Step | Hyp | Ref
| Expression |
1 | | ioorp 12251 |
. . . 4
⊢
(0(,)+∞) = ℝ+ |
2 | 1 | eqcomi 2631 |
. . 3
⊢
ℝ+ = (0(,)+∞) |
3 | | logdivsqrle.a |
. . 3
⊢ (𝜑 → 𝐴 ∈
ℝ+) |
4 | | logdivsqrle.b |
. . 3
⊢ (𝜑 → 𝐵 ∈
ℝ+) |
5 | | simpr 477 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈
ℝ+) |
6 | 5 | relogcld 24369 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
(log‘𝑥) ∈
ℝ) |
7 | 5 | rpsqrtcld 14150 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
(√‘𝑥) ∈
ℝ+) |
8 | 7 | rpred 11872 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
(√‘𝑥) ∈
ℝ) |
9 | | rpsqrtcl 14005 |
. . . . . . 7
⊢ (𝑥 ∈ ℝ+
→ (√‘𝑥)
∈ ℝ+) |
10 | | rpne0 11848 |
. . . . . . 7
⊢
((√‘𝑥)
∈ ℝ+ → (√‘𝑥) ≠ 0) |
11 | 9, 10 | syl 17 |
. . . . . 6
⊢ (𝑥 ∈ ℝ+
→ (√‘𝑥)
≠ 0) |
12 | 11 | adantl 482 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
(√‘𝑥) ≠
0) |
13 | 6, 8, 12 | redivcld 10853 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
((log‘𝑥) /
(√‘𝑥)) ∈
ℝ) |
14 | | eqid 2622 |
. . . 4
⊢ (𝑥 ∈ ℝ+
↦ ((log‘𝑥) /
(√‘𝑥))) =
(𝑥 ∈
ℝ+ ↦ ((log‘𝑥) / (√‘𝑥))) |
15 | 13, 14 | fmptd 6385 |
. . 3
⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦
((log‘𝑥) /
(√‘𝑥))):ℝ+⟶ℝ) |
16 | | rpcn 11841 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ℝ+
→ 𝑥 ∈
ℂ) |
17 | 16 | adantl 482 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈
ℂ) |
18 | | rpne0 11848 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ℝ+
→ 𝑥 ≠
0) |
19 | 18 | adantl 482 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑥 ≠ 0) |
20 | 17, 19 | logcld 24317 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
(log‘𝑥) ∈
ℂ) |
21 | 17 | sqrtcld 14176 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
(√‘𝑥) ∈
ℂ) |
22 | 20, 21, 12 | divrecd 10804 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
((log‘𝑥) /
(√‘𝑥)) =
((log‘𝑥) · (1
/ (√‘𝑥)))) |
23 | | 2cnd 11093 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 2 ∈
ℂ) |
24 | 23 | adantr 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 2 ∈
ℂ) |
25 | | 2ne0 11113 |
. . . . . . . . . . . . 13
⊢ 2 ≠
0 |
26 | 25 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 2 ≠
0) |
27 | 24, 26 | reccld 10794 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (1 / 2)
∈ ℂ) |
28 | 17, 19, 27 | cxpnegd 24461 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (𝑥↑𝑐-(1 /
2)) = (1 / (𝑥↑𝑐(1 /
2)))) |
29 | | cxpsqrt 24449 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℂ → (𝑥↑𝑐(1 /
2)) = (√‘𝑥)) |
30 | 17, 29 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (𝑥↑𝑐(1 /
2)) = (√‘𝑥)) |
31 | 30 | oveq2d 6666 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (1 /
(𝑥↑𝑐(1 / 2))) = (1 /
(√‘𝑥))) |
32 | 28, 31 | eqtrd 2656 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (𝑥↑𝑐-(1 /
2)) = (1 / (√‘𝑥))) |
33 | 32 | oveq2d 6666 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
((log‘𝑥) ·
(𝑥↑𝑐-(1 / 2))) =
((log‘𝑥) · (1
/ (√‘𝑥)))) |
34 | 22, 33 | eqtr4d 2659 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
((log‘𝑥) /
(√‘𝑥)) =
((log‘𝑥) ·
(𝑥↑𝑐-(1 /
2)))) |
35 | 34 | mpteq2dva 4744 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦
((log‘𝑥) /
(√‘𝑥))) =
(𝑥 ∈
ℝ+ ↦ ((log‘𝑥) · (𝑥↑𝑐-(1 /
2))))) |
36 | 35 | oveq2d 6666 |
. . . . 5
⊢ (𝜑 → (ℝ D (𝑥 ∈ ℝ+
↦ ((log‘𝑥) /
(√‘𝑥)))) =
(ℝ D (𝑥 ∈
ℝ+ ↦ ((log‘𝑥) · (𝑥↑𝑐-(1 /
2)))))) |
37 | | reelprrecn 10028 |
. . . . . . 7
⊢ ℝ
∈ {ℝ, ℂ} |
38 | 37 | a1i 11 |
. . . . . 6
⊢ (𝜑 → ℝ ∈ {ℝ,
ℂ}) |
39 | 5 | rpreccld 11882 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (1 /
𝑥) ∈
ℝ+) |
40 | | dvrelog 24383 |
. . . . . . 7
⊢ (ℝ
D (log ↾ ℝ+)) = (𝑥 ∈ ℝ+ ↦ (1 /
𝑥)) |
41 | | logf1o 24311 |
. . . . . . . . . . 11
⊢
log:(ℂ ∖ {0})–1-1-onto→ran
log |
42 | | f1of 6137 |
. . . . . . . . . . 11
⊢
(log:(ℂ ∖ {0})–1-1-onto→ran
log → log:(ℂ ∖ {0})⟶ran log) |
43 | 41, 42 | ax-mp 5 |
. . . . . . . . . 10
⊢
log:(ℂ ∖ {0})⟶ran log |
44 | 43 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → log:(ℂ ∖
{0})⟶ran log) |
45 | 16 | ssriv 3607 |
. . . . . . . . . . 11
⊢
ℝ+ ⊆ ℂ |
46 | | 0nrp 11865 |
. . . . . . . . . . 11
⊢ ¬ 0
∈ ℝ+ |
47 | | ssdifsn 4318 |
. . . . . . . . . . 11
⊢
(ℝ+ ⊆ (ℂ ∖ {0}) ↔
(ℝ+ ⊆ ℂ ∧ ¬ 0 ∈
ℝ+)) |
48 | 45, 46, 47 | mpbir2an 955 |
. . . . . . . . . 10
⊢
ℝ+ ⊆ (ℂ ∖ {0}) |
49 | 48 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → ℝ+
⊆ (ℂ ∖ {0})) |
50 | 44, 49 | feqresmpt 6250 |
. . . . . . . 8
⊢ (𝜑 → (log ↾
ℝ+) = (𝑥
∈ ℝ+ ↦ (log‘𝑥))) |
51 | 50 | oveq2d 6666 |
. . . . . . 7
⊢ (𝜑 → (ℝ D (log ↾
ℝ+)) = (ℝ D (𝑥 ∈ ℝ+ ↦
(log‘𝑥)))) |
52 | 40, 51 | syl5reqr 2671 |
. . . . . 6
⊢ (𝜑 → (ℝ D (𝑥 ∈ ℝ+
↦ (log‘𝑥))) =
(𝑥 ∈
ℝ+ ↦ (1 / 𝑥))) |
53 | | 1cnd 10056 |
. . . . . . . . . 10
⊢ (𝜑 → 1 ∈
ℂ) |
54 | 53 | halfcld 11277 |
. . . . . . . . 9
⊢ (𝜑 → (1 / 2) ∈
ℂ) |
55 | 54 | negcld 10379 |
. . . . . . . 8
⊢ (𝜑 → -(1 / 2) ∈
ℂ) |
56 | 55 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → -(1 / 2)
∈ ℂ) |
57 | 17, 56 | cxpcld 24454 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (𝑥↑𝑐-(1 /
2)) ∈ ℂ) |
58 | 53 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 1 ∈
ℂ) |
59 | 56, 58 | subcld 10392 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (-(1 / 2)
− 1) ∈ ℂ) |
60 | 17, 59 | cxpcld 24454 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (𝑥↑𝑐(-(1 /
2) − 1)) ∈ ℂ) |
61 | 56, 60 | mulcld 10060 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (-(1 / 2)
· (𝑥↑𝑐(-(1 / 2) −
1))) ∈ ℂ) |
62 | | dvcxp1 24481 |
. . . . . . 7
⊢ (-(1 / 2)
∈ ℂ → (ℝ D (𝑥 ∈ ℝ+ ↦ (𝑥↑𝑐-(1 /
2)))) = (𝑥 ∈
ℝ+ ↦ (-(1 / 2) · (𝑥↑𝑐(-(1 / 2) −
1))))) |
63 | 55, 62 | syl 17 |
. . . . . 6
⊢ (𝜑 → (ℝ D (𝑥 ∈ ℝ+
↦ (𝑥↑𝑐-(1 / 2)))) =
(𝑥 ∈
ℝ+ ↦ (-(1 / 2) · (𝑥↑𝑐(-(1 / 2) −
1))))) |
64 | 38, 20, 39, 52, 57, 61, 63 | dvmptmul 23724 |
. . . . 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))))) |
65 | 36, 64 | eqtrd 2656 |
. . . 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))))) |
66 | | ax-resscn 9993 |
. . . . . 6
⊢ ℝ
⊆ ℂ |
67 | 66 | a1i 11 |
. . . . 5
⊢ (𝜑 → ℝ ⊆
ℂ) |
68 | | eqid 2622 |
. . . . . 6
⊢
(TopOpen‘ℂfld) =
(TopOpen‘ℂfld) |
69 | 68 | addcn 22668 |
. . . . . . 7
⊢ + ∈
(((TopOpen‘ℂfld) ×t
(TopOpen‘ℂfld)) Cn
(TopOpen‘ℂfld)) |
70 | 69 | a1i 11 |
. . . . . 6
⊢ (𝜑 → + ∈
(((TopOpen‘ℂfld) ×t
(TopOpen‘ℂfld)) Cn
(TopOpen‘ℂfld))) |
71 | 45 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → ℝ+
⊆ ℂ) |
72 | | ssid 3624 |
. . . . . . . . . 10
⊢ ℂ
⊆ ℂ |
73 | 72 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → ℂ ⊆
ℂ) |
74 | | cncfmptc 22714 |
. . . . . . . . 9
⊢ ((1
∈ ℂ ∧ ℝ+ ⊆ ℂ ∧ ℂ ⊆
ℂ) → (𝑥 ∈
ℝ+ ↦ 1) ∈ (ℝ+–cn→ℂ)) |
75 | 53, 71, 73, 74 | syl3anc 1326 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ 1) ∈
(ℝ+–cn→ℂ)) |
76 | | difss 3737 |
. . . . . . . . 9
⊢ (ℂ
∖ {0}) ⊆ ℂ |
77 | | cncfmptid 22715 |
. . . . . . . . 9
⊢
((ℝ+ ⊆ (ℂ ∖ {0}) ∧ (ℂ
∖ {0}) ⊆ ℂ) → (𝑥 ∈ ℝ+ ↦ 𝑥) ∈
(ℝ+–cn→(ℂ ∖ {0}))) |
78 | 49, 76, 77 | sylancl 694 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ 𝑥) ∈
(ℝ+–cn→(ℂ ∖ {0}))) |
79 | 75, 78 | divcncf 23216 |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ (1 /
𝑥)) ∈
(ℝ+–cn→ℂ)) |
80 | | ax-1 6 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℝ+
→ (𝑥 ∈ ℝ
→ 𝑥 ∈
ℝ+)) |
81 | 16, 80 | jca 554 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ℝ+
→ (𝑥 ∈ ℂ
∧ (𝑥 ∈ ℝ
→ 𝑥 ∈
ℝ+))) |
82 | | eqid 2622 |
. . . . . . . . . . . 12
⊢ (ℂ
∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0)) |
83 | 82 | ellogdm 24385 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ (ℂ ∖
(-∞(,]0)) ↔ (𝑥
∈ ℂ ∧ (𝑥
∈ ℝ → 𝑥
∈ ℝ+))) |
84 | 81, 83 | sylibr 224 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℝ+
→ 𝑥 ∈ (ℂ
∖ (-∞(,]0))) |
85 | 84 | ssriv 3607 |
. . . . . . . . 9
⊢
ℝ+ ⊆ (ℂ ∖
(-∞(,]0)) |
86 | 85 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → ℝ+
⊆ (ℂ ∖ (-∞(,]0))) |
87 | 55, 86 | cxpcncf1 30673 |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ (𝑥↑𝑐-(1 /
2))) ∈ (ℝ+–cn→ℂ)) |
88 | 79, 87 | mulcncf 23215 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ ((1 /
𝑥) · (𝑥↑𝑐-(1 /
2)))) ∈ (ℝ+–cn→ℂ)) |
89 | | cncfmptc 22714 |
. . . . . . . . 9
⊢ ((-(1 /
2) ∈ ℂ ∧ ℝ+ ⊆ ℂ ∧ ℂ
⊆ ℂ) → (𝑥
∈ ℝ+ ↦ -(1 / 2)) ∈
(ℝ+–cn→ℂ)) |
90 | 55, 71, 73, 89 | syl3anc 1326 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ -(1 / 2))
∈ (ℝ+–cn→ℂ)) |
91 | 55, 53 | subcld 10392 |
. . . . . . . . 9
⊢ (𝜑 → (-(1 / 2) − 1)
∈ ℂ) |
92 | 91, 86 | cxpcncf1 30673 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ (𝑥↑𝑐(-(1 /
2) − 1))) ∈ (ℝ+–cn→ℂ)) |
93 | 90, 92 | mulcncf 23215 |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ (-(1 / 2)
· (𝑥↑𝑐(-(1 / 2) −
1)))) ∈ (ℝ+–cn→ℂ)) |
94 | | cncfss 22702 |
. . . . . . . . 9
⊢ ((ℝ
⊆ ℂ ∧ ℂ ⊆ ℂ) →
(ℝ+–cn→ℝ) ⊆
(ℝ+–cn→ℂ)) |
95 | 66, 72, 94 | mp2an 708 |
. . . . . . . 8
⊢
(ℝ+–cn→ℝ) ⊆
(ℝ+–cn→ℂ) |
96 | | relogcn 24384 |
. . . . . . . . 9
⊢ (log
↾ ℝ+) ∈ (ℝ+–cn→ℝ) |
97 | 50, 96 | syl6eqelr 2710 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦
(log‘𝑥)) ∈
(ℝ+–cn→ℝ)) |
98 | 95, 97 | sseldi 3601 |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦
(log‘𝑥)) ∈
(ℝ+–cn→ℂ)) |
99 | 93, 98 | mulcncf 23215 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ ((-(1 /
2) · (𝑥↑𝑐(-(1 / 2) −
1))) · (log‘𝑥))) ∈ (ℝ+–cn→ℂ)) |
100 | 68, 70, 88, 99 | cncfmpt2f 22717 |
. . . . 5
⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ (((1 /
𝑥) · (𝑥↑𝑐-(1 /
2))) + ((-(1 / 2) · (𝑥↑𝑐(-(1 / 2) −
1))) · (log‘𝑥)))) ∈ (ℝ+–cn→ℂ)) |
101 | | rpre 11839 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℝ+
→ 𝑥 ∈
ℝ) |
102 | 101, 18 | rereccld 10852 |
. . . . . . . . 9
⊢ (𝑥 ∈ ℝ+
→ (1 / 𝑥) ∈
ℝ) |
103 | | rpge0 11845 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℝ+
→ 0 ≤ 𝑥) |
104 | | halfre 11246 |
. . . . . . . . . . . 12
⊢ (1 / 2)
∈ ℝ |
105 | 104 | renegcli 10342 |
. . . . . . . . . . 11
⊢ -(1 / 2)
∈ ℝ |
106 | 105 | a1i 11 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℝ+
→ -(1 / 2) ∈ ℝ) |
107 | 101, 103,
106 | recxpcld 24469 |
. . . . . . . . 9
⊢ (𝑥 ∈ ℝ+
→ (𝑥↑𝑐-(1 / 2)) ∈
ℝ) |
108 | 102, 107 | remulcld 10070 |
. . . . . . . 8
⊢ (𝑥 ∈ ℝ+
→ ((1 / 𝑥) ·
(𝑥↑𝑐-(1 / 2))) ∈
ℝ) |
109 | | 1re 10039 |
. . . . . . . . . . . . 13
⊢ 1 ∈
ℝ |
110 | 105, 109 | resubcli 10343 |
. . . . . . . . . . . 12
⊢ (-(1 / 2)
− 1) ∈ ℝ |
111 | 110 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ℝ+
→ (-(1 / 2) − 1) ∈ ℝ) |
112 | 101, 103,
111 | recxpcld 24469 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℝ+
→ (𝑥↑𝑐(-(1 / 2) −
1)) ∈ ℝ) |
113 | 106, 112 | remulcld 10070 |
. . . . . . . . 9
⊢ (𝑥 ∈ ℝ+
→ (-(1 / 2) · (𝑥↑𝑐(-(1 / 2) −
1))) ∈ ℝ) |
114 | | relogcl 24322 |
. . . . . . . . 9
⊢ (𝑥 ∈ ℝ+
→ (log‘𝑥) ∈
ℝ) |
115 | 113, 114 | remulcld 10070 |
. . . . . . . 8
⊢ (𝑥 ∈ ℝ+
→ ((-(1 / 2) · (𝑥↑𝑐(-(1 / 2) −
1))) · (log‘𝑥)) ∈ ℝ) |
116 | 108, 115 | readdcld 10069 |
. . . . . . 7
⊢ (𝑥 ∈ ℝ+
→ (((1 / 𝑥) ·
(𝑥↑𝑐-(1 / 2))) + ((-(1
/ 2) · (𝑥↑𝑐(-(1 / 2) −
1))) · (log‘𝑥))) ∈ ℝ) |
117 | 116 | adantl 482 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (((1 /
𝑥) · (𝑥↑𝑐-(1 /
2))) + ((-(1 / 2) · (𝑥↑𝑐(-(1 / 2) −
1))) · (log‘𝑥))) ∈ ℝ) |
118 | | eqid 2622 |
. . . . . 6
⊢ (𝑥 ∈ ℝ+
↦ (((1 / 𝑥) ·
(𝑥↑𝑐-(1 / 2))) + ((-(1
/ 2) · (𝑥↑𝑐(-(1 / 2) −
1))) · (log‘𝑥)))) = (𝑥 ∈ ℝ+ ↦ (((1 /
𝑥) · (𝑥↑𝑐-(1 /
2))) + ((-(1 / 2) · (𝑥↑𝑐(-(1 / 2) −
1))) · (log‘𝑥)))) |
119 | 117, 118 | fmptd 6385 |
. . . . 5
⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ (((1 /
𝑥) · (𝑥↑𝑐-(1 /
2))) + ((-(1 / 2) · (𝑥↑𝑐(-(1 / 2) −
1))) · (log‘𝑥)))):ℝ+⟶ℝ) |
120 | | cncffvrn 22701 |
. . . . . 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)) |
121 | 120 | biimpar 502 |
. . . . 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)) |
122 | 67, 100, 119, 121 | syl21anc 1325 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ (((1 /
𝑥) · (𝑥↑𝑐-(1 /
2))) + ((-(1 / 2) · (𝑥↑𝑐(-(1 / 2) −
1))) · (log‘𝑥)))) ∈ (ℝ+–cn→ℝ)) |
123 | 65, 122 | eqeltrd 2701 |
. . 3
⊢ (𝜑 → (ℝ D (𝑥 ∈ ℝ+
↦ ((log‘𝑥) /
(√‘𝑥)))) ∈
(ℝ+–cn→ℝ)) |
124 | | logdivsqrle.2 |
. . 3
⊢ (𝜑 → 𝐴 ≤ 𝐵) |
125 | 65 | fveq1d 6193 |
. . . . 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)) |
126 | 125 | adantr 481 |
. . . 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)) |
127 | 58 | negcld 10379 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → -1 ∈
ℂ) |
128 | | cxpadd 24425 |
. . . . . . . . . . . 12
⊢ (((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) ∧ -(1 / 2) ∈
ℂ ∧ -1 ∈ ℂ) → (𝑥↑𝑐(-(1 / 2) + -1)) =
((𝑥↑𝑐-(1 / 2)) ·
(𝑥↑𝑐-1))) |
129 | 17, 19, 56, 127, 128 | syl211anc 1332 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (𝑥↑𝑐(-(1 /
2) + -1)) = ((𝑥↑𝑐-(1 / 2)) ·
(𝑥↑𝑐-1))) |
130 | 60 | mulid2d 10058 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (1
· (𝑥↑𝑐(-(1 / 2) −
1))) = (𝑥↑𝑐(-(1 / 2) −
1))) |
131 | 56, 58 | negsubd 10398 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (-(1 / 2)
+ -1) = (-(1 / 2) − 1)) |
132 | 131 | oveq2d 6666 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (𝑥↑𝑐(-(1 /
2) + -1)) = (𝑥↑𝑐(-(1 / 2) −
1))) |
133 | 130, 132 | eqtr4d 2659 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (1
· (𝑥↑𝑐(-(1 / 2) −
1))) = (𝑥↑𝑐(-(1 / 2) +
-1))) |
134 | 45, 39 | sseldi 3601 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (1 /
𝑥) ∈
ℂ) |
135 | 134, 57 | mulcomd 10061 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ((1 /
𝑥) · (𝑥↑𝑐-(1 /
2))) = ((𝑥↑𝑐-(1 / 2)) ·
(1 / 𝑥))) |
136 | | cxpneg 24427 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0 ∧ 1 ∈ ℂ)
→ (𝑥↑𝑐-1) = (1 / (𝑥↑𝑐1))) |
137 | 17, 19, 58, 136 | syl3anc 1326 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (𝑥↑𝑐-1) =
(1 / (𝑥↑𝑐1))) |
138 | 17 | cxp1d 24452 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (𝑥↑𝑐1) =
𝑥) |
139 | 138 | oveq2d 6666 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (1 /
(𝑥↑𝑐1)) = (1 / 𝑥)) |
140 | 137, 139 | eqtr2d 2657 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (1 /
𝑥) = (𝑥↑𝑐-1)) |
141 | 140 | oveq2d 6666 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ((𝑥↑𝑐-(1 /
2)) · (1 / 𝑥)) =
((𝑥↑𝑐-(1 / 2)) ·
(𝑥↑𝑐-1))) |
142 | 135, 141 | eqtrd 2656 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ((1 /
𝑥) · (𝑥↑𝑐-(1 /
2))) = ((𝑥↑𝑐-(1 / 2)) ·
(𝑥↑𝑐-1))) |
143 | 129, 133,
142 | 3eqtr4rd 2667 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ((1 /
𝑥) · (𝑥↑𝑐-(1 /
2))) = (1 · (𝑥↑𝑐(-(1 / 2) −
1)))) |
144 | 56, 60, 20 | mul32d 10246 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ((-(1 /
2) · (𝑥↑𝑐(-(1 / 2) −
1))) · (log‘𝑥)) = ((-(1 / 2) · (log‘𝑥)) · (𝑥↑𝑐(-(1 / 2) −
1)))) |
145 | 143, 144 | oveq12d 6668 |
. . . . . . . . 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))))) |
146 | 56, 20 | mulcld 10060 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (-(1 / 2)
· (log‘𝑥))
∈ ℂ) |
147 | 58, 146, 60 | adddird 10065 |
. . . . . . . . 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))))) |
148 | 145, 147 | eqtr4d 2659 |
. . . . . . . 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)))) |
149 | 148 | mpteq2dva 4744 |
. . . . . . 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))))) |
150 | 149 | fveq1d 6193 |
. . . . . 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)) |
151 | 150 | adantr 481 |
. . . . 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)) |
152 | | eqidd 2623 |
. . . . . . 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))))) |
153 | | simpr 477 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑦) → 𝑥 = 𝑦) |
154 | 153 | fveq2d 6195 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑦) → (log‘𝑥) = (log‘𝑦)) |
155 | 154 | oveq2d 6666 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑦) → (-(1 / 2) · (log‘𝑥)) = (-(1 / 2) ·
(log‘𝑦))) |
156 | 155 | oveq2d 6666 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑦) → (1 + (-(1 / 2) ·
(log‘𝑥))) = (1 + (-(1
/ 2) · (log‘𝑦)))) |
157 | 153 | oveq1d 6665 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑦) → (𝑥↑𝑐(-(1 / 2) −
1)) = (𝑦↑𝑐(-(1 / 2) −
1))) |
158 | 156, 157 | oveq12d 6668 |
. . . . . . 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)))) |
159 | | ioossicc 12259 |
. . . . . . . . . 10
⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵) |
160 | 159 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)) |
161 | 2, 3, 4 | fct2relem 30675 |
. . . . . . . . 9
⊢ (𝜑 → (𝐴[,]𝐵) ⊆
ℝ+) |
162 | 160, 161 | sstrd 3613 |
. . . . . . . 8
⊢ (𝜑 → (𝐴(,)𝐵) ⊆
ℝ+) |
163 | 162 | sselda 3603 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ ℝ+) |
164 | | ovexd 6680 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((1 + (-(1 / 2) ·
(log‘𝑦))) ·
(𝑦↑𝑐(-(1 / 2) −
1))) ∈ V) |
165 | 152, 158,
163, 164 | fvmptd 6288 |
. . . . . 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)))) |
166 | 109 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 1 ∈ ℝ) |
167 | 105 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → -(1 / 2) ∈
ℝ) |
168 | 163 | relogcld 24369 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (log‘𝑦) ∈ ℝ) |
169 | 167, 168 | remulcld 10070 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (-(1 / 2) ·
(log‘𝑦)) ∈
ℝ) |
170 | 166, 169 | readdcld 10069 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (1 + (-(1 / 2) ·
(log‘𝑦))) ∈
ℝ) |
171 | | 0red 10041 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 0 ∈ ℝ) |
172 | | rpcxpcl 24422 |
. . . . . . . . . 10
⊢ ((𝑦 ∈ ℝ+
∧ (-(1 / 2) − 1) ∈ ℝ) → (𝑦↑𝑐(-(1 / 2) −
1)) ∈ ℝ+) |
173 | 163, 110,
172 | sylancl 694 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝑦↑𝑐(-(1 / 2) −
1)) ∈ ℝ+) |
174 | 173 | rpred 11872 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝑦↑𝑐(-(1 / 2) −
1)) ∈ ℝ) |
175 | 173 | rpge0d 11876 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 0 ≤ (𝑦↑𝑐(-(1 / 2) −
1))) |
176 | | 2cn 11091 |
. . . . . . . . . . . . . 14
⊢ 2 ∈
ℂ |
177 | 176 | mulid2i 10043 |
. . . . . . . . . . . . 13
⊢ (1
· 2) = 2 |
178 | | 2re 11090 |
. . . . . . . . . . . . . . . . . 18
⊢ 2 ∈
ℝ |
179 | 178 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 2 ∈ ℝ) |
180 | 179 | reefcld 14818 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (exp‘2) ∈
ℝ) |
181 | 3 | rpred 11872 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝐴 ∈ ℝ) |
182 | 181 | adantr 481 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ) |
183 | 163 | rpred 11872 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ ℝ) |
184 | | logdivsqrle.1 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (exp‘2) ≤ 𝐴) |
185 | 184 | adantr 481 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (exp‘2) ≤ 𝐴) |
186 | | eliooord 12233 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 ∈ (𝐴(,)𝐵) → (𝐴 < 𝑦 ∧ 𝑦 < 𝐵)) |
187 | 186 | simpld 475 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 ∈ (𝐴(,)𝐵) → 𝐴 < 𝑦) |
188 | 187 | adantl 482 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐴 < 𝑦) |
189 | 182, 183,
188 | ltled 10185 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐴 ≤ 𝑦) |
190 | 180, 182,
183, 185, 189 | letrd 10194 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (exp‘2) ≤ 𝑦) |
191 | | reeflog 24327 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 ∈ ℝ+
→ (exp‘(log‘𝑦)) = 𝑦) |
192 | 163, 191 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (exp‘(log‘𝑦)) = 𝑦) |
193 | 190, 192 | breqtrrd 4681 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (exp‘2) ≤
(exp‘(log‘𝑦))) |
194 | | efle 14848 |
. . . . . . . . . . . . . . 15
⊢ ((2
∈ ℝ ∧ (log‘𝑦) ∈ ℝ) → (2 ≤
(log‘𝑦) ↔
(exp‘2) ≤ (exp‘(log‘𝑦)))) |
195 | 178, 168,
194 | sylancr 695 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (2 ≤ (log‘𝑦) ↔ (exp‘2) ≤
(exp‘(log‘𝑦)))) |
196 | 193, 195 | mpbird 247 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 2 ≤ (log‘𝑦)) |
197 | 177, 196 | syl5eqbr 4688 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (1 · 2) ≤
(log‘𝑦)) |
198 | | 2rp 11837 |
. . . . . . . . . . . . . 14
⊢ 2 ∈
ℝ+ |
199 | 198 | a1i 11 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 2 ∈
ℝ+) |
200 | 166, 168,
199 | lemuldivd 11921 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((1 · 2) ≤
(log‘𝑦) ↔ 1 ≤
((log‘𝑦) /
2))) |
201 | 197, 200 | mpbid 222 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 1 ≤ ((log‘𝑦) / 2)) |
202 | 66, 168 | sseldi 3601 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (log‘𝑦) ∈ ℂ) |
203 | 23 | adantr 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 2 ∈ ℂ) |
204 | 25 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 2 ≠ 0) |
205 | 202, 203,
204 | divrec2d 10805 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((log‘𝑦) / 2) = ((1 / 2) · (log‘𝑦))) |
206 | 201, 205 | breqtrd 4679 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 1 ≤ ((1 / 2) ·
(log‘𝑦))) |
207 | 54 | adantr 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (1 / 2) ∈
ℂ) |
208 | 207, 202 | mulneg1d 10483 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (-(1 / 2) ·
(log‘𝑦)) = -((1 / 2)
· (log‘𝑦))) |
209 | 208 | oveq2d 6666 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (0 − (-(1 / 2) ·
(log‘𝑦))) = (0
− -((1 / 2) · (log‘𝑦)))) |
210 | 66, 171 | sseldi 3601 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 0 ∈ ℂ) |
211 | 207, 202 | mulcld 10060 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((1 / 2) · (log‘𝑦)) ∈
ℂ) |
212 | 210, 211 | subnegd 10399 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (0 − -((1 / 2) ·
(log‘𝑦))) = (0 + ((1
/ 2) · (log‘𝑦)))) |
213 | 211 | addid2d 10237 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (0 + ((1 / 2) ·
(log‘𝑦))) = ((1 / 2)
· (log‘𝑦))) |
214 | 209, 212,
213 | 3eqtrd 2660 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (0 − (-(1 / 2) ·
(log‘𝑦))) = ((1 / 2)
· (log‘𝑦))) |
215 | 206, 214 | breqtrrd 4681 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 1 ≤ (0 − (-(1 / 2)
· (log‘𝑦)))) |
216 | | leaddsub 10504 |
. . . . . . . . . 10
⊢ ((1
∈ ℝ ∧ (-(1 / 2) · (log‘𝑦)) ∈ ℝ ∧ 0 ∈ ℝ)
→ ((1 + (-(1 / 2) · (log‘𝑦))) ≤ 0 ↔ 1 ≤ (0 − (-(1 / 2)
· (log‘𝑦))))) |
217 | 166, 169,
171, 216 | syl3anc 1326 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((1 + (-(1 / 2) ·
(log‘𝑦))) ≤ 0
↔ 1 ≤ (0 − (-(1 / 2) · (log‘𝑦))))) |
218 | 215, 217 | mpbird 247 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (1 + (-(1 / 2) ·
(log‘𝑦))) ≤
0) |
219 | 170, 171,
174, 175, 218 | lemul1ad 10963 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((1 + (-(1 / 2) ·
(log‘𝑦))) ·
(𝑦↑𝑐(-(1 / 2) −
1))) ≤ (0 · (𝑦↑𝑐(-(1 / 2) −
1)))) |
220 | 45, 173 | sseldi 3601 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝑦↑𝑐(-(1 / 2) −
1)) ∈ ℂ) |
221 | 220 | mul02d 10234 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (0 · (𝑦↑𝑐(-(1 / 2) −
1))) = 0) |
222 | 219, 221 | breqtrd 4679 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((1 + (-(1 / 2) ·
(log‘𝑦))) ·
(𝑦↑𝑐(-(1 / 2) −
1))) ≤ 0) |
223 | 165, 222 | eqbrtrd 4675 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((𝑥 ∈ ℝ+ ↦ ((1 +
(-(1 / 2) · (log‘𝑥))) · (𝑥↑𝑐(-(1 / 2) −
1))))‘𝑦) ≤
0) |
224 | 151, 223 | eqbrtrd 4675 |
. . . 4
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((𝑥 ∈ ℝ+ ↦ (((1 /
𝑥) · (𝑥↑𝑐-(1 /
2))) + ((-(1 / 2) · (𝑥↑𝑐(-(1 / 2) −
1))) · (log‘𝑥))))‘𝑦) ≤ 0) |
225 | 126, 224 | eqbrtrd 4675 |
. . 3
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((ℝ D (𝑥 ∈ ℝ+ ↦
((log‘𝑥) /
(√‘𝑥))))‘𝑦) ≤ 0) |
226 | 2, 3, 4, 15, 123, 124, 225 | fdvnegge 30680 |
. 2
⊢ (𝜑 → ((𝑥 ∈ ℝ+ ↦
((log‘𝑥) /
(√‘𝑥)))‘𝐵) ≤ ((𝑥 ∈ ℝ+ ↦
((log‘𝑥) /
(√‘𝑥)))‘𝐴)) |
227 | | eqidd 2623 |
. . 3
⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦
((log‘𝑥) /
(√‘𝑥))) =
(𝑥 ∈
ℝ+ ↦ ((log‘𝑥) / (√‘𝑥)))) |
228 | | simpr 477 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝑥 = 𝐵) |
229 | 228 | fveq2d 6195 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 = 𝐵) → (log‘𝑥) = (log‘𝐵)) |
230 | 228 | fveq2d 6195 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 = 𝐵) → (√‘𝑥) = (√‘𝐵)) |
231 | 229, 230 | oveq12d 6668 |
. . 3
⊢ ((𝜑 ∧ 𝑥 = 𝐵) → ((log‘𝑥) / (√‘𝑥)) = ((log‘𝐵) / (√‘𝐵))) |
232 | | ovex 6678 |
. . . 4
⊢
((log‘𝐵) /
(√‘𝐵)) ∈
V |
233 | 232 | a1i 11 |
. . 3
⊢ (𝜑 → ((log‘𝐵) / (√‘𝐵)) ∈ V) |
234 | 227, 231,
4, 233 | fvmptd 6288 |
. 2
⊢ (𝜑 → ((𝑥 ∈ ℝ+ ↦
((log‘𝑥) /
(√‘𝑥)))‘𝐵) = ((log‘𝐵) / (√‘𝐵))) |
235 | | simpr 477 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝑥 = 𝐴) |
236 | 235 | fveq2d 6195 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (log‘𝑥) = (log‘𝐴)) |
237 | 235 | fveq2d 6195 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (√‘𝑥) = (√‘𝐴)) |
238 | 236, 237 | oveq12d 6668 |
. . 3
⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((log‘𝑥) / (√‘𝑥)) = ((log‘𝐴) / (√‘𝐴))) |
239 | | ovex 6678 |
. . . 4
⊢
((log‘𝐴) /
(√‘𝐴)) ∈
V |
240 | 239 | a1i 11 |
. . 3
⊢ (𝜑 → ((log‘𝐴) / (√‘𝐴)) ∈ V) |
241 | 227, 238,
3, 240 | fvmptd 6288 |
. 2
⊢ (𝜑 → ((𝑥 ∈ ℝ+ ↦
((log‘𝑥) /
(√‘𝑥)))‘𝐴) = ((log‘𝐴) / (√‘𝐴))) |
242 | 226, 234,
241 | 3brtr3d 4684 |
1
⊢ (𝜑 → ((log‘𝐵) / (√‘𝐵)) ≤ ((log‘𝐴) / (√‘𝐴))) |