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Theorem ostth2 25326

Description: - Lemma for ostth 25328: 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 11761	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘2) ↔ (𝑁 ∈ ℕ ∧ 1 < 𝑁))
5	3, 4	sylib 208	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 1 < 𝑁))
6	5	simpld 475	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ)
7		nnq 11801	. . . . . . . . 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 25308	. . . . . . . . 9 ⊢ ℚ = (Base‘𝑄)
12	9, 11	abvcl 18824	. . . . . . . 8 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑁 ∈ ℚ) → (𝐹‘𝑁) ∈ ℝ)
13	2, 8, 12	syl2anc 693	. . . . . . 7 ⊢ (𝜑 → (𝐹‘𝑁) ∈ ℝ)
14		ostth2.3	. . . . . . 7 ⊢ (𝜑 → 1 < (𝐹‘𝑁))
15	13, 14	rplogcld 24375	. . . . . 6 ⊢ (𝜑 → (log‘(𝐹‘𝑁)) ∈ ℝ⁺)
16	6	nnred 11035	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℝ)
17	5	simprd 479	. . . . . . 7 ⊢ (𝜑 → 1 < 𝑁)
18	16, 17	rplogcld 24375	. . . . . 6 ⊢ (𝜑 → (log‘𝑁) ∈ ℝ⁺)
19	15, 18	rpdivcld 11889	. . . . 5 ⊢ (𝜑 → ((log‘(𝐹‘𝑁)) / (log‘𝑁)) ∈ ℝ⁺)
20	1, 19	syl5eqel 2705	. . . 4 ⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
21	20	rpred 11872	. . 3 ⊢ (𝜑 → 𝑅 ∈ ℝ)
22	20	rpgt0d 11875	. . 3 ⊢ (𝜑 → 0 < 𝑅)
23	6	nnnn0d 11351	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
24	10, 9	qabvle 25314	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑁 ∈ ℕ₀) → (𝐹‘𝑁) ≤ 𝑁)
25	2, 23, 24	syl2anc 693	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝑁) ≤ 𝑁)
26	6	nnne0d 11065	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ≠ 0)
27	10	qrng0 25310	. . . . . . . . . . . 12 ⊢ 0 = (0_g‘𝑄)
28	9, 11, 27	abvgt0 18828	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑁 ∈ ℚ ∧ 𝑁 ≠ 0) → 0 < (𝐹‘𝑁))
29	2, 8, 26, 28	syl3anc 1326	. . . . . . . . . 10 ⊢ (𝜑 → 0 < (𝐹‘𝑁))
30	13, 29	elrpd 11869	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘𝑁) ∈ ℝ⁺)
31	30	reeflogd 24370	. . . . . . . 8 ⊢ (𝜑 → (exp‘(log‘(𝐹‘𝑁))) = (𝐹‘𝑁))
32	6	nnrpd 11870	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℝ⁺)
33	32	reeflogd 24370	. . . . . . . 8 ⊢ (𝜑 → (exp‘(log‘𝑁)) = 𝑁)
34	25, 31, 33	3brtr4d 4685	. . . . . . 7 ⊢ (𝜑 → (exp‘(log‘(𝐹‘𝑁))) ≤ (exp‘(log‘𝑁)))
35	15	rpred 11872	. . . . . . . 8 ⊢ (𝜑 → (log‘(𝐹‘𝑁)) ∈ ℝ)
36	32	relogcld 24369	. . . . . . . 8 ⊢ (𝜑 → (log‘𝑁) ∈ ℝ)
37		efle 14848	. . . . . . . 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 693	. . . . . . 7 ⊢ (𝜑 → ((log‘(𝐹‘𝑁)) ≤ (log‘𝑁) ↔ (exp‘(log‘(𝐹‘𝑁))) ≤ (exp‘(log‘𝑁))))
39	34, 38	mpbird 247	. . . . . 6 ⊢ (𝜑 → (log‘(𝐹‘𝑁)) ≤ (log‘𝑁))
40	18	rpcnd 11874	. . . . . . 7 ⊢ (𝜑 → (log‘𝑁) ∈ ℂ)
41	40	mulid1d 10057	. . . . . 6 ⊢ (𝜑 → ((log‘𝑁) · 1) = (log‘𝑁))
42	39, 41	breqtrrd 4681	. . . . 5 ⊢ (𝜑 → (log‘(𝐹‘𝑁)) ≤ ((log‘𝑁) · 1))
43		1red 10055	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℝ)
44	35, 43, 18	ledivmuld 11925	. . . . 5 ⊢ (𝜑 → (((log‘(𝐹‘𝑁)) / (log‘𝑁)) ≤ 1 ↔ (log‘(𝐹‘𝑁)) ≤ ((log‘𝑁) · 1)))
45	42, 44	mpbird 247	. . . 4 ⊢ (𝜑 → ((log‘(𝐹‘𝑁)) / (log‘𝑁)) ≤ 1)
46	1, 45	syl5eqbr 4688	. . 3 ⊢ (𝜑 → 𝑅 ≤ 1)
47		0xr 10086	. . . 4 ⊢ 0 ∈ ℝ^*
48		1re 10039	. . . 4 ⊢ 1 ∈ ℝ
49		elioc2 12236	. . . 4 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ) → (𝑅 ∈ (0(,]1) ↔ (𝑅 ∈ ℝ ∧ 0 < 𝑅 ∧ 𝑅 ≤ 1)))
50	47, 48, 49	mp2an 708	. . 3 ⊢ (𝑅 ∈ (0(,]1) ↔ (𝑅 ∈ ℝ ∧ 0 < 𝑅 ∧ 𝑅 ≤ 1))
51	21, 22, 46, 50	syl3anbrc 1246	. 2 ⊢ (𝜑 → 𝑅 ∈ (0(,]1))
52	10, 9	qabsabv 25318	. . . 4 ⊢ (abs ↾ ℚ) ∈ 𝐴
53		fvres 6207	. . . . . . . 8 ⊢ (𝑦 ∈ ℚ → ((abs ↾ ℚ)‘𝑦) = (abs‘𝑦))
54	53	oveq1d 6665	. . . . . . 7 ⊢ (𝑦 ∈ ℚ → (((abs ↾ ℚ)‘𝑦)↑_𝑐𝑅) = ((abs‘𝑦)↑_𝑐𝑅))
55	54	mpteq2ia 4740	. . . . . 6 ⊢ (𝑦 ∈ ℚ ↦ (((abs ↾ ℚ)‘𝑦)↑_𝑐𝑅)) = (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅))
56	55	eqcomi 2631	. . . . 5 ⊢ (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅)) = (𝑦 ∈ ℚ ↦ (((abs ↾ ℚ)‘𝑦)↑_𝑐𝑅))
57	9, 11, 56	abvcxp 25304	. . . 4 ⊢ (((abs ↾ ℚ) ∈ 𝐴 ∧ 𝑅 ∈ (0(,]1)) → (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅)) ∈ 𝐴)
58	52, 51, 57	sylancr 695	. . 3 ⊢ (𝜑 → (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅)) ∈ 𝐴)
59		eluzelz 11697	. . . . . 6 ⊢ (𝑧 ∈ (ℤ_≥‘2) → 𝑧 ∈ ℤ)
60		zq 11794	. . . . . 6 ⊢ (𝑧 ∈ ℤ → 𝑧 ∈ ℚ)
61		fveq2 6191	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → (abs‘𝑦) = (abs‘𝑧))
62	61	oveq1d 6665	. . . . . . 7 ⊢ (𝑦 = 𝑧 → ((abs‘𝑦)↑_𝑐𝑅) = ((abs‘𝑧)↑_𝑐𝑅))
63		eqid 2622	. . . . . . 7 ⊢ (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅)) = (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅))
64		ovex 6678	. . . . . . 7 ⊢ ((abs‘𝑧)↑_𝑐𝑅) ∈ V
65	62, 63, 64	fvmpt 6282	. . . . . 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 477	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑧 ∈ (ℤ_≥‘2))
69		eluz2b2 11761	. . . . . . . . 9 ⊢ (𝑧 ∈ (ℤ_≥‘2) ↔ (𝑧 ∈ ℕ ∧ 1 < 𝑧))
70	68, 69	sylib 208	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (𝑧 ∈ ℕ ∧ 1 < 𝑧))
71	70	simpld 475	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑧 ∈ ℕ)
72	71	nnred 11035	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑧 ∈ ℝ)
73	71	nnnn0d 11351	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑧 ∈ ℕ₀)
74	73	nn0ge0d 11354	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 0 ≤ 𝑧)
75	72, 74	absidd 14161	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (abs‘𝑧) = 𝑧)
76	75	oveq1d 6665	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → ((abs‘𝑧)↑_𝑐𝑅) = (𝑧↑_𝑐𝑅))
77	72	recnd 10068	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑧 ∈ ℂ)
78	71	nnne0d 11065	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑧 ≠ 0)
79	20	rpcnd 11874	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ ℂ)
80	79	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑅 ∈ ℂ)
81	77, 78, 80	cxpefd 24458	. . . . 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 2622	. . . . . . . . . . 11 ⊢ ((log‘(𝐹‘𝑧)) / (log‘𝑧)) = ((log‘(𝐹‘𝑧)) / (log‘𝑧))
88		eqid 2622	. . . . . . . . . . 11 ⊢ if((𝐹‘𝑧) ≤ 1, 1, (𝐹‘𝑧)) = if((𝐹‘𝑧) ≤ 1, 1, (𝐹‘𝑧))
89		eqid 2622	. . . . . . . . . . 11 ⊢ ((log‘𝑁) / (log‘𝑧)) = ((log‘𝑁) / (log‘𝑧))
90	10, 9, 82, 83, 84, 85, 86, 1, 68, 87, 88, 89	ostth2lem4 25325	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (1 < (𝐹‘𝑧) ∧ 𝑅 ≤ ((log‘(𝐹‘𝑧)) / (log‘𝑧))))
91	90	simprd 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑅 ≤ ((log‘(𝐹‘𝑧)) / (log‘𝑧)))
92	90	simpld 475	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 1 < (𝐹‘𝑧))
93		eqid 2622	. . . . . . . . . . 11 ⊢ if((𝐹‘𝑁) ≤ 1, 1, (𝐹‘𝑁)) = if((𝐹‘𝑁) ≤ 1, 1, (𝐹‘𝑁))
94		eqid 2622	. . . . . . . . . . 11 ⊢ ((log‘𝑧) / (log‘𝑁)) = ((log‘𝑧) / (log‘𝑁))
95	10, 9, 82, 83, 84, 68, 92, 87, 85, 1, 93, 94	ostth2lem4 25325	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (1 < (𝐹‘𝑁) ∧ ((log‘(𝐹‘𝑧)) / (log‘𝑧)) ≤ 𝑅))
96	95	simprd 479	. . . . . . . . 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 18824	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑧 ∈ ℚ) → (𝐹‘𝑧) ∈ ℝ)
101	84, 99, 100	syl2anc 693	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (𝐹‘𝑧) ∈ ℝ)
102	9, 11, 27	abvgt0 18828	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑧 ∈ ℚ ∧ 𝑧 ≠ 0) → 0 < (𝐹‘𝑧))
103	84, 99, 78, 102	syl3anc 1326	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 0 < (𝐹‘𝑧))
104	101, 103	elrpd 11869	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (𝐹‘𝑧) ∈ ℝ⁺)
105	104	relogcld 24369	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (log‘(𝐹‘𝑧)) ∈ ℝ)
106	71	nnrpd 11870	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑧 ∈ ℝ⁺)
107	106	relogcld 24369	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (log‘𝑧) ∈ ℝ)
108		ef0 14821	. . . . . . . . . . . . . 14 ⊢ (exp‘0) = 1
109	70	simprd 479	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 1 < 𝑧)
110	106	reeflogd 24370	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (exp‘(log‘𝑧)) = 𝑧)
111	109, 110	breqtrrd 4681	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 1 < (exp‘(log‘𝑧)))
112	108, 111	syl5eqbr 4688	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (exp‘0) < (exp‘(log‘𝑧)))
113		0re 10040	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ
114		eflt 14847	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℝ ∧ (log‘𝑧) ∈ ℝ) → (0 < (log‘𝑧) ↔ (exp‘0) < (exp‘(log‘𝑧))))
115	113, 107, 114	sylancr 695	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (0 < (log‘𝑧) ↔ (exp‘0) < (exp‘(log‘𝑧))))
116	112, 115	mpbird 247	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 0 < (log‘𝑧))
117	116	gt0ne0d 10592	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (log‘𝑧) ≠ 0)
118	105, 107, 117	redivcld 10853	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → ((log‘(𝐹‘𝑧)) / (log‘𝑧)) ∈ ℝ)
119	97, 118	letri3d 10179	. . . . . . . . 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 957	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → 𝑅 = ((log‘(𝐹‘𝑧)) / (log‘𝑧)))
121	120	oveq1d 6665	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (𝑅 · (log‘𝑧)) = (((log‘(𝐹‘𝑧)) / (log‘𝑧)) · (log‘𝑧)))
122	105	recnd 10068	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (log‘(𝐹‘𝑧)) ∈ ℂ)
123	107	recnd 10068	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (log‘𝑧) ∈ ℂ)
124	122, 123, 117	divcan1d 10802	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (((log‘(𝐹‘𝑧)) / (log‘𝑧)) · (log‘𝑧)) = (log‘(𝐹‘𝑧)))
125	121, 124	eqtrd 2656	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (𝑅 · (log‘𝑧)) = (log‘(𝐹‘𝑧)))
126	125	fveq2d 6195	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (exp‘(𝑅 · (log‘𝑧))) = (exp‘(log‘(𝐹‘𝑧))))
127	104	reeflogd 24370	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (exp‘(log‘(𝐹‘𝑧))) = (𝐹‘𝑧))
128	81, 126, 127	3eqtrd 2660	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (𝑧↑_𝑐𝑅) = (𝐹‘𝑧))
129	67, 76, 128	3eqtrrd 2661	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘2)) → (𝐹‘𝑧) = ((𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅))‘𝑧))
130	10, 9, 2, 58, 129	ostthlem1 25316	. 2 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅)))
131		oveq2 6658	. . . . 5 ⊢ (𝑎 = 𝑅 → ((abs‘𝑦)↑_𝑐𝑎) = ((abs‘𝑦)↑_𝑐𝑅))
132	131	mpteq2dv 4745	. . . 4 ⊢ (𝑎 = 𝑅 → (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑎)) = (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅)))
133	132	eqeq2d 2632	. . 3 ⊢ (𝑎 = 𝑅 → (𝐹 = (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑎)) ↔ 𝐹 = (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅))))
134	133	rspcev 3309	. 2 ⊢ ((𝑅 ∈ (0(,]1) ∧ 𝐹 = (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑅))) → ∃𝑎 ∈ (0(,]1)𝐹 = (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑎)))
135	51, 130, 134	syl2anc 693	1 ⊢ (𝜑 → ∃𝑎 ∈ (0(,]1)𝐹 = (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑_𝑐𝑎)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∃wrex 2913 ifcif 4086 class class class wbr 4653 ↦ cmpt 4729 ↾ cres 5116 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 · cmul 9941 ℝ^*cxr 10073 < clt 10074 ≤ cle 10075 -cneg 10267 / cdiv 10684 ℕcn 11020 2c2 11070 ℕ₀cn0 11292 ℤcz 11377 ℤ_≥cuz 11687 ℚcq 11788 ℝ⁺crp 11832 (,]cioc 12176 ↑cexp 12860 abscabs 13974 expce 14792 ℙcprime 15385 pCnt cpc 15541 ↾_s cress 15858 AbsValcabv 18816 ℂ_fldccnfld 19746 logclog 24301 ↑_𝑐ccxp 24302

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-shft 13807 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-sum 14417 df-ef 14798 df-sin 14800 df-cos 14801 df-pi 14803 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-grp 17425 df-minusg 17426 df-mulg 17541 df-subg 17591 df-cntz 17750 df-cmn 18195 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-dvr 18683 df-drng 18749 df-subrg 18778 df-abv 18817 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-limc 23630 df-dv 23631 df-log 24303 df-cxp 24304

This theorem is referenced by: ostth 25328

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