ostth2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ostth2		Structured version Visualization version GIF version

Theorem ostth2 27832

Description: - Lemma for ostth 27834: regular case. (Contributed by Mario Carneiro, 10-Sep-2014.)

**Hypotheses**
Ref	Expression
qrng.q	⊢ 𝑄 = (ℂ_fld ↾_s ℚ)
qabsabv.a	⊢ 𝐴 = (AbsVal‘𝑄)
padic.j	⊢ 𝐽 = (𝑞 ∈ ℙ ↦ (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, (𝑞↑-(𝑞 pCnt 𝑥)))))
ostth.k	⊢ 𝐾 = (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, 1))
ostth.1	⊢ (𝜑 → 𝐹 ∈ 𝐴)
ostth2.2	⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘2))
ostth2.3	⊢ (𝜑 → 1 < (𝐹‘𝑁))
ostth2.4	⊢ 𝑅 = ((log‘(𝐹‘𝑁)) / (log‘𝑁))

**Assertion**
Ref	Expression
ostth2	⊢ (𝜑 → ∃𝑎 ∈ (0(,]1)𝐹 = (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑎)))

Distinct variable groups: 𝑞,𝑎,𝑥,𝑦,𝜑 𝐽,𝑎,𝑦 𝐴,𝑎,𝑞,𝑥,𝑦 𝑥,𝑁,𝑦 𝑥,𝑄,𝑦 𝐹,𝑎,𝑞,𝑦 𝑅,𝑎,𝑞,𝑦 𝑥,𝐹 Allowed substitution hints: 𝑄(𝑞, 𝑎) 𝑅(𝑥) 𝐽(𝑥, 𝑞) 𝐾(𝑥, 𝑦, 𝑞, 𝑎) 𝑁(𝑞, 𝑎)

Proof of Theorem ostth2 Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ostth2.4	. . . . 5 ⊢ 𝑅 = ((log‘(𝐹‘𝑁)) / (log‘𝑁))
2		ostth.1	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ 𝐴)
3		ostth2.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘2))
4		eluz2b2 12957	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘2) ↔ (𝑁 ∈ ℕ ∧ 1 < 𝑁))
5	3, 4	sylib 221	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 1 < 𝑁))
6	5	simpld 500	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ)
7		nnq 12998	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℚ)
8	6, 7	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℚ)
9		qabsabv.a	. . . . . . . . 9 ⊢ 𝐴 = (AbsVal‘𝑄)
10		qrng.q	. . . . . . . . . 10 ⊢ 𝑄 = (ℂ_fld ↾_s ℚ)
11	10	qrngbas 27814	. . . . . . . . 9 ⊢ ℚ = (Base‘𝑄)
12	9, 11	abvcl 20949	. . . . . . . 8 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑁 ∈ ℚ) → (𝐹‘𝑁) ∈ ℝ)
13	2, 8, 12	syl2anc 596	. . . . . . 7 ⊢ (𝜑 → (𝐹‘𝑁) ∈ ℝ)
14		ostth2.3	. . . . . . 7 ⊢ (𝜑 → 1 < (𝐹‘𝑁))
15	13, 14	rplogcld 26825	. . . . . 6 ⊢ (𝜑 → (log‘(𝐹‘𝑁)) ∈ ℝ⁺)
16	6	nnred 12259	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℝ)
17	5	simprd 501	. . . . . . 7 ⊢ (𝜑 → 1 < 𝑁)
18	16, 17	rplogcld 26825	. . . . . 6 ⊢ (𝜑 → (log‘𝑁) ∈ ℝ⁺)
19	15, 18	rpdivcld 13089	. . . . 5 ⊢ (𝜑 → ((log‘(𝐹‘𝑁)) / (log‘𝑁)) ∈ ℝ⁺)
20	1, 19	eqeltrid 2869	. . . 4 ⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
21	20	rpred 13072	. . 3 ⊢ (𝜑 → 𝑅 ∈ ℝ)
22	20	rpgt0d 13075	. . 3 ⊢ (𝜑 → 0 < 𝑅)
23	6	nnnn0d 12576	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
24	10, 9	qabvle 27820	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑁 ∈ ℕ₀) → (𝐹‘𝑁) ≤ 𝑁)
25	2, 23, 24	syl2anc 596	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝑁) ≤ 𝑁)
26	6	nnne0d 12297	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ≠ 0)
27	10	qrng0 27816	. . . . . . . . . . . 12 ⊢ 0 = (0_g‘𝑄)
28	9, 11, 27	abvgt0 20953	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑁 ∈ ℚ ∧ 𝑁 ≠ 0) → 0 < (𝐹‘𝑁))
29	2, 8, 26, 28	syl3anc 1398	. . . . . . . . . 10 ⊢ (𝜑 → 0 < (𝐹‘𝑁))
30	13, 29	elrpd 13069	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘𝑁) ∈ ℝ⁺)
31	30	reeflogd 26820	. . . . . . . 8 ⊢ (𝜑 → (exp‘(log‘(𝐹‘𝑁))) = (𝐹‘𝑁))
32	6	nnrpd 13070	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℝ⁺)
33	32	reeflogd 26820	. . . . . . . 8 ⊢ (𝜑 → (exp‘(log‘𝑁)) = 𝑁)
34	25, 31, 33	3brtr4d 5145	. . . . . . 7 ⊢ (𝜑 → (exp‘(log‘(𝐹‘𝑁))) ≤ (exp‘(log‘𝑁)))
35	15	rpred 13072	. . . . . . . 8 ⊢ (𝜑 → (log‘(𝐹‘𝑁)) ∈ ℝ)
36	32	relogcld 26819	. . . . . . . 8 ⊢ (𝜑 → (log‘𝑁) ∈ ℝ)
37		efle 16192	. . . . . . . 8 ⊢ (((log‘(𝐹‘𝑁)) ∈ ℝ ∧ (log‘𝑁) ∈ ℝ) → ((log‘(𝐹‘𝑁)) ≤ (log‘𝑁) ↔ (exp‘(log‘(𝐹‘𝑁))) ≤ (exp‘(log‘𝑁))))
38	35, 36, 37	syl2anc 596	. . . . . . 7 ⊢ (𝜑 → ((log‘(𝐹‘𝑁)) ≤ (log‘𝑁) ↔ (exp‘(log‘(𝐹‘𝑁))) ≤ (exp‘(log‘𝑁))))
39	34, 38	mpbird 260	. . . . . 6 ⊢ (𝜑 → (log‘(𝐹‘𝑁)) ≤ (log‘𝑁))
40	18	rpcnd 13074	. . . . . . 7 ⊢ (𝜑 → (log‘𝑁) ∈ ℂ)
41	40	mulridd 11237	. . . . . 6 ⊢ (𝜑 → ((log‘𝑁) · 1) = (log‘𝑁))
42	39, 41	breqtrrd 5141	. . . . 5 ⊢ (𝜑 → (log‘(𝐹‘𝑁)) ≤ ((log‘𝑁) · 1))
43		1red 11220	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℝ)
44	35, 43, 18	ledivmuld 13125	. . . . 5 ⊢ (𝜑 → (((log‘(𝐹‘𝑁)) / (log‘𝑁)) ≤ 1 ↔ (log‘(𝐹‘𝑁)) ≤ ((log‘𝑁) · 1)))
45	42, 44	mpbird 260	. . . 4 ⊢ (𝜑 → ((log‘(𝐹‘𝑁)) / (log‘𝑁)) ≤ 1)
46	1, 45	eqbrtrid 5148	. . 3 ⊢ (𝜑 → 𝑅 ≤ 1)
47		0xr 11267	. . . 4 ⊢ 0 ∈ ℝ^*
48		1re 11219	. . . 4 ⊢ 1 ∈ ℝ
49		elioc2 13448	. . . 4 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ) → (𝑅 ∈ (0(,]1) ↔ (𝑅 ∈ ℝ ∧ 0 < 𝑅 ∧ 𝑅 ≤ 1)))
50	47, 48, 49	mp2an 705	. . 3 ⊢ (𝑅 ∈ (0(,]1) ↔ (𝑅 ∈ ℝ ∧ 0 < 𝑅 ∧ 𝑅 ≤ 1))
51	21, 22, 46, 50	syl3anbrc 1362	. 2 ⊢ (𝜑 → 𝑅 ∈ (0(,]1))
52	10, 9	qabsabv 27824	. . . 4 ⊢ (abs ↾ ℚ) ∈ 𝐴
53		fvres 6904	. . . . . . . 8 ⊢ (𝑦 ∈ ℚ → ((abs ↾ ℚ)‘𝑦) = (abs‘𝑦))
54	53	oveq1d 7431	. . . . . . 7 ⊢ (𝑦 ∈ ℚ → (((abs ↾ ℚ)‘𝑦)↑_𝑐𝑅) = ((abs‘𝑦)↑_𝑐𝑅))
55	54	mpteq2ia 5208	. . . . . 6 ⊢ (𝑦 ∈ ℚ ↦ (((abs ↾ ℚ)‘𝑦)↑_𝑐𝑅)) = (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅))
56	55	eqcomi 2774	. . . . 5 ⊢ (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅)) = (𝑦 ∈ ℚ ↦ (((abs ↾ ℚ)‘𝑦)↑_𝑐𝑅))
57	9, 11, 56	abvcxp 27810	. . . 4 ⊢ (((abs ↾ ℚ) ∈ 𝐴 ∧ 𝑅 ∈ (0(,]1)) → (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅)) ∈ 𝐴)
58	52, 51, 57	sylancr 599	. . 3 ⊢ (𝜑 → (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅)) ∈ 𝐴)
59		eluzelz 12884	. . . . . 6 ⊢ (𝑧 ∈ (ℤ_≥‘2) → 𝑧 ∈ ℤ)
60		zq 12990	. . . . . 6 ⊢ (𝑧 ∈ ℤ → 𝑧 ∈ ℚ)
61		fveq2 6885	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → (abs‘𝑦) = (abs‘𝑧))
62	61	oveq1d 7431	. . . . . . 7 ⊢ (𝑦 = 𝑧 → ((abs‘𝑦)↑_𝑐𝑅) = ((abs‘𝑧)↑_𝑐𝑅))
63		eqid 2765	. . . . . . 7 ⊢ (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅)) = (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅))
64		ovex 7449	. . . . . . 7 ⊢ ((abs‘𝑧)↑_𝑐𝑅) ∈ V
65	62, 63, 64	fvmpt 6993	. . . . . 6 ⊢ (𝑧 ∈ ℚ → ((𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅))‘𝑧) = ((abs‘𝑧)↑_𝑐𝑅))
66	59, 60, 65	3syl 19	. . . . 5 ⊢ (𝑧 ∈ (ℤ_≥‘2) → ((𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅))‘𝑧) = ((abs‘𝑧)↑_𝑐𝑅))
67	66	adantl 487	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → ((𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅))‘𝑧) = ((abs‘𝑧)↑_𝑐𝑅))
68		simpr 490	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑧 ∈ (ℤ_≥‘2))
69		eluz2b2 12957	. . . . . . . . 9 ⊢ (𝑧 ∈ (ℤ_≥‘2) ↔ (𝑧 ∈ ℕ ∧ 1 < 𝑧))
70	68, 69	sylib 221	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (𝑧 ∈ ℕ ∧ 1 < 𝑧))
71	70	simpld 500	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑧 ∈ ℕ)
72	71	nnred 12259	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑧 ∈ ℝ)
73	71	nnnn0d 12576	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑧 ∈ ℕ₀)
74	73	nn0ge0d 12579	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 0 ≤ 𝑧)
75	72, 74	absidd 15494	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (abs‘𝑧) = 𝑧)
76	75	oveq1d 7431	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → ((abs‘𝑧)↑_𝑐𝑅) = (𝑧↑_𝑐𝑅))
77	72	recnd 11248	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑧 ∈ ℂ)
78	71	nnne0d 12297	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑧 ≠ 0)
79	20	rpcnd 13074	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ ℂ)
80	79	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑅 ∈ ℂ)
81	77, 78, 80	cxpefd 26908	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (𝑧↑_𝑐𝑅) = (exp‘(𝑅 · (log‘𝑧))))
82		padic.j	. . . . . . . . . . 11 ⊢ 𝐽 = (𝑞 ∈ ℙ ↦ (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, (𝑞↑-(𝑞 pCnt 𝑥)))))
83		ostth.k	. . . . . . . . . . 11 ⊢ 𝐾 = (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, 1))
84	2	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝐹 ∈ 𝐴)
85	3	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑁 ∈ (ℤ_≥‘2))
86	14	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 1 < (𝐹‘𝑁))
87		eqid 2765	. . . . . . . . . . 11 ⊢ ((log‘(𝐹‘𝑧)) / (log‘𝑧)) = ((log‘(𝐹‘𝑧)) / (log‘𝑧))
88		eqid 2765	. . . . . . . . . . 11 ⊢ if((𝐹‘𝑧) ≤ 1, 1, (𝐹‘𝑧)) = if((𝐹‘𝑧) ≤ 1, 1, (𝐹‘𝑧))
89		eqid 2765	. . . . . . . . . . 11 ⊢ ((log‘𝑁) / (log‘𝑧)) = ((log‘𝑁) / (log‘𝑧))
90	10, 9, 82, 83, 84, 85, 86, 1, 68, 87, 88, 89	ostth2lem4 27831	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (1 < (𝐹‘𝑧) ∧ 𝑅 ≤ ((log‘(𝐹‘𝑧)) / (log‘𝑧))))
91	90	simprd 501	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑅 ≤ ((log‘(𝐹‘𝑧)) / (log‘𝑧)))
92	90	simpld 500	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 1 < (𝐹‘𝑧))
93		eqid 2765	. . . . . . . . . . 11 ⊢ if((𝐹‘𝑁) ≤ 1, 1, (𝐹‘𝑁)) = if((𝐹‘𝑁) ≤ 1, 1, (𝐹‘𝑁))
94		eqid 2765	. . . . . . . . . . 11 ⊢ ((log‘𝑧) / (log‘𝑁)) = ((log‘𝑧) / (log‘𝑁))
95	10, 9, 82, 83, 84, 68, 92, 87, 85, 1, 93, 94	ostth2lem4 27831	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (1 < (𝐹‘𝑁) ∧ ((log‘(𝐹‘𝑧)) / (log‘𝑧)) ≤ 𝑅))
96	95	simprd 501	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → ((log‘(𝐹‘𝑧)) / (log‘𝑧)) ≤ 𝑅)
97	21	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑅 ∈ ℝ)
98	59	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑧 ∈ ℤ)
99	98, 60	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑧 ∈ ℚ)
100	9, 11	abvcl 20949	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑧 ∈ ℚ) → (𝐹‘𝑧) ∈ ℝ)
101	84, 99, 100	syl2anc 596	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (𝐹‘𝑧) ∈ ℝ)
102	9, 11, 27	abvgt0 20953	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑧 ∈ ℚ ∧ 𝑧 ≠ 0) → 0 < (𝐹‘𝑧))
103	84, 99, 78, 102	syl3anc 1398	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 0 < (𝐹‘𝑧))
104	101, 103	elrpd 13069	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (𝐹‘𝑧) ∈ ℝ⁺)
105	104	relogcld 26819	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (log‘(𝐹‘𝑧)) ∈ ℝ)
106	71	nnrpd 13070	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑧 ∈ ℝ⁺)
107	106	relogcld 26819	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (log‘𝑧) ∈ ℝ)
108		ef0 16163	. . . . . . . . . . . . . 14 ⊢ (exp‘0) = 1
109	70	simprd 501	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 1 < 𝑧)
110	106	reeflogd 26820	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (exp‘(log‘𝑧)) = 𝑧)
111	109, 110	breqtrrd 5141	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 1 < (exp‘(log‘𝑧)))
112	108, 111	eqbrtrid 5148	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (exp‘0) < (exp‘(log‘𝑧)))
113		0re 11221	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ
114		eflt 16191	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℝ ∧ (log‘𝑧) ∈ ℝ) → (0 < (log‘𝑧) ↔ (exp‘0) < (exp‘(log‘𝑧))))
115	113, 107, 114	sylancr 599	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (0 < (log‘𝑧) ↔ (exp‘0) < (exp‘(log‘𝑧))))
116	112, 115	mpbird 260	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 0 < (log‘𝑧))
117	116	gt0ne0d 11789	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (log‘𝑧) ≠ 0)
118	105, 107, 117	redivcld 12054	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → ((log‘(𝐹‘𝑧)) / (log‘𝑧)) ∈ ℝ)
119	97, 118	letri3d 11363	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (𝑅 = ((log‘(𝐹‘𝑧)) / (log‘𝑧)) ↔ (𝑅 ≤ ((log‘(𝐹‘𝑧)) / (log‘𝑧)) ∧ ((log‘(𝐹‘𝑧)) / (log‘𝑧)) ≤ 𝑅)))
120	91, 96, 119	mpbir2and 726	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑅 = ((log‘(𝐹‘𝑧)) / (log‘𝑧)))
121	120	oveq1d 7431	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (𝑅 · (log‘𝑧)) = (((log‘(𝐹‘𝑧)) / (log‘𝑧)) · (log‘𝑧)))
122	105	recnd 11248	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (log‘(𝐹‘𝑧)) ∈ ℂ)
123	107	recnd 11248	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (log‘𝑧) ∈ ℂ)
124	122, 123, 117	divcan1d 12003	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (((log‘(𝐹‘𝑧)) / (log‘𝑧)) · (log‘𝑧)) = (log‘(𝐹‘𝑧)))
125	121, 124	eqtrd 2800	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (𝑅 · (log‘𝑧)) = (log‘(𝐹‘𝑧)))
126	125	fveq2d 6889	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (exp‘(𝑅 · (log‘𝑧))) = (exp‘(log‘(𝐹‘𝑧))))
127	104	reeflogd 26820	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (exp‘(log‘(𝐹‘𝑧))) = (𝐹‘𝑧))
128	81, 126, 127	3eqtrd 2804	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (𝑧↑_𝑐𝑅) = (𝐹‘𝑧))
129	67, 76, 128	3eqtrrd 2805	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (𝐹‘𝑧) = ((𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅))‘𝑧))
130	10, 9, 2, 58, 129	ostthlem1 27822	. 2 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅)))
131		oveq2 7424	. . . 4 ⊢ (𝑎 = 𝑅 → ((abs‘𝑦)↑_𝑐𝑎) = ((abs‘𝑦)↑_𝑐𝑅))
132	131	mpteq2dv 5207	. . 3 ⊢ (𝑎 = 𝑅 → (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑎)) = (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅)))
133	132	rspceeqv 3606	. 2 ⊢ ((𝑅 ∈ (0(,]1) ∧ 𝐹 = (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅))) → ∃𝑎 ∈ (0(,]1)𝐹 = (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑎)))
134	51, 130, 133	syl2anc 596	1 ⊢ (𝜑 → ∃𝑎 ∈ (0(,]1)𝐹 = (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑎)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 ≠ wne 2960 ∃wrex 3091 ifcif 4489 class class class wbr 5111 ↦ cmpt 5194 ↾ cres 5665 ‘cfv 6540 (class class class)co 7416 ℂcc 11109 ℝcr 11110 0cc0 11111 1c1 11112 · cmul 11116 ℝ^*cxr 11253 < clt 11254 ≤ cle 11255 -cneg 11453 / cdiv 11882 ℕcn 12244 2c2 12306 ℕ₀cn0 12515 ℤcz 12602 ℤ_≥cuz 12874 ℚcq 12984 ℝ⁺crp 13028 (,]cioc 13385 ↑cexp 14111 abscabs 15305 expce 16133 ℙcprime 16747 pCnt cpc 16914 ↾_s cress 17308 AbsValcabv 20941 ℂ_fldccnfld 21552 logclog 26750 ↑_𝑐ccxp 26751

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7738 ax-inf2 9613 ax-cnex 11167 ax-resscn 11168 ax-1cn 11169 ax-icn 11170 ax-addcl 11171 ax-addrcl 11172 ax-mulcl 11173 ax-mulrcl 11174 ax-mulcom 11175 ax-addass 11176 ax-mulass 11177 ax-distr 11178 ax-i2m1 11179 ax-1ne0 11180 ax-1rid 11181 ax-rnegex 11182 ax-rrecex 11183 ax-cnre 11184 ax-pre-lttri 11185 ax-pre-lttrn 11186 ax-pre-ltadd 11187 ax-pre-mulgt0 11188 ax-pre-sup 11189 ax-addf 11190 ax-mulf 11191

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7680 df-om 7865 df-1st 7988 df-2nd 7989 df-supp 8159 df-tpos 8224 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-2o 8456 df-er 8696 df-map 8828 df-pm 8829 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fsupp 9325 df-fi 9374 df-sup 9405 df-inf 9406 df-oi 9475 df-card 9937 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11454 df-neg 11455 df-div 11883 df-nn 12245 df-2 12314 df-3 12315 df-4 12316 df-5 12317 df-6 12318 df-7 12319 df-8 12320 df-9 12321 df-n0 12516 df-z 12603 df-dec 12724 df-uz 12875 df-q 12985 df-rp 13029 df-xneg 13149 df-xadd 13150 df-xmul 13151 df-ioo 13388 df-ioc 13389 df-ico 13390 df-icc 13391 df-fz 13548 df-fzo 13696 df-fl 13839 df-mod 13917 df-seq 14052 df-exp 14112 df-fac 14324 df-bc 14353 df-hash 14381 df-shft 15124 df-cj 15170 df-re 15171 df-im 15172 df-sqrt 15306 df-abs 15307 df-limsup 15542 df-clim 15559 df-rlim 15560 df-sum 15758 df-ef 16139 df-sin 16141 df-cos 16142 df-pi 16144 df-struct 17225 df-sets 17242 df-slot 17260 df-ndx 17272 df-base 17288 df-ress 17309 df-plusg 17341 df-mulr 17342 df-starv 17343 df-sca 17344 df-vsca 17345 df-ip 17346 df-tset 17347 df-ple 17348 df-ds 17350 df-unif 17351 df-hom 17352 df-cco 17353 df-rest 17493 df-topn 17494 df-0g 17512 df-gsum 17513 df-topgen 17514 df-pt 17515 df-prds 17518 df-xrs 17574 df-qtop 17579 df-imas 17580 df-xps 17582 df-mre 17656 df-mrc 17657 df-acs 17659 df-mgm 18716 df-sgrp 18799 df-mnd 18815 df-submnd 18866 df-grp 19027 df-minusg 19028 df-mulg 19158 df-subg 19213 df-cntz 19411 df-cmn 19876 df-abl 19877 df-mgp 20241 df-rng 20255 df-ur 20288 df-ring 20341 df-cring 20342 df-oppr 20445 df-dvdsr 20465 df-unit 20466 df-invr 20496 df-dvr 20509 df-subrng 20675 df-subrg 20699 df-drng 20859 df-abv 20942 df-psmet 21544 df-xmet 21545 df-met 21546 df-bl 21547 df-mopn 21548 df-fbas 21549 df-fg 21550 df-cnfld 21553 df-top 23081 df-topon 23098 df-topsp 23120 df-bases 23133 df-cld 23206 df-ntr 23207 df-cls 23208 df-nei 23285 df-lp 23323 df-perf 23324 df-cn 23414 df-cnp 23415 df-haus 23502 df-tx 23750 df-hmeo 23943 df-fil 24034 df-fm 24126 df-flim 24127 df-flf 24128 df-xms 24508 df-ms 24509 df-tms 24510 df-cncf 25068 df-limc 26056 df-dv 26057 df-log 26752 df-cxp 26753

This theorem is used by: ostth 27834

W3C validator