logdivsqrle - Mathbox for Thierry Arnoux

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Theorem logdivsqrle 34413

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 13437	. . . 4 ⊢ (0(,)+∞) = ℝ⁺
2	1	eqcomi 2734	. . 3 ⊢ ℝ⁺ = (0(,)+∞)
3		logdivsqrle.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
4		logdivsqrle.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
5		simpr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ⁺)
6	5	relogcld 26602	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℝ)
7	5	rpsqrtcld 15394	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (√‘𝑥) ∈ ℝ⁺)
8	7	rpred 13051	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (√‘𝑥) ∈ ℝ)
9		rpsqrtcl 15247	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (√‘𝑥) ∈ ℝ⁺)
10		rpne0 13025	. . . . . . 7 ⊢ ((√‘𝑥) ∈ ℝ⁺ → (√‘𝑥) ≠ 0)
11	9, 10	syl 17	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (√‘𝑥) ≠ 0)
12	11	adantl 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (√‘𝑥) ≠ 0)
13	6, 8, 12	redivcld 12075	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) / (√‘𝑥)) ∈ ℝ)
14	13	fmpttd 7124	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (√‘𝑥))):ℝ⁺⟶ℝ)
15		rpcn 13019	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
16	15	adantl 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℂ)
17		rpne0 13025	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≠ 0)
18	17	adantl 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ≠ 0)
19	16, 18	logcld 26549	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℂ)
20	16	sqrtcld 15420	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (√‘𝑥) ∈ ℂ)
21	19, 20, 12	divrecd 12026	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) / (√‘𝑥)) = ((log‘𝑥) · (1 / (√‘𝑥))))
22		2cnd 12323	. . . . . . . . . . . . 13 ⊢ (𝜑 → 2 ∈ ℂ)
23	22	adantr 479	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 2 ∈ ℂ)
24		2ne0 12349	. . . . . . . . . . . . 13 ⊢ 2 ≠ 0
25	24	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 2 ≠ 0)
26	23, 25	reccld 12016	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1 / 2) ∈ ℂ)
27	16, 18, 26	cxpnegd 26694	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐-(1 / 2)) = (1 / (𝑥↑_𝑐(1 / 2))))
28		cxpsqrt 26682	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ → (𝑥↑_𝑐(1 / 2)) = (√‘𝑥))
29	16, 28	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐(1 / 2)) = (√‘𝑥))
30	29	oveq2d 7435	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1 / (𝑥↑_𝑐(1 / 2))) = (1 / (√‘𝑥)))
31	27, 30	eqtrd 2765	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐-(1 / 2)) = (1 / (√‘𝑥)))
32	31	oveq2d 7435	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) · (𝑥↑_𝑐-(1 / 2))) = ((log‘𝑥) · (1 / (√‘𝑥))))
33	21, 32	eqtr4d 2768	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) / (√‘𝑥)) = ((log‘𝑥) · (𝑥↑_𝑐-(1 / 2))))
34	33	mpteq2dva 5249	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (√‘𝑥))) = (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) · (𝑥↑_𝑐-(1 / 2)))))
35	34	oveq2d 7435	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (√‘𝑥)))) = (ℝ D (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) · (𝑥↑_𝑐-(1 / 2))))))
36		reelprrecn 11232	. . . . . . 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 26543	. . . . . . . . . . 11 ⊢ log:(ℂ ∖ {0})–1-1-onto→ran log
40		f1of 6838	. . . . . . . . . . 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 3980	. . . . . . . . . . 11 ⊢ ℝ⁺ ⊆ ℂ
44		0nrp 13044	. . . . . . . . . . 11 ⊢ ¬ 0 ∈ ℝ⁺
45		ssdifsn 4793	. . . . . . . . . . 11 ⊢ (ℝ⁺ ⊆ (ℂ ∖ {0}) ↔ (ℝ⁺ ⊆ ℂ ∧ ¬ 0 ∈ ℝ⁺))
46	43, 44, 45	mpbir2an 709	. . . . . . . . . 10 ⊢ ℝ⁺ ⊆ (ℂ ∖ {0})
47	46	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ℝ⁺ ⊆ (ℂ ∖ {0}))
48	42, 47	feqresmpt 6967	. . . . . . . 8 ⊢ (𝜑 → (log ↾ ℝ⁺) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)))
49	48	oveq2d 7435	. . . . . . 7 ⊢ (𝜑 → (ℝ D (log ↾ ℝ⁺)) = (ℝ D (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))))
50		dvrelog 26616	. . . . . . 7 ⊢ (ℝ D (log ↾ ℝ⁺)) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥))
51	49, 50	eqtr3di 2780	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)))
52		1cnd 11241	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℂ)
53	52	halfcld 12490	. . . . . . . . 9 ⊢ (𝜑 → (1 / 2) ∈ ℂ)
54	53	negcld 11590	. . . . . . . 8 ⊢ (𝜑 → -(1 / 2) ∈ ℂ)
55	54	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → -(1 / 2) ∈ ℂ)
56	16, 55	cxpcld 26687	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐-(1 / 2)) ∈ ℂ)
57	52	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 1 ∈ ℂ)
58	55, 57	subcld 11603	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (-(1 / 2) − 1) ∈ ℂ)
59	16, 58	cxpcld 26687	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐(-(1 / 2) − 1)) ∈ ℂ)
60	55, 59	mulcld 11266	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1))) ∈ ℂ)
61		dvcxp1 26719	. . . . . . 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 17	. . . . . 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 25937	. . . . 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 2765	. . . 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 11197	. . . . . 6 ⊢ ℝ ⊆ ℂ
66	65	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ⊆ ℂ)
67		eqid 2725	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
68	67	addcn 24825	. . . . . . 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 3999	. . . . . . . . . 10 ⊢ ℂ ⊆ ℂ
72	71	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ℂ ⊆ ℂ)
73		cncfmptc 24876	. . . . . . . . 9 ⊢ ((1 ∈ ℂ ∧ ℝ⁺ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ ℝ⁺ ↦ 1) ∈ (ℝ⁺–cn→ℂ))
74	52, 70, 72, 73	syl3anc 1368	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ 1) ∈ (ℝ⁺–cn→ℂ))
75		difss 4128	. . . . . . . . 9 ⊢ (ℂ ∖ {0}) ⊆ ℂ
76		cncfmptid 24877	. . . . . . . . 9 ⊢ ((ℝ⁺ ⊆ (ℂ ∖ {0}) ∧ (ℂ ∖ {0}) ⊆ ℂ) → (𝑥 ∈ ℝ⁺ ↦ 𝑥) ∈ (ℝ⁺–cn→(ℂ ∖ {0})))
77	47, 75, 76	sylancl 584	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ 𝑥) ∈ (ℝ⁺–cn→(ℂ ∖ {0})))
78	74, 77	divcncf 25420	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)) ∈ (ℝ⁺–cn→ℂ))
79		ax-1 6	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ → 𝑥 ∈ ℝ⁺))
80	15, 79	jca 510	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℂ ∧ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ⁺)))
81		eqid 2725	. . . . . . . . . . . 12 ⊢ (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0))
82	81	ellogdm 26618	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (ℂ ∖ (-∞(,]0)) ↔ (𝑥 ∈ ℂ ∧ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ⁺)))
83	80, 82	sylibr 233	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ (ℂ ∖ (-∞(,]0)))
84	83	ssriv 3980	. . . . . . . . 9 ⊢ ℝ⁺ ⊆ (ℂ ∖ (-∞(,]0))
85	84	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ⁺ ⊆ (ℂ ∖ (-∞(,]0)))
86	54, 85	cxpcncf1 34358	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (𝑥↑_𝑐-(1 / 2))) ∈ (ℝ⁺–cn→ℂ))
87	78, 86	mulcncf 25418	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((1 / 𝑥) · (𝑥↑_𝑐-(1 / 2)))) ∈ (ℝ⁺–cn→ℂ))
88		cncfmptc 24876	. . . . . . . . 9 ⊢ ((-(1 / 2) ∈ ℂ ∧ ℝ⁺ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ ℝ⁺ ↦ -(1 / 2)) ∈ (ℝ⁺–cn→ℂ))
89	54, 70, 72, 88	syl3anc 1368	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ -(1 / 2)) ∈ (ℝ⁺–cn→ℂ))
90	54, 52	subcld 11603	. . . . . . . . 9 ⊢ (𝜑 → (-(1 / 2) − 1) ∈ ℂ)
91	90, 85	cxpcncf1 34358	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (𝑥↑_𝑐(-(1 / 2) − 1))) ∈ (ℝ⁺–cn→ℂ))
92	89, 91	mulcncf 25418	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1)))) ∈ (ℝ⁺–cn→ℂ))
93		cncfss 24863	. . . . . . . . 9 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (ℝ⁺–cn→ℝ) ⊆ (ℝ⁺–cn→ℂ))
94	65, 71, 93	mp2an 690	. . . . . . . 8 ⊢ (ℝ⁺–cn→ℝ) ⊆ (ℝ⁺–cn→ℂ)
95		relogcn 26617	. . . . . . . . 9 ⊢ (log ↾ ℝ⁺) ∈ (ℝ⁺–cn→ℝ)
96	48, 95	eqeltrrdi 2834	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)) ∈ (ℝ⁺–cn→ℝ))
97	94, 96	sselid 3974	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)) ∈ (ℝ⁺–cn→ℂ))
98	92, 97	mulcncf 25418	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1))) · (log‘𝑥))) ∈ (ℝ⁺–cn→ℂ))
99	67, 69, 87, 98	cncfmpt2f 24879	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (((1 / 𝑥) · (𝑥↑_𝑐-(1 / 2))) + ((-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1))) · (log‘𝑥)))) ∈ (ℝ⁺–cn→ℂ))
100		rpre 13017	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
101	100, 17	rereccld 12074	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (1 / 𝑥) ∈ ℝ)
102		rpge0 13022	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → 0 ≤ 𝑥)
103		halfre 12459	. . . . . . . . . . . 12 ⊢ (1 / 2) ∈ ℝ
104	103	renegcli 11553	. . . . . . . . . . 11 ⊢ -(1 / 2) ∈ ℝ
105	104	a1i 11	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → -(1 / 2) ∈ ℝ)
106	100, 102, 105	recxpcld 26702	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥↑_𝑐-(1 / 2)) ∈ ℝ)
107	101, 106	remulcld 11276	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → ((1 / 𝑥) · (𝑥↑_𝑐-(1 / 2))) ∈ ℝ)
108		1re 11246	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ
109	104, 108	resubcli 11554	. . . . . . . . . . . 12 ⊢ (-(1 / 2) − 1) ∈ ℝ
110	109	a1i 11	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (-(1 / 2) − 1) ∈ ℝ)
111	100, 102, 110	recxpcld 26702	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥↑_𝑐(-(1 / 2) − 1)) ∈ ℝ)
112	105, 111	remulcld 11276	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1))) ∈ ℝ)
113		relogcl 26554	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
114	112, 113	remulcld 11276	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → ((-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1))) · (log‘𝑥)) ∈ ℝ)
115	107, 114	readdcld 11275	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (((1 / 𝑥) · (𝑥↑_𝑐-(1 / 2))) + ((-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1))) · (log‘𝑥))) ∈ ℝ)
116	115	adantl 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (((1 / 𝑥) · (𝑥↑_𝑐-(1 / 2))) + ((-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1))) · (log‘𝑥))) ∈ ℝ)
117	116	fmpttd 7124	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (((1 / 𝑥) · (𝑥↑_𝑐-(1 / 2))) + ((-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1))) · (log‘𝑥)))):ℝ⁺⟶ℝ)
118		cncfcdm 24862	. . . . . 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 476	. . . . 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 836	. . . 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 2825	. . 3 ⊢ (𝜑 → (ℝ D (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (√‘𝑥)))) ∈ (ℝ⁺–cn→ℝ))
122		logdivsqrle.2	. . 3 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
123	64	fveq1d 6898	. . . . 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 479	. . . 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 11590	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → -1 ∈ ℂ)
126		cxpadd 26658	. . . . . . . . . . . 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 1373	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐(-(1 / 2) + -1)) = ((𝑥↑_𝑐-(1 / 2)) · (𝑥↑_𝑐-1)))
128	59	mullidd 11264	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1 · (𝑥↑_𝑐(-(1 / 2) − 1))) = (𝑥↑_𝑐(-(1 / 2) − 1)))
129	55, 57	negsubd 11609	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (-(1 / 2) + -1) = (-(1 / 2) − 1))
130	129	oveq2d 7435	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐(-(1 / 2) + -1)) = (𝑥↑_𝑐(-(1 / 2) − 1)))
131	128, 130	eqtr4d 2768	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1 · (𝑥↑_𝑐(-(1 / 2) − 1))) = (𝑥↑_𝑐(-(1 / 2) + -1)))
132	43, 38	sselid 3974	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ ℂ)
133	132, 56	mulcomd 11267	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((1 / 𝑥) · (𝑥↑_𝑐-(1 / 2))) = ((𝑥↑_𝑐-(1 / 2)) · (1 / 𝑥)))
134		cxpneg 26660	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0 ∧ 1 ∈ ℂ) → (𝑥↑_𝑐-1) = (1 / (𝑥↑_𝑐1)))
135	16, 18, 57, 134	syl3anc 1368	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐-1) = (1 / (𝑥↑_𝑐1)))
136	16	cxp1d 26685	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐1) = 𝑥)
137	136	oveq2d 7435	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1 / (𝑥↑_𝑐1)) = (1 / 𝑥))
138	135, 137	eqtr2d 2766	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) = (𝑥↑_𝑐-1))
139	138	oveq2d 7435	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((𝑥↑_𝑐-(1 / 2)) · (1 / 𝑥)) = ((𝑥↑_𝑐-(1 / 2)) · (𝑥↑_𝑐-1)))
140	133, 139	eqtrd 2765	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((1 / 𝑥) · (𝑥↑_𝑐-(1 / 2))) = ((𝑥↑_𝑐-(1 / 2)) · (𝑥↑_𝑐-1)))
141	127, 131, 140	3eqtr4rd 2776	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((1 / 𝑥) · (𝑥↑_𝑐-(1 / 2))) = (1 · (𝑥↑_𝑐(-(1 / 2) − 1))))
142	55, 59, 19	mul32d 11456	. . . . . . . . . 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 7437	. . . . . . . . 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 11266	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (-(1 / 2) · (log‘𝑥)) ∈ ℂ)
145	57, 144, 59	adddird 11271	. . . . . . . . 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 2768	. . . . . . . 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 5249	. . . . . . 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 6898	. . . . . 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 479	. . . . 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 2726	. . . . . . 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 483	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑦) → 𝑥 = 𝑦)
152	151	fveq2d 6900	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑦) → (log‘𝑥) = (log‘𝑦))
153	152	oveq2d 7435	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑦) → (-(1 / 2) · (log‘𝑥)) = (-(1 / 2) · (log‘𝑦)))
154	153	oveq2d 7435	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑦) → (1 + (-(1 / 2) · (log‘𝑥))) = (1 + (-(1 / 2) · (log‘𝑦))))
155	151	oveq1d 7434	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑦) → (𝑥↑_𝑐(-(1 / 2) − 1)) = (𝑦↑_𝑐(-(1 / 2) − 1)))
156	154, 155	oveq12d 7437	. . . . . . 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 13445	. . . . . . . . . 10 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
158	157	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵))
159	2, 3, 4	fct2relem 34360	. . . . . . . . 9 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ⁺)
160	158, 159	sstrd 3987	. . . . . . . 8 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ⁺)
161	160	sselda 3976	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ ℝ⁺)
162		ovexd 7454	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((1 + (-(1 / 2) · (log‘𝑦))) · (𝑦↑_𝑐(-(1 / 2) − 1))) ∈ V)
163	150, 156, 161, 162	fvmptd 7011	. . . . . 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 26602	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (log‘𝑦) ∈ ℝ)
167	165, 166	remulcld 11276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (-(1 / 2) · (log‘𝑦)) ∈ ℝ)
168	164, 167	readdcld 11275	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (1 + (-(1 / 2) · (log‘𝑦))) ∈ ℝ)
169		0red 11249	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 0 ∈ ℝ)
170		rpcxpcl 26655	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℝ⁺ ∧ (-(1 / 2) − 1) ∈ ℝ) → (𝑦↑_𝑐(-(1 / 2) − 1)) ∈ ℝ⁺)
171	161, 109, 170	sylancl 584	. . . . . . . . 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 12320	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℂ
175	174	mullidi 11251	. . . . . . . . . . . . 13 ⊢ (1 · 2) = 2
176		2re 12319	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ ℝ
177	176	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 2 ∈ ℝ)
178	177	reefcld 16068	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (exp‘2) ∈ ℝ)
179	3	rpred 13051	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐴 ∈ ℝ)
180	179	adantr 479	. . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (exp‘2) ≤ 𝐴)
184		eliooord 13418	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ (𝐴(,)𝐵) → (𝐴 < 𝑦 ∧ 𝑦 < 𝐵))
185	184	simpld 493	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (𝐴(,)𝐵) → 𝐴 < 𝑦)
186	185	adantl 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐴 < 𝑦)
187	180, 181, 186	ltled 11394	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐴 ≤ 𝑦)
188	178, 180, 181, 183, 187	letrd 11403	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (exp‘2) ≤ 𝑦)
189		reeflog 26559	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℝ⁺ → (exp‘(log‘𝑦)) = 𝑦)
190	161, 189	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (exp‘(log‘𝑦)) = 𝑦)
191	188, 190	breqtrrd 5177	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (exp‘2) ≤ (exp‘(log‘𝑦)))
192		efle 16098	. . . . . . . . . . . . . . 15 ⊢ ((2 ∈ ℝ ∧ (log‘𝑦) ∈ ℝ) → (2 ≤ (log‘𝑦) ↔ (exp‘2) ≤ (exp‘(log‘𝑦))))
193	176, 166, 192	sylancr 585	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (2 ≤ (log‘𝑦) ↔ (exp‘2) ≤ (exp‘(log‘𝑦))))
194	191, 193	mpbird 256	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 2 ≤ (log‘𝑦))
195	175, 194	eqbrtrid 5184	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (1 · 2) ≤ (log‘𝑦))
196		2rp 13014	. . . . . . . . . . . . . 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 231	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 1 ≤ ((log‘𝑦) / 2))
200	65, 166	sselid 3974	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (log‘𝑦) ∈ ℂ)
201	22	adantr 479	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 2 ∈ ℂ)
202	24	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 2 ≠ 0)
203	200, 201, 202	divrec2d 12027	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((log‘𝑦) / 2) = ((1 / 2) · (log‘𝑦)))
204	199, 203	breqtrd 5175	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 1 ≤ ((1 / 2) · (log‘𝑦)))
205	53	adantr 479	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (1 / 2) ∈ ℂ)
206	205, 200	mulneg1d 11699	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (-(1 / 2) · (log‘𝑦)) = -((1 / 2) · (log‘𝑦)))
207	206	oveq2d 7435	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (0 − (-(1 / 2) · (log‘𝑦))) = (0 − -((1 / 2) · (log‘𝑦))))
208	65, 169	sselid 3974	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 0 ∈ ℂ)
209	205, 200	mulcld 11266	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((1 / 2) · (log‘𝑦)) ∈ ℂ)
210	208, 209	subnegd 11610	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (0 − -((1 / 2) · (log‘𝑦))) = (0 + ((1 / 2) · (log‘𝑦))))
211	209	addlidd 11447	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (0 + ((1 / 2) · (log‘𝑦))) = ((1 / 2) · (log‘𝑦)))
212	207, 210, 211	3eqtrd 2769	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (0 − (-(1 / 2) · (log‘𝑦))) = ((1 / 2) · (log‘𝑦)))
213	204, 212	breqtrrd 5177	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 1 ≤ (0 − (-(1 / 2) · (log‘𝑦))))
214		leaddsub 11722	. . . . . . . . . 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 1368	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((1 + (-(1 / 2) · (log‘𝑦))) ≤ 0 ↔ 1 ≤ (0 − (-(1 / 2) · (log‘𝑦)))))
216	213, 215	mpbird 256	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (1 + (-(1 / 2) · (log‘𝑦))) ≤ 0)
217	168, 169, 172, 173, 216	lemul1ad 12186	. . . . . . 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 3974	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝑦↑_𝑐(-(1 / 2) − 1)) ∈ ℂ)
219	218	mul02d 11444	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (0 · (𝑦↑_𝑐(-(1 / 2) − 1))) = 0)
220	217, 219	breqtrd 5175	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((1 + (-(1 / 2) · (log‘𝑦))) · (𝑦↑_𝑐(-(1 / 2) − 1))) ≤ 0)
221	163, 220	eqbrtrd 5171	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((𝑥 ∈ ℝ⁺ ↦ ((1 + (-(1 / 2) · (log‘𝑥))) · (𝑥↑_𝑐(-(1 / 2) − 1))))‘𝑦) ≤ 0)
222	149, 221	eqbrtrd 5171	. . . 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 5171	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((ℝ D (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (√‘𝑥))))‘𝑦) ≤ 0)
224	2, 3, 4, 14, 121, 122, 223	fdvnegge 34365	. 2 ⊢ (𝜑 → ((𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (√‘𝑥)))‘𝐵) ≤ ((𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (√‘𝑥)))‘𝐴))
225		eqidd 2726	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (√‘𝑥))) = (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (√‘𝑥))))
226		simpr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝑥 = 𝐵)
227	226	fveq2d 6900	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → (log‘𝑥) = (log‘𝐵))
228	226	fveq2d 6900	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → (√‘𝑥) = (√‘𝐵))
229	227, 228	oveq12d 7437	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → ((log‘𝑥) / (√‘𝑥)) = ((log‘𝐵) / (√‘𝐵)))
230		ovex 7452	. . . 4 ⊢ ((log‘𝐵) / (√‘𝐵)) ∈ V
231	230	a1i 11	. . 3 ⊢ (𝜑 → ((log‘𝐵) / (√‘𝐵)) ∈ V)
232	225, 229, 4, 231	fvmptd 7011	. 2 ⊢ (𝜑 → ((𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (√‘𝑥)))‘𝐵) = ((log‘𝐵) / (√‘𝐵)))
233		simpr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝑥 = 𝐴)
234	233	fveq2d 6900	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (log‘𝑥) = (log‘𝐴))
235	233	fveq2d 6900	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (√‘𝑥) = (√‘𝐴))
236	234, 235	oveq12d 7437	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((log‘𝑥) / (√‘𝑥)) = ((log‘𝐴) / (√‘𝐴)))
237		ovex 7452	. . . 4 ⊢ ((log‘𝐴) / (√‘𝐴)) ∈ V
238	237	a1i 11	. . 3 ⊢ (𝜑 → ((log‘𝐴) / (√‘𝐴)) ∈ V)
239	225, 236, 3, 238	fvmptd 7011	. 2 ⊢ (𝜑 → ((𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (√‘𝑥)))‘𝐴) = ((log‘𝐴) / (√‘𝐴)))
240	224, 232, 239	3brtr3d 5180	1 ⊢ (𝜑 → ((log‘𝐵) / (√‘𝐵)) ≤ ((log‘𝐴) / (√‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 Vcvv 3461 ∖ cdif 3941 ⊆ wss 3944 {csn 4630 {cpr 4632 class class class wbr 5149 ↦ cmpt 5232 ran crn 5679 ↾ cres 5680 ⟶wf 6545 –1-1-onto→wf1o 6548 ‘cfv 6549 (class class class)co 7419 ℂcc 11138 ℝcr 11139 0cc0 11140 1c1 11141 + caddc 11143 · cmul 11145 +∞cpnf 11277 -∞cmnf 11278 < clt 11280 ≤ cle 11281 − cmin 11476 -cneg 11477 / cdiv 11903 2c2 12300 ℝ⁺crp 13009 (,)cioo 13359 (,]cioc 13360 [,]cicc 13362 √csqrt 15216 expce 16041 TopOpenctopn 17406 ℂ_fldccnfld 21296 Cn ccn 23172 ×_t ctx 23508 –cn→ccncf 24840 D cdv 25836 logclog 26533 ↑_𝑐ccxp 26534

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-inf2 9666 ax-cc 10460 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 ax-addf 11219

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-symdif 4241 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-disj 5115 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-ofr 7686 df-om 7872 df-1st 7994 df-2nd 7995 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-oadd 8491 df-omul 8492 df-er 8725 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9388 df-fi 9436 df-sup 9467 df-inf 9468 df-oi 9535 df-dju 9926 df-card 9964 df-acn 9967 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-z 12592 df-dec 12711 df-uz 12856 df-q 12966 df-rp 13010 df-xneg 13127 df-xadd 13128 df-xmul 13129 df-ioo 13363 df-ioc 13364 df-ico 13365 df-icc 13366 df-fz 13520 df-fzo 13663 df-fl 13793 df-mod 13871 df-seq 14003 df-exp 14063 df-fac 14269 df-bc 14298 df-hash 14326 df-shft 15050 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 df-limsup 15451 df-clim 15468 df-rlim 15469 df-sum 15669 df-ef 16047 df-sin 16049 df-cos 16050 df-tan 16051 df-pi 16052 df-struct 17119 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-mulr 17250 df-starv 17251 df-sca 17252 df-vsca 17253 df-ip 17254 df-tset 17255 df-ple 17256 df-ds 17258 df-unif 17259 df-hom 17260 df-cco 17261 df-rest 17407 df-topn 17408 df-0g 17426 df-gsum 17427 df-topgen 17428 df-pt 17429 df-prds 17432 df-xrs 17487 df-qtop 17492 df-imas 17493 df-xps 17495 df-mre 17569 df-mrc 17570 df-acs 17572 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-submnd 18744 df-mulg 19032 df-cntz 19280 df-cmn 19749 df-psmet 21288 df-xmet 21289 df-met 21290 df-bl 21291 df-mopn 21292 df-fbas 21293 df-fg 21294 df-cnfld 21297 df-top 22840 df-topon 22857 df-topsp 22879 df-bases 22893 df-cld 22967 df-ntr 22968 df-cls 22969 df-nei 23046 df-lp 23084 df-perf 23085 df-cn 23175 df-cnp 23176 df-haus 23263 df-cmp 23335 df-tx 23510 df-hmeo 23703 df-fil 23794 df-fm 23886 df-flim 23887 df-flf 23888 df-xms 24270 df-ms 24271 df-tms 24272 df-cncf 24842 df-ovol 25437 df-vol 25438 df-mbf 25592 df-itg1 25593 df-itg2 25594 df-ibl 25595 df-itg 25596 df-0p 25643 df-limc 25839 df-dv 25840 df-log 26535 df-cxp 26536

This theorem is referenced by: hgt750lem 34414

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