logdivsqrle - Mathbox for Thierry Arnoux

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Theorem logdivsqrle 34653

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 13317	. . . 4 ⊢ (0(,)+∞) = ℝ⁺
2	1	eqcomi 2739	. . 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 26552	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℝ)
7	5	rpsqrtcld 15311	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (√‘𝑥) ∈ ℝ⁺)
8	7	rpred 12926	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (√‘𝑥) ∈ ℝ)
9		rpsqrtcl 15163	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (√‘𝑥) ∈ ℝ⁺)
10		rpne0 12899	. . . . . . 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 11941	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) / (√‘𝑥)) ∈ ℝ)
14	13	fmpttd 7043	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (√‘𝑥))):ℝ⁺⟶ℝ)
15		rpcn 12893	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
16	15	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℂ)
17		rpne0 12899	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≠ 0)
18	17	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ≠ 0)
19	16, 18	logcld 26499	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℂ)
20	16	sqrtcld 15339	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (√‘𝑥) ∈ ℂ)
21	19, 20, 12	divrecd 11892	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) / (√‘𝑥)) = ((log‘𝑥) · (1 / (√‘𝑥))))
22		2cnd 12195	. . . . . . . . . . . . 13 ⊢ (𝜑 → 2 ∈ ℂ)
23	22	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 2 ∈ ℂ)
24		2ne0 12221	. . . . . . . . . . . . 13 ⊢ 2 ≠ 0
25	24	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 2 ≠ 0)
26	23, 25	reccld 11882	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1 / 2) ∈ ℂ)
27	16, 18, 26	cxpnegd 26644	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐-(1 / 2)) = (1 / (𝑥↑_𝑐(1 / 2))))
28		cxpsqrt 26632	. . . . . . . . . . . 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 7357	. . . . . . . . . 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 7357	. . . . . . . 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 5182	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (√‘𝑥))) = (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) · (𝑥↑_𝑐-(1 / 2)))))
35	34	oveq2d 7357	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (√‘𝑥)))) = (ℝ D (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) · (𝑥↑_𝑐-(1 / 2))))))
36		reelprrecn 11090	. . . . . . 7 ⊢ ℝ ∈ {ℝ, ℂ}
37	36	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
38	5	rpreccld 12936	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ ℝ⁺)
39		logf1o 26493	. . . . . . . . . . 11 ⊢ log:(ℂ ∖ {0})–1-1-onto→ran log
40		f1of 6759	. . . . . . . . . . 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 3936	. . . . . . . . . . 11 ⊢ ℝ⁺ ⊆ ℂ
44		0nrp 12919	. . . . . . . . . . 11 ⊢ ¬ 0 ∈ ℝ⁺
45		ssdifsn 4738	. . . . . . . . . . 11 ⊢ (ℝ⁺ ⊆ (ℂ ∖ {0}) ↔ (ℝ⁺ ⊆ ℂ ∧ ¬ 0 ∈ ℝ⁺))
46	43, 44, 45	mpbir2an 711	. . . . . . . . . 10 ⊢ ℝ⁺ ⊆ (ℂ ∖ {0})
47	46	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ℝ⁺ ⊆ (ℂ ∖ {0}))
48	42, 47	feqresmpt 6886	. . . . . . . 8 ⊢ (𝜑 → (log ↾ ℝ⁺) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)))
49	48	oveq2d 7357	. . . . . . 7 ⊢ (𝜑 → (ℝ D (log ↾ ℝ⁺)) = (ℝ D (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))))
50		dvrelog 26566	. . . . . . 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 11099	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℂ)
53	52	halfcld 12358	. . . . . . . . 9 ⊢ (𝜑 → (1 / 2) ∈ ℂ)
54	53	negcld 11451	. . . . . . . 8 ⊢ (𝜑 → -(1 / 2) ∈ ℂ)
55	54	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → -(1 / 2) ∈ ℂ)
56	16, 55	cxpcld 26637	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐-(1 / 2)) ∈ ℂ)
57	52	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 1 ∈ ℂ)
58	55, 57	subcld 11464	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (-(1 / 2) − 1) ∈ ℂ)
59	16, 58	cxpcld 26637	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐(-(1 / 2) − 1)) ∈ ℂ)
60	55, 59	mulcld 11124	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1))) ∈ ℂ)
61		dvcxp1 26669	. . . . . . 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 25885	. . . . 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 11055	. . . . . 6 ⊢ ℝ ⊆ ℂ
66	65	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ⊆ ℂ)
67		eqid 2730	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
68	67	addcn 24774	. . . . . . 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 3955	. . . . . . . . . 10 ⊢ ℂ ⊆ ℂ
72	71	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ℂ ⊆ ℂ)
73		cncfmptc 24825	. . . . . . . . 9 ⊢ ((1 ∈ ℂ ∧ ℝ⁺ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ ℝ⁺ ↦ 1) ∈ (ℝ⁺–cn→ℂ))
74	52, 70, 72, 73	syl3anc 1373	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ 1) ∈ (ℝ⁺–cn→ℂ))
75		difss 4084	. . . . . . . . 9 ⊢ (ℂ ∖ {0}) ⊆ ℂ
76		cncfmptid 24826	. . . . . . . . 9 ⊢ ((ℝ⁺ ⊆ (ℂ ∖ {0}) ∧ (ℂ ∖ {0}) ⊆ ℂ) → (𝑥 ∈ ℝ⁺ ↦ 𝑥) ∈ (ℝ⁺–cn→(ℂ ∖ {0})))
77	47, 75, 76	sylancl 586	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ 𝑥) ∈ (ℝ⁺–cn→(ℂ ∖ {0})))
78	74, 77	divcncf 25368	. . . . . . 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 2730	. . . . . . . . . . . 12 ⊢ (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0))
82	81	ellogdm 26568	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (ℂ ∖ (-∞(,]0)) ↔ (𝑥 ∈ ℂ ∧ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ⁺)))
83	80, 82	sylibr 234	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ (ℂ ∖ (-∞(,]0)))
84	83	ssriv 3936	. . . . . . . . 9 ⊢ ℝ⁺ ⊆ (ℂ ∖ (-∞(,]0))
85	84	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ⁺ ⊆ (ℂ ∖ (-∞(,]0)))
86	54, 85	cxpcncf1 34598	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (𝑥↑_𝑐-(1 / 2))) ∈ (ℝ⁺–cn→ℂ))
87	78, 86	mulcncf 25366	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((1 / 𝑥) · (𝑥↑_𝑐-(1 / 2)))) ∈ (ℝ⁺–cn→ℂ))
88		cncfmptc 24825	. . . . . . . . 9 ⊢ ((-(1 / 2) ∈ ℂ ∧ ℝ⁺ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ ℝ⁺ ↦ -(1 / 2)) ∈ (ℝ⁺–cn→ℂ))
89	54, 70, 72, 88	syl3anc 1373	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ -(1 / 2)) ∈ (ℝ⁺–cn→ℂ))
90	54, 52	subcld 11464	. . . . . . . . 9 ⊢ (𝜑 → (-(1 / 2) − 1) ∈ ℂ)
91	90, 85	cxpcncf1 34598	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (𝑥↑_𝑐(-(1 / 2) − 1))) ∈ (ℝ⁺–cn→ℂ))
92	89, 91	mulcncf 25366	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1)))) ∈ (ℝ⁺–cn→ℂ))
93		cncfss 24812	. . . . . . . . 9 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (ℝ⁺–cn→ℝ) ⊆ (ℝ⁺–cn→ℂ))
94	65, 71, 93	mp2an 692	. . . . . . . 8 ⊢ (ℝ⁺–cn→ℝ) ⊆ (ℝ⁺–cn→ℂ)
95		relogcn 26567	. . . . . . . . 9 ⊢ (log ↾ ℝ⁺) ∈ (ℝ⁺–cn→ℝ)
96	48, 95	eqeltrrdi 2838	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)) ∈ (ℝ⁺–cn→ℝ))
97	94, 96	sselid 3930	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)) ∈ (ℝ⁺–cn→ℂ))
98	92, 97	mulcncf 25366	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1))) · (log‘𝑥))) ∈ (ℝ⁺–cn→ℂ))
99	67, 69, 87, 98	cncfmpt2f 24828	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (((1 / 𝑥) · (𝑥↑_𝑐-(1 / 2))) + ((-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1))) · (log‘𝑥)))) ∈ (ℝ⁺–cn→ℂ))
100		rpre 12891	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
101	100, 17	rereccld 11940	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (1 / 𝑥) ∈ ℝ)
102		rpge0 12896	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → 0 ≤ 𝑥)
103		halfre 12326	. . . . . . . . . . . 12 ⊢ (1 / 2) ∈ ℝ
104	103	renegcli 11414	. . . . . . . . . . 11 ⊢ -(1 / 2) ∈ ℝ
105	104	a1i 11	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → -(1 / 2) ∈ ℝ)
106	100, 102, 105	recxpcld 26652	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥↑_𝑐-(1 / 2)) ∈ ℝ)
107	101, 106	remulcld 11134	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → ((1 / 𝑥) · (𝑥↑_𝑐-(1 / 2))) ∈ ℝ)
108		1re 11104	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ
109	104, 108	resubcli 11415	. . . . . . . . . . . 12 ⊢ (-(1 / 2) − 1) ∈ ℝ
110	109	a1i 11	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (-(1 / 2) − 1) ∈ ℝ)
111	100, 102, 110	recxpcld 26652	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥↑_𝑐(-(1 / 2) − 1)) ∈ ℝ)
112	105, 111	remulcld 11134	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1))) ∈ ℝ)
113		relogcl 26504	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
114	112, 113	remulcld 11134	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → ((-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1))) · (log‘𝑥)) ∈ ℝ)
115	107, 114	readdcld 11133	. . . . . . 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 7043	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (((1 / 𝑥) · (𝑥↑_𝑐-(1 / 2))) + ((-(1 / 2) · (𝑥↑_𝑐(-(1 / 2) − 1))) · (log‘𝑥)))):ℝ⁺⟶ℝ)
118		cncfcdm 24811	. . . . . 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 837	. . . 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 2829	. . 3 ⊢ (𝜑 → (ℝ D (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (√‘𝑥)))) ∈ (ℝ⁺–cn→ℝ))
122		logdivsqrle.2	. . 3 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
123	64	fveq1d 6819	. . . . 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 11451	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → -1 ∈ ℂ)
126		cxpadd 26608	. . . . . . . . . . . 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 1378	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐(-(1 / 2) + -1)) = ((𝑥↑_𝑐-(1 / 2)) · (𝑥↑_𝑐-1)))
128	59	mullidd 11122	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1 · (𝑥↑_𝑐(-(1 / 2) − 1))) = (𝑥↑_𝑐(-(1 / 2) − 1)))
129	55, 57	negsubd 11470	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (-(1 / 2) + -1) = (-(1 / 2) − 1))
130	129	oveq2d 7357	. . . . . . . . . . . 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 3930	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ ℂ)
133	132, 56	mulcomd 11125	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((1 / 𝑥) · (𝑥↑_𝑐-(1 / 2))) = ((𝑥↑_𝑐-(1 / 2)) · (1 / 𝑥)))
134		cxpneg 26610	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0 ∧ 1 ∈ ℂ) → (𝑥↑_𝑐-1) = (1 / (𝑥↑_𝑐1)))
135	16, 18, 57, 134	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐-1) = (1 / (𝑥↑_𝑐1)))
136	16	cxp1d 26635	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐1) = 𝑥)
137	136	oveq2d 7357	. . . . . . . . . . . . . 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 7357	. . . . . . . . . . . 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 11315	. . . . . . . . . 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 7359	. . . . . . . . 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 11124	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (-(1 / 2) · (log‘𝑥)) ∈ ℂ)
145	57, 144, 59	adddird 11129	. . . . . . . . 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 5182	. . . . . . 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 6819	. . . . . 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 2731	. . . . . . 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 6821	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑦) → (log‘𝑥) = (log‘𝑦))
153	152	oveq2d 7357	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑦) → (-(1 / 2) · (log‘𝑥)) = (-(1 / 2) · (log‘𝑦)))
154	153	oveq2d 7357	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑦) → (1 + (-(1 / 2) · (log‘𝑥))) = (1 + (-(1 / 2) · (log‘𝑦))))
155	151	oveq1d 7356	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑦) → (𝑥↑_𝑐(-(1 / 2) − 1)) = (𝑦↑_𝑐(-(1 / 2) − 1)))
156	154, 155	oveq12d 7359	. . . . . . 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 13325	. . . . . . . . . 10 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
158	157	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵))
159	2, 3, 4	fct2relem 34600	. . . . . . . . 9 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ⁺)
160	158, 159	sstrd 3943	. . . . . . . 8 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ⁺)
161	160	sselda 3932	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ ℝ⁺)
162		ovexd 7376	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((1 + (-(1 / 2) · (log‘𝑦))) · (𝑦↑_𝑐(-(1 / 2) − 1))) ∈ V)
163	150, 156, 161, 162	fvmptd 6931	. . . . . 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 26552	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (log‘𝑦) ∈ ℝ)
167	165, 166	remulcld 11134	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (-(1 / 2) · (log‘𝑦)) ∈ ℝ)
168	164, 167	readdcld 11133	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (1 + (-(1 / 2) · (log‘𝑦))) ∈ ℝ)
169		0red 11107	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 0 ∈ ℝ)
170		rpcxpcl 26605	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℝ⁺ ∧ (-(1 / 2) − 1) ∈ ℝ) → (𝑦↑_𝑐(-(1 / 2) − 1)) ∈ ℝ⁺)
171	161, 109, 170	sylancl 586	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝑦↑_𝑐(-(1 / 2) − 1)) ∈ ℝ⁺)
172	171	rpred 12926	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝑦↑_𝑐(-(1 / 2) − 1)) ∈ ℝ)
173	171	rpge0d 12930	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 0 ≤ (𝑦↑_𝑐(-(1 / 2) − 1)))
174		2cn 12192	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℂ
175	174	mullidi 11109	. . . . . . . . . . . . 13 ⊢ (1 · 2) = 2
176		2re 12191	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ ℝ
177	176	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 2 ∈ ℝ)
178	177	reefcld 15987	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (exp‘2) ∈ ℝ)
179	3	rpred 12926	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐴 ∈ ℝ)
180	179	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ)
181	161	rpred 12926	. . . . . . . . . . . . . . . 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 13297	. . . . . . . . . . . . . . . . . . 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 11253	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐴 ≤ 𝑦)
188	178, 180, 181, 183, 187	letrd 11262	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (exp‘2) ≤ 𝑦)
189		reeflog 26509	. . . . . . . . . . . . . . . 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 5117	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (exp‘2) ≤ (exp‘(log‘𝑦)))
192		efle 16019	. . . . . . . . . . . . . . 15 ⊢ ((2 ∈ ℝ ∧ (log‘𝑦) ∈ ℝ) → (2 ≤ (log‘𝑦) ↔ (exp‘2) ≤ (exp‘(log‘𝑦))))
193	176, 166, 192	sylancr 587	. . . . . . . . . . . . . 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 5124	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (1 · 2) ≤ (log‘𝑦))
196		2rp 12887	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℝ⁺
197	196	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 2 ∈ ℝ⁺)
198	164, 166, 197	lemuldivd 12975	. . . . . . . . . . . 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 3930	. . . . . . . . . . . 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 11893	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((log‘𝑦) / 2) = ((1 / 2) · (log‘𝑦)))
204	199, 203	breqtrd 5115	. . . . . . . . . 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 11562	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (-(1 / 2) · (log‘𝑦)) = -((1 / 2) · (log‘𝑦)))
207	206	oveq2d 7357	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (0 − (-(1 / 2) · (log‘𝑦))) = (0 − -((1 / 2) · (log‘𝑦))))
208	65, 169	sselid 3930	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 0 ∈ ℂ)
209	205, 200	mulcld 11124	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((1 / 2) · (log‘𝑦)) ∈ ℂ)
210	208, 209	subnegd 11471	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (0 − -((1 / 2) · (log‘𝑦))) = (0 + ((1 / 2) · (log‘𝑦))))
211	209	addlidd 11306	. . . . . . . . . . 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 5117	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 1 ≤ (0 − (-(1 / 2) · (log‘𝑦))))
214		leaddsub 11585	. . . . . . . . . 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 1373	. . . . . . . . 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 12053	. . . . . . 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 3930	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝑦↑_𝑐(-(1 / 2) − 1)) ∈ ℂ)
219	218	mul02d 11303	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (0 · (𝑦↑_𝑐(-(1 / 2) − 1))) = 0)
220	217, 219	breqtrd 5115	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((1 + (-(1 / 2) · (log‘𝑦))) · (𝑦↑_𝑐(-(1 / 2) − 1))) ≤ 0)
221	163, 220	eqbrtrd 5111	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((𝑥 ∈ ℝ⁺ ↦ ((1 + (-(1 / 2) · (log‘𝑥))) · (𝑥↑_𝑐(-(1 / 2) − 1))))‘𝑦) ≤ 0)
222	149, 221	eqbrtrd 5111	. . . 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 5111	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((ℝ D (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (√‘𝑥))))‘𝑦) ≤ 0)
224	2, 3, 4, 14, 121, 122, 223	fdvnegge 34605	. 2 ⊢ (𝜑 → ((𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (√‘𝑥)))‘𝐵) ≤ ((𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (√‘𝑥)))‘𝐴))
225		eqidd 2731	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (√‘𝑥))) = (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (√‘𝑥))))
226		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝑥 = 𝐵)
227	226	fveq2d 6821	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → (log‘𝑥) = (log‘𝐵))
228	226	fveq2d 6821	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → (√‘𝑥) = (√‘𝐵))
229	227, 228	oveq12d 7359	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → ((log‘𝑥) / (√‘𝑥)) = ((log‘𝐵) / (√‘𝐵)))
230		ovex 7374	. . . 4 ⊢ ((log‘𝐵) / (√‘𝐵)) ∈ V
231	230	a1i 11	. . 3 ⊢ (𝜑 → ((log‘𝐵) / (√‘𝐵)) ∈ V)
232	225, 229, 4, 231	fvmptd 6931	. 2 ⊢ (𝜑 → ((𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (√‘𝑥)))‘𝐵) = ((log‘𝐵) / (√‘𝐵)))
233		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝑥 = 𝐴)
234	233	fveq2d 6821	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (log‘𝑥) = (log‘𝐴))
235	233	fveq2d 6821	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (√‘𝑥) = (√‘𝐴))
236	234, 235	oveq12d 7359	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((log‘𝑥) / (√‘𝑥)) = ((log‘𝐴) / (√‘𝐴)))
237		ovex 7374	. . . 4 ⊢ ((log‘𝐴) / (√‘𝐴)) ∈ V
238	237	a1i 11	. . 3 ⊢ (𝜑 → ((log‘𝐴) / (√‘𝐴)) ∈ V)
239	225, 236, 3, 238	fvmptd 6931	. 2 ⊢ (𝜑 → ((𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (√‘𝑥)))‘𝐴) = ((log‘𝐴) / (√‘𝐴)))
240	224, 232, 239	3brtr3d 5120	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 1541 ∈ wcel 2110 ≠ wne 2926 Vcvv 3434 ∖ cdif 3897 ⊆ wss 3900 {csn 4574 {cpr 4576 class class class wbr 5089 ↦ cmpt 5170 ran crn 5615 ↾ cres 5616 ⟶wf 6473 –1-1-onto→wf1o 6476 ‘cfv 6477 (class class class)co 7341 ℂcc 10996 ℝcr 10997 0cc0 10998 1c1 10999 + caddc 11001 · cmul 11003 +∞cpnf 11135 -∞cmnf 11136 < clt 11138 ≤ cle 11139 − cmin 11336 -cneg 11337 / cdiv 11766 2c2 12172 ℝ⁺crp 12882 (,)cioo 13237 (,]cioc 13238 [,]cicc 13240 √csqrt 15132 expce 15960 TopOpenctopn 17317 ℂ_fldccnfld 21284 Cn ccn 23132 ×_t ctx 23468 –cn→ccncf 24789 D cdv 25784 logclog 26483 ↑_𝑐ccxp 26484

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2112 ax-9 2120 ax-10 2143 ax-11 2159 ax-12 2179 ax-ext 2702 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7663 ax-inf2 9526 ax-cc 10318 ax-cnex 11054 ax-resscn 11055 ax-1cn 11056 ax-icn 11057 ax-addcl 11058 ax-addrcl 11059 ax-mulcl 11060 ax-mulrcl 11061 ax-mulcom 11062 ax-addass 11063 ax-mulass 11064 ax-distr 11065 ax-i2m1 11066 ax-1ne0 11067 ax-1rid 11068 ax-rnegex 11069 ax-rrecex 11070 ax-cnre 11071 ax-pre-lttri 11072 ax-pre-lttrn 11073 ax-pre-ltadd 11074 ax-pre-mulgt0 11075 ax-pre-sup 11076 ax-addf 11077

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3344 df-reu 3345 df-rab 3394 df-v 3436 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-pss 3920 df-symdif 4201 df-nul 4282 df-if 4474 df-pw 4550 df-sn 4575 df-pr 4577 df-tp 4579 df-op 4581 df-uni 4858 df-int 4896 df-iun 4941 df-iin 4942 df-disj 5057 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6244 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6433 df-fun 6479 df-fn 6480 df-f 6481 df-f1 6482 df-fo 6483 df-f1o 6484 df-fv 6485 df-isom 6486 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-of 7605 df-ofr 7606 df-om 7792 df-1st 7916 df-2nd 7917 df-supp 8086 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-2o 8381 df-oadd 8384 df-omul 8385 df-er 8617 df-map 8747 df-pm 8748 df-ixp 8817 df-en 8865 df-dom 8866 df-sdom 8867 df-fin 8868 df-fsupp 9241 df-fi 9290 df-sup 9321 df-inf 9322 df-oi 9391 df-dju 9786 df-card 9824 df-acn 9827 df-pnf 11140 df-mnf 11141 df-xr 11142 df-ltxr 11143 df-le 11144 df-sub 11338 df-neg 11339 df-div 11767 df-nn 12118 df-2 12180 df-3 12181 df-4 12182 df-5 12183 df-6 12184 df-7 12185 df-8 12186 df-9 12187 df-n0 12374 df-z 12461 df-dec 12581 df-uz 12725 df-q 12839 df-rp 12883 df-xneg 13003 df-xadd 13004 df-xmul 13005 df-ioo 13241 df-ioc 13242 df-ico 13243 df-icc 13244 df-fz 13400 df-fzo 13547 df-fl 13688 df-mod 13766 df-seq 13901 df-exp 13961 df-fac 14173 df-bc 14202 df-hash 14230 df-shft 14966 df-cj 14998 df-re 14999 df-im 15000 df-sqrt 15134 df-abs 15135 df-limsup 15370 df-clim 15387 df-rlim 15388 df-sum 15586 df-ef 15966 df-sin 15968 df-cos 15969 df-tan 15970 df-pi 15971 df-struct 17050 df-sets 17067 df-slot 17085 df-ndx 17097 df-base 17113 df-ress 17134 df-plusg 17166 df-mulr 17167 df-starv 17168 df-sca 17169 df-vsca 17170 df-ip 17171 df-tset 17172 df-ple 17173 df-ds 17175 df-unif 17176 df-hom 17177 df-cco 17178 df-rest 17318 df-topn 17319 df-0g 17337 df-gsum 17338 df-topgen 17339 df-pt 17340 df-prds 17343 df-xrs 17398 df-qtop 17403 df-imas 17404 df-xps 17406 df-mre 17480 df-mrc 17481 df-acs 17483 df-mgm 18540 df-sgrp 18619 df-mnd 18635 df-submnd 18684 df-mulg 18973 df-cntz 19222 df-cmn 19687 df-psmet 21276 df-xmet 21277 df-met 21278 df-bl 21279 df-mopn 21280 df-fbas 21281 df-fg 21282 df-cnfld 21285 df-top 22802 df-topon 22819 df-topsp 22841 df-bases 22854 df-cld 22927 df-ntr 22928 df-cls 22929 df-nei 23006 df-lp 23044 df-perf 23045 df-cn 23135 df-cnp 23136 df-haus 23223 df-cmp 23295 df-tx 23470 df-hmeo 23663 df-fil 23754 df-fm 23846 df-flim 23847 df-flf 23848 df-xms 24228 df-ms 24229 df-tms 24230 df-cncf 24791 df-ovol 25385 df-vol 25386 df-mbf 25540 df-itg1 25541 df-itg2 25542 df-ibl 25543 df-itg 25544 df-0p 25591 df-limc 25787 df-dv 25788 df-log 26485 df-cxp 26486

This theorem is referenced by: hgt750lem 34654

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