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 27957

Description: - Lemma for ostth 27959: 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 13041	. . . . . . . . . . 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 13082	. . . . . . . . 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 27939	. . . . . . . . 9 ⊢ ℚ = (Base‘𝑄)
12	9, 11	abvcl 21066	. . . . . . . 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 26950	. . . . . 6 ⊢ (𝜑 → (log‘(𝐹‘𝑁)) ∈ ℝ⁺)
16	6	nnred 12343	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℝ)
17	5	simprd 501	. . . . . . 7 ⊢ (𝜑 → 1 < 𝑁)
18	16, 17	rplogcld 26950	. . . . . 6 ⊢ (𝜑 → (log‘𝑁) ∈ ℝ⁺)
19	15, 18	rpdivcld 13174	. . . . 5 ⊢ (𝜑 → ((log‘(𝐹‘𝑁)) / (log‘𝑁)) ∈ ℝ⁺)
20	1, 19	eqeltrid 2865	. . . 4 ⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
21	20	rpred 13157	. . 3 ⊢ (𝜑 → 𝑅 ∈ ℝ)
22	20	rpgt0d 13160	. . 3 ⊢ (𝜑 → 0 < 𝑅)
23	6	nnnn0d 12660	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
24	10, 9	qabvle 27945	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑁 ∈ ℕ₀) → (𝐹‘𝑁) ≤ 𝑁)
25	2, 23, 24	syl2anc 596	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝑁) ≤ 𝑁)
26	6	nnne0d 12381	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ≠ 0)
27	10	qrng0 27941	. . . . . . . . . . . 12 ⊢ 0 = (0_g‘𝑄)
28	9, 11, 27	abvgt0 21070	. . . . . . . . . . 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 13154	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘𝑁) ∈ ℝ⁺)
31	30	reeflogd 26945	. . . . . . . 8 ⊢ (𝜑 → (exp‘(log‘(𝐹‘𝑁))) = (𝐹‘𝑁))
32	6	nnrpd 13155	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℝ⁺)
33	32	reeflogd 26945	. . . . . . . 8 ⊢ (𝜑 → (exp‘(log‘𝑁)) = 𝑁)
34	25, 31, 33	3brtr4d 5137	. . . . . . 7 ⊢ (𝜑 → (exp‘(log‘(𝐹‘𝑁))) ≤ (exp‘(log‘𝑁)))
35	15	rpred 13157	. . . . . . . 8 ⊢ (𝜑 → (log‘(𝐹‘𝑁)) ∈ ℝ)
36	32	relogcld 26944	. . . . . . . 8 ⊢ (𝜑 → (log‘𝑁) ∈ ℝ)
37		efle 16279	. . . . . . . 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 13159	. . . . . . 7 ⊢ (𝜑 → (log‘𝑁) ∈ ℂ)
41	40	mulridd 11319	. . . . . 6 ⊢ (𝜑 → ((log‘𝑁) · 1) = (log‘𝑁))
42	39, 41	breqtrrd 5133	. . . . 5 ⊢ (𝜑 → (log‘(𝐹‘𝑁)) ≤ ((log‘𝑁) · 1))
43		1red 11302	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℝ)
44	35, 43, 18	ledivmuld 13210	. . . . 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 5140	. . 3 ⊢ (𝜑 → 𝑅 ≤ 1)
47		0xr 11349	. . . 4 ⊢ 0 ∈ ℝ^*
48		1re 11301	. . . 4 ⊢ 1 ∈ ℝ
49		elioc2 13533	. . . 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 27949	. . . 4 ⊢ (abs ↾ ℚ) ∈ 𝐴
53		fvres 6902	. . . . . . . 8 ⊢ (𝑦 ∈ ℚ → ((abs ↾ ℚ)‘𝑦) = (abs‘𝑦))
54	53	oveq1d 7433	. . . . . . 7 ⊢ (𝑦 ∈ ℚ → (((abs ↾ ℚ)‘𝑦)↑_𝑐𝑅) = ((abs‘𝑦)↑_𝑐𝑅))
55	54	mpteq2ia 5200	. . . . . 6 ⊢ (𝑦 ∈ ℚ ↦ (((abs ↾ ℚ)‘𝑦)↑_𝑐𝑅)) = (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅))
56	55	eqcomi 2770	. . . . 5 ⊢ (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅)) = (𝑦 ∈ ℚ ↦ (((abs ↾ ℚ)‘𝑦)↑_𝑐𝑅))
57	9, 11, 56	abvcxp 27935	. . . 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 12968	. . . . . 6 ⊢ (𝑧 ∈ (ℤ_≥‘2) → 𝑧 ∈ ℤ)
60		zq 13074	. . . . . 6 ⊢ (𝑧 ∈ ℤ → 𝑧 ∈ ℚ)
61		fveq2 6883	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → (abs‘𝑦) = (abs‘𝑧))
62	61	oveq1d 7433	. . . . . . 7 ⊢ (𝑦 = 𝑧 → ((abs‘𝑦)↑_𝑐𝑅) = ((abs‘𝑧)↑_𝑐𝑅))
63		eqid 2761	. . . . . . 7 ⊢ (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅)) = (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅))
64		ovex 7451	. . . . . . 7 ⊢ ((abs‘𝑧)↑_𝑐𝑅) ∈ V
65	62, 63, 64	fvmpt 6991	. . . . . 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 13041	. . . . . . . . 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 12343	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑧 ∈ ℝ)
73	71	nnnn0d 12660	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑧 ∈ ℕ₀)
74	73	nn0ge0d 12663	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 0 ≤ 𝑧)
75	72, 74	absidd 15583	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (abs‘𝑧) = 𝑧)
76	75	oveq1d 7433	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → ((abs‘𝑧)↑_𝑐𝑅) = (𝑧↑_𝑐𝑅))
77	72	recnd 11330	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑧 ∈ ℂ)
78	71	nnne0d 12381	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑧 ≠ 0)
79	20	rpcnd 13159	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ ℂ)
80	79	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑅 ∈ ℂ)
81	77, 78, 80	cxpefd 27033	. . . . 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 2761	. . . . . . . . . . 11 ⊢ ((log‘(𝐹‘𝑧)) / (log‘𝑧)) = ((log‘(𝐹‘𝑧)) / (log‘𝑧))
88		eqid 2761	. . . . . . . . . . 11 ⊢ if((𝐹‘𝑧) ≤ 1, 1, (𝐹‘𝑧)) = if((𝐹‘𝑧) ≤ 1, 1, (𝐹‘𝑧))
89		eqid 2761	. . . . . . . . . . 11 ⊢ ((log‘𝑁) / (log‘𝑧)) = ((log‘𝑁) / (log‘𝑧))
90	10, 9, 82, 83, 84, 85, 86, 1, 68, 87, 88, 89	ostth2lem4 27956	. . . . . . . . . 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 2761	. . . . . . . . . . 11 ⊢ if((𝐹‘𝑁) ≤ 1, 1, (𝐹‘𝑁)) = if((𝐹‘𝑁) ≤ 1, 1, (𝐹‘𝑁))
94		eqid 2761	. . . . . . . . . . 11 ⊢ ((log‘𝑧) / (log‘𝑁)) = ((log‘𝑧) / (log‘𝑁))
95	10, 9, 82, 83, 84, 68, 92, 87, 85, 1, 93, 94	ostth2lem4 27956	. . . . . . . . . 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 21066	. . . . . . . . . . . . . 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 21070	. . . . . . . . . . . . . 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 13154	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (𝐹‘𝑧) ∈ ℝ⁺)
105	104	relogcld 26944	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (log‘(𝐹‘𝑧)) ∈ ℝ)
106	71	nnrpd 13155	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑧 ∈ ℝ⁺)
107	106	relogcld 26944	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (log‘𝑧) ∈ ℝ)
108		ef0 16250	. . . . . . . . . . . . . 14 ⊢ (exp‘0) = 1
109	70	simprd 501	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 1 < 𝑧)
110	106	reeflogd 26945	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (exp‘(log‘𝑧)) = 𝑧)
111	109, 110	breqtrrd 5133	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 1 < (exp‘(log‘𝑧)))
112	108, 111	eqbrtrid 5140	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (exp‘0) < (exp‘(log‘𝑧)))
113		0re 11303	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ
114		eflt 16278	. . . . . . . . . . . . . 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 11873	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (log‘𝑧) ≠ 0)
118	105, 107, 117	redivcld 12138	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → ((log‘(𝐹‘𝑧)) / (log‘𝑧)) ∈ ℝ)
119	97, 118	letri3d 11445	. . . . . . . . 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 7433	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (𝑅 · (log‘𝑧)) = (((log‘(𝐹‘𝑧)) / (log‘𝑧)) · (log‘𝑧)))
122	105	recnd 11330	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (log‘(𝐹‘𝑧)) ∈ ℂ)
123	107	recnd 11330	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (log‘𝑧) ∈ ℂ)
124	122, 123, 117	divcan1d 12087	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (((log‘(𝐹‘𝑧)) / (log‘𝑧)) · (log‘𝑧)) = (log‘(𝐹‘𝑧)))
125	121, 124	eqtrd 2796	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (𝑅 · (log‘𝑧)) = (log‘(𝐹‘𝑧)))
126	125	fveq2d 6887	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (exp‘(𝑅 · (log‘𝑧))) = (exp‘(log‘(𝐹‘𝑧))))
127	104	reeflogd 26945	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (exp‘(log‘(𝐹‘𝑧))) = (𝐹‘𝑧))
128	81, 126, 127	3eqtrd 2800	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (𝑧↑_𝑐𝑅) = (𝐹‘𝑧))
129	67, 76, 128	3eqtrrd 2801	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (𝐹‘𝑧) = ((𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅))‘𝑧))
130	10, 9, 2, 58, 129	ostthlem1 27947	. 2 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅)))
131		oveq2 7426	. . . 4 ⊢ (𝑎 = 𝑅 → ((abs‘𝑦)↑_𝑐𝑎) = ((abs‘𝑦)↑_𝑐𝑅))
132	131	mpteq2dv 5199	. . 3 ⊢ (𝑎 = 𝑅 → (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑎)) = (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅)))
133	132	rspceeqv 3599	. 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 2145 ≠ wne 2956 ∃wrex 3087 ifcif 4482 class class class wbr 5103 ↦ cmpt 5186 ↾ cres 5653 ‘cfv 6537 (class class class)co 7418 ℂcc 11191 ℝcr 11192 0cc0 11193 1c1 11194 · cmul 11198 ℝ^*cxr 11335 < clt 11336 ≤ cle 11337 -cneg 11535 / cdiv 11966 ℕcn 12328 2c2 12390 ℕ₀cn0 12599 ℤcz 12686 ℤ_≥cuz 12958 ℚcq 13068 ℝ⁺crp 13113 (,]cioc 13470 ↑cexp 14197 abscabs 15394 expce 16220 ℙcprime 16839 pCnt cpc 17007 ↾_s cress 17401 AbsValcabv 21058 ℂ_fldccnfld 21671 logclog 26875 ↑_𝑐ccxp 26876

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 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7749 ax-inf2 9635 ax-cnex 11249 ax-resscn 11250 ax-1cn 11251 ax-icn 11252 ax-addcl 11253 ax-addrcl 11254 ax-mulcl 11255 ax-mulrcl 11256 ax-mulcom 11257 ax-addass 11258 ax-mulass 11259 ax-distr 11260 ax-i2m1 11261 ax-1ne0 11262 ax-1rid 11263 ax-rnegex 11264 ax-rrecex 11265 ax-cnre 11266 ax-pre-lttri 11267 ax-pre-lttrn 11268 ax-pre-ltadd 11269 ax-pre-mulgt0 11270 ax-pre-sup 11271 ax-addf 11272 ax-mulf 11273

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 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 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 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7375 df-ov 7421 df-oprab 7422 df-mpo 7423 df-of 7691 df-om 7876 df-1st 7999 df-2nd 8000 df-supp 8171 df-tpos 8236 df-frecs 8292 df-wrecs 8323 df-recs 8372 df-rdg 8411 df-1o 8469 df-2o 8470 df-er 8710 df-map 8842 df-pm 8843 df-ixp 8919 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-fsupp 9347 df-fi 9396 df-sup 9427 df-inf 9428 df-oi 9497 df-card 10013 df-pnf 11338 df-mnf 11339 df-xr 11340 df-ltxr 11341 df-le 11342 df-sub 11536 df-neg 11537 df-div 11967 df-nn 12329 df-2 12398 df-3 12399 df-4 12400 df-5 12401 df-6 12402 df-7 12403 df-8 12404 df-9 12405 df-n0 12600 df-z 12687 df-dec 12808 df-uz 12959 df-q 13069 df-rp 13114 df-xneg 13234 df-xadd 13235 df-xmul 13236 df-ioo 13473 df-ioc 13474 df-ico 13475 df-icc 13476 df-fz 13633 df-fzo 13782 df-fl 13925 df-mod 14003 df-seq 14138 df-exp 14198 df-fac 14411 df-bc 14440 df-hash 14468 df-shft 15213 df-cj 15259 df-re 15260 df-im 15261 df-sqrt 15395 df-abs 15396 df-limsup 15631 df-clim 15648 df-rlim 15649 df-sum 15847 df-ef 16226 df-sin 16228 df-cos 16229 df-pi 16231 df-struct 17318 df-sets 17335 df-slot 17353 df-ndx 17365 df-base 17381 df-ress 17402 df-plusg 17434 df-mulr 17435 df-starv 17436 df-sca 17437 df-vsca 17438 df-ip 17439 df-tset 17440 df-ple 17441 df-ds 17443 df-unif 17444 df-hom 17445 df-cco 17446 df-rest 17586 df-topn 17587 df-0g 17605 df-gsum 17606 df-topgen 17607 df-pt 17608 df-prds 17611 df-xrs 17667 df-qtop 17672 df-imas 17673 df-xps 17675 df-mre 17749 df-mrc 17750 df-acs 17752 df-mgm 18809 df-sgrp 18901 df-mnd 18917 df-submnd 18972 df-grp 19140 df-minusg 19141 df-mulg 19271 df-subg 19326 df-cntz 19524 df-cmn 19989 df-abl 19990 df-mgp 20354 df-rng 20368 df-ur 20401 df-ring 20454 df-cring 20455 df-oppr 20560 df-dvdsr 20580 df-unit 20581 df-invr 20611 df-dvr 20624 df-subrng 20791 df-subrg 20815 df-drng 20975 df-abv 21059 df-psmet 21663 df-xmet 21664 df-met 21665 df-bl 21666 df-mopn 21667 df-fbas 21668 df-fg 21669 df-cnfld 21672 df-top 23205 df-topon 23222 df-topsp 23244 df-bases 23257 df-cld 23330 df-ntr 23331 df-cls 23332 df-nei 23409 df-lp 23447 df-perf 23448 df-cn 23538 df-cnp 23539 df-haus 23626 df-tx 23874 df-hmeo 24067 df-fil 24158 df-fm 24250 df-flim 24251 df-flf 24252 df-xms 24632 df-ms 24633 df-tms 24634 df-cncf 25192 df-limc 26179 df-dv 26180 df-log 26877 df-cxp 26878

This theorem is used by: ostth 27959

W3C validator