Proof of Theorem jensenlem2
Step | Hyp | Ref
| Expression |
1 | | cnfld0 20632 |
. . . . . . 7
⊢ 0 =
(0g‘ℂfld) |
2 | | cnring 20630 |
. . . . . . . 8
⊢
ℂfld ∈ Ring |
3 | | ringabl 19829 |
. . . . . . . 8
⊢
(ℂfld ∈ Ring → ℂfld ∈
Abel) |
4 | 2, 3 | mp1i 13 |
. . . . . . 7
⊢ (𝜑 → ℂfld
∈ Abel) |
5 | | jensen.4 |
. . . . . . . 8
⊢ (𝜑 → 𝐴 ∈ Fin) |
6 | | jensenlem.2 |
. . . . . . . . 9
⊢ (𝜑 → (𝐵 ∪ {𝑧}) ⊆ 𝐴) |
7 | 6 | unssad 4120 |
. . . . . . . 8
⊢ (𝜑 → 𝐵 ⊆ 𝐴) |
8 | 5, 7 | ssfid 9029 |
. . . . . . 7
⊢ (𝜑 → 𝐵 ∈ Fin) |
9 | | resubdrg 20823 |
. . . . . . . . 9
⊢ (ℝ
∈ (SubRing‘ℂfld) ∧ ℝfld ∈
DivRing) |
10 | 9 | simpli 484 |
. . . . . . . 8
⊢ ℝ
∈ (SubRing‘ℂfld) |
11 | | subrgsubg 20040 |
. . . . . . . 8
⊢ (ℝ
∈ (SubRing‘ℂfld) → ℝ ∈
(SubGrp‘ℂfld)) |
12 | 10, 11 | mp1i 13 |
. . . . . . 7
⊢ (𝜑 → ℝ ∈
(SubGrp‘ℂfld)) |
13 | | remulcl 10966 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 · 𝑦) ∈ ℝ) |
14 | 13 | adantl 482 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 · 𝑦) ∈ ℝ) |
15 | | jensen.5 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑇:𝐴⟶(0[,)+∞)) |
16 | | rge0ssre 13198 |
. . . . . . . . . 10
⊢
(0[,)+∞) ⊆ ℝ |
17 | | fss 6609 |
. . . . . . . . . 10
⊢ ((𝑇:𝐴⟶(0[,)+∞) ∧ (0[,)+∞)
⊆ ℝ) → 𝑇:𝐴⟶ℝ) |
18 | 15, 16, 17 | sylancl 586 |
. . . . . . . . 9
⊢ (𝜑 → 𝑇:𝐴⟶ℝ) |
19 | | jensen.6 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑋:𝐴⟶𝐷) |
20 | | jensen.1 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐷 ⊆ ℝ) |
21 | 19, 20 | fssd 6610 |
. . . . . . . . 9
⊢ (𝜑 → 𝑋:𝐴⟶ℝ) |
22 | | inidm 4152 |
. . . . . . . . 9
⊢ (𝐴 ∩ 𝐴) = 𝐴 |
23 | 14, 18, 21, 5, 5, 22 | off 7541 |
. . . . . . . 8
⊢ (𝜑 → (𝑇 ∘f · 𝑋):𝐴⟶ℝ) |
24 | 23, 7 | fssresd 6633 |
. . . . . . 7
⊢ (𝜑 → ((𝑇 ∘f · 𝑋) ↾ 𝐵):𝐵⟶ℝ) |
25 | | c0ex 10979 |
. . . . . . . . 9
⊢ 0 ∈
V |
26 | 25 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → 0 ∈
V) |
27 | 24, 8, 26 | fdmfifsupp 9125 |
. . . . . . 7
⊢ (𝜑 → ((𝑇 ∘f · 𝑋) ↾ 𝐵) finSupp 0) |
28 | 1, 4, 8, 12, 24, 27 | gsumsubgcl 19531 |
. . . . . 6
⊢ (𝜑 → (ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) ∈ ℝ) |
29 | 28 | recnd 11013 |
. . . . 5
⊢ (𝜑 → (ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) ∈ ℂ) |
30 | | ax-resscn 10938 |
. . . . . . . 8
⊢ ℝ
⊆ ℂ |
31 | 16, 30 | sstri 3929 |
. . . . . . 7
⊢
(0[,)+∞) ⊆ ℂ |
32 | 6 | unssbd 4121 |
. . . . . . . . 9
⊢ (𝜑 → {𝑧} ⊆ 𝐴) |
33 | | vex 3433 |
. . . . . . . . . 10
⊢ 𝑧 ∈ V |
34 | 33 | snss 4719 |
. . . . . . . . 9
⊢ (𝑧 ∈ 𝐴 ↔ {𝑧} ⊆ 𝐴) |
35 | 32, 34 | sylibr 233 |
. . . . . . . 8
⊢ (𝜑 → 𝑧 ∈ 𝐴) |
36 | 15, 35 | ffvelrnd 6954 |
. . . . . . 7
⊢ (𝜑 → (𝑇‘𝑧) ∈ (0[,)+∞)) |
37 | 31, 36 | sselid 3918 |
. . . . . 6
⊢ (𝜑 → (𝑇‘𝑧) ∈ ℂ) |
38 | 19, 35 | ffvelrnd 6954 |
. . . . . . . 8
⊢ (𝜑 → (𝑋‘𝑧) ∈ 𝐷) |
39 | 20, 38 | sseldd 3921 |
. . . . . . 7
⊢ (𝜑 → (𝑋‘𝑧) ∈ ℝ) |
40 | 39 | recnd 11013 |
. . . . . 6
⊢ (𝜑 → (𝑋‘𝑧) ∈ ℂ) |
41 | 37, 40 | mulcld 11005 |
. . . . 5
⊢ (𝜑 → ((𝑇‘𝑧) · (𝑋‘𝑧)) ∈ ℂ) |
42 | | jensen.2 |
. . . . . . . 8
⊢ (𝜑 → 𝐹:𝐷⟶ℝ) |
43 | | jensen.3 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑎 ∈ 𝐷 ∧ 𝑏 ∈ 𝐷)) → (𝑎[,]𝑏) ⊆ 𝐷) |
44 | | jensen.7 |
. . . . . . . 8
⊢ (𝜑 → 0 <
(ℂfld Σg 𝑇)) |
45 | | jensen.8 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷 ∧ 𝑡 ∈ (0[,]1))) → (𝐹‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ≤ ((𝑡 · (𝐹‘𝑥)) + ((1 − 𝑡) · (𝐹‘𝑦)))) |
46 | | jensenlem.1 |
. . . . . . . 8
⊢ (𝜑 → ¬ 𝑧 ∈ 𝐵) |
47 | | jensenlem.s |
. . . . . . . 8
⊢ 𝑆 = (ℂfld
Σg (𝑇 ↾ 𝐵)) |
48 | | jensenlem.l |
. . . . . . . 8
⊢ 𝐿 = (ℂfld
Σg (𝑇 ↾ (𝐵 ∪ {𝑧}))) |
49 | 20, 42, 43, 5, 15, 19, 44, 45, 46, 6, 47, 48 | jensenlem1 26146 |
. . . . . . 7
⊢ (𝜑 → 𝐿 = (𝑆 + (𝑇‘𝑧))) |
50 | | jensenlem.3 |
. . . . . . . . 9
⊢ (𝜑 → 𝑆 ∈
ℝ+) |
51 | 50 | rpred 12782 |
. . . . . . . 8
⊢ (𝜑 → 𝑆 ∈ ℝ) |
52 | | elrege0 13196 |
. . . . . . . . . 10
⊢ ((𝑇‘𝑧) ∈ (0[,)+∞) ↔ ((𝑇‘𝑧) ∈ ℝ ∧ 0 ≤ (𝑇‘𝑧))) |
53 | 52 | simplbi 498 |
. . . . . . . . 9
⊢ ((𝑇‘𝑧) ∈ (0[,)+∞) → (𝑇‘𝑧) ∈ ℝ) |
54 | 36, 53 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → (𝑇‘𝑧) ∈ ℝ) |
55 | 51, 54 | readdcld 11014 |
. . . . . . 7
⊢ (𝜑 → (𝑆 + (𝑇‘𝑧)) ∈ ℝ) |
56 | 49, 55 | eqeltrd 2839 |
. . . . . 6
⊢ (𝜑 → 𝐿 ∈ ℝ) |
57 | 56 | recnd 11013 |
. . . . 5
⊢ (𝜑 → 𝐿 ∈ ℂ) |
58 | | 0red 10988 |
. . . . . . 7
⊢ (𝜑 → 0 ∈
ℝ) |
59 | 50 | rpgt0d 12785 |
. . . . . . 7
⊢ (𝜑 → 0 < 𝑆) |
60 | 52 | simprbi 497 |
. . . . . . . . . 10
⊢ ((𝑇‘𝑧) ∈ (0[,)+∞) → 0 ≤ (𝑇‘𝑧)) |
61 | 36, 60 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 0 ≤ (𝑇‘𝑧)) |
62 | 51, 54 | addge01d 11573 |
. . . . . . . . 9
⊢ (𝜑 → (0 ≤ (𝑇‘𝑧) ↔ 𝑆 ≤ (𝑆 + (𝑇‘𝑧)))) |
63 | 61, 62 | mpbid 231 |
. . . . . . . 8
⊢ (𝜑 → 𝑆 ≤ (𝑆 + (𝑇‘𝑧))) |
64 | 63, 49 | breqtrrd 5101 |
. . . . . . 7
⊢ (𝜑 → 𝑆 ≤ 𝐿) |
65 | 58, 51, 56, 59, 64 | ltletrd 11145 |
. . . . . 6
⊢ (𝜑 → 0 < 𝐿) |
66 | 65 | gt0ne0d 11549 |
. . . . 5
⊢ (𝜑 → 𝐿 ≠ 0) |
67 | 29, 41, 57, 66 | divdird 11799 |
. . . 4
⊢ (𝜑 → (((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) + ((𝑇‘𝑧) · (𝑋‘𝑧))) / 𝐿) = (((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝐿) + (((𝑇‘𝑧) · (𝑋‘𝑧)) / 𝐿))) |
68 | | cnfldbas 20611 |
. . . . . . 7
⊢ ℂ =
(Base‘ℂfld) |
69 | | cnfldadd 20612 |
. . . . . . 7
⊢ + =
(+g‘ℂfld) |
70 | | ringcmn 19830 |
. . . . . . . 8
⊢
(ℂfld ∈ Ring → ℂfld ∈
CMnd) |
71 | 2, 70 | mp1i 13 |
. . . . . . 7
⊢ (𝜑 → ℂfld
∈ CMnd) |
72 | 7 | sselda 3920 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐴) |
73 | 15 | ffvelrnda 6953 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑇‘𝑥) ∈ (0[,)+∞)) |
74 | 72, 73 | syldan 591 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑇‘𝑥) ∈ (0[,)+∞)) |
75 | 31, 74 | sselid 3918 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑇‘𝑥) ∈ ℂ) |
76 | 20 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐷 ⊆ ℝ) |
77 | 19 | ffvelrnda 6953 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑋‘𝑥) ∈ 𝐷) |
78 | 72, 77 | syldan 591 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑋‘𝑥) ∈ 𝐷) |
79 | 76, 78 | sseldd 3921 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑋‘𝑥) ∈ ℝ) |
80 | 79 | recnd 11013 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑋‘𝑥) ∈ ℂ) |
81 | 75, 80 | mulcld 11005 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝑇‘𝑥) · (𝑋‘𝑥)) ∈ ℂ) |
82 | | fveq2 6766 |
. . . . . . . 8
⊢ (𝑥 = 𝑧 → (𝑇‘𝑥) = (𝑇‘𝑧)) |
83 | | fveq2 6766 |
. . . . . . . 8
⊢ (𝑥 = 𝑧 → (𝑋‘𝑥) = (𝑋‘𝑧)) |
84 | 82, 83 | oveq12d 7285 |
. . . . . . 7
⊢ (𝑥 = 𝑧 → ((𝑇‘𝑥) · (𝑋‘𝑥)) = ((𝑇‘𝑧) · (𝑋‘𝑧))) |
85 | 68, 69, 71, 8, 81, 35, 46, 41, 84 | gsumunsn 19571 |
. . . . . 6
⊢ (𝜑 → (ℂfld
Σg (𝑥 ∈ (𝐵 ∪ {𝑧}) ↦ ((𝑇‘𝑥) · (𝑋‘𝑥)))) = ((ℂfld
Σg (𝑥 ∈ 𝐵 ↦ ((𝑇‘𝑥) · (𝑋‘𝑥)))) + ((𝑇‘𝑧) · (𝑋‘𝑧)))) |
86 | 15 | feqmptd 6829 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑇 = (𝑥 ∈ 𝐴 ↦ (𝑇‘𝑥))) |
87 | 19 | feqmptd 6829 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑋 = (𝑥 ∈ 𝐴 ↦ (𝑋‘𝑥))) |
88 | 5, 73, 77, 86, 87 | offval2 7543 |
. . . . . . . . 9
⊢ (𝜑 → (𝑇 ∘f · 𝑋) = (𝑥 ∈ 𝐴 ↦ ((𝑇‘𝑥) · (𝑋‘𝑥)))) |
89 | 88 | reseq1d 5883 |
. . . . . . . 8
⊢ (𝜑 → ((𝑇 ∘f · 𝑋) ↾ (𝐵 ∪ {𝑧})) = ((𝑥 ∈ 𝐴 ↦ ((𝑇‘𝑥) · (𝑋‘𝑥))) ↾ (𝐵 ∪ {𝑧}))) |
90 | 6 | resmptd 5941 |
. . . . . . . 8
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ ((𝑇‘𝑥) · (𝑋‘𝑥))) ↾ (𝐵 ∪ {𝑧})) = (𝑥 ∈ (𝐵 ∪ {𝑧}) ↦ ((𝑇‘𝑥) · (𝑋‘𝑥)))) |
91 | 89, 90 | eqtrd 2778 |
. . . . . . 7
⊢ (𝜑 → ((𝑇 ∘f · 𝑋) ↾ (𝐵 ∪ {𝑧})) = (𝑥 ∈ (𝐵 ∪ {𝑧}) ↦ ((𝑇‘𝑥) · (𝑋‘𝑥)))) |
92 | 91 | oveq2d 7283 |
. . . . . 6
⊢ (𝜑 → (ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ (𝐵 ∪ {𝑧}))) = (ℂfld
Σg (𝑥 ∈ (𝐵 ∪ {𝑧}) ↦ ((𝑇‘𝑥) · (𝑋‘𝑥))))) |
93 | 88 | reseq1d 5883 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑇 ∘f · 𝑋) ↾ 𝐵) = ((𝑥 ∈ 𝐴 ↦ ((𝑇‘𝑥) · (𝑋‘𝑥))) ↾ 𝐵)) |
94 | 7 | resmptd 5941 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ ((𝑇‘𝑥) · (𝑋‘𝑥))) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ ((𝑇‘𝑥) · (𝑋‘𝑥)))) |
95 | 93, 94 | eqtrd 2778 |
. . . . . . . 8
⊢ (𝜑 → ((𝑇 ∘f · 𝑋) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ ((𝑇‘𝑥) · (𝑋‘𝑥)))) |
96 | 95 | oveq2d 7283 |
. . . . . . 7
⊢ (𝜑 → (ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) = (ℂfld
Σg (𝑥 ∈ 𝐵 ↦ ((𝑇‘𝑥) · (𝑋‘𝑥))))) |
97 | 96 | oveq1d 7282 |
. . . . . 6
⊢ (𝜑 → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) + ((𝑇‘𝑧) · (𝑋‘𝑧))) = ((ℂfld
Σg (𝑥 ∈ 𝐵 ↦ ((𝑇‘𝑥) · (𝑋‘𝑥)))) + ((𝑇‘𝑧) · (𝑋‘𝑧)))) |
98 | 85, 92, 97 | 3eqtr4d 2788 |
. . . . 5
⊢ (𝜑 → (ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ (𝐵 ∪ {𝑧}))) = ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) + ((𝑇‘𝑧) · (𝑋‘𝑧)))) |
99 | 98 | oveq1d 7282 |
. . . 4
⊢ (𝜑 → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ (𝐵 ∪ {𝑧}))) / 𝐿) = (((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) + ((𝑇‘𝑧) · (𝑋‘𝑧))) / 𝐿)) |
100 | 51 | recnd 11013 |
. . . . . 6
⊢ (𝜑 → 𝑆 ∈ ℂ) |
101 | 50 | rpne0d 12787 |
. . . . . 6
⊢ (𝜑 → 𝑆 ≠ 0) |
102 | 29, 100, 57, 101, 66 | dmdcand 11790 |
. . . . 5
⊢ (𝜑 → ((𝑆 / 𝐿) · ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆)) = ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝐿)) |
103 | 57, 100, 57, 66 | divsubdird 11800 |
. . . . . . . 8
⊢ (𝜑 → ((𝐿 − 𝑆) / 𝐿) = ((𝐿 / 𝐿) − (𝑆 / 𝐿))) |
104 | 100, 37, 49 | mvrladdd 11398 |
. . . . . . . . 9
⊢ (𝜑 → (𝐿 − 𝑆) = (𝑇‘𝑧)) |
105 | 104 | oveq1d 7282 |
. . . . . . . 8
⊢ (𝜑 → ((𝐿 − 𝑆) / 𝐿) = ((𝑇‘𝑧) / 𝐿)) |
106 | 57, 66 | dividd 11759 |
. . . . . . . . 9
⊢ (𝜑 → (𝐿 / 𝐿) = 1) |
107 | 106 | oveq1d 7282 |
. . . . . . . 8
⊢ (𝜑 → ((𝐿 / 𝐿) − (𝑆 / 𝐿)) = (1 − (𝑆 / 𝐿))) |
108 | 103, 105,
107 | 3eqtr3rd 2787 |
. . . . . . 7
⊢ (𝜑 → (1 − (𝑆 / 𝐿)) = ((𝑇‘𝑧) / 𝐿)) |
109 | 108 | oveq1d 7282 |
. . . . . 6
⊢ (𝜑 → ((1 − (𝑆 / 𝐿)) · (𝑋‘𝑧)) = (((𝑇‘𝑧) / 𝐿) · (𝑋‘𝑧))) |
110 | 37, 40, 57, 66 | div23d 11798 |
. . . . . 6
⊢ (𝜑 → (((𝑇‘𝑧) · (𝑋‘𝑧)) / 𝐿) = (((𝑇‘𝑧) / 𝐿) · (𝑋‘𝑧))) |
111 | 109, 110 | eqtr4d 2781 |
. . . . 5
⊢ (𝜑 → ((1 − (𝑆 / 𝐿)) · (𝑋‘𝑧)) = (((𝑇‘𝑧) · (𝑋‘𝑧)) / 𝐿)) |
112 | 102, 111 | oveq12d 7285 |
. . . 4
⊢ (𝜑 → (((𝑆 / 𝐿) · ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − (𝑆 / 𝐿)) · (𝑋‘𝑧))) = (((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝐿) + (((𝑇‘𝑧) · (𝑋‘𝑧)) / 𝐿))) |
113 | 67, 99, 112 | 3eqtr4d 2788 |
. . 3
⊢ (𝜑 → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ (𝐵 ∪ {𝑧}))) / 𝐿) = (((𝑆 / 𝐿) · ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − (𝑆 / 𝐿)) · (𝑋‘𝑧)))) |
114 | | jensenlem.4 |
. . . . 5
⊢ (𝜑 → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆) ∈ 𝐷) |
115 | 51, 56, 66 | redivcld 11813 |
. . . . . 6
⊢ (𝜑 → (𝑆 / 𝐿) ∈ ℝ) |
116 | 50 | rpge0d 12786 |
. . . . . . 7
⊢ (𝜑 → 0 ≤ 𝑆) |
117 | | divge0 11854 |
. . . . . . 7
⊢ (((𝑆 ∈ ℝ ∧ 0 ≤
𝑆) ∧ (𝐿 ∈ ℝ ∧ 0 < 𝐿)) → 0 ≤ (𝑆 / 𝐿)) |
118 | 51, 116, 56, 65, 117 | syl22anc 836 |
. . . . . 6
⊢ (𝜑 → 0 ≤ (𝑆 / 𝐿)) |
119 | 57 | mulid1d 11002 |
. . . . . . . 8
⊢ (𝜑 → (𝐿 · 1) = 𝐿) |
120 | 64, 119 | breqtrrd 5101 |
. . . . . . 7
⊢ (𝜑 → 𝑆 ≤ (𝐿 · 1)) |
121 | | 1red 10986 |
. . . . . . . 8
⊢ (𝜑 → 1 ∈
ℝ) |
122 | | ledivmul 11861 |
. . . . . . . 8
⊢ ((𝑆 ∈ ℝ ∧ 1 ∈
ℝ ∧ (𝐿 ∈
ℝ ∧ 0 < 𝐿))
→ ((𝑆 / 𝐿) ≤ 1 ↔ 𝑆 ≤ (𝐿 · 1))) |
123 | 51, 121, 56, 65, 122 | syl112anc 1373 |
. . . . . . 7
⊢ (𝜑 → ((𝑆 / 𝐿) ≤ 1 ↔ 𝑆 ≤ (𝐿 · 1))) |
124 | 120, 123 | mpbird 256 |
. . . . . 6
⊢ (𝜑 → (𝑆 / 𝐿) ≤ 1) |
125 | | elicc01 13208 |
. . . . . 6
⊢ ((𝑆 / 𝐿) ∈ (0[,]1) ↔ ((𝑆 / 𝐿) ∈ ℝ ∧ 0 ≤ (𝑆 / 𝐿) ∧ (𝑆 / 𝐿) ≤ 1)) |
126 | 115, 118,
124, 125 | syl3anbrc 1342 |
. . . . 5
⊢ (𝜑 → (𝑆 / 𝐿) ∈ (0[,]1)) |
127 | 114, 38, 126 | 3jca 1127 |
. . . 4
⊢ (𝜑 → (((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆) ∈ 𝐷 ∧ (𝑋‘𝑧) ∈ 𝐷 ∧ (𝑆 / 𝐿) ∈ (0[,]1))) |
128 | 20, 43 | cvxcl 26144 |
. . . 4
⊢ ((𝜑 ∧ (((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆) ∈ 𝐷 ∧ (𝑋‘𝑧) ∈ 𝐷 ∧ (𝑆 / 𝐿) ∈ (0[,]1))) → (((𝑆 / 𝐿) · ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − (𝑆 / 𝐿)) · (𝑋‘𝑧))) ∈ 𝐷) |
129 | 127, 128 | mpdan 684 |
. . 3
⊢ (𝜑 → (((𝑆 / 𝐿) · ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − (𝑆 / 𝐿)) · (𝑋‘𝑧))) ∈ 𝐷) |
130 | 113, 129 | eqeltrd 2839 |
. 2
⊢ (𝜑 → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ (𝐵 ∪ {𝑧}))) / 𝐿) ∈ 𝐷) |
131 | 42, 129 | ffvelrnd 6954 |
. . . 4
⊢ (𝜑 → (𝐹‘(((𝑆 / 𝐿) · ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − (𝑆 / 𝐿)) · (𝑋‘𝑧)))) ∈ ℝ) |
132 | 42, 114 | ffvelrnd 6954 |
. . . . . 6
⊢ (𝜑 → (𝐹‘((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆)) ∈ ℝ) |
133 | 115, 132 | remulcld 11015 |
. . . . 5
⊢ (𝜑 → ((𝑆 / 𝐿) · (𝐹‘((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆))) ∈ ℝ) |
134 | 42, 38 | ffvelrnd 6954 |
. . . . . . 7
⊢ (𝜑 → (𝐹‘(𝑋‘𝑧)) ∈ ℝ) |
135 | 54, 134 | remulcld 11015 |
. . . . . 6
⊢ (𝜑 → ((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧))) ∈ ℝ) |
136 | 135, 56, 66 | redivcld 11813 |
. . . . 5
⊢ (𝜑 → (((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧))) / 𝐿) ∈ ℝ) |
137 | 133, 136 | readdcld 11014 |
. . . 4
⊢ (𝜑 → (((𝑆 / 𝐿) · (𝐹‘((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆))) + (((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧))) / 𝐿)) ∈ ℝ) |
138 | | fco 6616 |
. . . . . . . . . . 11
⊢ ((𝐹:𝐷⟶ℝ ∧ 𝑋:𝐴⟶𝐷) → (𝐹 ∘ 𝑋):𝐴⟶ℝ) |
139 | 42, 19, 138 | syl2anc 584 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐹 ∘ 𝑋):𝐴⟶ℝ) |
140 | 14, 18, 139, 5, 5, 22 | off 7541 |
. . . . . . . . 9
⊢ (𝜑 → (𝑇 ∘f · (𝐹 ∘ 𝑋)):𝐴⟶ℝ) |
141 | 140, 7 | fssresd 6633 |
. . . . . . . 8
⊢ (𝜑 → ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐵):𝐵⟶ℝ) |
142 | 141, 8, 26 | fdmfifsupp 9125 |
. . . . . . . 8
⊢ (𝜑 → ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐵) finSupp 0) |
143 | 1, 4, 8, 12, 141, 142 | gsumsubgcl 19531 |
. . . . . . 7
⊢ (𝜑 → (ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) ∈ ℝ) |
144 | 143, 51, 101 | redivcld 11813 |
. . . . . 6
⊢ (𝜑 → ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) / 𝑆) ∈ ℝ) |
145 | 115, 144 | remulcld 11015 |
. . . . 5
⊢ (𝜑 → ((𝑆 / 𝐿) · ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) / 𝑆)) ∈ ℝ) |
146 | | 1re 10985 |
. . . . . . 7
⊢ 1 ∈
ℝ |
147 | | resubcl 11295 |
. . . . . . 7
⊢ ((1
∈ ℝ ∧ (𝑆 /
𝐿) ∈ ℝ) →
(1 − (𝑆 / 𝐿)) ∈
ℝ) |
148 | 146, 115,
147 | sylancr 587 |
. . . . . 6
⊢ (𝜑 → (1 − (𝑆 / 𝐿)) ∈ ℝ) |
149 | 148, 134 | remulcld 11015 |
. . . . 5
⊢ (𝜑 → ((1 − (𝑆 / 𝐿)) · (𝐹‘(𝑋‘𝑧))) ∈ ℝ) |
150 | 145, 149 | readdcld 11014 |
. . . 4
⊢ (𝜑 → (((𝑆 / 𝐿) · ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) / 𝑆)) + ((1 − (𝑆 / 𝐿)) · (𝐹‘(𝑋‘𝑧)))) ∈ ℝ) |
151 | | oveq2 7275 |
. . . . . . . . . . 11
⊢ (𝑥 = ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆) → (𝑡 · 𝑥) = (𝑡 · ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆))) |
152 | 151 | fvoveq1d 7289 |
. . . . . . . . . 10
⊢ (𝑥 = ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆) → (𝐹‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) = (𝐹‘((𝑡 · ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − 𝑡) · 𝑦)))) |
153 | | fveq2 6766 |
. . . . . . . . . . . 12
⊢ (𝑥 = ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆) → (𝐹‘𝑥) = (𝐹‘((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆))) |
154 | 153 | oveq2d 7283 |
. . . . . . . . . . 11
⊢ (𝑥 = ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆) → (𝑡 · (𝐹‘𝑥)) = (𝑡 · (𝐹‘((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆)))) |
155 | 154 | oveq1d 7282 |
. . . . . . . . . 10
⊢ (𝑥 = ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆) → ((𝑡 · (𝐹‘𝑥)) + ((1 − 𝑡) · (𝐹‘𝑦))) = ((𝑡 · (𝐹‘((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆))) + ((1 − 𝑡) · (𝐹‘𝑦)))) |
156 | 152, 155 | breq12d 5086 |
. . . . . . . . 9
⊢ (𝑥 = ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆) → ((𝐹‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ≤ ((𝑡 · (𝐹‘𝑥)) + ((1 − 𝑡) · (𝐹‘𝑦))) ↔ (𝐹‘((𝑡 · ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − 𝑡) · 𝑦))) ≤ ((𝑡 · (𝐹‘((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆))) + ((1 − 𝑡) · (𝐹‘𝑦))))) |
157 | 156 | imbi2d 341 |
. . . . . . . 8
⊢ (𝑥 = ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆) → ((𝜑 → (𝐹‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ≤ ((𝑡 · (𝐹‘𝑥)) + ((1 − 𝑡) · (𝐹‘𝑦)))) ↔ (𝜑 → (𝐹‘((𝑡 · ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − 𝑡) · 𝑦))) ≤ ((𝑡 · (𝐹‘((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆))) + ((1 − 𝑡) · (𝐹‘𝑦)))))) |
158 | | oveq2 7275 |
. . . . . . . . . . . 12
⊢ (𝑦 = (𝑋‘𝑧) → ((1 − 𝑡) · 𝑦) = ((1 − 𝑡) · (𝑋‘𝑧))) |
159 | 158 | oveq2d 7283 |
. . . . . . . . . . 11
⊢ (𝑦 = (𝑋‘𝑧) → ((𝑡 · ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − 𝑡) · 𝑦)) = ((𝑡 · ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − 𝑡) · (𝑋‘𝑧)))) |
160 | 159 | fveq2d 6770 |
. . . . . . . . . 10
⊢ (𝑦 = (𝑋‘𝑧) → (𝐹‘((𝑡 · ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − 𝑡) · 𝑦))) = (𝐹‘((𝑡 · ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − 𝑡) · (𝑋‘𝑧))))) |
161 | | fveq2 6766 |
. . . . . . . . . . . 12
⊢ (𝑦 = (𝑋‘𝑧) → (𝐹‘𝑦) = (𝐹‘(𝑋‘𝑧))) |
162 | 161 | oveq2d 7283 |
. . . . . . . . . . 11
⊢ (𝑦 = (𝑋‘𝑧) → ((1 − 𝑡) · (𝐹‘𝑦)) = ((1 − 𝑡) · (𝐹‘(𝑋‘𝑧)))) |
163 | 162 | oveq2d 7283 |
. . . . . . . . . 10
⊢ (𝑦 = (𝑋‘𝑧) → ((𝑡 · (𝐹‘((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆))) + ((1 − 𝑡) · (𝐹‘𝑦))) = ((𝑡 · (𝐹‘((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆))) + ((1 − 𝑡) · (𝐹‘(𝑋‘𝑧))))) |
164 | 160, 163 | breq12d 5086 |
. . . . . . . . 9
⊢ (𝑦 = (𝑋‘𝑧) → ((𝐹‘((𝑡 · ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − 𝑡) · 𝑦))) ≤ ((𝑡 · (𝐹‘((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆))) + ((1 − 𝑡) · (𝐹‘𝑦))) ↔ (𝐹‘((𝑡 · ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − 𝑡) · (𝑋‘𝑧)))) ≤ ((𝑡 · (𝐹‘((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆))) + ((1 − 𝑡) · (𝐹‘(𝑋‘𝑧)))))) |
165 | 164 | imbi2d 341 |
. . . . . . . 8
⊢ (𝑦 = (𝑋‘𝑧) → ((𝜑 → (𝐹‘((𝑡 · ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − 𝑡) · 𝑦))) ≤ ((𝑡 · (𝐹‘((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆))) + ((1 − 𝑡) · (𝐹‘𝑦)))) ↔ (𝜑 → (𝐹‘((𝑡 · ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − 𝑡) · (𝑋‘𝑧)))) ≤ ((𝑡 · (𝐹‘((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆))) + ((1 − 𝑡) · (𝐹‘(𝑋‘𝑧))))))) |
166 | | oveq1 7274 |
. . . . . . . . . . . 12
⊢ (𝑡 = (𝑆 / 𝐿) → (𝑡 · ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆)) = ((𝑆 / 𝐿) · ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆))) |
167 | | oveq2 7275 |
. . . . . . . . . . . . 13
⊢ (𝑡 = (𝑆 / 𝐿) → (1 − 𝑡) = (1 − (𝑆 / 𝐿))) |
168 | 167 | oveq1d 7282 |
. . . . . . . . . . . 12
⊢ (𝑡 = (𝑆 / 𝐿) → ((1 − 𝑡) · (𝑋‘𝑧)) = ((1 − (𝑆 / 𝐿)) · (𝑋‘𝑧))) |
169 | 166, 168 | oveq12d 7285 |
. . . . . . . . . . 11
⊢ (𝑡 = (𝑆 / 𝐿) → ((𝑡 · ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − 𝑡) · (𝑋‘𝑧))) = (((𝑆 / 𝐿) · ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − (𝑆 / 𝐿)) · (𝑋‘𝑧)))) |
170 | 169 | fveq2d 6770 |
. . . . . . . . . 10
⊢ (𝑡 = (𝑆 / 𝐿) → (𝐹‘((𝑡 · ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − 𝑡) · (𝑋‘𝑧)))) = (𝐹‘(((𝑆 / 𝐿) · ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − (𝑆 / 𝐿)) · (𝑋‘𝑧))))) |
171 | | oveq1 7274 |
. . . . . . . . . . 11
⊢ (𝑡 = (𝑆 / 𝐿) → (𝑡 · (𝐹‘((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆))) = ((𝑆 / 𝐿) · (𝐹‘((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆)))) |
172 | 167 | oveq1d 7282 |
. . . . . . . . . . 11
⊢ (𝑡 = (𝑆 / 𝐿) → ((1 − 𝑡) · (𝐹‘(𝑋‘𝑧))) = ((1 − (𝑆 / 𝐿)) · (𝐹‘(𝑋‘𝑧)))) |
173 | 171, 172 | oveq12d 7285 |
. . . . . . . . . 10
⊢ (𝑡 = (𝑆 / 𝐿) → ((𝑡 · (𝐹‘((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆))) + ((1 − 𝑡) · (𝐹‘(𝑋‘𝑧)))) = (((𝑆 / 𝐿) · (𝐹‘((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆))) + ((1 − (𝑆 / 𝐿)) · (𝐹‘(𝑋‘𝑧))))) |
174 | 170, 173 | breq12d 5086 |
. . . . . . . . 9
⊢ (𝑡 = (𝑆 / 𝐿) → ((𝐹‘((𝑡 · ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − 𝑡) · (𝑋‘𝑧)))) ≤ ((𝑡 · (𝐹‘((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆))) + ((1 − 𝑡) · (𝐹‘(𝑋‘𝑧)))) ↔ (𝐹‘(((𝑆 / 𝐿) · ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − (𝑆 / 𝐿)) · (𝑋‘𝑧)))) ≤ (((𝑆 / 𝐿) · (𝐹‘((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆))) + ((1 − (𝑆 / 𝐿)) · (𝐹‘(𝑋‘𝑧)))))) |
175 | 174 | imbi2d 341 |
. . . . . . . 8
⊢ (𝑡 = (𝑆 / 𝐿) → ((𝜑 → (𝐹‘((𝑡 · ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − 𝑡) · (𝑋‘𝑧)))) ≤ ((𝑡 · (𝐹‘((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆))) + ((1 − 𝑡) · (𝐹‘(𝑋‘𝑧))))) ↔ (𝜑 → (𝐹‘(((𝑆 / 𝐿) · ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − (𝑆 / 𝐿)) · (𝑋‘𝑧)))) ≤ (((𝑆 / 𝐿) · (𝐹‘((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆))) + ((1 − (𝑆 / 𝐿)) · (𝐹‘(𝑋‘𝑧))))))) |
176 | 45 | expcom 414 |
. . . . . . . 8
⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷 ∧ 𝑡 ∈ (0[,]1)) → (𝜑 → (𝐹‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ≤ ((𝑡 · (𝐹‘𝑥)) + ((1 − 𝑡) · (𝐹‘𝑦))))) |
177 | 157, 165,
175, 176 | vtocl3ga 3514 |
. . . . . . 7
⊢
((((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆) ∈ 𝐷 ∧ (𝑋‘𝑧) ∈ 𝐷 ∧ (𝑆 / 𝐿) ∈ (0[,]1)) → (𝜑 → (𝐹‘(((𝑆 / 𝐿) · ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − (𝑆 / 𝐿)) · (𝑋‘𝑧)))) ≤ (((𝑆 / 𝐿) · (𝐹‘((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆))) + ((1 − (𝑆 / 𝐿)) · (𝐹‘(𝑋‘𝑧)))))) |
178 | 114, 38, 126, 177 | syl3anc 1370 |
. . . . . 6
⊢ (𝜑 → (𝜑 → (𝐹‘(((𝑆 / 𝐿) · ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − (𝑆 / 𝐿)) · (𝑋‘𝑧)))) ≤ (((𝑆 / 𝐿) · (𝐹‘((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆))) + ((1 − (𝑆 / 𝐿)) · (𝐹‘(𝑋‘𝑧)))))) |
179 | 178 | pm2.43i 52 |
. . . . 5
⊢ (𝜑 → (𝐹‘(((𝑆 / 𝐿) · ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − (𝑆 / 𝐿)) · (𝑋‘𝑧)))) ≤ (((𝑆 / 𝐿) · (𝐹‘((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆))) + ((1 − (𝑆 / 𝐿)) · (𝐹‘(𝑋‘𝑧))))) |
180 | 108 | oveq1d 7282 |
. . . . . . 7
⊢ (𝜑 → ((1 − (𝑆 / 𝐿)) · (𝐹‘(𝑋‘𝑧))) = (((𝑇‘𝑧) / 𝐿) · (𝐹‘(𝑋‘𝑧)))) |
181 | 134 | recnd 11013 |
. . . . . . . 8
⊢ (𝜑 → (𝐹‘(𝑋‘𝑧)) ∈ ℂ) |
182 | 37, 181, 57, 66 | div23d 11798 |
. . . . . . 7
⊢ (𝜑 → (((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧))) / 𝐿) = (((𝑇‘𝑧) / 𝐿) · (𝐹‘(𝑋‘𝑧)))) |
183 | 180, 182 | eqtr4d 2781 |
. . . . . 6
⊢ (𝜑 → ((1 − (𝑆 / 𝐿)) · (𝐹‘(𝑋‘𝑧))) = (((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧))) / 𝐿)) |
184 | 183 | oveq2d 7283 |
. . . . 5
⊢ (𝜑 → (((𝑆 / 𝐿) · (𝐹‘((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆))) + ((1 − (𝑆 / 𝐿)) · (𝐹‘(𝑋‘𝑧)))) = (((𝑆 / 𝐿) · (𝐹‘((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆))) + (((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧))) / 𝐿))) |
185 | 179, 184 | breqtrd 5099 |
. . . 4
⊢ (𝜑 → (𝐹‘(((𝑆 / 𝐿) · ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − (𝑆 / 𝐿)) · (𝑋‘𝑧)))) ≤ (((𝑆 / 𝐿) · (𝐹‘((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆))) + (((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧))) / 𝐿))) |
186 | 182, 180 | eqtr4d 2781 |
. . . . . 6
⊢ (𝜑 → (((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧))) / 𝐿) = ((1 − (𝑆 / 𝐿)) · (𝐹‘(𝑋‘𝑧)))) |
187 | 186 | oveq2d 7283 |
. . . . 5
⊢ (𝜑 → (((𝑆 / 𝐿) · (𝐹‘((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆))) + (((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧))) / 𝐿)) = (((𝑆 / 𝐿) · (𝐹‘((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆))) + ((1 − (𝑆 / 𝐿)) · (𝐹‘(𝑋‘𝑧))))) |
188 | | jensenlem.5 |
. . . . . . 7
⊢ (𝜑 → (𝐹‘((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆)) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) / 𝑆)) |
189 | 51, 56, 59, 65 | divgt0d 11920 |
. . . . . . . 8
⊢ (𝜑 → 0 < (𝑆 / 𝐿)) |
190 | | lemul2 11838 |
. . . . . . . 8
⊢ (((𝐹‘((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆)) ∈ ℝ ∧
((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) / 𝑆) ∈ ℝ ∧ ((𝑆 / 𝐿) ∈ ℝ ∧ 0 < (𝑆 / 𝐿))) → ((𝐹‘((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆)) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) / 𝑆) ↔ ((𝑆 / 𝐿) · (𝐹‘((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆))) ≤ ((𝑆 / 𝐿) · ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) / 𝑆)))) |
191 | 132, 144,
115, 189, 190 | syl112anc 1373 |
. . . . . . 7
⊢ (𝜑 → ((𝐹‘((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆)) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) / 𝑆) ↔ ((𝑆 / 𝐿) · (𝐹‘((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆))) ≤ ((𝑆 / 𝐿) · ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) / 𝑆)))) |
192 | 188, 191 | mpbid 231 |
. . . . . 6
⊢ (𝜑 → ((𝑆 / 𝐿) · (𝐹‘((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆))) ≤ ((𝑆 / 𝐿) · ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) / 𝑆))) |
193 | 133, 145,
149, 192 | leadd1dd 11599 |
. . . . 5
⊢ (𝜑 → (((𝑆 / 𝐿) · (𝐹‘((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆))) + ((1 − (𝑆 / 𝐿)) · (𝐹‘(𝑋‘𝑧)))) ≤ (((𝑆 / 𝐿) · ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) / 𝑆)) + ((1 − (𝑆 / 𝐿)) · (𝐹‘(𝑋‘𝑧))))) |
194 | 187, 193 | eqbrtrd 5095 |
. . . 4
⊢ (𝜑 → (((𝑆 / 𝐿) · (𝐹‘((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆))) + (((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧))) / 𝐿)) ≤ (((𝑆 / 𝐿) · ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) / 𝑆)) + ((1 − (𝑆 / 𝐿)) · (𝐹‘(𝑋‘𝑧))))) |
195 | 131, 137,
150, 185, 194 | letrd 11142 |
. . 3
⊢ (𝜑 → (𝐹‘(((𝑆 / 𝐿) · ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − (𝑆 / 𝐿)) · (𝑋‘𝑧)))) ≤ (((𝑆 / 𝐿) · ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) / 𝑆)) + ((1 − (𝑆 / 𝐿)) · (𝐹‘(𝑋‘𝑧))))) |
196 | 113 | fveq2d 6770 |
. . 3
⊢ (𝜑 → (𝐹‘((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ (𝐵 ∪ {𝑧}))) / 𝐿)) = (𝐹‘(((𝑆 / 𝐿) · ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − (𝑆 / 𝐿)) · (𝑋‘𝑧))))) |
197 | 143 | recnd 11013 |
. . . . 5
⊢ (𝜑 → (ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) ∈ ℂ) |
198 | 135 | recnd 11013 |
. . . . 5
⊢ (𝜑 → ((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧))) ∈ ℂ) |
199 | 197, 198,
57, 66 | divdird 11799 |
. . . 4
⊢ (𝜑 → (((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) + ((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧)))) / 𝐿) = (((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) / 𝐿) + (((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧))) / 𝐿))) |
200 | 16, 73 | sselid 3918 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑇‘𝑥) ∈ ℝ) |
201 | 42 | ffvelrnda 6953 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑋‘𝑥) ∈ 𝐷) → (𝐹‘(𝑋‘𝑥)) ∈ ℝ) |
202 | 77, 201 | syldan 591 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘(𝑋‘𝑥)) ∈ ℝ) |
203 | 200, 202 | remulcld 11015 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥))) ∈ ℝ) |
204 | 203 | recnd 11013 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥))) ∈ ℂ) |
205 | 72, 204 | syldan 591 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥))) ∈ ℂ) |
206 | 83 | fveq2d 6770 |
. . . . . . . 8
⊢ (𝑥 = 𝑧 → (𝐹‘(𝑋‘𝑥)) = (𝐹‘(𝑋‘𝑧))) |
207 | 82, 206 | oveq12d 7285 |
. . . . . . 7
⊢ (𝑥 = 𝑧 → ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥))) = ((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧)))) |
208 | 68, 69, 71, 8, 205, 35, 46, 198, 207 | gsumunsn 19571 |
. . . . . 6
⊢ (𝜑 → (ℂfld
Σg (𝑥 ∈ (𝐵 ∪ {𝑧}) ↦ ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥))))) = ((ℂfld
Σg (𝑥 ∈ 𝐵 ↦ ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥))))) + ((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧))))) |
209 | 42 | feqmptd 6829 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐹 = (𝑦 ∈ 𝐷 ↦ (𝐹‘𝑦))) |
210 | | fveq2 6766 |
. . . . . . . . . . 11
⊢ (𝑦 = (𝑋‘𝑥) → (𝐹‘𝑦) = (𝐹‘(𝑋‘𝑥))) |
211 | 77, 87, 209, 210 | fmptco 6993 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐹 ∘ 𝑋) = (𝑥 ∈ 𝐴 ↦ (𝐹‘(𝑋‘𝑥)))) |
212 | 5, 73, 202, 86, 211 | offval2 7543 |
. . . . . . . . 9
⊢ (𝜑 → (𝑇 ∘f · (𝐹 ∘ 𝑋)) = (𝑥 ∈ 𝐴 ↦ ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥))))) |
213 | 212 | reseq1d 5883 |
. . . . . . . 8
⊢ (𝜑 → ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝐵 ∪ {𝑧})) = ((𝑥 ∈ 𝐴 ↦ ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥)))) ↾ (𝐵 ∪ {𝑧}))) |
214 | 6 | resmptd 5941 |
. . . . . . . 8
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥)))) ↾ (𝐵 ∪ {𝑧})) = (𝑥 ∈ (𝐵 ∪ {𝑧}) ↦ ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥))))) |
215 | 213, 214 | eqtrd 2778 |
. . . . . . 7
⊢ (𝜑 → ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝐵 ∪ {𝑧})) = (𝑥 ∈ (𝐵 ∪ {𝑧}) ↦ ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥))))) |
216 | 215 | oveq2d 7283 |
. . . . . 6
⊢ (𝜑 → (ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝐵 ∪ {𝑧}))) = (ℂfld
Σg (𝑥 ∈ (𝐵 ∪ {𝑧}) ↦ ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥)))))) |
217 | 212 | reseq1d 5883 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐵) = ((𝑥 ∈ 𝐴 ↦ ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥)))) ↾ 𝐵)) |
218 | 7 | resmptd 5941 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥)))) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥))))) |
219 | 217, 218 | eqtrd 2778 |
. . . . . . . 8
⊢ (𝜑 → ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥))))) |
220 | 219 | oveq2d 7283 |
. . . . . . 7
⊢ (𝜑 → (ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) = (ℂfld
Σg (𝑥 ∈ 𝐵 ↦ ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥)))))) |
221 | 220 | oveq1d 7282 |
. . . . . 6
⊢ (𝜑 → ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) + ((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧)))) = ((ℂfld
Σg (𝑥 ∈ 𝐵 ↦ ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥))))) + ((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧))))) |
222 | 208, 216,
221 | 3eqtr4d 2788 |
. . . . 5
⊢ (𝜑 → (ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝐵 ∪ {𝑧}))) = ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) + ((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧))))) |
223 | 222 | oveq1d 7282 |
. . . 4
⊢ (𝜑 → ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝐵 ∪ {𝑧}))) / 𝐿) = (((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) + ((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧)))) / 𝐿)) |
224 | 197, 100,
57, 101, 66 | dmdcand 11790 |
. . . . 5
⊢ (𝜑 → ((𝑆 / 𝐿) · ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) / 𝑆)) = ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) / 𝐿)) |
225 | 224, 183 | oveq12d 7285 |
. . . 4
⊢ (𝜑 → (((𝑆 / 𝐿) · ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) / 𝑆)) + ((1 − (𝑆 / 𝐿)) · (𝐹‘(𝑋‘𝑧)))) = (((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) / 𝐿) + (((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧))) / 𝐿))) |
226 | 199, 223,
225 | 3eqtr4d 2788 |
. . 3
⊢ (𝜑 → ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝐵 ∪ {𝑧}))) / 𝐿) = (((𝑆 / 𝐿) · ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) / 𝑆)) + ((1 − (𝑆 / 𝐿)) · (𝐹‘(𝑋‘𝑧))))) |
227 | 195, 196,
226 | 3brtr4d 5105 |
. 2
⊢ (𝜑 → (𝐹‘((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ (𝐵 ∪ {𝑧}))) / 𝐿)) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝐵 ∪ {𝑧}))) / 𝐿)) |
228 | 130, 227 | jca 512 |
1
⊢ (𝜑 → (((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ (𝐵 ∪ {𝑧}))) / 𝐿) ∈ 𝐷 ∧ (𝐹‘((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ (𝐵 ∪ {𝑧}))) / 𝐿)) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝐵 ∪ {𝑧}))) / 𝐿))) |