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Theorem ostth2 27618

Description: - Lemma for ostth 27620: 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 12862	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘2) ↔ (𝑁 ∈ ℕ ∧ 1 < 𝑁))
5	3, 4	sylib 219	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 1 < 𝑁))
6	5	simpld 495	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ)
7		nnq 12903	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℚ)
8	6, 7	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℚ)
9		qabsabv.a	. . . . . . . . 9 ⊢ 𝐴 = (AbsVal‘𝑄)
10		qrng.q	. . . . . . . . . 10 ⊢ 𝑄 = (ℂ_fld ↾_s ℚ)
11	10	qrngbas 27600	. . . . . . . . 9 ⊢ ℚ = (Base‘𝑄)
12	9, 11	abvcl 20788	. . . . . . . 8 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑁 ∈ ℚ) → (𝐹‘𝑁) ∈ ℝ)
13	2, 8, 12	syl2anc 590	. . . . . . 7 ⊢ (𝜑 → (𝐹‘𝑁) ∈ ℝ)
14		ostth2.3	. . . . . . 7 ⊢ (𝜑 → 1 < (𝐹‘𝑁))
15	13, 14	rplogcld 26611	. . . . . 6 ⊢ (𝜑 → (log‘(𝐹‘𝑁)) ∈ ℝ⁺)
16	6	nnred 12180	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℝ)
17	5	simprd 496	. . . . . . 7 ⊢ (𝜑 → 1 < 𝑁)
18	16, 17	rplogcld 26611	. . . . . 6 ⊢ (𝜑 → (log‘𝑁) ∈ ℝ⁺)
19	15, 18	rpdivcld 12994	. . . . 5 ⊢ (𝜑 → ((log‘(𝐹‘𝑁)) / (log‘𝑁)) ∈ ℝ⁺)
20	1, 19	eqeltrid 2843	. . . 4 ⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
21	20	rpred 12977	. . 3 ⊢ (𝜑 → 𝑅 ∈ ℝ)
22	20	rpgt0d 12980	. . 3 ⊢ (𝜑 → 0 < 𝑅)
23	6	nnnn0d 12489	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
24	10, 9	qabvle 27606	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑁 ∈ ℕ₀) → (𝐹‘𝑁) ≤ 𝑁)
25	2, 23, 24	syl2anc 590	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝑁) ≤ 𝑁)
26	6	nnne0d 12218	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ≠ 0)
27	10	qrng0 27602	. . . . . . . . . . . 12 ⊢ 0 = (0_g‘𝑄)
28	9, 11, 27	abvgt0 20792	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑁 ∈ ℚ ∧ 𝑁 ≠ 0) → 0 < (𝐹‘𝑁))
29	2, 8, 26, 28	syl3anc 1379	. . . . . . . . . 10 ⊢ (𝜑 → 0 < (𝐹‘𝑁))
30	13, 29	elrpd 12974	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘𝑁) ∈ ℝ⁺)
31	30	reeflogd 26606	. . . . . . . 8 ⊢ (𝜑 → (exp‘(log‘(𝐹‘𝑁))) = (𝐹‘𝑁))
32	6	nnrpd 12975	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℝ⁺)
33	32	reeflogd 26606	. . . . . . . 8 ⊢ (𝜑 → (exp‘(log‘𝑁)) = 𝑁)
34	25, 31, 33	3brtr4d 5104	. . . . . . 7 ⊢ (𝜑 → (exp‘(log‘(𝐹‘𝑁))) ≤ (exp‘(log‘𝑁)))
35	15	rpred 12977	. . . . . . . 8 ⊢ (𝜑 → (log‘(𝐹‘𝑁)) ∈ ℝ)
36	32	relogcld 26605	. . . . . . . 8 ⊢ (𝜑 → (log‘𝑁) ∈ ℝ)
37		efle 16076	. . . . . . . 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 590	. . . . . . 7 ⊢ (𝜑 → ((log‘(𝐹‘𝑁)) ≤ (log‘𝑁) ↔ (exp‘(log‘(𝐹‘𝑁))) ≤ (exp‘(log‘𝑁))))
39	34, 38	mpbird 258	. . . . . 6 ⊢ (𝜑 → (log‘(𝐹‘𝑁)) ≤ (log‘𝑁))
40	18	rpcnd 12979	. . . . . . 7 ⊢ (𝜑 → (log‘𝑁) ∈ ℂ)
41	40	mulridd 11153	. . . . . 6 ⊢ (𝜑 → ((log‘𝑁) · 1) = (log‘𝑁))
42	39, 41	breqtrrd 5100	. . . . 5 ⊢ (𝜑 → (log‘(𝐹‘𝑁)) ≤ ((log‘𝑁) · 1))
43		1red 11136	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℝ)
44	35, 43, 18	ledivmuld 13030	. . . . 5 ⊢ (𝜑 → (((log‘(𝐹‘𝑁)) / (log‘𝑁)) ≤ 1 ↔ (log‘(𝐹‘𝑁)) ≤ ((log‘𝑁) · 1)))
45	42, 44	mpbird 258	. . . 4 ⊢ (𝜑 → ((log‘(𝐹‘𝑁)) / (log‘𝑁)) ≤ 1)
46	1, 45	eqbrtrid 5107	. . 3 ⊢ (𝜑 → 𝑅 ≤ 1)
47		0xr 11183	. . . 4 ⊢ 0 ∈ ℝ^*
48		1re 11135	. . . 4 ⊢ 1 ∈ ℝ
49		elioc2 13353	. . . 4 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ) → (𝑅 ∈ (0(,]1) ↔ (𝑅 ∈ ℝ ∧ 0 < 𝑅 ∧ 𝑅 ≤ 1)))
50	47, 48, 49	mp2an 698	. . 3 ⊢ (𝑅 ∈ (0(,]1) ↔ (𝑅 ∈ ℝ ∧ 0 < 𝑅 ∧ 𝑅 ≤ 1))
51	21, 22, 46, 50	syl3anbrc 1350	. 2 ⊢ (𝜑 → 𝑅 ∈ (0(,]1))
52	10, 9	qabsabv 27610	. . . 4 ⊢ (abs ↾ ℚ) ∈ 𝐴
53		fvres 6846	. . . . . . . 8 ⊢ (𝑦 ∈ ℚ → ((abs ↾ ℚ)‘𝑦) = (abs‘𝑦))
54	53	oveq1d 7371	. . . . . . 7 ⊢ (𝑦 ∈ ℚ → (((abs ↾ ℚ)‘𝑦)↑_𝑐𝑅) = ((abs‘𝑦)↑_𝑐𝑅))
55	54	mpteq2ia 5167	. . . . . 6 ⊢ (𝑦 ∈ ℚ ↦ (((abs ↾ ℚ)‘𝑦)↑_𝑐𝑅)) = (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅))
56	55	eqcomi 2748	. . . . 5 ⊢ (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅)) = (𝑦 ∈ ℚ ↦ (((abs ↾ ℚ)‘𝑦)↑_𝑐𝑅))
57	9, 11, 56	abvcxp 27596	. . . 4 ⊢ (((abs ↾ ℚ) ∈ 𝐴 ∧ 𝑅 ∈ (0(,]1)) → (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅)) ∈ 𝐴)
58	52, 51, 57	sylancr 593	. . 3 ⊢ (𝜑 → (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅)) ∈ 𝐴)
59		eluzelz 12789	. . . . . 6 ⊢ (𝑧 ∈ (ℤ_≥‘2) → 𝑧 ∈ ℤ)
60		zq 12895	. . . . . 6 ⊢ (𝑧 ∈ ℤ → 𝑧 ∈ ℚ)
61		fveq2 6827	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → (abs‘𝑦) = (abs‘𝑧))
62	61	oveq1d 7371	. . . . . . 7 ⊢ (𝑦 = 𝑧 → ((abs‘𝑦)↑_𝑐𝑅) = ((abs‘𝑧)↑_𝑐𝑅))
63		eqid 2739	. . . . . . 7 ⊢ (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅)) = (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅))
64		ovex 7389	. . . . . . 7 ⊢ ((abs‘𝑧)↑_𝑐𝑅) ∈ V
65	62, 63, 64	fvmpt 6935	. . . . . 6 ⊢ (𝑧 ∈ ℚ → ((𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅))‘𝑧) = ((abs‘𝑧)↑_𝑐𝑅))
66	59, 60, 65	3syl 18	. . . . 5 ⊢ (𝑧 ∈ (ℤ_≥‘2) → ((𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅))‘𝑧) = ((abs‘𝑧)↑_𝑐𝑅))
67	66	adantl 482	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → ((𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅))‘𝑧) = ((abs‘𝑧)↑_𝑐𝑅))
68		simpr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑧 ∈ (ℤ_≥‘2))
69		eluz2b2 12862	. . . . . . . . 9 ⊢ (𝑧 ∈ (ℤ_≥‘2) ↔ (𝑧 ∈ ℕ ∧ 1 < 𝑧))
70	68, 69	sylib 219	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (𝑧 ∈ ℕ ∧ 1 < 𝑧))
71	70	simpld 495	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑧 ∈ ℕ)
72	71	nnred 12180	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑧 ∈ ℝ)
73	71	nnnn0d 12489	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑧 ∈ ℕ₀)
74	73	nn0ge0d 12492	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 0 ≤ 𝑧)
75	72, 74	absidd 15376	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (abs‘𝑧) = 𝑧)
76	75	oveq1d 7371	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → ((abs‘𝑧)↑_𝑐𝑅) = (𝑧↑_𝑐𝑅))
77	72	recnd 11164	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑧 ∈ ℂ)
78	71	nnne0d 12218	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑧 ≠ 0)
79	20	rpcnd 12979	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ ℂ)
80	79	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑅 ∈ ℂ)
81	77, 78, 80	cxpefd 26694	. . . . 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 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝐹 ∈ 𝐴)
85	3	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑁 ∈ (ℤ_≥‘2))
86	14	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 1 < (𝐹‘𝑁))
87		eqid 2739	. . . . . . . . . . 11 ⊢ ((log‘(𝐹‘𝑧)) / (log‘𝑧)) = ((log‘(𝐹‘𝑧)) / (log‘𝑧))
88		eqid 2739	. . . . . . . . . . 11 ⊢ if((𝐹‘𝑧) ≤ 1, 1, (𝐹‘𝑧)) = if((𝐹‘𝑧) ≤ 1, 1, (𝐹‘𝑧))
89		eqid 2739	. . . . . . . . . . 11 ⊢ ((log‘𝑁) / (log‘𝑧)) = ((log‘𝑁) / (log‘𝑧))
90	10, 9, 82, 83, 84, 85, 86, 1, 68, 87, 88, 89	ostth2lem4 27617	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (1 < (𝐹‘𝑧) ∧ 𝑅 ≤ ((log‘(𝐹‘𝑧)) / (log‘𝑧))))
91	90	simprd 496	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑅 ≤ ((log‘(𝐹‘𝑧)) / (log‘𝑧)))
92	90	simpld 495	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 1 < (𝐹‘𝑧))
93		eqid 2739	. . . . . . . . . . 11 ⊢ if((𝐹‘𝑁) ≤ 1, 1, (𝐹‘𝑁)) = if((𝐹‘𝑁) ≤ 1, 1, (𝐹‘𝑁))
94		eqid 2739	. . . . . . . . . . 11 ⊢ ((log‘𝑧) / (log‘𝑁)) = ((log‘𝑧) / (log‘𝑁))
95	10, 9, 82, 83, 84, 68, 92, 87, 85, 1, 93, 94	ostth2lem4 27617	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (1 < (𝐹‘𝑁) ∧ ((log‘(𝐹‘𝑧)) / (log‘𝑧)) ≤ 𝑅))
96	95	simprd 496	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → ((log‘(𝐹‘𝑧)) / (log‘𝑧)) ≤ 𝑅)
97	21	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑅 ∈ ℝ)
98	59	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑧 ∈ ℤ)
99	98, 60	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑧 ∈ ℚ)
100	9, 11	abvcl 20788	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑧 ∈ ℚ) → (𝐹‘𝑧) ∈ ℝ)
101	84, 99, 100	syl2anc 590	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (𝐹‘𝑧) ∈ ℝ)
102	9, 11, 27	abvgt0 20792	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑧 ∈ ℚ ∧ 𝑧 ≠ 0) → 0 < (𝐹‘𝑧))
103	84, 99, 78, 102	syl3anc 1379	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 0 < (𝐹‘𝑧))
104	101, 103	elrpd 12974	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (𝐹‘𝑧) ∈ ℝ⁺)
105	104	relogcld 26605	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (log‘(𝐹‘𝑧)) ∈ ℝ)
106	71	nnrpd 12975	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑧 ∈ ℝ⁺)
107	106	relogcld 26605	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (log‘𝑧) ∈ ℝ)
108		ef0 16047	. . . . . . . . . . . . . 14 ⊢ (exp‘0) = 1
109	70	simprd 496	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 1 < 𝑧)
110	106	reeflogd 26606	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (exp‘(log‘𝑧)) = 𝑧)
111	109, 110	breqtrrd 5100	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 1 < (exp‘(log‘𝑧)))
112	108, 111	eqbrtrid 5107	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (exp‘0) < (exp‘(log‘𝑧)))
113		0re 11137	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ
114		eflt 16075	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℝ ∧ (log‘𝑧) ∈ ℝ) → (0 < (log‘𝑧) ↔ (exp‘0) < (exp‘(log‘𝑧))))
115	113, 107, 114	sylancr 593	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (0 < (log‘𝑧) ↔ (exp‘0) < (exp‘(log‘𝑧))))
116	112, 115	mpbird 258	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 0 < (log‘𝑧))
117	116	gt0ne0d 11705	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (log‘𝑧) ≠ 0)
118	105, 107, 117	redivcld 11974	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → ((log‘(𝐹‘𝑧)) / (log‘𝑧)) ∈ ℝ)
119	97, 118	letri3d 11279	. . . . . . . . 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 719	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑅 = ((log‘(𝐹‘𝑧)) / (log‘𝑧)))
121	120	oveq1d 7371	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (𝑅 · (log‘𝑧)) = (((log‘(𝐹‘𝑧)) / (log‘𝑧)) · (log‘𝑧)))
122	105	recnd 11164	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (log‘(𝐹‘𝑧)) ∈ ℂ)
123	107	recnd 11164	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (log‘𝑧) ∈ ℂ)
124	122, 123, 117	divcan1d 11923	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (((log‘(𝐹‘𝑧)) / (log‘𝑧)) · (log‘𝑧)) = (log‘(𝐹‘𝑧)))
125	121, 124	eqtrd 2774	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (𝑅 · (log‘𝑧)) = (log‘(𝐹‘𝑧)))
126	125	fveq2d 6831	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (exp‘(𝑅 · (log‘𝑧))) = (exp‘(log‘(𝐹‘𝑧))))
127	104	reeflogd 26606	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (exp‘(log‘(𝐹‘𝑧))) = (𝐹‘𝑧))
128	81, 126, 127	3eqtrd 2778	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (𝑧↑_𝑐𝑅) = (𝐹‘𝑧))
129	67, 76, 128	3eqtrrd 2779	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (𝐹‘𝑧) = ((𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅))‘𝑧))
130	10, 9, 2, 58, 129	ostthlem1 27608	. 2 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅)))
131		oveq2 7364	. . . 4 ⊢ (𝑎 = 𝑅 → ((abs‘𝑦)↑_𝑐𝑎) = ((abs‘𝑦)↑_𝑐𝑅))
132	131	mpteq2dv 5166	. . 3 ⊢ (𝑎 = 𝑅 → (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑎)) = (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅)))
133	132	rspceeqv 3583	. 2 ⊢ ((𝑅 ∈ (0(,]1) ∧ 𝐹 = (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅))) → ∃𝑎 ∈ (0(,]1)𝐹 = (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑎)))
134	51, 130, 133	syl2anc 590	1 ⊢ (𝜑 → ∃𝑎 ∈ (0(,]1)𝐹 = (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑎)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 ≠ wne 2934 ∃wrex 3063 ifcif 4454 class class class wbr 5072 ↦ cmpt 5153 ↾ cres 5620 ‘cfv 6485 (class class class)co 7356 ℂcc 11027 ℝcr 11028 0cc0 11029 1c1 11030 · cmul 11034 ℝ^*cxr 11169 < clt 11170 ≤ cle 11171 -cneg 11369 / cdiv 11798 ℕcn 12165 2c2 12227 ℕ₀cn0 12428 ℤcz 12515 ℤ_≥cuz 12779 ℚcq 12889 ℝ⁺crp 12933 (,]cioc 13290 ↑cexp 14014 abscabs 15187 expce 16017 ℙcprime 16631 pCnt cpc 16798 ↾_s cress 17191 AbsValcabv 20780 ℂ_fldccnfld 21347 logclog 26536 ↑_𝑐ccxp 26537

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-inf2 9553 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107 ax-addf 11108 ax-mulf 11109

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-iin 4924 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-se 5572 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-isom 6494 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-tpos 8166 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8633 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-q 12890 df-rp 12934 df-xneg 13054 df-xadd 13055 df-xmul 13056 df-ioo 13293 df-ioc 13294 df-ico 13295 df-icc 13296 df-fz 13453 df-fzo 13600 df-fl 13742 df-mod 13820 df-seq 13955 df-exp 14015 df-fac 14227 df-bc 14256 df-hash 14284 df-shft 15020 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-limsup 15424 df-clim 15441 df-rlim 15442 df-sum 15640 df-ef 16023 df-sin 16025 df-cos 16026 df-pi 16028 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-starv 17226 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-hom 17235 df-cco 17236 df-rest 17376 df-topn 17377 df-0g 17395 df-gsum 17396 df-topgen 17397 df-pt 17398 df-prds 17401 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18743 df-grp 18903 df-minusg 18904 df-mulg 19035 df-subg 19090 df-cntz 19283 df-cmn 19748 df-abl 19749 df-mgp 20113 df-rng 20125 df-ur 20154 df-ring 20207 df-cring 20208 df-oppr 20308 df-dvdsr 20328 df-unit 20329 df-invr 20359 df-dvr 20372 df-subrng 20518 df-subrg 20542 df-drng 20703 df-abv 20781 df-psmet 21339 df-xmet 21340 df-met 21341 df-bl 21342 df-mopn 21343 df-fbas 21344 df-fg 21345 df-cnfld 21348 df-top 22877 df-topon 22894 df-topsp 22916 df-bases 22929 df-cld 23002 df-ntr 23003 df-cls 23004 df-nei 23081 df-lp 23119 df-perf 23120 df-cn 23210 df-cnp 23211 df-haus 23298 df-tx 23545 df-hmeo 23738 df-fil 23829 df-fm 23921 df-flim 23922 df-flf 23923 df-xms 24303 df-ms 24304 df-tms 24305 df-cncf 24863 df-limc 25851 df-dv 25852 df-log 26538 df-cxp 26539

This theorem is referenced by: ostth 27620

W3C validator