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 26690

Description: - Lemma for ostth 26692: 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 12590	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘2) ↔ (𝑁 ∈ ℕ ∧ 1 < 𝑁))
5	3, 4	sylib 217	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 1 < 𝑁))
6	5	simpld 494	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ)
7		nnq 12631	. . . . . . . . 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 26672	. . . . . . . . 9 ⊢ ℚ = (Base‘𝑄)
12	9, 11	abvcl 19999	. . . . . . . 8 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑁 ∈ ℚ) → (𝐹‘𝑁) ∈ ℝ)
13	2, 8, 12	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → (𝐹‘𝑁) ∈ ℝ)
14		ostth2.3	. . . . . . 7 ⊢ (𝜑 → 1 < (𝐹‘𝑁))
15	13, 14	rplogcld 25689	. . . . . 6 ⊢ (𝜑 → (log‘(𝐹‘𝑁)) ∈ ℝ⁺)
16	6	nnred 11918	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℝ)
17	5	simprd 495	. . . . . . 7 ⊢ (𝜑 → 1 < 𝑁)
18	16, 17	rplogcld 25689	. . . . . 6 ⊢ (𝜑 → (log‘𝑁) ∈ ℝ⁺)
19	15, 18	rpdivcld 12718	. . . . 5 ⊢ (𝜑 → ((log‘(𝐹‘𝑁)) / (log‘𝑁)) ∈ ℝ⁺)
20	1, 19	eqeltrid 2843	. . . 4 ⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
21	20	rpred 12701	. . 3 ⊢ (𝜑 → 𝑅 ∈ ℝ)
22	20	rpgt0d 12704	. . 3 ⊢ (𝜑 → 0 < 𝑅)
23	6	nnnn0d 12223	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
24	10, 9	qabvle 26678	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑁 ∈ ℕ₀) → (𝐹‘𝑁) ≤ 𝑁)
25	2, 23, 24	syl2anc 583	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝑁) ≤ 𝑁)
26	6	nnne0d 11953	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ≠ 0)
27	10	qrng0 26674	. . . . . . . . . . . 12 ⊢ 0 = (0_g‘𝑄)
28	9, 11, 27	abvgt0 20003	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑁 ∈ ℚ ∧ 𝑁 ≠ 0) → 0 < (𝐹‘𝑁))
29	2, 8, 26, 28	syl3anc 1369	. . . . . . . . . 10 ⊢ (𝜑 → 0 < (𝐹‘𝑁))
30	13, 29	elrpd 12698	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘𝑁) ∈ ℝ⁺)
31	30	reeflogd 25684	. . . . . . . 8 ⊢ (𝜑 → (exp‘(log‘(𝐹‘𝑁))) = (𝐹‘𝑁))
32	6	nnrpd 12699	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℝ⁺)
33	32	reeflogd 25684	. . . . . . . 8 ⊢ (𝜑 → (exp‘(log‘𝑁)) = 𝑁)
34	25, 31, 33	3brtr4d 5102	. . . . . . 7 ⊢ (𝜑 → (exp‘(log‘(𝐹‘𝑁))) ≤ (exp‘(log‘𝑁)))
35	15	rpred 12701	. . . . . . . 8 ⊢ (𝜑 → (log‘(𝐹‘𝑁)) ∈ ℝ)
36	32	relogcld 25683	. . . . . . . 8 ⊢ (𝜑 → (log‘𝑁) ∈ ℝ)
37		efle 15755	. . . . . . . 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 583	. . . . . . 7 ⊢ (𝜑 → ((log‘(𝐹‘𝑁)) ≤ (log‘𝑁) ↔ (exp‘(log‘(𝐹‘𝑁))) ≤ (exp‘(log‘𝑁))))
39	34, 38	mpbird 256	. . . . . 6 ⊢ (𝜑 → (log‘(𝐹‘𝑁)) ≤ (log‘𝑁))
40	18	rpcnd 12703	. . . . . . 7 ⊢ (𝜑 → (log‘𝑁) ∈ ℂ)
41	40	mulid1d 10923	. . . . . 6 ⊢ (𝜑 → ((log‘𝑁) · 1) = (log‘𝑁))
42	39, 41	breqtrrd 5098	. . . . 5 ⊢ (𝜑 → (log‘(𝐹‘𝑁)) ≤ ((log‘𝑁) · 1))
43		1red 10907	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℝ)
44	35, 43, 18	ledivmuld 12754	. . . . 5 ⊢ (𝜑 → (((log‘(𝐹‘𝑁)) / (log‘𝑁)) ≤ 1 ↔ (log‘(𝐹‘𝑁)) ≤ ((log‘𝑁) · 1)))
45	42, 44	mpbird 256	. . . 4 ⊢ (𝜑 → ((log‘(𝐹‘𝑁)) / (log‘𝑁)) ≤ 1)
46	1, 45	eqbrtrid 5105	. . 3 ⊢ (𝜑 → 𝑅 ≤ 1)
47		0xr 10953	. . . 4 ⊢ 0 ∈ ℝ^*
48		1re 10906	. . . 4 ⊢ 1 ∈ ℝ
49		elioc2 13071	. . . 4 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ) → (𝑅 ∈ (0(,]1) ↔ (𝑅 ∈ ℝ ∧ 0 < 𝑅 ∧ 𝑅 ≤ 1)))
50	47, 48, 49	mp2an 688	. . 3 ⊢ (𝑅 ∈ (0(,]1) ↔ (𝑅 ∈ ℝ ∧ 0 < 𝑅 ∧ 𝑅 ≤ 1))
51	21, 22, 46, 50	syl3anbrc 1341	. 2 ⊢ (𝜑 → 𝑅 ∈ (0(,]1))
52	10, 9	qabsabv 26682	. . . 4 ⊢ (abs ↾ ℚ) ∈ 𝐴
53		fvres 6775	. . . . . . . 8 ⊢ (𝑦 ∈ ℚ → ((abs ↾ ℚ)‘𝑦) = (abs‘𝑦))
54	53	oveq1d 7270	. . . . . . 7 ⊢ (𝑦 ∈ ℚ → (((abs ↾ ℚ)‘𝑦)↑_𝑐𝑅) = ((abs‘𝑦)↑_𝑐𝑅))
55	54	mpteq2ia 5173	. . . . . 6 ⊢ (𝑦 ∈ ℚ ↦ (((abs ↾ ℚ)‘𝑦)↑_𝑐𝑅)) = (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅))
56	55	eqcomi 2747	. . . . 5 ⊢ (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅)) = (𝑦 ∈ ℚ ↦ (((abs ↾ ℚ)‘𝑦)↑_𝑐𝑅))
57	9, 11, 56	abvcxp 26668	. . . 4 ⊢ (((abs ↾ ℚ) ∈ 𝐴 ∧ 𝑅 ∈ (0(,]1)) → (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅)) ∈ 𝐴)
58	52, 51, 57	sylancr 586	. . 3 ⊢ (𝜑 → (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅)) ∈ 𝐴)
59		eluzelz 12521	. . . . . 6 ⊢ (𝑧 ∈ (ℤ_≥‘2) → 𝑧 ∈ ℤ)
60		zq 12623	. . . . . 6 ⊢ (𝑧 ∈ ℤ → 𝑧 ∈ ℚ)
61		fveq2 6756	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → (abs‘𝑦) = (abs‘𝑧))
62	61	oveq1d 7270	. . . . . . 7 ⊢ (𝑦 = 𝑧 → ((abs‘𝑦)↑_𝑐𝑅) = ((abs‘𝑧)↑_𝑐𝑅))
63		eqid 2738	. . . . . . 7 ⊢ (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅)) = (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅))
64		ovex 7288	. . . . . . 7 ⊢ ((abs‘𝑧)↑_𝑐𝑅) ∈ V
65	62, 63, 64	fvmpt 6857	. . . . . 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 481	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → ((𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅))‘𝑧) = ((abs‘𝑧)↑_𝑐𝑅))
68		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑧 ∈ (ℤ_≥‘2))
69		eluz2b2 12590	. . . . . . . . 9 ⊢ (𝑧 ∈ (ℤ_≥‘2) ↔ (𝑧 ∈ ℕ ∧ 1 < 𝑧))
70	68, 69	sylib 217	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (𝑧 ∈ ℕ ∧ 1 < 𝑧))
71	70	simpld 494	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑧 ∈ ℕ)
72	71	nnred 11918	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑧 ∈ ℝ)
73	71	nnnn0d 12223	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑧 ∈ ℕ₀)
74	73	nn0ge0d 12226	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 0 ≤ 𝑧)
75	72, 74	absidd 15062	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (abs‘𝑧) = 𝑧)
76	75	oveq1d 7270	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → ((abs‘𝑧)↑_𝑐𝑅) = (𝑧↑_𝑐𝑅))
77	72	recnd 10934	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑧 ∈ ℂ)
78	71	nnne0d 11953	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑧 ≠ 0)
79	20	rpcnd 12703	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ ℂ)
80	79	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑅 ∈ ℂ)
81	77, 78, 80	cxpefd 25772	. . . . 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 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝐹 ∈ 𝐴)
85	3	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑁 ∈ (ℤ_≥‘2))
86	14	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 1 < (𝐹‘𝑁))
87		eqid 2738	. . . . . . . . . . 11 ⊢ ((log‘(𝐹‘𝑧)) / (log‘𝑧)) = ((log‘(𝐹‘𝑧)) / (log‘𝑧))
88		eqid 2738	. . . . . . . . . . 11 ⊢ if((𝐹‘𝑧) ≤ 1, 1, (𝐹‘𝑧)) = if((𝐹‘𝑧) ≤ 1, 1, (𝐹‘𝑧))
89		eqid 2738	. . . . . . . . . . 11 ⊢ ((log‘𝑁) / (log‘𝑧)) = ((log‘𝑁) / (log‘𝑧))
90	10, 9, 82, 83, 84, 85, 86, 1, 68, 87, 88, 89	ostth2lem4 26689	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (1 < (𝐹‘𝑧) ∧ 𝑅 ≤ ((log‘(𝐹‘𝑧)) / (log‘𝑧))))
91	90	simprd 495	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑅 ≤ ((log‘(𝐹‘𝑧)) / (log‘𝑧)))
92	90	simpld 494	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 1 < (𝐹‘𝑧))
93		eqid 2738	. . . . . . . . . . 11 ⊢ if((𝐹‘𝑁) ≤ 1, 1, (𝐹‘𝑁)) = if((𝐹‘𝑁) ≤ 1, 1, (𝐹‘𝑁))
94		eqid 2738	. . . . . . . . . . 11 ⊢ ((log‘𝑧) / (log‘𝑁)) = ((log‘𝑧) / (log‘𝑁))
95	10, 9, 82, 83, 84, 68, 92, 87, 85, 1, 93, 94	ostth2lem4 26689	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (1 < (𝐹‘𝑁) ∧ ((log‘(𝐹‘𝑧)) / (log‘𝑧)) ≤ 𝑅))
96	95	simprd 495	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → ((log‘(𝐹‘𝑧)) / (log‘𝑧)) ≤ 𝑅)
97	21	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑅 ∈ ℝ)
98	59	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑧 ∈ ℤ)
99	98, 60	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑧 ∈ ℚ)
100	9, 11	abvcl 19999	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑧 ∈ ℚ) → (𝐹‘𝑧) ∈ ℝ)
101	84, 99, 100	syl2anc 583	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (𝐹‘𝑧) ∈ ℝ)
102	9, 11, 27	abvgt0 20003	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑧 ∈ ℚ ∧ 𝑧 ≠ 0) → 0 < (𝐹‘𝑧))
103	84, 99, 78, 102	syl3anc 1369	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 0 < (𝐹‘𝑧))
104	101, 103	elrpd 12698	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (𝐹‘𝑧) ∈ ℝ⁺)
105	104	relogcld 25683	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (log‘(𝐹‘𝑧)) ∈ ℝ)
106	71	nnrpd 12699	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑧 ∈ ℝ⁺)
107	106	relogcld 25683	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (log‘𝑧) ∈ ℝ)
108		ef0 15728	. . . . . . . . . . . . . 14 ⊢ (exp‘0) = 1
109	70	simprd 495	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 1 < 𝑧)
110	106	reeflogd 25684	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (exp‘(log‘𝑧)) = 𝑧)
111	109, 110	breqtrrd 5098	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 1 < (exp‘(log‘𝑧)))
112	108, 111	eqbrtrid 5105	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (exp‘0) < (exp‘(log‘𝑧)))
113		0re 10908	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ
114		eflt 15754	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℝ ∧ (log‘𝑧) ∈ ℝ) → (0 < (log‘𝑧) ↔ (exp‘0) < (exp‘(log‘𝑧))))
115	113, 107, 114	sylancr 586	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (0 < (log‘𝑧) ↔ (exp‘0) < (exp‘(log‘𝑧))))
116	112, 115	mpbird 256	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 0 < (log‘𝑧))
117	116	gt0ne0d 11469	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (log‘𝑧) ≠ 0)
118	105, 107, 117	redivcld 11733	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → ((log‘(𝐹‘𝑧)) / (log‘𝑧)) ∈ ℝ)
119	97, 118	letri3d 11047	. . . . . . . . 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 709	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑅 = ((log‘(𝐹‘𝑧)) / (log‘𝑧)))
121	120	oveq1d 7270	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (𝑅 · (log‘𝑧)) = (((log‘(𝐹‘𝑧)) / (log‘𝑧)) · (log‘𝑧)))
122	105	recnd 10934	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (log‘(𝐹‘𝑧)) ∈ ℂ)
123	107	recnd 10934	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (log‘𝑧) ∈ ℂ)
124	122, 123, 117	divcan1d 11682	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (((log‘(𝐹‘𝑧)) / (log‘𝑧)) · (log‘𝑧)) = (log‘(𝐹‘𝑧)))
125	121, 124	eqtrd 2778	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (𝑅 · (log‘𝑧)) = (log‘(𝐹‘𝑧)))
126	125	fveq2d 6760	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (exp‘(𝑅 · (log‘𝑧))) = (exp‘(log‘(𝐹‘𝑧))))
127	104	reeflogd 25684	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (exp‘(log‘(𝐹‘𝑧))) = (𝐹‘𝑧))
128	81, 126, 127	3eqtrd 2782	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (𝑧↑_𝑐𝑅) = (𝐹‘𝑧))
129	67, 76, 128	3eqtrrd 2783	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (𝐹‘𝑧) = ((𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅))‘𝑧))
130	10, 9, 2, 58, 129	ostthlem1 26680	. 2 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅)))
131		oveq2 7263	. . . 4 ⊢ (𝑎 = 𝑅 → ((abs‘𝑦)↑_𝑐𝑎) = ((abs‘𝑦)↑_𝑐𝑅))
132	131	mpteq2dv 5172	. . 3 ⊢ (𝑎 = 𝑅 → (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑎)) = (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅)))
133	132	rspceeqv 3567	. 2 ⊢ ((𝑅 ∈ (0(,]1) ∧ 𝐹 = (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅))) → ∃𝑎 ∈ (0(,]1)𝐹 = (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑎)))
134	51, 130, 133	syl2anc 583	1 ⊢ (𝜑 → ∃𝑎 ∈ (0(,]1)𝐹 = (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑎)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∃wrex 3064 ifcif 4456 class class class wbr 5070 ↦ cmpt 5153 ↾ cres 5582 ‘cfv 6418 (class class class)co 7255 ℂcc 10800 ℝcr 10801 0cc0 10802 1c1 10803 · cmul 10807 ℝ^*cxr 10939 < clt 10940 ≤ cle 10941 -cneg 11136 / cdiv 11562 ℕcn 11903 2c2 11958 ℕ₀cn0 12163 ℤcz 12249 ℤ_≥cuz 12511 ℚcq 12617 ℝ⁺crp 12659 (,]cioc 13009 ↑cexp 13710 abscabs 14873 expce 15699 ℙcprime 16304 pCnt cpc 16465 ↾_s cress 16867 AbsValcabv 19991 ℂ_fldccnfld 20510 logclog 25615 ↑_𝑐ccxp 25616

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-tpos 8013 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-ioc 13013 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-fac 13916 df-bc 13945 df-hash 13973 df-shft 14706 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-limsup 15108 df-clim 15125 df-rlim 15126 df-sum 15326 df-ef 15705 df-sin 15707 df-cos 15708 df-pi 15710 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-pt 17072 df-prds 17075 df-xrs 17130 df-qtop 17135 df-imas 17136 df-xps 17138 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-grp 18495 df-minusg 18496 df-mulg 18616 df-subg 18667 df-cntz 18838 df-cmn 19303 df-mgp 19636 df-ur 19653 df-ring 19700 df-cring 19701 df-oppr 19777 df-dvdsr 19798 df-unit 19799 df-invr 19829 df-dvr 19840 df-drng 19908 df-subrg 19937 df-abv 19992 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-fbas 20507 df-fg 20508 df-cnfld 20511 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-cld 22078 df-ntr 22079 df-cls 22080 df-nei 22157 df-lp 22195 df-perf 22196 df-cn 22286 df-cnp 22287 df-haus 22374 df-tx 22621 df-hmeo 22814 df-fil 22905 df-fm 22997 df-flim 22998 df-flf 22999 df-xms 23381 df-ms 23382 df-tms 23383 df-cncf 23947 df-limc 24935 df-dv 24936 df-log 25617 df-cxp 25618

This theorem is referenced by: ostth 26692

W3C validator