logdivsqrle - Mathbox for Thierry Arnoux

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Theorem logdivsqrle 34954

Description: Conditions for ((log x ) / ( sqrt 𝑥)) to be decreasing. (Contributed by Thierry Arnoux, 20-Dec-2021.)

**Hypotheses**
Ref	Expression
logdivsqrle.a	⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
logdivsqrle.b	⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
logdivsqrle.1	⊢ (𝜑 → (exp‘2) ≤ 𝐴)
logdivsqrle.2	⊢ (𝜑 → 𝐴 ≤ 𝐵)

**Assertion**
Ref	Expression
logdivsqrle	⊢ (𝜑 → ((log‘𝐵) / (√‘𝐵)) ≤ ((log‘𝐴) / (√‘𝐴)))

Proof of Theorem logdivsqrle Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ioorp 13443	. . . 4 ⊢ (0(,)+∞) = ℝ⁺
2	1	eqcomi 2774	. . 3 ⊢ ℝ⁺ = (0(,)+∞)
3		logdivsqrle.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
4		logdivsqrle.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
5		simpr 489	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ⁺)
6	5	relogcld 26746	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℝ)
7	5	rpsqrtcld 15453	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (√‘𝑥) ∈ ℝ⁺)
8	7	rpred 13051	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (√‘𝑥) ∈ ℝ)
9		rpsqrtcl 15305	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (√‘𝑥) ∈ ℝ⁺)
10		rpne0 13024	. . . . . . 7 ⊢ ((√‘𝑥) ∈ ℝ⁺ → (√‘𝑥) ≠ 0)
11	9, 10	syl 18	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (√‘𝑥) ≠ 0)
12	11	adantl 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (√‘𝑥) ≠ 0)
13	6, 8, 12	redivcld 12034	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) / (√‘𝑥)) ∈ ℝ)
14	13	fmpttd 7100	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (√‘𝑥))):ℝ⁺⟶ℝ)
15		rpcn 13018	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
16	15	adantl 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℂ)
17		rpne0 13024	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≠ 0)
18	17	adantl 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ≠ 0)
19	16, 18	logcld 26693	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℂ)
20	16	sqrtcld 15481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (√‘𝑥) ∈ ℂ)
21	19, 20, 12	divrecd 11985	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) / (√‘𝑥)) = ((log‘𝑥) · (1 / (√‘𝑥))))
22		2cnd 12310	. . . . . . . . . . . . 13 ⊢ (𝜑 → 2 ∈ ℂ)
23	22	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 2 ∈ ℂ)
24		2ne0 12338	. . . . . . . . . . . . 13 ⊢ 2 ≠ 0
25	24	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 2 ≠ 0)
26	23, 25	reccld 11975	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1 / 2) ∈ ℂ)
27	16, 18, 26	cxpnegd 26838	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐-(1 / 2)) = (1 / (𝑥↑_𝑐(1 / 2))))
28		cxpsqrt 26826	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ → (𝑥↑_𝑐(1 / 2)) = (√‘𝑥))
29	16, 28	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐(1 / 2)) = (√‘𝑥))
30	29	oveq2d 7416	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1 / (𝑥↑_𝑐(1 / 2))) = (1 / (√‘𝑥)))
31	27, 30	eqtrd 2800	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐-(1 / 2)) = (1 / (√‘𝑥)))
32	31	oveq2d 7416	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) · (𝑥↑_𝑐-(1 / 2))) = ((log‘𝑥) · (1 / (√‘𝑥))))
33	21, 32	eqtr4d 2803	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) / (√‘𝑥)) = ((log‘𝑥) · (𝑥↑_𝑐-(1 / 2))))
34	33	mpteq2dva 5198	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (√‘𝑥))) = (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) · (𝑥↑_𝑐-(1 / 2)))))
35	34	oveq2d 7416	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (√‘𝑥)))) = (ℝ D (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) · (𝑥↑_𝑐-(1 / 2))))))
36		reelprrecn 11180	. . . . . . 7 ⊢ ℝ ∈ {ℝ, ℂ}
37	36	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
38	5	rpreccld 13061	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ ℝ⁺)
39		logf1o 26687	. . . . . . . . . . 11 ⊢ log:(ℂ ∖ {0})–1-1-onto→ran log
40		f1of 6810	. . . . . . . . . . 11 ⊢ (log:(ℂ ∖ {0})–1-1-onto→ran log → log:(ℂ ∖ {0})⟶ran log)
41	39, 40	ax-mp 5	. . . . . . . . . 10 ⊢ log:(ℂ ∖ {0})⟶ran log
42	41	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → log:(ℂ ∖ {0})⟶ran log)
43	15	ssriv 3943	. . . . . . . . . . 11 ⊢ ℝ⁺ ⊆ ℂ
44		0nrp 13044	. . . . . . . . . . 11 ⊢ ¬ 0 ∈ ℝ⁺
45		ssdifsn 4751	. . . . . . . . . . 11 ⊢ (ℝ⁺ ⊆ (ℂ ∖ {0}) ↔ (ℝ⁺ ⊆ ℂ ∧ ¬ 0 ∈ ℝ⁺))
46	43, 44, 45	mpbir2an 723	. . . . . . . . . 10 ⊢ ℝ⁺ ⊆ (ℂ ∖ {0})
47	46	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ℝ⁺ ⊆ (ℂ ∖ {0}))
48	42, 47	feqresmpt 6940	. . . . . . . 8 ⊢ (𝜑 → (log ↾ ℝ⁺) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)))
49	48	oveq2d 7416	. . . . . . 7 ⊢ (𝜑 → (ℝ D (log ↾ ℝ⁺)) = (ℝ D (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))))
50		dvrelog 26760	. . . . . . 7 ⊢ (ℝ D (log ↾ ℝ⁺)) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥))
51	49, 50	eqtr3di 2815	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)))
52		1cnd 11190	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℂ)
53	52	halfcld 12480	. . . . . . . . 9 ⊢ (𝜑 → (1 / 2) ∈ ℂ)
54	53	negcld 11544	. . . . . . . 8 ⊢ (𝜑 → -(1 / 2) ∈ ℂ)
55	54	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → -(1 / 2) ∈ ℂ)
56	16, 55	cxpcld 26831	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐-(1 / 2)) ∈ ℂ)
57	52	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 1 ∈ ℂ)
58	55, 57	subcld 11557	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (-(1 / 2) − 1) ∈ ℂ)
59	16, 58	cxpcld 26831	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐(-(1 / 2) − 1)) ∈ ℂ)
60	55, 59	mulcld 11217	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1))) ∈ ℂ)
61		dvcxp1 26863	. . . . . . 7 ⊢ (-(1 / 2) ∈ ℂ → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (𝑥↑_𝑐-(1 / 2)))) = (𝑥 ∈ ℝ⁺ ↦ (-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1)))))
62	54, 61	syl 18	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (𝑥↑_𝑐-(1 / 2)))) = (𝑥 ∈ ℝ⁺ ↦ (-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1)))))
63	37, 19, 38, 51, 56, 60, 62	dvmptmul 26081	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) · (𝑥↑_𝑐-(1 / 2))))) = (𝑥 ∈ ℝ⁺ ↦ (((1 / 𝑥) · (𝑥↑_𝑐-(1 / 2))) + ((-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1))) · (log‘𝑥)))))
64	35, 63	eqtrd 2800	. . . 4 ⊢ (𝜑 → (ℝ D (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (√‘𝑥)))) = (𝑥 ∈ ℝ⁺ ↦ (((1 / 𝑥) · (𝑥↑_𝑐-(1 / 2))) + ((-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1))) · (log‘𝑥)))))
65		ax-resscn 11145	. . . . . 6 ⊢ ℝ ⊆ ℂ
66	65	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ⊆ ℂ)
67		eqid 2765	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
68	67	addcn 24984	. . . . . . 7 ⊢ + ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld))
69	68	a1i 11	. . . . . 6 ⊢ (𝜑 → + ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld)))
70	43	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ℝ⁺ ⊆ ℂ)
71		ssid 3961	. . . . . . . . . 10 ⊢ ℂ ⊆ ℂ
72	71	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ℂ ⊆ ℂ)
73		cncfmptc 25032	. . . . . . . . 9 ⊢ ((1 ∈ ℂ ∧ ℝ⁺ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ ℝ⁺ ↦ 1) ∈ (ℝ⁺–cn→ℂ))
74	52, 70, 72, 73	syl3anc 1394	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ 1) ∈ (ℝ⁺–cn→ℂ))
75		difss 4092	. . . . . . . . 9 ⊢ (ℂ ∖ {0}) ⊆ ℂ
76		cncfmptid 25033	. . . . . . . . 9 ⊢ ((ℝ⁺ ⊆ (ℂ ∖ {0}) ∧ (ℂ ∖ {0}) ⊆ ℂ) → (𝑥 ∈ ℝ⁺ ↦ 𝑥) ∈ (ℝ⁺–cn→(ℂ ∖ {0})))
77	47, 75, 76	sylancl 597	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ 𝑥) ∈ (ℝ⁺–cn→(ℂ ∖ {0})))
78	74, 77	divcncf 25567	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)) ∈ (ℝ⁺–cn→ℂ))
79		ax-1 6	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ → 𝑥 ∈ ℝ⁺))
80	15, 79	jca 520	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℂ ∧ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ⁺)))
81		eqid 2765	. . . . . . . . . . . 12 ⊢ (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0))
82	81	ellogdm 26762	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (ℂ ∖ (-∞(,]0)) ↔ (𝑥 ∈ ℂ ∧ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ⁺)))
83	80, 82	sylibr 237	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ (ℂ ∖ (-∞(,]0)))
84	83	ssriv 3943	. . . . . . . . 9 ⊢ ℝ⁺ ⊆ (ℂ ∖ (-∞(,]0))
85	84	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ⁺ ⊆ (ℂ ∖ (-∞(,]0)))
86	54, 85	cxpcncf1 34899	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (𝑥↑_𝑐-(1 / 2))) ∈ (ℝ⁺–cn→ℂ))
87	78, 86	mulcncf 25566	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((1 / 𝑥) · (𝑥↑_𝑐-(1 / 2)))) ∈ (ℝ⁺–cn→ℂ))
88		cncfmptc 25032	. . . . . . . . 9 ⊢ ((-(1 / 2) ∈ ℂ ∧ ℝ⁺ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ ℝ⁺ ↦ -(1 / 2)) ∈ (ℝ⁺–cn→ℂ))
89	54, 70, 72, 88	syl3anc 1394	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ -(1 / 2)) ∈ (ℝ⁺–cn→ℂ))
90	54, 52	subcld 11557	. . . . . . . . 9 ⊢ (𝜑 → (-(1 / 2) − 1) ∈ ℂ)
91	90, 85	cxpcncf1 34899	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (𝑥↑_𝑐(-(1 / 2) − 1))) ∈ (ℝ⁺–cn→ℂ))
92	89, 91	mulcncf 25566	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1)))) ∈ (ℝ⁺–cn→ℂ))
93		cncfss 25019	. . . . . . . . 9 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (ℝ⁺–cn→ℝ) ⊆ (ℝ⁺–cn→ℂ))
94	65, 71, 93	mp2an 704	. . . . . . . 8 ⊢ (ℝ⁺–cn→ℝ) ⊆ (ℝ⁺–cn→ℂ)
95		relogcn 26761	. . . . . . . . 9 ⊢ (log ↾ ℝ⁺) ∈ (ℝ⁺–cn→ℝ)
96	48, 95	eqeltrrdi 2874	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)) ∈ (ℝ⁺–cn→ℝ))
97	94, 96	sselid 3937	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)) ∈ (ℝ⁺–cn→ℂ))
98	92, 97	mulcncf 25566	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1))) · (log‘𝑥))) ∈ (ℝ⁺–cn→ℂ))
99	67, 69, 87, 98	cncfmpt2f 25035	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (((1 / 𝑥) · (𝑥↑_𝑐-(1 / 2))) + ((-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1))) · (log‘𝑥)))) ∈ (ℝ⁺–cn→ℂ))
100		rpre 13016	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
101	100, 17	rereccld 12033	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (1 / 𝑥) ∈ ℝ)
102		rpge0 13021	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → 0 ≤ 𝑥)
103		halfre 12448	. . . . . . . . . . . 12 ⊢ (1 / 2) ∈ ℝ
104	103	renegcli 11507	. . . . . . . . . . 11 ⊢ -(1 / 2) ∈ ℝ
105	104	a1i 11	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → -(1 / 2) ∈ ℝ)
106	100, 102, 105	recxpcld 26846	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥↑_𝑐-(1 / 2)) ∈ ℝ)
107	101, 106	remulcld 11227	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → ((1 / 𝑥) · (𝑥↑_𝑐-(1 / 2))) ∈ ℝ)
108		1re 11196	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ
109	104, 108	resubcli 11508	. . . . . . . . . . . 12 ⊢ (-(1 / 2) − 1) ∈ ℝ
110	109	a1i 11	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (-(1 / 2) − 1) ∈ ℝ)
111	100, 102, 110	recxpcld 26846	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥↑_𝑐(-(1 / 2) − 1)) ∈ ℝ)
112	105, 111	remulcld 11227	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1))) ∈ ℝ)
113		relogcl 26698	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
114	112, 113	remulcld 11227	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → ((-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1))) · (log‘𝑥)) ∈ ℝ)
115	107, 114	readdcld 11226	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (((1 / 𝑥) · (𝑥↑_𝑐-(1 / 2))) + ((-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1))) · (log‘𝑥))) ∈ ℝ)
116	115	adantl 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (((1 / 𝑥) · (𝑥↑_𝑐-(1 / 2))) + ((-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1))) · (log‘𝑥))) ∈ ℝ)
117	116	fmpttd 7100	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (((1 / 𝑥) · (𝑥↑_𝑐-(1 / 2))) + ((-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1))) · (log‘𝑥)))):ℝ⁺⟶ℝ)
118		cncfcdm 25018	. . . . . 6 ⊢ ((ℝ ⊆ ℂ ∧ (𝑥 ∈ ℝ⁺ ↦ (((1 / 𝑥) · (𝑥↑_𝑐-(1 / 2))) + ((-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1))) · (log‘𝑥)))) ∈ (ℝ⁺–cn→ℂ)) → ((𝑥 ∈ ℝ⁺ ↦ (((1 / 𝑥) · (𝑥↑_𝑐-(1 / 2))) + ((-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1))) · (log‘𝑥)))) ∈ (ℝ⁺–cn→ℝ) ↔ (𝑥 ∈ ℝ⁺ ↦ (((1 / 𝑥) · (𝑥↑_𝑐-(1 / 2))) + ((-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1))) · (log‘𝑥)))):ℝ⁺⟶ℝ))
119	118	biimpar 482	. . . . 5 ⊢ (((ℝ ⊆ ℂ ∧ (𝑥 ∈ ℝ⁺ ↦ (((1 / 𝑥) · (𝑥↑_𝑐-(1 / 2))) + ((-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1))) · (log‘𝑥)))) ∈ (ℝ⁺–cn→ℂ)) ∧ (𝑥 ∈ ℝ⁺ ↦ (((1 / 𝑥) · (𝑥↑_𝑐-(1 / 2))) + ((-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1))) · (log‘𝑥)))):ℝ⁺⟶ℝ) → (𝑥 ∈ ℝ⁺ ↦ (((1 / 𝑥) · (𝑥↑_𝑐-(1 / 2))) + ((-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1))) · (log‘𝑥)))) ∈ (ℝ⁺–cn→ℝ))
120	66, 99, 117, 119	syl21anc 850	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (((1 / 𝑥) · (𝑥↑_𝑐-(1 / 2))) + ((-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1))) · (log‘𝑥)))) ∈ (ℝ⁺–cn→ℝ))
121	64, 120	eqeltrd 2865	. . 3 ⊢ (𝜑 → (ℝ D (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (√‘𝑥)))) ∈ (ℝ⁺–cn→ℝ))
122		logdivsqrle.2	. . 3 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
123	64	fveq1d 6873	. . . . 5 ⊢ (𝜑 → ((ℝ D (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (√‘𝑥))))‘𝑦) = ((𝑥 ∈ ℝ⁺ ↦ (((1 / 𝑥) · (𝑥↑_𝑐-(1 / 2))) + ((-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1))) · (log‘𝑥))))‘𝑦))
124	123	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((ℝ D (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (√‘𝑥))))‘𝑦) = ((𝑥 ∈ ℝ⁺ ↦ (((1 / 𝑥) · (𝑥↑_𝑐-(1 / 2))) + ((-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1))) · (log‘𝑥))))‘𝑦))
125	57	negcld 11544	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → -1 ∈ ℂ)
126		cxpadd 26802	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) ∧ -(1 / 2) ∈ ℂ ∧ -1 ∈ ℂ) → (𝑥↑_𝑐(-(1 / 2) + -1)) = ((𝑥↑_𝑐-(1 / 2)) · (𝑥↑_𝑐-1)))
127	16, 18, 55, 125, 126	syl211anc 1399	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐(-(1 / 2) + -1)) = ((𝑥↑_𝑐-(1 / 2)) · (𝑥↑_𝑐-1)))
128	59	mullidd 11215	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1 · (𝑥↑_𝑐(-(1 / 2) − 1))) = (𝑥↑_𝑐(-(1 / 2) − 1)))
129	55, 57	negsubd 11563	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (-(1 / 2) + -1) = (-(1 / 2) − 1))
130	129	oveq2d 7416	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐(-(1 / 2) + -1)) = (𝑥↑_𝑐(-(1 / 2) − 1)))
131	128, 130	eqtr4d 2803	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1 · (𝑥↑_𝑐(-(1 / 2) − 1))) = (𝑥↑_𝑐(-(1 / 2) + -1)))
132	43, 38	sselid 3937	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ ℂ)
133	132, 56	mulcomd 11218	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((1 / 𝑥) · (𝑥↑_𝑐-(1 / 2))) = ((𝑥↑_𝑐-(1 / 2)) · (1 / 𝑥)))
134		cxpneg 26804	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0 ∧ 1 ∈ ℂ) → (𝑥↑_𝑐-1) = (1 / (𝑥↑_𝑐1)))
135	16, 18, 57, 134	syl3anc 1394	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐-1) = (1 / (𝑥↑_𝑐1)))
136	16	cxp1d 26829	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐1) = 𝑥)
137	136	oveq2d 7416	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1 / (𝑥↑_𝑐1)) = (1 / 𝑥))
138	135, 137	eqtr2d 2801	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) = (𝑥↑_𝑐-1))
139	138	oveq2d 7416	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((𝑥↑_𝑐-(1 / 2)) · (1 / 𝑥)) = ((𝑥↑_𝑐-(1 / 2)) · (𝑥↑_𝑐-1)))
140	133, 139	eqtrd 2800	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((1 / 𝑥) · (𝑥↑_𝑐-(1 / 2))) = ((𝑥↑_𝑐-(1 / 2)) · (𝑥↑_𝑐-1)))
141	127, 131, 140	3eqtr4rd 2811	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((1 / 𝑥) · (𝑥↑_𝑐-(1 / 2))) = (1 · (𝑥↑_𝑐(-(1 / 2) − 1))))
142	55, 59, 19	mul32d 11408	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1))) · (log‘𝑥)) = ((-(1 / 2) · (log‘𝑥)) · (𝑥↑_𝑐(-(1 / 2) − 1))))
143	141, 142	oveq12d 7418	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (((1 / 𝑥) · (𝑥↑_𝑐-(1 / 2))) + ((-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1))) · (log‘𝑥))) = ((1 · (𝑥↑_𝑐(-(1 / 2) − 1))) + ((-(1 / 2) · (log‘𝑥)) · (𝑥↑_𝑐(-(1 / 2) − 1)))))
144	55, 19	mulcld 11217	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (-(1 / 2) · (log‘𝑥)) ∈ ℂ)
145	57, 144, 59	adddird 11222	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((1 + (-(1 / 2) · (log‘𝑥))) · (𝑥↑_𝑐(-(1 / 2) − 1))) = ((1 · (𝑥↑_𝑐(-(1 / 2) − 1))) + ((-(1 / 2) · (log‘𝑥)) · (𝑥↑_𝑐(-(1 / 2) − 1)))))
146	143, 145	eqtr4d 2803	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (((1 / 𝑥) · (𝑥↑_𝑐-(1 / 2))) + ((-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1))) · (log‘𝑥))) = ((1 + (-(1 / 2) · (log‘𝑥))) · (𝑥↑_𝑐(-(1 / 2) − 1))))
147	146	mpteq2dva 5198	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (((1 / 𝑥) · (𝑥↑_𝑐-(1 / 2))) + ((-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1))) · (log‘𝑥)))) = (𝑥 ∈ ℝ⁺ ↦ ((1 + (-(1 / 2) · (log‘𝑥))) · (𝑥↑_𝑐(-(1 / 2) − 1)))))
148	147	fveq1d 6873	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ ℝ⁺ ↦ (((1 / 𝑥) · (𝑥↑_𝑐-(1 / 2))) + ((-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1))) · (log‘𝑥))))‘𝑦) = ((𝑥 ∈ ℝ⁺ ↦ ((1 + (-(1 / 2) · (log‘𝑥))) · (𝑥↑_𝑐(-(1 / 2) − 1))))‘𝑦))
149	148	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((𝑥 ∈ ℝ⁺ ↦ (((1 / 𝑥) · (𝑥↑_𝑐-(1 / 2))) + ((-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1))) · (log‘𝑥))))‘𝑦) = ((𝑥 ∈ ℝ⁺ ↦ ((1 + (-(1 / 2) · (log‘𝑥))) · (𝑥↑_𝑐(-(1 / 2) − 1))))‘𝑦))
150		eqidd 2766	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝑥 ∈ ℝ⁺ ↦ ((1 + (-(1 / 2) · (log‘𝑥))) · (𝑥↑_𝑐(-(1 / 2) − 1)))) = (𝑥 ∈ ℝ⁺ ↦ ((1 + (-(1 / 2) · (log‘𝑥))) · (𝑥↑_𝑐(-(1 / 2) − 1)))))
151		simpr 489	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑦) → 𝑥 = 𝑦)
152	151	fveq2d 6875	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑦) → (log‘𝑥) = (log‘𝑦))
153	152	oveq2d 7416	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑦) → (-(1 / 2) · (log‘𝑥)) = (-(1 / 2) · (log‘𝑦)))
154	153	oveq2d 7416	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑦) → (1 + (-(1 / 2) · (log‘𝑥))) = (1 + (-(1 / 2) · (log‘𝑦))))
155	151	oveq1d 7415	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑦) → (𝑥↑_𝑐(-(1 / 2) − 1)) = (𝑦↑_𝑐(-(1 / 2) − 1)))
156	154, 155	oveq12d 7418	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑦) → ((1 + (-(1 / 2) · (log‘𝑥))) · (𝑥↑_𝑐(-(1 / 2) − 1))) = ((1 + (-(1 / 2) · (log‘𝑦))) · (𝑦↑_𝑐(-(1 / 2) − 1))))
157		ioossicc 13451	. . . . . . . . . 10 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
158	157	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵))
159	2, 3, 4	fct2relem 34901	. . . . . . . . 9 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ⁺)
160	158, 159	sstrd 3949	. . . . . . . 8 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ⁺)
161	160	sselda 3939	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ ℝ⁺)
162		ovexd 7435	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((1 + (-(1 / 2) · (log‘𝑦))) · (𝑦↑_𝑐(-(1 / 2) − 1))) ∈ V)
163	150, 156, 161, 162	fvmptd 6987	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((𝑥 ∈ ℝ⁺ ↦ ((1 + (-(1 / 2) · (log‘𝑥))) · (𝑥↑_𝑐(-(1 / 2) − 1))))‘𝑦) = ((1 + (-(1 / 2) · (log‘𝑦))) · (𝑦↑_𝑐(-(1 / 2) − 1))))
164	108	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 1 ∈ ℝ)
165	104	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → -(1 / 2) ∈ ℝ)
166	161	relogcld 26746	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (log‘𝑦) ∈ ℝ)
167	165, 166	remulcld 11227	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (-(1 / 2) · (log‘𝑦)) ∈ ℝ)
168	164, 167	readdcld 11226	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (1 + (-(1 / 2) · (log‘𝑦))) ∈ ℝ)
169		0red 11199	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 0 ∈ ℝ)
170		rpcxpcl 26799	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℝ⁺ ∧ (-(1 / 2) − 1) ∈ ℝ) → (𝑦↑_𝑐(-(1 / 2) − 1)) ∈ ℝ⁺)
171	161, 109, 170	sylancl 597	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝑦↑_𝑐(-(1 / 2) − 1)) ∈ ℝ⁺)
172	171	rpred 13051	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝑦↑_𝑐(-(1 / 2) − 1)) ∈ ℝ)
173	171	rpge0d 13055	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 0 ≤ (𝑦↑_𝑐(-(1 / 2) − 1)))
174		2cn 12307	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℂ
175	174	mullidi 11202	. . . . . . . . . . . . 13 ⊢ (1 · 2) = 2
176		2re 12306	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ ℝ
177	176	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 2 ∈ ℝ)
178	177	reefcld 16132	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (exp‘2) ∈ ℝ)
179	3	rpred 13051	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐴 ∈ ℝ)
180	179	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ)
181	161	rpred 13051	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ ℝ)
182		logdivsqrle.1	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (exp‘2) ≤ 𝐴)
183	182	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (exp‘2) ≤ 𝐴)
184		eliooord 13423	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ (𝐴(,)𝐵) → (𝐴 < 𝑦 ∧ 𝑦 < 𝐵))
185	184	simpld 499	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (𝐴(,)𝐵) → 𝐴 < 𝑦)
186	185	adantl 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐴 < 𝑦)
187	180, 181, 186	ltled 11346	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐴 ≤ 𝑦)
188	178, 180, 181, 183, 187	letrd 11355	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (exp‘2) ≤ 𝑦)
189		reeflog 26703	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℝ⁺ → (exp‘(log‘𝑦)) = 𝑦)
190	161, 189	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (exp‘(log‘𝑦)) = 𝑦)
191	188, 190	breqtrrd 5133	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (exp‘2) ≤ (exp‘(log‘𝑦)))
192		efle 16164	. . . . . . . . . . . . . . 15 ⊢ ((2 ∈ ℝ ∧ (log‘𝑦) ∈ ℝ) → (2 ≤ (log‘𝑦) ↔ (exp‘2) ≤ (exp‘(log‘𝑦))))
193	176, 166, 192	sylancr 598	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (2 ≤ (log‘𝑦) ↔ (exp‘2) ≤ (exp‘(log‘𝑦))))
194	191, 193	mpbird 260	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 2 ≤ (log‘𝑦))
195	175, 194	eqbrtrid 5140	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (1 · 2) ≤ (log‘𝑦))
196		2rp 13012	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℝ⁺
197	196	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 2 ∈ ℝ⁺)
198	164, 166, 197	lemuldivd 13100	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((1 · 2) ≤ (log‘𝑦) ↔ 1 ≤ ((log‘𝑦) / 2)))
199	195, 198	mpbid 235	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 1 ≤ ((log‘𝑦) / 2))
200	65, 166	sselid 3937	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (log‘𝑦) ∈ ℂ)
201	22	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 2 ∈ ℂ)
202	24	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 2 ≠ 0)
203	200, 201, 202	divrec2d 11986	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((log‘𝑦) / 2) = ((1 / 2) · (log‘𝑦)))
204	199, 203	breqtrd 5131	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 1 ≤ ((1 / 2) · (log‘𝑦)))
205	53	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (1 / 2) ∈ ℂ)
206	205, 200	mulneg1d 11655	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (-(1 / 2) · (log‘𝑦)) = -((1 / 2) · (log‘𝑦)))
207	206	oveq2d 7416	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (0 − (-(1 / 2) · (log‘𝑦))) = (0 − -((1 / 2) · (log‘𝑦))))
208	65, 169	sselid 3937	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 0 ∈ ℂ)
209	205, 200	mulcld 11217	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((1 / 2) · (log‘𝑦)) ∈ ℂ)
210	208, 209	subnegd 11564	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (0 − -((1 / 2) · (log‘𝑦))) = (0 + ((1 / 2) · (log‘𝑦))))
211	209	addlidd 11399	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (0 + ((1 / 2) · (log‘𝑦))) = ((1 / 2) · (log‘𝑦)))
212	207, 210, 211	3eqtrd 2804	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (0 − (-(1 / 2) · (log‘𝑦))) = ((1 / 2) · (log‘𝑦)))
213	204, 212	breqtrrd 5133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 1 ≤ (0 − (-(1 / 2) · (log‘𝑦))))
214		leaddsub 11678	. . . . . . . . . 10 ⊢ ((1 ∈ ℝ ∧ (-(1 / 2) · (log‘𝑦)) ∈ ℝ ∧ 0 ∈ ℝ) → ((1 + (-(1 / 2) · (log‘𝑦))) ≤ 0 ↔ 1 ≤ (0 − (-(1 / 2) · (log‘𝑦)))))
215	164, 167, 169, 214	syl3anc 1394	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((1 + (-(1 / 2) · (log‘𝑦))) ≤ 0 ↔ 1 ≤ (0 − (-(1 / 2) · (log‘𝑦)))))
216	213, 215	mpbird 260	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (1 + (-(1 / 2) · (log‘𝑦))) ≤ 0)
217	168, 169, 172, 173, 216	lemul1ad 12145	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((1 + (-(1 / 2) · (log‘𝑦))) · (𝑦↑_𝑐(-(1 / 2) − 1))) ≤ (0 · (𝑦↑_𝑐(-(1 / 2) − 1))))
218	43, 171	sselid 3937	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝑦↑_𝑐(-(1 / 2) − 1)) ∈ ℂ)
219	218	mul02d 11396	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (0 · (𝑦↑_𝑐(-(1 / 2) − 1))) = 0)
220	217, 219	breqtrd 5131	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((1 + (-(1 / 2) · (log‘𝑦))) · (𝑦↑_𝑐(-(1 / 2) − 1))) ≤ 0)
221	163, 220	eqbrtrd 5127	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((𝑥 ∈ ℝ⁺ ↦ ((1 + (-(1 / 2) · (log‘𝑥))) · (𝑥↑_𝑐(-(1 / 2) − 1))))‘𝑦) ≤ 0)
222	149, 221	eqbrtrd 5127	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((𝑥 ∈ ℝ⁺ ↦ (((1 / 𝑥) · (𝑥↑_𝑐-(1 / 2))) + ((-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1))) · (log‘𝑥))))‘𝑦) ≤ 0)
223	124, 222	eqbrtrd 5127	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((ℝ D (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (√‘𝑥))))‘𝑦) ≤ 0)
224	2, 3, 4, 14, 121, 122, 223	fdvnegge 34906	. 2 ⊢ (𝜑 → ((𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (√‘𝑥)))‘𝐵) ≤ ((𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (√‘𝑥)))‘𝐴))
225		eqidd 2766	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (√‘𝑥))) = (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (√‘𝑥))))
226		simpr 489	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝑥 = 𝐵)
227	226	fveq2d 6875	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → (log‘𝑥) = (log‘𝐵))
228	226	fveq2d 6875	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → (√‘𝑥) = (√‘𝐵))
229	227, 228	oveq12d 7418	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → ((log‘𝑥) / (√‘𝑥)) = ((log‘𝐵) / (√‘𝐵)))
230		ovex 7433	. . . 4 ⊢ ((log‘𝐵) / (√‘𝐵)) ∈ V
231	230	a1i 11	. . 3 ⊢ (𝜑 → ((log‘𝐵) / (√‘𝐵)) ∈ V)
232	225, 229, 4, 231	fvmptd 6987	. 2 ⊢ (𝜑 → ((𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (√‘𝑥)))‘𝐵) = ((log‘𝐵) / (√‘𝐵)))
233		simpr 489	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝑥 = 𝐴)
234	233	fveq2d 6875	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (log‘𝑥) = (log‘𝐴))
235	233	fveq2d 6875	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (√‘𝑥) = (√‘𝐴))
236	234, 235	oveq12d 7418	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((log‘𝑥) / (√‘𝑥)) = ((log‘𝐴) / (√‘𝐴)))
237		ovex 7433	. . . 4 ⊢ ((log‘𝐴) / (√‘𝐴)) ∈ V
238	237	a1i 11	. . 3 ⊢ (𝜑 → ((log‘𝐴) / (√‘𝐴)) ∈ V)
239	225, 236, 3, 238	fvmptd 6987	. 2 ⊢ (𝜑 → ((𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (√‘𝑥)))‘𝐴) = ((log‘𝐴) / (√‘𝐴)))
240	224, 232, 239	3brtr3d 5136	1 ⊢ (𝜑 → ((log‘𝐵) / (√‘𝐵)) ≤ ((log‘𝐴) / (√‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1563 ∈ wcel 2145 ≠ wne 2960 Vcvv 3457 ∖ cdif 3904 ⊆ wss 3907 {csn 4585 {cpr 4587 class class class wbr 5105 ↦ cmpt 5186 ran crn 5653 ↾ cres 5654 ⟶wf 6521 –1-1-onto→wf1o 6524 ‘cfv 6525 (class class class)co 7400 ℂcc 11086 ℝcr 11087 0cc0 11088 1c1 11089 + caddc 11091 · cmul 11093 +∞cpnf 11228 -∞cmnf 11229 < clt 11231 ≤ cle 11232 − cmin 11429 -cneg 11430 / cdiv 11859 2c2 12286 ℝ⁺crp 13007 (,)cioo 13363 (,]cioc 13364 [,]cicc 13366 √csqrt 15274 expce 16105 TopOpenctopn 17464 ℂ_fldccnfld 21482 Cn ccn 23342 ×_t ctx 23678 –cn→ccncf 24996 D cdv 25983 logclog 26677 ↑_𝑐ccxp 26678

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5232 ax-sep 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 ax-un 7722 ax-inf2 9598 ax-cc 10407 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165 ax-pre-sup 11166 ax-addf 11167

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-symdif 4208 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4869 df-int 4909 df-iun 4954 df-iin 4955 df-disj 5073 df-br 5106 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5547 df-eprel 5552 df-po 5560 df-so 5561 df-fr 5605 df-se 5606 df-we 5607 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6292 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-isom 6534 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-of 7664 df-ofr 7665 df-om 7851 df-1st 7974 df-2nd 7975 df-supp 8145 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-1o 8441 df-2o 8442 df-oadd 8445 df-omul 8446 df-er 8682 df-map 8814 df-pm 8815 df-ixp 8884 df-en 8932 df-dom 8933 df-sdom 8934 df-fin 8935 df-fsupp 9310 df-fi 9359 df-sup 9390 df-inf 9391 df-oi 9460 df-dju 9875 df-card 9913 df-acn 9916 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-div 11860 df-nn 12225 df-2 12294 df-3 12295 df-4 12296 df-5 12297 df-6 12298 df-7 12299 df-8 12300 df-9 12301 df-n0 12496 df-z 12583 df-dec 12703 df-uz 12854 df-q 12964 df-rp 13008 df-xneg 13128 df-xadd 13129 df-xmul 13130 df-ioo 13367 df-ioc 13368 df-ico 13369 df-icc 13370 df-fz 13527 df-fzo 13674 df-fl 13816 df-mod 13894 df-seq 14029 df-exp 14089 df-fac 14301 df-bc 14330 df-hash 14358 df-shft 15094 df-cj 15140 df-re 15141 df-im 15142 df-sqrt 15276 df-abs 15277 df-limsup 15512 df-clim 15529 df-rlim 15530 df-sum 15728 df-ef 16111 df-sin 16113 df-cos 16114 df-tan 16115 df-pi 16116 df-struct 17197 df-sets 17214 df-slot 17232 df-ndx 17244 df-base 17260 df-ress 17281 df-plusg 17313 df-mulr 17314 df-starv 17315 df-sca 17316 df-vsca 17317 df-ip 17318 df-tset 17319 df-ple 17320 df-ds 17322 df-unif 17323 df-hom 17324 df-cco 17325 df-rest 17465 df-topn 17466 df-0g 17484 df-gsum 17485 df-topgen 17486 df-pt 17487 df-prds 17490 df-xrs 17546 df-qtop 17551 df-imas 17552 df-xps 17554 df-mre 17628 df-mrc 17629 df-acs 17631 df-mgm 18688 df-sgrp 18767 df-mnd 18783 df-submnd 18832 df-mulg 19125 df-cntz 19378 df-cmn 19843 df-psmet 21474 df-xmet 21475 df-met 21476 df-bl 21477 df-mopn 21478 df-fbas 21479 df-fg 21480 df-cnfld 21483 df-top 23012 df-topon 23029 df-topsp 23051 df-bases 23064 df-cld 23137 df-ntr 23138 df-cls 23139 df-nei 23216 df-lp 23254 df-perf 23255 df-cn 23345 df-cnp 23346 df-haus 23433 df-cmp 23505 df-tx 23680 df-hmeo 23873 df-fil 23964 df-fm 24056 df-flim 24057 df-flf 24058 df-xms 24438 df-ms 24439 df-tms 24440 df-cncf 24998 df-ovol 25584 df-vol 25585 df-mbf 25739 df-itg1 25740 df-itg2 25741 df-ibl 25742 df-itg 25743 df-0p 25790 df-limc 25986 df-dv 25987 df-log 26679 df-cxp 26680

This theorem is referenced by: hgt750lem 34955

W3C validator