stirlinglem8 - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > stirlinglem8		Structured version Visualization version GIF version

Theorem stirlinglem8 42356

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 5155	. . . 4 ⊢ Ⅎ𝑛(𝑛 ∈ ℕ ↦ ((𝐴‘𝑛)↑4))
4	2, 3	nfcxfr 2973	. . 3 ⊢ Ⅎ𝑛𝐿
5		stirlinglem8.8	. . . 4 ⊢ 𝑀 = (𝑛 ∈ ℕ ↦ ((𝐷‘𝑛)↑2))
6		nfmpt1 5155	. . . 4 ⊢ Ⅎ𝑛(𝑛 ∈ ℕ ↦ ((𝐷‘𝑛)↑2))
7	5, 6	nfcxfr 2973	. . 3 ⊢ Ⅎ𝑛𝑀
8		stirlinglem8.6	. . . 4 ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (((𝐴‘𝑛)↑4) / ((𝐷‘𝑛)↑2)))
9		nfmpt1 5155	. . . 4 ⊢ Ⅎ𝑛(𝑛 ∈ ℕ ↦ (((𝐴‘𝑛)↑4) / ((𝐷‘𝑛)↑2)))
10	8, 9	nfcxfr 2973	. . 3 ⊢ Ⅎ𝑛𝐹
11		nnuz 12273	. . 3 ⊢ ℕ = (ℤ_≥‘1)
12		1zzd 12005	. . 3 ⊢ (𝜑 → 1 ∈ ℤ)
13		stirlinglem8.2	. . . 4 ⊢ Ⅎ𝑛𝐴
14		stirlinglem8.5	. . . . 5 ⊢ (𝜑 → 𝐴:ℕ⟶ℝ⁺)
15		rrpsscn 41858	. . . . 5 ⊢ ℝ⁺ ⊆ ℂ
16		fss 6520	. . . . 5 ⊢ ((𝐴:ℕ⟶ℝ⁺ ∧ ℝ⁺ ⊆ ℂ) → 𝐴:ℕ⟶ℂ)
17	14, 15, 16	sylancl 588	. . . 4 ⊢ (𝜑 → 𝐴:ℕ⟶ℂ)
18		stirlinglem8.11	. . . 4 ⊢ (𝜑 → 𝐴 ⇝ 𝐶)
19		4nn0 11908	. . . . 5 ⊢ 4 ∈ ℕ₀
20	19	a1i 11	. . . 4 ⊢ (𝜑 → 4 ∈ ℕ₀)
21		nnex 11636	. . . . . . 7 ⊢ ℕ ∈ V
22	21	mptex 6978	. . . . . 6 ⊢ (𝑛 ∈ ℕ ↦ ((𝐴‘𝑛)↑4)) ∈ V
23	2, 22	eqeltri 2907	. . . . 5 ⊢ 𝐿 ∈ V
24	23	a1i 11	. . . 4 ⊢ (𝜑 → 𝐿 ∈ V)
25		simpr 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
26	14	ffvelrnda 6844	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ∈ ℝ⁺)
27	26	rpcnd 12425	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ∈ ℂ)
28	19	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 4 ∈ ℕ₀)
29	27, 28	expcld 13502	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐴‘𝑛)↑4) ∈ ℂ)
30	2	fvmpt2 6772	. . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ ((𝐴‘𝑛)↑4) ∈ ℂ) → (𝐿‘𝑛) = ((𝐴‘𝑛)↑4))
31	25, 29, 30	syl2anc 586	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐿‘𝑛) = ((𝐴‘𝑛)↑4))
32	1, 13, 4, 11, 12, 17, 18, 20, 24, 31	climexp 41875	. . 3 ⊢ (𝜑 → 𝐿 ⇝ (𝐶↑4))
33	21	mptex 6978	. . . . 5 ⊢ (𝑛 ∈ ℕ ↦ (((𝐴‘𝑛)↑4) / ((𝐷‘𝑛)↑2))) ∈ V
34	8, 33	eqeltri 2907	. . . 4 ⊢ 𝐹 ∈ V
35	34	a1i 11	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
36		stirlinglem8.3	. . . 4 ⊢ Ⅎ𝑛𝐷
37	17	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴:ℕ⟶ℂ)
38		2nn 11702	. . . . . . . . 9 ⊢ 2 ∈ ℕ
39	38	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → 2 ∈ ℕ)
40		id 22	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ)
41	39, 40	nnmulcld 11682	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (2 · 𝑛) ∈ ℕ)
42	41	adantl 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2 · 𝑛) ∈ ℕ)
43	37, 42	ffvelrnd 6845	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘(2 · 𝑛)) ∈ ℂ)
44		stirlinglem8.4	. . . . 5 ⊢ 𝐷 = (𝑛 ∈ ℕ ↦ (𝐴‘(2 · 𝑛)))
45	1, 43, 44	fmptdf 6874	. . . 4 ⊢ (𝜑 → 𝐷:ℕ⟶ℂ)
46		nfmpt1 5155	. . . . 5 ⊢ Ⅎ𝑛(𝑛 ∈ ℕ ↦ (2 · 𝑛))
47		fex 6981	. . . . . 6 ⊢ ((𝐴:ℕ⟶ℂ ∧ ℕ ∈ V) → 𝐴 ∈ V)
48	17, 21, 47	sylancl 588	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ V)
49		1nn 11641	. . . . . . 7 ⊢ 1 ∈ ℕ
50		2cnd 11707	. . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℂ)
51		1cnd 10628	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℂ)
52	50, 51	mulcld 10653	. . . . . . 7 ⊢ (𝜑 → (2 · 1) ∈ ℂ)
53		oveq2 7156	. . . . . . . 8 ⊢ (𝑛 = 1 → (2 · 𝑛) = (2 · 1))
54		eqid 2819	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ ↦ (2 · 𝑛)) = (𝑛 ∈ ℕ ↦ (2 · 𝑛))
55	53, 54	fvmptg 6759	. . . . . . 7 ⊢ ((1 ∈ ℕ ∧ (2 · 1) ∈ ℂ) → ((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘1) = (2 · 1))
56	49, 52, 55	sylancr 589	. . . . . 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 11682	. . . . . 6 ⊢ (𝜑 → (2 · 1) ∈ ℕ)
60	56, 59	eqeltrd 2911	. . . . 5 ⊢ (𝜑 → ((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘1) ∈ ℕ)
61		1red 10634	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → 1 ∈ ℝ)
62	39	nnred 11645	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → 2 ∈ ℝ)
63	41	nnred 11645	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (2 · 𝑛) ∈ ℝ)
64	39	nnge1d 11677	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → 1 ≤ 2)
65	61, 62, 63, 64	leadd2dd 11247	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → ((2 · 𝑛) + 1) ≤ ((2 · 𝑛) + 2))
66	54	fvmpt2 6772	. . . . . . . . . 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 7163	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘𝑛) + 1) = ((2 · 𝑛) + 1))
69		oveq2 7156	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (2 · 𝑛) = (2 · 𝑘))
70	69	cbvmptv 5160	. . . . . . . . . . 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 487	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 = (𝑛 + 1)) → 𝑘 = (𝑛 + 1))
73	72	oveq2d 7164	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 = (𝑛 + 1)) → (2 · 𝑘) = (2 · (𝑛 + 1)))
74		peano2nn 11642	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
75	39, 74	nnmulcld 11682	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (2 · (𝑛 + 1)) ∈ ℕ)
76	71, 73, 74, 75	fvmptd 6768	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘(𝑛 + 1)) = (2 · (𝑛 + 1)))
77		2cnd 11707	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 2 ∈ ℂ)
78		nncn 11638	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
79		1cnd 10628	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 1 ∈ ℂ)
80	77, 78, 79	adddid 10657	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (2 · (𝑛 + 1)) = ((2 · 𝑛) + (2 · 1)))
81	77	mulid1d 10650	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (2 · 1) = 2)
82	81	oveq2d 7164	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ((2 · 𝑛) + (2 · 1)) = ((2 · 𝑛) + 2))
83	76, 80, 82	3eqtrd 2858	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘(𝑛 + 1)) = ((2 · 𝑛) + 2))
84	65, 68, 83	3brtr4d 5089	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘𝑛) + 1) ≤ ((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘(𝑛 + 1)))
85	41	nnzd 12078	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (2 · 𝑛) ∈ ℤ)
86	67, 85	eqeltrd 2911	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘𝑛) ∈ ℤ)
87	86	peano2zd 12082	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘𝑛) + 1) ∈ ℤ)
88	75	nnzd 12078	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (2 · (𝑛 + 1)) ∈ ℤ)
89	76, 88	eqeltrd 2911	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘(𝑛 + 1)) ∈ ℤ)
90		eluz 12249	. . . . . . . 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 586	. . . . . . 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 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘(𝑛 + 1)) ∈ (ℤ_≥‘(((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘𝑛) + 1)))
94	21	mptex 6978	. . . . . . 7 ⊢ (𝑛 ∈ ℕ ↦ (𝐴‘(2 · 𝑛))) ∈ V
95	44, 94	eqeltri 2907	. . . . . 6 ⊢ 𝐷 ∈ V
96	95	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ V)
97	44	fvmpt2 6772	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ ∧ (𝐴‘(2 · 𝑛)) ∈ ℂ) → (𝐷‘𝑛) = (𝐴‘(2 · 𝑛)))
98	25, 43, 97	syl2anc 586	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) = (𝐴‘(2 · 𝑛)))
99	67	adantl 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘𝑛) = (2 · 𝑛))
100	99	eqcomd 2825	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2 · 𝑛) = ((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘𝑛))
101	100	fveq2d 6667	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘(2 · 𝑛)) = (𝐴‘((𝑛 ∈ ℕ ↦ (2 · 𝑛))‘𝑛)))
102	98, 101	eqtrd 2854	. . . . 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 41878	. . . 4 ⊢ (𝜑 → 𝐷 ⇝ 𝐶)
104		2nn0 11906	. . . . 5 ⊢ 2 ∈ ℕ₀
105	104	a1i 11	. . . 4 ⊢ (𝜑 → 2 ∈ ℕ₀)
106	21	mptex 6978	. . . . . 6 ⊢ (𝑛 ∈ ℕ ↦ ((𝐷‘𝑛)↑2)) ∈ V
107	5, 106	eqeltri 2907	. . . . 5 ⊢ 𝑀 ∈ V
108	107	a1i 11	. . . 4 ⊢ (𝜑 → 𝑀 ∈ V)
109		stirlinglem8.9	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) ∈ ℝ⁺)
110	109	rpcnd 12425	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) ∈ ℂ)
111	110	sqcld 13500	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐷‘𝑛)↑2) ∈ ℂ)
112	5	fvmpt2 6772	. . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ ((𝐷‘𝑛)↑2) ∈ ℂ) → (𝑀‘𝑛) = ((𝐷‘𝑛)↑2))
113	25, 111, 112	syl2anc 586	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘𝑛) = ((𝐷‘𝑛)↑2))
114	1, 36, 7, 11, 12, 45, 103, 105, 108, 113	climexp 41875	. . 3 ⊢ (𝜑 → 𝑀 ⇝ (𝐶↑2))
115		stirlinglem8.10	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ⁺)
116	115	rpcnd 12425	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℂ)
117	115	rpne0d 12428	. . . 4 ⊢ (𝜑 → 𝐶 ≠ 0)
118		2z 12006	. . . . 5 ⊢ 2 ∈ ℤ
119	118	a1i 11	. . . 4 ⊢ (𝜑 → 2 ∈ ℤ)
120	116, 117, 119	expne0d 13508	. . 3 ⊢ (𝜑 → (𝐶↑2) ≠ 0)
121	1, 29, 2	fmptdf 6874	. . . 4 ⊢ (𝜑 → 𝐿:ℕ⟶ℂ)
122	121	ffvelrnda 6844	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐿‘𝑛) ∈ ℂ)
123	113, 111	eqeltrd 2911	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘𝑛) ∈ ℂ)
124	98	oveq1d 7163	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐷‘𝑛)↑2) = ((𝐴‘(2 · 𝑛))↑2))
125	113, 124	eqtrd 2854	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘𝑛) = ((𝐴‘(2 · 𝑛))↑2))
126	98, 109	eqeltrrd 2912	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘(2 · 𝑛)) ∈ ℝ⁺)
127	118	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 2 ∈ ℤ)
128	126, 127	rpexpcld 13600	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐴‘(2 · 𝑛))↑2) ∈ ℝ⁺)
129	125, 128	eqeltrd 2911	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘𝑛) ∈ ℝ⁺)
130	129	rpne0d 12428	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘𝑛) ≠ 0)
131	130	neneqd 3019	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ¬ (𝑀‘𝑛) = 0)
132		0cn 10625	. . . . . 6 ⊢ 0 ∈ ℂ
133		elsn2g 4595	. . . . . 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 3945	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘𝑛) ∈ (ℂ ∖ {0}))
137	28	nn0zd 12077	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 4 ∈ ℤ)
138	26, 137	rpexpcld 13600	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐴‘𝑛)↑4) ∈ ℝ⁺)
139	109, 127	rpexpcld 13600	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐷‘𝑛)↑2) ∈ ℝ⁺)
140	138, 139	rpdivcld 12440	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((𝐴‘𝑛)↑4) / ((𝐷‘𝑛)↑2)) ∈ ℝ⁺)
141	8	fvmpt2 6772	. . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ (((𝐴‘𝑛)↑4) / ((𝐷‘𝑛)↑2)) ∈ ℝ⁺) → (𝐹‘𝑛) = (((𝐴‘𝑛)↑4) / ((𝐷‘𝑛)↑2)))
142	25, 140, 141	syl2anc 586	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) = (((𝐴‘𝑛)↑4) / ((𝐷‘𝑛)↑2)))
143	2	fvmpt2 6772	. . . . . 6 ⊢ ((𝑛 ∈ ℕ ∧ ((𝐴‘𝑛)↑4) ∈ ℝ⁺) → (𝐿‘𝑛) = ((𝐴‘𝑛)↑4))
144	25, 138, 143	syl2anc 586	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐿‘𝑛) = ((𝐴‘𝑛)↑4))
145	144, 113	oveq12d 7166	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐿‘𝑛) / (𝑀‘𝑛)) = (((𝐴‘𝑛)↑4) / ((𝐷‘𝑛)↑2)))
146	142, 145	eqtr4d 2857	. . 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 41882	. 2 ⊢ (𝜑 → 𝐹 ⇝ ((𝐶↑4) / (𝐶↑2)))
148		2cn 11704	. . . . . 6 ⊢ 2 ∈ ℂ
149		2p2e4 11764	. . . . . 6 ⊢ (2 + 2) = 4
150	148, 148, 149	mvlladdi 10896	. . . . 5 ⊢ 2 = (4 − 2)
151	150	a1i 11	. . . 4 ⊢ (𝜑 → 2 = (4 − 2))
152	151	oveq2d 7164	. . 3 ⊢ (𝜑 → (𝐶↑2) = (𝐶↑(4 − 2)))
153	20	nn0zd 12077	. . . 4 ⊢ (𝜑 → 4 ∈ ℤ)
154	116, 117, 119, 153	expsubd 13513	. . 3 ⊢ (𝜑 → (𝐶↑(4 − 2)) = ((𝐶↑4) / (𝐶↑2)))
155	152, 154	eqtrd 2854	. 2 ⊢ (𝜑 → (𝐶↑2) = ((𝐶↑4) / (𝐶↑2)))
156	147, 155	breqtrrd 5085	1 ⊢ (𝜑 → 𝐹 ⇝ (𝐶↑2))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1531 Ⅎwnf 1778 ∈ wcel 2108 Ⅎwnfc 2959 Vcvv 3493 ⊆ wss 3934 {csn 4559 class class class wbr 5057 ↦ cmpt 5137 ⟶wf 6344 ‘cfv 6348 (class class class)co 7148 ℂcc 10527 0cc0 10529 1c1 10530 + caddc 10532 · cmul 10534 ≤ cle 10668 − cmin 10862 / cdiv 11289 ℕcn 11630 2c2 11684 4c4 11686 ℕ₀cn0 11889 ℤcz 11973 ℤ_≥cuz 12235 ℝ⁺crp 12381 ↑cexp 13421 ⇝ cli 14833

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 ax-pre-sup 10607 ax-mulf 10609

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-of 7401 df-om 7573 df-1st 7681 df-2nd 7682 df-supp 7823 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-2o 8095 df-oadd 8098 df-er 8281 df-map 8400 df-ixp 8454 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-fsupp 8826 df-fi 8867 df-sup 8898 df-inf 8899 df-oi 8966 df-card 9360 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-div 11290 df-nn 11631 df-2 11692 df-3 11693 df-4 11694 df-5 11695 df-6 11696 df-7 11697 df-8 11698 df-9 11699 df-n0 11890 df-z 11974 df-dec 12091 df-uz 12236 df-q 12341 df-rp 12382 df-xneg 12499 df-xadd 12500 df-xmul 12501 df-icc 12737 df-fz 12885 df-fzo 13026 df-seq 13362 df-exp 13422 df-hash 13683 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20529 df-xmet 20530 df-met 20531 df-bl 20532 df-mopn 20533 df-cnfld 20538 df-top 21494 df-topon 21511 df-topsp 21533 df-bases 21546 df-cn 21827 df-cnp 21828 df-tx 22162 df-hmeo 22355 df-xms 22922 df-ms 22923 df-tms 22924 df-cncf 23478

This theorem is referenced by: stirlinglem15 42363

W3C validator