‘𝑛)↑2) ∈ ℂ) → (𝑀‘𝑛) = ((𝐷‘𝑛)↑2))
113	25, 111, 112	syl2anc 593	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘𝑛) = ((𝐷‘𝑛)↑2))
114	1, 36, 7, 11, 12, 45, 103, 105, 108, 113	climexp 46186	. . 3 ⊢ (𝜑 → 𝑀 ⇝ (𝐶↑2))
115		stirlinglem8.10	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ⁺)
116	115	rpcnd 13041	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℂ)
117	115	rpne0d 13044	. . . 4 ⊢ (𝜑 → 𝐶 ≠ 0)
118		2z 12605	. . . . 5 ⊢ 2 ∈ ℤ
119	118	a1i 11	. . . 4 ⊢ (𝜑 → 2 ∈ ℤ)
120	116, 117, 119	expne0d 14167	. . 3 ⊢ (𝜑 → (𝐶↑2) ≠ 0)
121	1, 29, 2	fmptdf 7100	. . . 4 ⊢ (𝜑 → 𝐿:ℕ⟶ℂ)
122	121	ffvelcdmda 7067	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐿‘𝑛) ∈ ℂ)
123	113, 111	eqeltrd 2864	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘𝑛) ∈ ℂ)
124	98	oveq1d 7413	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐷‘𝑛)↑2) = ((𝐴‘(2 · 𝑛))↑2))
125	113, 124	eqtrd 2799	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘𝑛) = ((𝐴‘(2 · 𝑛))↑2))
126	98, 109	eqeltrrd 2865	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘(2 · 𝑛)) ∈ ℝ⁺)
127	118	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 2 ∈ ℤ)
128	126, 127	rpexpcld 14262	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐴‘(2 · 𝑛))↑2) ∈ ℝ⁺)
129	125, 128	eqeltrd 2864	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘𝑛) ∈ ℝ⁺)
130	129	rpne0d 13044	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘𝑛) ≠ 0)
131	130	neneqd 2964	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ¬ (𝑀‘𝑛) = 0)
132		0cn 11173	. . . . . 6 ⊢ 0 ∈ ℂ
133		elsn2g 4625	. . . . . 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 331	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ¬ (𝑀‘𝑛) ∈ {0})
136	123, 135	eldifd 3917	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘𝑛) ∈ (ℂ ∖ {0}))
137	28	nn0zd 12595	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 4 ∈ ℤ)
138	26, 137	rpexpcld 14262	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐴‘𝑛)↑4) ∈ ℝ⁺)
139	109, 127	rpexpcld 14262	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐷‘𝑛)↑2) ∈ ℝ⁺)
140	138, 139	rpdivcld 13056	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((𝐴‘𝑛)↑4) / ((𝐷‘𝑛)↑2)) ∈ ℝ⁺)
141	8	fvmpt2 6989	. . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ (((𝐴‘𝑛)↑4) / ((𝐷‘𝑛)↑2)) ∈ ℝ⁺) → (𝐹‘𝑛) = (((𝐴‘𝑛)↑4) / ((𝐷‘𝑛)↑2)))
142	25, 140, 141	syl2anc 593	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) = (((𝐴‘𝑛)↑4) / ((𝐷‘𝑛)↑2)))
143	2	fvmpt2 6989	. . . . . 6 ⊢ ((𝑛 ∈ ℕ ∧ ((𝐴‘𝑛)↑4) ∈ ℝ⁺) → (𝐿‘𝑛) = ((𝐴‘𝑛)↑4))
144	25, 138, 143	syl2anc 593	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐿‘𝑛) = ((𝐴‘𝑛)↑4))
145	144, 113	oveq12d 7416	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐿‘𝑛) / (𝑀‘𝑛)) = (((𝐴‘𝑛)↑4) / ((𝐷‘𝑛)↑2)))
146	142, 145	eqtr4d 2802	. . 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 46193	. 2 ⊢ (𝜑 → 𝐹 ⇝ ((𝐶↑4) / (𝐶↑2)))
148		2cn 12295	. . . . . 6 ⊢ 2 ∈ ℂ
149		2p2e4 12354	. . . . . 6 ⊢ (2 + 2) = 4
150	148, 148, 149	mvlladdi 11451	. . . . 5 ⊢ 2 = (4 − 2)
151	150	a1i 11	. . . 4 ⊢ (𝜑 → 2 = (4 − 2))
152	151	oveq2d 7414	. . 3 ⊢ (𝜑 → (𝐶↑2) = (𝐶↑(4 − 2)))
153	20	nn0zd 12595	. . . 4 ⊢ (𝜑 → 4 ∈ ℤ)
154	116, 117, 119, 153	expsubd 14172	. . 3 ⊢ (𝜑 → (𝐶↑(4 − 2)) = ((𝐶↑4) / (𝐶↑2)))
155	152, 154	eqtrd 2799	. 2 ⊢ (𝜑 → (𝐶↑2) = ((𝐶↑4) / (𝐶↑2)))
156	147, 155	breqtrrd 5130	1 ⊢ (𝜑 → 𝐹 ⇝ (𝐶↑2))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 399 = wceq 1562 Ⅎwnf 1805 ∈ wcel 2144 Ⅎwnfc 2911 Vcvv 3456 ⊆ wss 3906 {csn 4584 class class class wbr 5102 ↦ cmpt 5183 ⟶wf 6519 ‘cfv 6523 (class class class)co 7398 ℂcc 11073 0cc0 11075 1c1 11076 + caddc 11078 · cmul 11080 ≤ cle 11219 − cmin 11416 / cdiv 11846 ℕcn 12212 2c2 12274 4c4 12276 ℕ₀cn0 12483 ℤcz 12570 ℤ_≥cuz 12841 ℝ⁺crp 12995 ↑cexp 14076 ⇝ cli 15513

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1817 ax-4 1831 ax-5 1932 ax-6 1989 ax-7 2030 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2214 ax-ext 2736 ax-rep 5229 ax-sep 5248 ax-nul 5258 ax-pow 5324 ax-pr 5392 ax-un 7720 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1565 df-fal 1575 df-ex 1802 df-nf 1806 df-sb 2093 df-mo 2568 df-eu 2598 df-clab 2743 df-cleq 2756 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3458 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5544 df-eprel 5549 df-po 5557 df-so 5558 df-fr 5602 df-se 5603 df-we 5604 df-xp 5655 df-rel 5656 df-cnv 5657 df-co 5658 df-dm 5659 df-rn 5660 df-res 5661 df-ima 5662 df-pred 6290 df-ord 6351 df-on 6352 df-lim 6353 df-suc 6354 df-iota 6479 df-fun 6525 df-fn 6526 df-f 6527 df-f1 6528 df-fo 6529 df-f1o 6530 df-fv 6531 df-isom 6532 df-riota 7355 df-ov 7401 df-oprab 7402 df-mpo 7403 df-of 7662 df-om 7849 df-1st 7972 df-2nd 7973 df-supp 8143 df-frecs 8264 df-wrecs 8295 df-recs 8344 df-rdg 8383 df-1o 8439 df-2o 8440 df-er 8680 df-map 8812 df-ixp 8882 df-en 8930 df-dom 8931 df-sdom 8932 df-fin 8933 df-fsupp 9310 df-fi 9359 df-sup 9390 df-inf 9391 df-oi 9460 df-card 9899 df-pnf 11220 df-mnf 11221 df-xr 11222 df-ltxr 11223 df-le 11224 df-sub 11418 df-neg 11419 df-div 11847 df-nn 12213 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12484 df-z 12571 df-dec 12691 df-uz 12842 df-q 12952 df-rp 12996 df-xneg 13116 df-xadd 13117 df-xmul 13118 df-icc 13358 df-fz 13515 df-fzo 13662 df-seq 14017 df-exp 14077 df-hash 14346 df-cj 15128 df-re 15129 df-im 15130 df-sqrt 15264 df-abs 15265 df-clim 15517 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17248 df-ress 17269 df-plusg 17301 df-mulr 17302 df-starv 17303 df-sca 17304 df-vsca 17305 df-ip 17306 df-tset 17307 df-ple 17308 df-ds 17310 df-unif 17311 df-hom 17312 df-cco 17313 df-rest 17453 df-topn 17454 df-0g 17472 df-gsum 17473 df-topgen 17474 df-pt 17475 df-prds 17478 df-xrs 17534 df-qtop 17539 df-imas 17540 df-xps 17542 df-mre 17616 df-mrc 17617 df-acs 17619 df-mgm 18676 df-sgrp 18755 df-mnd 18771 df-submnd 18820 df-mulg 19112 df-cntz 19359 df-cmn 19824 df-psmet 21418 df-xmet 21419 df-met 21420 df-bl 21421 df-mopn 21422 df-cnfld 21427 df-top 22956 df-topon 22973 df-topsp 22995 df-bases 23008 df-cn 23289 df-cnp 23290 df-tx 23624 df-hmeo 23817 df-xms 24382 df-ms 24383 df-tms 24384 df-cncf 24942

This theorem is referenced by: stirlinglem15 46667

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