logdivsqrle - Mathbox for Thierry Arnoux

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Theorem logdivsqrle 34827

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 13353	. . . 4 ⊢ (0(,)+∞) = ℝ⁺
2	1	eqcomi 2746	. . 3 ⊢ ℝ⁺ = (0(,)+∞)
3		logdivsqrle.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
4		logdivsqrle.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
5		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ⁺)
6	5	relogcld 26600	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℝ)
7	5	rpsqrtcld 15347	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (√‘𝑥) ∈ ℝ⁺)
8	7	rpred 12961	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (√‘𝑥) ∈ ℝ)
9		rpsqrtcl 15199	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (√‘𝑥) ∈ ℝ⁺)
10		rpne0 12934	. . . . . . 7 ⊢ ((√‘𝑥) ∈ ℝ⁺ → (√‘𝑥) ≠ 0)
11	9, 10	syl 17	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (√‘𝑥) ≠ 0)
12	11	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (√‘𝑥) ≠ 0)
13	6, 8, 12	redivcld 11981	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) / (√‘𝑥)) ∈ ℝ)
14	13	fmpttd 7069	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (√‘𝑥))):ℝ⁺⟶ℝ)
15		rpcn 12928	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
16	15	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℂ)
17		rpne0 12934	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≠ 0)
18	17	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ≠ 0)
19	16, 18	logcld 26547	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℂ)
20	16	sqrtcld 15375	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (√‘𝑥) ∈ ℂ)
21	19, 20, 12	divrecd 11932	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) / (√‘𝑥)) = ((log‘𝑥) · (1 / (√‘𝑥))))
22		2cnd 12235	. . . . . . . . . . . . 13 ⊢ (𝜑 → 2 ∈ ℂ)
23	22	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 2 ∈ ℂ)
24		2ne0 12261	. . . . . . . . . . . . 13 ⊢ 2 ≠ 0
25	24	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 2 ≠ 0)
26	23, 25	reccld 11922	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1 / 2) ∈ ℂ)
27	16, 18, 26	cxpnegd 26692	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐-(1 / 2)) = (1 / (𝑥↑_𝑐(1 / 2))))
28		cxpsqrt 26680	. . . . . . . . . . . 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 7384	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1 / (𝑥↑_𝑐(1 / 2))) = (1 / (√‘𝑥)))
31	27, 30	eqtrd 2772	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐-(1 / 2)) = (1 / (√‘𝑥)))
32	31	oveq2d 7384	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) · (𝑥↑_𝑐-(1 / 2))) = ((log‘𝑥) · (1 / (√‘𝑥))))
33	21, 32	eqtr4d 2775	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) / (√‘𝑥)) = ((log‘𝑥) · (𝑥↑_𝑐-(1 / 2))))
34	33	mpteq2dva 5193	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (√‘𝑥))) = (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) · (𝑥↑_𝑐-(1 / 2)))))
35	34	oveq2d 7384	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (√‘𝑥)))) = (ℝ D (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) · (𝑥↑_𝑐-(1 / 2))))))
36		reelprrecn 11130	. . . . . . 7 ⊢ ℝ ∈ {ℝ, ℂ}
37	36	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
38	5	rpreccld 12971	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ ℝ⁺)
39		logf1o 26541	. . . . . . . . . . 11 ⊢ log:(ℂ ∖ {0})–1-1-onto→ran log
40		f1of 6782	. . . . . . . . . . 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 3939	. . . . . . . . . . 11 ⊢ ℝ⁺ ⊆ ℂ
44		0nrp 12954	. . . . . . . . . . 11 ⊢ ¬ 0 ∈ ℝ⁺
45		ssdifsn 4746	. . . . . . . . . . 11 ⊢ (ℝ⁺ ⊆ (ℂ ∖ {0}) ↔ (ℝ⁺ ⊆ ℂ ∧ ¬ 0 ∈ ℝ⁺))
46	43, 44, 45	mpbir2an 712	. . . . . . . . . 10 ⊢ ℝ⁺ ⊆ (ℂ ∖ {0})
47	46	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ℝ⁺ ⊆ (ℂ ∖ {0}))
48	42, 47	feqresmpt 6911	. . . . . . . 8 ⊢ (𝜑 → (log ↾ ℝ⁺) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)))
49	48	oveq2d 7384	. . . . . . 7 ⊢ (𝜑 → (ℝ D (log ↾ ℝ⁺)) = (ℝ D (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))))
50		dvrelog 26614	. . . . . . 7 ⊢ (ℝ D (log ↾ ℝ⁺)) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥))
51	49, 50	eqtr3di 2787	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)))
52		1cnd 11139	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℂ)
53	52	halfcld 12398	. . . . . . . . 9 ⊢ (𝜑 → (1 / 2) ∈ ℂ)
54	53	negcld 11491	. . . . . . . 8 ⊢ (𝜑 → -(1 / 2) ∈ ℂ)
55	54	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → -(1 / 2) ∈ ℂ)
56	16, 55	cxpcld 26685	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐-(1 / 2)) ∈ ℂ)
57	52	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 1 ∈ ℂ)
58	55, 57	subcld 11504	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (-(1 / 2) − 1) ∈ ℂ)
59	16, 58	cxpcld 26685	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐(-(1 / 2) − 1)) ∈ ℂ)
60	55, 59	mulcld 11164	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1))) ∈ ℂ)
61		dvcxp1 26717	. . . . . . 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 25933	. . . . 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 2772	. . . 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 11095	. . . . . 6 ⊢ ℝ ⊆ ℂ
66	65	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ⊆ ℂ)
67		eqid 2737	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
68	67	addcn 24822	. . . . . . 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 3958	. . . . . . . . . 10 ⊢ ℂ ⊆ ℂ
72	71	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ℂ ⊆ ℂ)
73		cncfmptc 24873	. . . . . . . . 9 ⊢ ((1 ∈ ℂ ∧ ℝ⁺ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ ℝ⁺ ↦ 1) ∈ (ℝ⁺–cn→ℂ))
74	52, 70, 72, 73	syl3anc 1374	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ 1) ∈ (ℝ⁺–cn→ℂ))
75		difss 4090	. . . . . . . . 9 ⊢ (ℂ ∖ {0}) ⊆ ℂ
76		cncfmptid 24874	. . . . . . . . 9 ⊢ ((ℝ⁺ ⊆ (ℂ ∖ {0}) ∧ (ℂ ∖ {0}) ⊆ ℂ) → (𝑥 ∈ ℝ⁺ ↦ 𝑥) ∈ (ℝ⁺–cn→(ℂ ∖ {0})))
77	47, 75, 76	sylancl 587	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ 𝑥) ∈ (ℝ⁺–cn→(ℂ ∖ {0})))
78	74, 77	divcncf 25416	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)) ∈ (ℝ⁺–cn→ℂ))
79		ax-1 6	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ → 𝑥 ∈ ℝ⁺))
80	15, 79	jca 511	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℂ ∧ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ⁺)))
81		eqid 2737	. . . . . . . . . . . 12 ⊢ (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0))
82	81	ellogdm 26616	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (ℂ ∖ (-∞(,]0)) ↔ (𝑥 ∈ ℂ ∧ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ⁺)))
83	80, 82	sylibr 234	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ (ℂ ∖ (-∞(,]0)))
84	83	ssriv 3939	. . . . . . . . 9 ⊢ ℝ⁺ ⊆ (ℂ ∖ (-∞(,]0))
85	84	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ⁺ ⊆ (ℂ ∖ (-∞(,]0)))
86	54, 85	cxpcncf1 34772	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (𝑥↑_𝑐-(1 / 2))) ∈ (ℝ⁺–cn→ℂ))
87	78, 86	mulcncf 25414	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((1 / 𝑥) · (𝑥↑_𝑐-(1 / 2)))) ∈ (ℝ⁺–cn→ℂ))
88		cncfmptc 24873	. . . . . . . . 9 ⊢ ((-(1 / 2) ∈ ℂ ∧ ℝ⁺ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ ℝ⁺ ↦ -(1 / 2)) ∈ (ℝ⁺–cn→ℂ))
89	54, 70, 72, 88	syl3anc 1374	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ -(1 / 2)) ∈ (ℝ⁺–cn→ℂ))
90	54, 52	subcld 11504	. . . . . . . . 9 ⊢ (𝜑 → (-(1 / 2) − 1) ∈ ℂ)
91	90, 85	cxpcncf1 34772	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (𝑥↑_𝑐(-(1 / 2) − 1))) ∈ (ℝ⁺–cn→ℂ))
92	89, 91	mulcncf 25414	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1)))) ∈ (ℝ⁺–cn→ℂ))
93		cncfss 24860	. . . . . . . . 9 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (ℝ⁺–cn→ℝ) ⊆ (ℝ⁺–cn→ℂ))
94	65, 71, 93	mp2an 693	. . . . . . . 8 ⊢ (ℝ⁺–cn→ℝ) ⊆ (ℝ⁺–cn→ℂ)
95		relogcn 26615	. . . . . . . . 9 ⊢ (log ↾ ℝ⁺) ∈ (ℝ⁺–cn→ℝ)
96	48, 95	eqeltrrdi 2846	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)) ∈ (ℝ⁺–cn→ℝ))
97	94, 96	sselid 3933	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)) ∈ (ℝ⁺–cn→ℂ))
98	92, 97	mulcncf 25414	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1))) · (log‘𝑥))) ∈ (ℝ⁺–cn→ℂ))
99	67, 69, 87, 98	cncfmpt2f 24876	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (((1 / 𝑥) · (𝑥↑_𝑐-(1 / 2))) + ((-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1))) · (log‘𝑥)))) ∈ (ℝ⁺–cn→ℂ))
100		rpre 12926	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
101	100, 17	rereccld 11980	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (1 / 𝑥) ∈ ℝ)
102		rpge0 12931	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → 0 ≤ 𝑥)
103		halfre 12366	. . . . . . . . . . . 12 ⊢ (1 / 2) ∈ ℝ
104	103	renegcli 11454	. . . . . . . . . . 11 ⊢ -(1 / 2) ∈ ℝ
105	104	a1i 11	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → -(1 / 2) ∈ ℝ)
106	100, 102, 105	recxpcld 26700	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥↑_𝑐-(1 / 2)) ∈ ℝ)
107	101, 106	remulcld 11174	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → ((1 / 𝑥) · (𝑥↑_𝑐-(1 / 2))) ∈ ℝ)
108		1re 11144	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ
109	104, 108	resubcli 11455	. . . . . . . . . . . 12 ⊢ (-(1 / 2) − 1) ∈ ℝ
110	109	a1i 11	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (-(1 / 2) − 1) ∈ ℝ)
111	100, 102, 110	recxpcld 26700	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥↑_𝑐(-(1 / 2) − 1)) ∈ ℝ)
112	105, 111	remulcld 11174	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1))) ∈ ℝ)
113		relogcl 26552	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
114	112, 113	remulcld 11174	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → ((-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1))) · (log‘𝑥)) ∈ ℝ)
115	107, 114	readdcld 11173	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (((1 / 𝑥) · (𝑥↑_𝑐-(1 / 2))) + ((-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1))) · (log‘𝑥))) ∈ ℝ)
116	115	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (((1 / 𝑥) · (𝑥↑_𝑐-(1 / 2))) + ((-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1))) · (log‘𝑥))) ∈ ℝ)
117	116	fmpttd 7069	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (((1 / 𝑥) · (𝑥↑_𝑐-(1 / 2))) + ((-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1))) · (log‘𝑥)))):ℝ⁺⟶ℝ)
118		cncfcdm 24859	. . . . . 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 477	. . . . 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 838	. . . 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 2837	. . 3 ⊢ (𝜑 → (ℝ D (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (√‘𝑥)))) ∈ (ℝ⁺–cn→ℝ))
122		logdivsqrle.2	. . 3 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
123	64	fveq1d 6844	. . . . 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 480	. . . 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 11491	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → -1 ∈ ℂ)
126		cxpadd 26656	. . . . . . . . . . . 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 1379	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐(-(1 / 2) + -1)) = ((𝑥↑_𝑐-(1 / 2)) · (𝑥↑_𝑐-1)))
128	59	mullidd 11162	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1 · (𝑥↑_𝑐(-(1 / 2) − 1))) = (𝑥↑_𝑐(-(1 / 2) − 1)))
129	55, 57	negsubd 11510	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (-(1 / 2) + -1) = (-(1 / 2) − 1))
130	129	oveq2d 7384	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐(-(1 / 2) + -1)) = (𝑥↑_𝑐(-(1 / 2) − 1)))
131	128, 130	eqtr4d 2775	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1 · (𝑥↑_𝑐(-(1 / 2) − 1))) = (𝑥↑_𝑐(-(1 / 2) + -1)))
132	43, 38	sselid 3933	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ ℂ)
133	132, 56	mulcomd 11165	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((1 / 𝑥) · (𝑥↑_𝑐-(1 / 2))) = ((𝑥↑_𝑐-(1 / 2)) · (1 / 𝑥)))
134		cxpneg 26658	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0 ∧ 1 ∈ ℂ) → (𝑥↑_𝑐-1) = (1 / (𝑥↑_𝑐1)))
135	16, 18, 57, 134	syl3anc 1374	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐-1) = (1 / (𝑥↑_𝑐1)))
136	16	cxp1d 26683	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐1) = 𝑥)
137	136	oveq2d 7384	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1 / (𝑥↑_𝑐1)) = (1 / 𝑥))
138	135, 137	eqtr2d 2773	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) = (𝑥↑_𝑐-1))
139	138	oveq2d 7384	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((𝑥↑_𝑐-(1 / 2)) · (1 / 𝑥)) = ((𝑥↑_𝑐-(1 / 2)) · (𝑥↑_𝑐-1)))
140	133, 139	eqtrd 2772	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((1 / 𝑥) · (𝑥↑_𝑐-(1 / 2))) = ((𝑥↑_𝑐-(1 / 2)) · (𝑥↑_𝑐-1)))
141	127, 131, 140	3eqtr4rd 2783	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((1 / 𝑥) · (𝑥↑_𝑐-(1 / 2))) = (1 · (𝑥↑_𝑐(-(1 / 2) − 1))))
142	55, 59, 19	mul32d 11355	. . . . . . . . . 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 7386	. . . . . . . . 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 11164	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (-(1 / 2) · (log‘𝑥)) ∈ ℂ)
145	57, 144, 59	adddird 11169	. . . . . . . . 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 2775	. . . . . . . 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 5193	. . . . . . 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 6844	. . . . . 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 480	. . . . 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 2738	. . . . . . 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 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑦) → 𝑥 = 𝑦)
152	151	fveq2d 6846	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑦) → (log‘𝑥) = (log‘𝑦))
153	152	oveq2d 7384	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑦) → (-(1 / 2) · (log‘𝑥)) = (-(1 / 2) · (log‘𝑦)))
154	153	oveq2d 7384	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑦) → (1 + (-(1 / 2) · (log‘𝑥))) = (1 + (-(1 / 2) · (log‘𝑦))))
155	151	oveq1d 7383	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑦) → (𝑥↑_𝑐(-(1 / 2) − 1)) = (𝑦↑_𝑐(-(1 / 2) − 1)))
156	154, 155	oveq12d 7386	. . . . . . 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 13361	. . . . . . . . . 10 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
158	157	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵))
159	2, 3, 4	fct2relem 34774	. . . . . . . . 9 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ⁺)
160	158, 159	sstrd 3946	. . . . . . . 8 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ⁺)
161	160	sselda 3935	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ ℝ⁺)
162		ovexd 7403	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((1 + (-(1 / 2) · (log‘𝑦))) · (𝑦↑_𝑐(-(1 / 2) − 1))) ∈ V)
163	150, 156, 161, 162	fvmptd 6957	. . . . . 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 26600	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (log‘𝑦) ∈ ℝ)
167	165, 166	remulcld 11174	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (-(1 / 2) · (log‘𝑦)) ∈ ℝ)
168	164, 167	readdcld 11173	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (1 + (-(1 / 2) · (log‘𝑦))) ∈ ℝ)
169		0red 11147	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 0 ∈ ℝ)
170		rpcxpcl 26653	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℝ⁺ ∧ (-(1 / 2) − 1) ∈ ℝ) → (𝑦↑_𝑐(-(1 / 2) − 1)) ∈ ℝ⁺)
171	161, 109, 170	sylancl 587	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝑦↑_𝑐(-(1 / 2) − 1)) ∈ ℝ⁺)
172	171	rpred 12961	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝑦↑_𝑐(-(1 / 2) − 1)) ∈ ℝ)
173	171	rpge0d 12965	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 0 ≤ (𝑦↑_𝑐(-(1 / 2) − 1)))
174		2cn 12232	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℂ
175	174	mullidi 11149	. . . . . . . . . . . . 13 ⊢ (1 · 2) = 2
176		2re 12231	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ ℝ
177	176	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 2 ∈ ℝ)
178	177	reefcld 16023	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (exp‘2) ∈ ℝ)
179	3	rpred 12961	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐴 ∈ ℝ)
180	179	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ)
181	161	rpred 12961	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ ℝ)
182		logdivsqrle.1	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (exp‘2) ≤ 𝐴)
183	182	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (exp‘2) ≤ 𝐴)
184		eliooord 13333	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ (𝐴(,)𝐵) → (𝐴 < 𝑦 ∧ 𝑦 < 𝐵))
185	184	simpld 494	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (𝐴(,)𝐵) → 𝐴 < 𝑦)
186	185	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐴 < 𝑦)
187	180, 181, 186	ltled 11293	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐴 ≤ 𝑦)
188	178, 180, 181, 183, 187	letrd 11302	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (exp‘2) ≤ 𝑦)
189		reeflog 26557	. . . . . . . . . . . . . . . 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 5128	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (exp‘2) ≤ (exp‘(log‘𝑦)))
192		efle 16055	. . . . . . . . . . . . . . 15 ⊢ ((2 ∈ ℝ ∧ (log‘𝑦) ∈ ℝ) → (2 ≤ (log‘𝑦) ↔ (exp‘2) ≤ (exp‘(log‘𝑦))))
193	176, 166, 192	sylancr 588	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (2 ≤ (log‘𝑦) ↔ (exp‘2) ≤ (exp‘(log‘𝑦))))
194	191, 193	mpbird 257	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 2 ≤ (log‘𝑦))
195	175, 194	eqbrtrid 5135	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (1 · 2) ≤ (log‘𝑦))
196		2rp 12922	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℝ⁺
197	196	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 2 ∈ ℝ⁺)
198	164, 166, 197	lemuldivd 13010	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((1 · 2) ≤ (log‘𝑦) ↔ 1 ≤ ((log‘𝑦) / 2)))
199	195, 198	mpbid 232	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 1 ≤ ((log‘𝑦) / 2))
200	65, 166	sselid 3933	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (log‘𝑦) ∈ ℂ)
201	22	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 2 ∈ ℂ)
202	24	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 2 ≠ 0)
203	200, 201, 202	divrec2d 11933	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((log‘𝑦) / 2) = ((1 / 2) · (log‘𝑦)))
204	199, 203	breqtrd 5126	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 1 ≤ ((1 / 2) · (log‘𝑦)))
205	53	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (1 / 2) ∈ ℂ)
206	205, 200	mulneg1d 11602	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (-(1 / 2) · (log‘𝑦)) = -((1 / 2) · (log‘𝑦)))
207	206	oveq2d 7384	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (0 − (-(1 / 2) · (log‘𝑦))) = (0 − -((1 / 2) · (log‘𝑦))))
208	65, 169	sselid 3933	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 0 ∈ ℂ)
209	205, 200	mulcld 11164	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((1 / 2) · (log‘𝑦)) ∈ ℂ)
210	208, 209	subnegd 11511	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (0 − -((1 / 2) · (log‘𝑦))) = (0 + ((1 / 2) · (log‘𝑦))))
211	209	addlidd 11346	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (0 + ((1 / 2) · (log‘𝑦))) = ((1 / 2) · (log‘𝑦)))
212	207, 210, 211	3eqtrd 2776	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (0 − (-(1 / 2) · (log‘𝑦))) = ((1 / 2) · (log‘𝑦)))
213	204, 212	breqtrrd 5128	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 1 ≤ (0 − (-(1 / 2) · (log‘𝑦))))
214		leaddsub 11625	. . . . . . . . . 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 1374	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((1 + (-(1 / 2) · (log‘𝑦))) ≤ 0 ↔ 1 ≤ (0 − (-(1 / 2) · (log‘𝑦)))))
216	213, 215	mpbird 257	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (1 + (-(1 / 2) · (log‘𝑦))) ≤ 0)
217	168, 169, 172, 173, 216	lemul1ad 12093	. . . . . . 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 3933	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝑦↑_𝑐(-(1 / 2) − 1)) ∈ ℂ)
219	218	mul02d 11343	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (0 · (𝑦↑_𝑐(-(1 / 2) − 1))) = 0)
220	217, 219	breqtrd 5126	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((1 + (-(1 / 2) · (log‘𝑦))) · (𝑦↑_𝑐(-(1 / 2) − 1))) ≤ 0)
221	163, 220	eqbrtrd 5122	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((𝑥 ∈ ℝ⁺ ↦ ((1 + (-(1 / 2) · (log‘𝑥))) · (𝑥↑_𝑐(-(1 / 2) − 1))))‘𝑦) ≤ 0)
222	149, 221	eqbrtrd 5122	. . . 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 5122	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((ℝ D (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (√‘𝑥))))‘𝑦) ≤ 0)
224	2, 3, 4, 14, 121, 122, 223	fdvnegge 34779	. 2 ⊢ (𝜑 → ((𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (√‘𝑥)))‘𝐵) ≤ ((𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (√‘𝑥)))‘𝐴))
225		eqidd 2738	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (√‘𝑥))) = (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (√‘𝑥))))
226		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝑥 = 𝐵)
227	226	fveq2d 6846	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → (log‘𝑥) = (log‘𝐵))
228	226	fveq2d 6846	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → (√‘𝑥) = (√‘𝐵))
229	227, 228	oveq12d 7386	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → ((log‘𝑥) / (√‘𝑥)) = ((log‘𝐵) / (√‘𝐵)))
230		ovex 7401	. . . 4 ⊢ ((log‘𝐵) / (√‘𝐵)) ∈ V
231	230	a1i 11	. . 3 ⊢ (𝜑 → ((log‘𝐵) / (√‘𝐵)) ∈ V)
232	225, 229, 4, 231	fvmptd 6957	. 2 ⊢ (𝜑 → ((𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (√‘𝑥)))‘𝐵) = ((log‘𝐵) / (√‘𝐵)))
233		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝑥 = 𝐴)
234	233	fveq2d 6846	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (log‘𝑥) = (log‘𝐴))
235	233	fveq2d 6846	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (√‘𝑥) = (√‘𝐴))
236	234, 235	oveq12d 7386	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((log‘𝑥) / (√‘𝑥)) = ((log‘𝐴) / (√‘𝐴)))
237		ovex 7401	. . . 4 ⊢ ((log‘𝐴) / (√‘𝐴)) ∈ V
238	237	a1i 11	. . 3 ⊢ (𝜑 → ((log‘𝐴) / (√‘𝐴)) ∈ V)
239	225, 236, 3, 238	fvmptd 6957	. 2 ⊢ (𝜑 → ((𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (√‘𝑥)))‘𝐴) = ((log‘𝐴) / (√‘𝐴)))
240	224, 232, 239	3brtr3d 5131	1 ⊢ (𝜑 → ((log‘𝐵) / (√‘𝐵)) ≤ ((log‘𝐴) / (√‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 Vcvv 3442 ∖ cdif 3900 ⊆ wss 3903 {csn 4582 {cpr 4584 class class class wbr 5100 ↦ cmpt 5181 ran crn 5633 ↾ cres 5634 ⟶wf 6496 –1-1-onto→wf1o 6499 ‘cfv 6500 (class class class)co 7368 ℂcc 11036 ℝcr 11037 0cc0 11038 1c1 11039 + caddc 11041 · cmul 11043 +∞cpnf 11175 -∞cmnf 11176 < clt 11178 ≤ cle 11179 − cmin 11376 -cneg 11377 / cdiv 11806 2c2 12212 ℝ⁺crp 12917 (,)cioo 13273 (,]cioc 13274 [,]cicc 13276 √csqrt 15168 expce 15996 TopOpenctopn 17353 ℂ_fldccnfld 21321 Cn ccn 23180 ×_t ctx 23516 –cn→ccncf 24837 D cdv 25832 logclog 26531 ↑_𝑐ccxp 26532

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-inf2 9562 ax-cc 10357 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-symdif 4207 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-disj 5068 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-ofr 7633 df-om 7819 df-1st 7943 df-2nd 7944 df-supp 8113 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-oadd 8411 df-omul 8412 df-er 8645 df-map 8777 df-pm 8778 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9277 df-fi 9326 df-sup 9357 df-inf 9358 df-oi 9427 df-dju 9825 df-card 9863 df-acn 9866 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-z 12501 df-dec 12620 df-uz 12764 df-q 12874 df-rp 12918 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-ioo 13277 df-ioc 13278 df-ico 13279 df-icc 13280 df-fz 13436 df-fzo 13583 df-fl 13724 df-mod 13802 df-seq 13937 df-exp 13997 df-fac 14209 df-bc 14238 df-hash 14266 df-shft 15002 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-limsup 15406 df-clim 15423 df-rlim 15424 df-sum 15622 df-ef 16002 df-sin 16004 df-cos 16005 df-tan 16006 df-pi 16007 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-starv 17204 df-sca 17205 df-vsca 17206 df-ip 17207 df-tset 17208 df-ple 17209 df-ds 17211 df-unif 17212 df-hom 17213 df-cco 17214 df-rest 17354 df-topn 17355 df-0g 17373 df-gsum 17374 df-topgen 17375 df-pt 17376 df-prds 17379 df-xrs 17435 df-qtop 17440 df-imas 17441 df-xps 17443 df-mre 17517 df-mrc 17518 df-acs 17520 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-submnd 18721 df-mulg 19010 df-cntz 19258 df-cmn 19723 df-psmet 21313 df-xmet 21314 df-met 21315 df-bl 21316 df-mopn 21317 df-fbas 21318 df-fg 21319 df-cnfld 21322 df-top 22850 df-topon 22867 df-topsp 22889 df-bases 22902 df-cld 22975 df-ntr 22976 df-cls 22977 df-nei 23054 df-lp 23092 df-perf 23093 df-cn 23183 df-cnp 23184 df-haus 23271 df-cmp 23343 df-tx 23518 df-hmeo 23711 df-fil 23802 df-fm 23894 df-flim 23895 df-flf 23896 df-xms 24276 df-ms 24277 df-tms 24278 df-cncf 24839 df-ovol 25433 df-vol 25434 df-mbf 25588 df-itg1 25589 df-itg2 25590 df-ibl 25591 df-itg 25592 df-0p 25639 df-limc 25835 df-dv 25836 df-log 26533 df-cxp 26534

This theorem is referenced by: hgt750lem 34828

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