Step | Hyp | Ref
| Expression |
1 | | jensen.7 |
. . . . . 6
⊢ (𝜑 → 0 <
(ℂfld Σg 𝑇)) |
2 | | jensen.5 |
. . . . . . . . 9
⊢ (𝜑 → 𝑇:𝐴⟶(0[,)+∞)) |
3 | 2 | ffnd 6505 |
. . . . . . . 8
⊢ (𝜑 → 𝑇 Fn 𝐴) |
4 | | fnresdm 6455 |
. . . . . . . 8
⊢ (𝑇 Fn 𝐴 → (𝑇 ↾ 𝐴) = 𝑇) |
5 | 3, 4 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (𝑇 ↾ 𝐴) = 𝑇) |
6 | 5 | oveq2d 7186 |
. . . . . 6
⊢ (𝜑 → (ℂfld
Σg (𝑇 ↾ 𝐴)) = (ℂfld
Σg 𝑇)) |
7 | 1, 6 | breqtrrd 5058 |
. . . . 5
⊢ (𝜑 → 0 <
(ℂfld Σg (𝑇 ↾ 𝐴))) |
8 | | ssid 3899 |
. . . . 5
⊢ 𝐴 ⊆ 𝐴 |
9 | 7, 8 | jctil 523 |
. . . 4
⊢ (𝜑 → (𝐴 ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ 𝐴)))) |
10 | | jensen.4 |
. . . . 5
⊢ (𝜑 → 𝐴 ∈ Fin) |
11 | | sseq1 3902 |
. . . . . . . . 9
⊢ (𝑎 = ∅ → (𝑎 ⊆ 𝐴 ↔ ∅ ⊆ 𝐴)) |
12 | | reseq2 5820 |
. . . . . . . . . . . . 13
⊢ (𝑎 = ∅ → (𝑇 ↾ 𝑎) = (𝑇 ↾ ∅)) |
13 | | res0 5829 |
. . . . . . . . . . . . 13
⊢ (𝑇 ↾ ∅) =
∅ |
14 | 12, 13 | eqtrdi 2789 |
. . . . . . . . . . . 12
⊢ (𝑎 = ∅ → (𝑇 ↾ 𝑎) = ∅) |
15 | 14 | oveq2d 7186 |
. . . . . . . . . . 11
⊢ (𝑎 = ∅ →
(ℂfld Σg (𝑇 ↾ 𝑎)) = (ℂfld
Σg ∅)) |
16 | | cnfld0 20241 |
. . . . . . . . . . . 12
⊢ 0 =
(0g‘ℂfld) |
17 | 16 | gsum0 18010 |
. . . . . . . . . . 11
⊢
(ℂfld Σg ∅) =
0 |
18 | 15, 17 | eqtrdi 2789 |
. . . . . . . . . 10
⊢ (𝑎 = ∅ →
(ℂfld Σg (𝑇 ↾ 𝑎)) = 0) |
19 | 18 | breq2d 5042 |
. . . . . . . . 9
⊢ (𝑎 = ∅ → (0 <
(ℂfld Σg (𝑇 ↾ 𝑎)) ↔ 0 < 0)) |
20 | 11, 19 | anbi12d 634 |
. . . . . . . 8
⊢ (𝑎 = ∅ → ((𝑎 ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ 𝑎))) ↔ (∅ ⊆ 𝐴 ∧ 0 < 0))) |
21 | | reseq2 5820 |
. . . . . . . . . . 11
⊢ (𝑎 = ∅ → ((𝑇 ∘f ·
𝑋) ↾ 𝑎) = ((𝑇 ∘f · 𝑋) ↾
∅)) |
22 | 21 | oveq2d 7186 |
. . . . . . . . . 10
⊢ (𝑎 = ∅ →
(ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) = (ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾
∅))) |
23 | 22, 18 | oveq12d 7188 |
. . . . . . . . 9
⊢ (𝑎 = ∅ →
((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) / (ℂfld
Σg (𝑇 ↾ 𝑎))) = ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ ∅)) /
0)) |
24 | | reseq2 5820 |
. . . . . . . . . . . . 13
⊢ (𝑎 = ∅ → ((𝑇 ∘f ·
(𝐹 ∘ 𝑋)) ↾ 𝑎) = ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ ∅)) |
25 | 24 | oveq2d 7186 |
. . . . . . . . . . . 12
⊢ (𝑎 = ∅ →
(ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) = (ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ ∅))) |
26 | 25, 18 | oveq12d 7188 |
. . . . . . . . . . 11
⊢ (𝑎 = ∅ →
((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) / (ℂfld
Σg (𝑇 ↾ 𝑎))) = ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ ∅)) / 0)) |
27 | 26 | breq2d 5042 |
. . . . . . . . . 10
⊢ (𝑎 = ∅ → ((𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) / (ℂfld
Σg (𝑇 ↾ 𝑎))) ↔ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ ∅)) / 0))) |
28 | 27 | rabbidv 3381 |
. . . . . . . . 9
⊢ (𝑎 = ∅ → {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) / (ℂfld
Σg (𝑇 ↾ 𝑎)))} = {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ ∅)) / 0)}) |
29 | 23, 28 | eleq12d 2827 |
. . . . . . . 8
⊢ (𝑎 = ∅ →
(((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) / (ℂfld
Σg (𝑇 ↾ 𝑎))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) / (ℂfld
Σg (𝑇 ↾ 𝑎)))} ↔ ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ ∅)) / 0) ∈
{𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ ∅)) /
0)})) |
30 | 20, 29 | imbi12d 348 |
. . . . . . 7
⊢ (𝑎 = ∅ → (((𝑎 ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ 𝑎))) → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) / (ℂfld
Σg (𝑇 ↾ 𝑎))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) / (ℂfld
Σg (𝑇 ↾ 𝑎)))}) ↔ ((∅ ⊆ 𝐴 ∧ 0 < 0) →
((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ ∅)) / 0) ∈
{𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ ∅)) /
0)}))) |
31 | 30 | imbi2d 344 |
. . . . . 6
⊢ (𝑎 = ∅ → ((𝜑 → ((𝑎 ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ 𝑎))) → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) / (ℂfld
Σg (𝑇 ↾ 𝑎))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) / (ℂfld
Σg (𝑇 ↾ 𝑎)))})) ↔ (𝜑 → ((∅ ⊆ 𝐴 ∧ 0 < 0) →
((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ ∅)) / 0) ∈
{𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ ∅)) /
0)})))) |
32 | | sseq1 3902 |
. . . . . . . . 9
⊢ (𝑎 = 𝑘 → (𝑎 ⊆ 𝐴 ↔ 𝑘 ⊆ 𝐴)) |
33 | | reseq2 5820 |
. . . . . . . . . . 11
⊢ (𝑎 = 𝑘 → (𝑇 ↾ 𝑎) = (𝑇 ↾ 𝑘)) |
34 | 33 | oveq2d 7186 |
. . . . . . . . . 10
⊢ (𝑎 = 𝑘 → (ℂfld
Σg (𝑇 ↾ 𝑎)) = (ℂfld
Σg (𝑇 ↾ 𝑘))) |
35 | 34 | breq2d 5042 |
. . . . . . . . 9
⊢ (𝑎 = 𝑘 → (0 < (ℂfld
Σg (𝑇 ↾ 𝑎)) ↔ 0 < (ℂfld
Σg (𝑇 ↾ 𝑘)))) |
36 | 32, 35 | anbi12d 634 |
. . . . . . . 8
⊢ (𝑎 = 𝑘 → ((𝑎 ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ 𝑎))) ↔ (𝑘 ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ 𝑘))))) |
37 | | reseq2 5820 |
. . . . . . . . . . 11
⊢ (𝑎 = 𝑘 → ((𝑇 ∘f · 𝑋) ↾ 𝑎) = ((𝑇 ∘f · 𝑋) ↾ 𝑘)) |
38 | 37 | oveq2d 7186 |
. . . . . . . . . 10
⊢ (𝑎 = 𝑘 → (ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) = (ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘))) |
39 | 38, 34 | oveq12d 7188 |
. . . . . . . . 9
⊢ (𝑎 = 𝑘 → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) / (ℂfld
Σg (𝑇 ↾ 𝑎))) = ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘)))) |
40 | | reseq2 5820 |
. . . . . . . . . . . . 13
⊢ (𝑎 = 𝑘 → ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎) = ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) |
41 | 40 | oveq2d 7186 |
. . . . . . . . . . . 12
⊢ (𝑎 = 𝑘 → (ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) = (ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘))) |
42 | 41, 34 | oveq12d 7188 |
. . . . . . . . . . 11
⊢ (𝑎 = 𝑘 → ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) / (ℂfld
Σg (𝑇 ↾ 𝑎))) = ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘)))) |
43 | 42 | breq2d 5042 |
. . . . . . . . . 10
⊢ (𝑎 = 𝑘 → ((𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) / (ℂfld
Σg (𝑇 ↾ 𝑎))) ↔ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘))))) |
44 | 43 | rabbidv 3381 |
. . . . . . . . 9
⊢ (𝑎 = 𝑘 → {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) / (ℂfld
Σg (𝑇 ↾ 𝑎)))} = {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘)))}) |
45 | 39, 44 | eleq12d 2827 |
. . . . . . . 8
⊢ (𝑎 = 𝑘 → (((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) / (ℂfld
Σg (𝑇 ↾ 𝑎))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) / (ℂfld
Σg (𝑇 ↾ 𝑎)))} ↔ ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘)))})) |
46 | 36, 45 | imbi12d 348 |
. . . . . . 7
⊢ (𝑎 = 𝑘 → (((𝑎 ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ 𝑎))) → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) / (ℂfld
Σg (𝑇 ↾ 𝑎))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) / (ℂfld
Σg (𝑇 ↾ 𝑎)))}) ↔ ((𝑘 ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ 𝑘))) → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘)))}))) |
47 | 46 | imbi2d 344 |
. . . . . 6
⊢ (𝑎 = 𝑘 → ((𝜑 → ((𝑎 ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ 𝑎))) → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) / (ℂfld
Σg (𝑇 ↾ 𝑎))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) / (ℂfld
Σg (𝑇 ↾ 𝑎)))})) ↔ (𝜑 → ((𝑘 ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ 𝑘))) → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘)))})))) |
48 | | sseq1 3902 |
. . . . . . . . 9
⊢ (𝑎 = (𝑘 ∪ {𝑐}) → (𝑎 ⊆ 𝐴 ↔ (𝑘 ∪ {𝑐}) ⊆ 𝐴)) |
49 | | reseq2 5820 |
. . . . . . . . . . 11
⊢ (𝑎 = (𝑘 ∪ {𝑐}) → (𝑇 ↾ 𝑎) = (𝑇 ↾ (𝑘 ∪ {𝑐}))) |
50 | 49 | oveq2d 7186 |
. . . . . . . . . 10
⊢ (𝑎 = (𝑘 ∪ {𝑐}) → (ℂfld
Σg (𝑇 ↾ 𝑎)) = (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) |
51 | 50 | breq2d 5042 |
. . . . . . . . 9
⊢ (𝑎 = (𝑘 ∪ {𝑐}) → (0 < (ℂfld
Σg (𝑇 ↾ 𝑎)) ↔ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) |
52 | 48, 51 | anbi12d 634 |
. . . . . . . 8
⊢ (𝑎 = (𝑘 ∪ {𝑐}) → ((𝑎 ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ 𝑎))) ↔ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))))) |
53 | | reseq2 5820 |
. . . . . . . . . . 11
⊢ (𝑎 = (𝑘 ∪ {𝑐}) → ((𝑇 ∘f · 𝑋) ↾ 𝑎) = ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) |
54 | 53 | oveq2d 7186 |
. . . . . . . . . 10
⊢ (𝑎 = (𝑘 ∪ {𝑐}) → (ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) = (ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐})))) |
55 | 54, 50 | oveq12d 7188 |
. . . . . . . . 9
⊢ (𝑎 = (𝑘 ∪ {𝑐}) → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) / (ℂfld
Σg (𝑇 ↾ 𝑎))) = ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) |
56 | | reseq2 5820 |
. . . . . . . . . . . . 13
⊢ (𝑎 = (𝑘 ∪ {𝑐}) → ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎) = ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) |
57 | 56 | oveq2d 7186 |
. . . . . . . . . . . 12
⊢ (𝑎 = (𝑘 ∪ {𝑐}) → (ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) = (ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐})))) |
58 | 57, 50 | oveq12d 7188 |
. . . . . . . . . . 11
⊢ (𝑎 = (𝑘 ∪ {𝑐}) → ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) / (ℂfld
Σg (𝑇 ↾ 𝑎))) = ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) |
59 | 58 | breq2d 5042 |
. . . . . . . . . 10
⊢ (𝑎 = (𝑘 ∪ {𝑐}) → ((𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) / (ℂfld
Σg (𝑇 ↾ 𝑎))) ↔ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))))) |
60 | 59 | rabbidv 3381 |
. . . . . . . . 9
⊢ (𝑎 = (𝑘 ∪ {𝑐}) → {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) / (ℂfld
Σg (𝑇 ↾ 𝑎)))} = {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))}) |
61 | 55, 60 | eleq12d 2827 |
. . . . . . . 8
⊢ (𝑎 = (𝑘 ∪ {𝑐}) → (((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) / (ℂfld
Σg (𝑇 ↾ 𝑎))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) / (ℂfld
Σg (𝑇 ↾ 𝑎)))} ↔ ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))})) |
62 | 52, 61 | imbi12d 348 |
. . . . . . 7
⊢ (𝑎 = (𝑘 ∪ {𝑐}) → (((𝑎 ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ 𝑎))) → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) / (ℂfld
Σg (𝑇 ↾ 𝑎))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) / (ℂfld
Σg (𝑇 ↾ 𝑎)))}) ↔ (((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))}))) |
63 | 62 | imbi2d 344 |
. . . . . 6
⊢ (𝑎 = (𝑘 ∪ {𝑐}) → ((𝜑 → ((𝑎 ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ 𝑎))) → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) / (ℂfld
Σg (𝑇 ↾ 𝑎))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) / (ℂfld
Σg (𝑇 ↾ 𝑎)))})) ↔ (𝜑 → (((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))})))) |
64 | | sseq1 3902 |
. . . . . . . . 9
⊢ (𝑎 = 𝐴 → (𝑎 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴)) |
65 | | reseq2 5820 |
. . . . . . . . . . 11
⊢ (𝑎 = 𝐴 → (𝑇 ↾ 𝑎) = (𝑇 ↾ 𝐴)) |
66 | 65 | oveq2d 7186 |
. . . . . . . . . 10
⊢ (𝑎 = 𝐴 → (ℂfld
Σg (𝑇 ↾ 𝑎)) = (ℂfld
Σg (𝑇 ↾ 𝐴))) |
67 | 66 | breq2d 5042 |
. . . . . . . . 9
⊢ (𝑎 = 𝐴 → (0 < (ℂfld
Σg (𝑇 ↾ 𝑎)) ↔ 0 < (ℂfld
Σg (𝑇 ↾ 𝐴)))) |
68 | 64, 67 | anbi12d 634 |
. . . . . . . 8
⊢ (𝑎 = 𝐴 → ((𝑎 ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ 𝑎))) ↔ (𝐴 ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ 𝐴))))) |
69 | | reseq2 5820 |
. . . . . . . . . . 11
⊢ (𝑎 = 𝐴 → ((𝑇 ∘f · 𝑋) ↾ 𝑎) = ((𝑇 ∘f · 𝑋) ↾ 𝐴)) |
70 | 69 | oveq2d 7186 |
. . . . . . . . . 10
⊢ (𝑎 = 𝐴 → (ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) = (ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐴))) |
71 | 70, 66 | oveq12d 7188 |
. . . . . . . . 9
⊢ (𝑎 = 𝐴 → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) / (ℂfld
Σg (𝑇 ↾ 𝑎))) = ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐴)) / (ℂfld
Σg (𝑇 ↾ 𝐴)))) |
72 | | reseq2 5820 |
. . . . . . . . . . . . 13
⊢ (𝑎 = 𝐴 → ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎) = ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐴)) |
73 | 72 | oveq2d 7186 |
. . . . . . . . . . . 12
⊢ (𝑎 = 𝐴 → (ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) = (ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐴))) |
74 | 73, 66 | oveq12d 7188 |
. . . . . . . . . . 11
⊢ (𝑎 = 𝐴 → ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) / (ℂfld
Σg (𝑇 ↾ 𝑎))) = ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐴)) / (ℂfld
Σg (𝑇 ↾ 𝐴)))) |
75 | 74 | breq2d 5042 |
. . . . . . . . . 10
⊢ (𝑎 = 𝐴 → ((𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) / (ℂfld
Σg (𝑇 ↾ 𝑎))) ↔ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐴)) / (ℂfld
Σg (𝑇 ↾ 𝐴))))) |
76 | 75 | rabbidv 3381 |
. . . . . . . . 9
⊢ (𝑎 = 𝐴 → {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) / (ℂfld
Σg (𝑇 ↾ 𝑎)))} = {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐴)) / (ℂfld
Σg (𝑇 ↾ 𝐴)))}) |
77 | 71, 76 | eleq12d 2827 |
. . . . . . . 8
⊢ (𝑎 = 𝐴 → (((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) / (ℂfld
Σg (𝑇 ↾ 𝑎))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) / (ℂfld
Σg (𝑇 ↾ 𝑎)))} ↔ ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐴)) / (ℂfld
Σg (𝑇 ↾ 𝐴))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐴)) / (ℂfld
Σg (𝑇 ↾ 𝐴)))})) |
78 | 68, 77 | imbi12d 348 |
. . . . . . 7
⊢ (𝑎 = 𝐴 → (((𝑎 ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ 𝑎))) → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) / (ℂfld
Σg (𝑇 ↾ 𝑎))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) / (ℂfld
Σg (𝑇 ↾ 𝑎)))}) ↔ ((𝐴 ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ 𝐴))) → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐴)) / (ℂfld
Σg (𝑇 ↾ 𝐴))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐴)) / (ℂfld
Σg (𝑇 ↾ 𝐴)))}))) |
79 | 78 | imbi2d 344 |
. . . . . 6
⊢ (𝑎 = 𝐴 → ((𝜑 → ((𝑎 ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ 𝑎))) → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) / (ℂfld
Σg (𝑇 ↾ 𝑎))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) / (ℂfld
Σg (𝑇 ↾ 𝑎)))})) ↔ (𝜑 → ((𝐴 ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ 𝐴))) → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐴)) / (ℂfld
Σg (𝑇 ↾ 𝐴))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐴)) / (ℂfld
Σg (𝑇 ↾ 𝐴)))})))) |
80 | | 0re 10721 |
. . . . . . . . . 10
⊢ 0 ∈
ℝ |
81 | 80 | ltnri 10827 |
. . . . . . . . 9
⊢ ¬ 0
< 0 |
82 | 81 | pm2.21i 119 |
. . . . . . . 8
⊢ (0 < 0
→ ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ ∅)) / 0) ∈
{𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ ∅)) / 0)}) |
83 | 82 | adantl 485 |
. . . . . . 7
⊢ ((∅
⊆ 𝐴 ∧ 0 < 0)
→ ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ ∅)) / 0) ∈
{𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ ∅)) / 0)}) |
84 | 83 | a1i 11 |
. . . . . 6
⊢ (𝜑 → ((∅ ⊆ 𝐴 ∧ 0 < 0) →
((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ ∅)) / 0) ∈
{𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ ∅)) /
0)})) |
85 | | impexp 454 |
. . . . . . . . . . . 12
⊢ (((𝑘 ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ 𝑘))) → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘)))}) ↔ (𝑘 ⊆ 𝐴 → (0 < (ℂfld
Σg (𝑇 ↾ 𝑘)) → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘)))}))) |
86 | | simprl 771 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (𝑘 ∪ {𝑐}) ⊆ 𝐴) |
87 | 86 | unssad 4077 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → 𝑘 ⊆ 𝐴) |
88 | | simpr 488 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 < (ℂfld
Σg (𝑇 ↾ 𝑘))) → 0 < (ℂfld
Σg (𝑇 ↾ 𝑘))) |
89 | | jensen.1 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝐷 ⊆ ℝ) |
90 | 89 | ad3antrrr 730 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld
Σg (𝑇 ↾ 𝑘)) ∧ ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘)))})) → 𝐷 ⊆ ℝ) |
91 | | jensen.2 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝐹:𝐷⟶ℝ) |
92 | 91 | ad3antrrr 730 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld
Σg (𝑇 ↾ 𝑘)) ∧ ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘)))})) → 𝐹:𝐷⟶ℝ) |
93 | | simplll 775 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld
Σg (𝑇 ↾ 𝑘)) ∧ ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘)))})) → 𝜑) |
94 | | jensen.3 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (𝑎 ∈ 𝐷 ∧ 𝑏 ∈ 𝐷)) → (𝑎[,]𝑏) ⊆ 𝐷) |
95 | 93, 94 | sylan 583 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝜑 ∧ ¬
𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld
Σg (𝑇 ↾ 𝑘)) ∧ ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘)))})) ∧ (𝑎 ∈ 𝐷 ∧ 𝑏 ∈ 𝐷)) → (𝑎[,]𝑏) ⊆ 𝐷) |
96 | 93, 10 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld
Σg (𝑇 ↾ 𝑘)) ∧ ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘)))})) → 𝐴 ∈ Fin) |
97 | 93, 2 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld
Σg (𝑇 ↾ 𝑘)) ∧ ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘)))})) → 𝑇:𝐴⟶(0[,)+∞)) |
98 | | jensen.6 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑋:𝐴⟶𝐷) |
99 | 93, 98 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld
Σg (𝑇 ↾ 𝑘)) ∧ ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘)))})) → 𝑋:𝐴⟶𝐷) |
100 | 1 | ad3antrrr 730 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld
Σg (𝑇 ↾ 𝑘)) ∧ ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘)))})) → 0 < (ℂfld
Σg 𝑇)) |
101 | | jensen.8 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷 ∧ 𝑡 ∈ (0[,]1))) → (𝐹‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ≤ ((𝑡 · (𝐹‘𝑥)) + ((1 − 𝑡) · (𝐹‘𝑦)))) |
102 | 93, 101 | sylan 583 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝜑 ∧ ¬
𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld
Σg (𝑇 ↾ 𝑘)) ∧ ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘)))})) ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷 ∧ 𝑡 ∈ (0[,]1))) → (𝐹‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ≤ ((𝑡 · (𝐹‘𝑥)) + ((1 − 𝑡) · (𝐹‘𝑦)))) |
103 | | simpllr 776 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld
Σg (𝑇 ↾ 𝑘)) ∧ ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘)))})) → ¬ 𝑐 ∈ 𝑘) |
104 | 86 | adantr 484 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld
Σg (𝑇 ↾ 𝑘)) ∧ ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘)))})) → (𝑘 ∪ {𝑐}) ⊆ 𝐴) |
105 | | eqid 2738 |
. . . . . . . . . . . . . . . . . 18
⊢
(ℂfld Σg (𝑇 ↾ 𝑘)) = (ℂfld
Σg (𝑇 ↾ 𝑘)) |
106 | | eqid 2738 |
. . . . . . . . . . . . . . . . . 18
⊢
(ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))) = (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))) |
107 | | cnring 20239 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
ℂfld ∈ Ring |
108 | | ringcmn 19453 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(ℂfld ∈ Ring → ℂfld ∈
CMnd) |
109 | 107, 108 | mp1i 13 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → ℂfld ∈
CMnd) |
110 | 10 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → 𝐴 ∈ Fin) |
111 | 110, 87 | ssfid 8819 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → 𝑘 ∈ Fin) |
112 | | rege0subm 20273 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(0[,)+∞) ∈
(SubMnd‘ℂfld) |
113 | 112 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (0[,)+∞) ∈
(SubMnd‘ℂfld)) |
114 | 2 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → 𝑇:𝐴⟶(0[,)+∞)) |
115 | 114, 87 | fssresd 6545 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (𝑇 ↾ 𝑘):𝑘⟶(0[,)+∞)) |
116 | | c0ex 10713 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ 0 ∈
V |
117 | 116 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → 0 ∈ V) |
118 | 115, 111,
117 | fdmfifsupp 8916 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (𝑇 ↾ 𝑘) finSupp 0) |
119 | 16, 109, 111, 113, 115, 118 | gsumsubmcl 19158 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (ℂfld
Σg (𝑇 ↾ 𝑘)) ∈ (0[,)+∞)) |
120 | | elrege0 12928 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((ℂfld Σg (𝑇 ↾ 𝑘)) ∈ (0[,)+∞) ↔
((ℂfld Σg (𝑇 ↾ 𝑘)) ∈ ℝ ∧ 0 ≤
(ℂfld Σg (𝑇 ↾ 𝑘)))) |
121 | 120 | simplbi 501 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((ℂfld Σg (𝑇 ↾ 𝑘)) ∈ (0[,)+∞) →
(ℂfld Σg (𝑇 ↾ 𝑘)) ∈ ℝ) |
122 | 119, 121 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (ℂfld
Σg (𝑇 ↾ 𝑘)) ∈ ℝ) |
123 | 122 | adantr 484 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld
Σg (𝑇 ↾ 𝑘)) ∧ ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘)))})) → (ℂfld
Σg (𝑇 ↾ 𝑘)) ∈ ℝ) |
124 | | simprl 771 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld
Σg (𝑇 ↾ 𝑘)) ∧ ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘)))})) → 0 < (ℂfld
Σg (𝑇 ↾ 𝑘))) |
125 | 123, 124 | elrpd 12511 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld
Σg (𝑇 ↾ 𝑘)) ∧ ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘)))})) → (ℂfld
Σg (𝑇 ↾ 𝑘)) ∈
ℝ+) |
126 | | simprr 773 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld
Σg (𝑇 ↾ 𝑘)) ∧ ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘)))})) → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘)))}) |
127 | | fveq2 6674 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑤 = ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘))) → (𝐹‘𝑤) = (𝐹‘((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘))))) |
128 | 127 | breq1d 5040 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑤 = ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘))) → ((𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘))) ↔ (𝐹‘((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘)))) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘))))) |
129 | 128 | elrab 3588 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘)))} ↔ (((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘))) ∈ 𝐷 ∧ (𝐹‘((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘)))) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘))))) |
130 | 126, 129 | sylib 221 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld
Σg (𝑇 ↾ 𝑘)) ∧ ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘)))})) → (((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘))) ∈ 𝐷 ∧ (𝐹‘((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘)))) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘))))) |
131 | 130 | simpld 498 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld
Σg (𝑇 ↾ 𝑘)) ∧ ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘)))})) → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘))) ∈ 𝐷) |
132 | 130 | simprd 499 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld
Σg (𝑇 ↾ 𝑘)) ∧ ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘)))})) → (𝐹‘((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘)))) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘)))) |
133 | 90, 92, 95, 96, 97, 99, 100, 102, 103, 104, 105, 106, 125, 131, 132 | jensenlem2 25725 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld
Σg (𝑇 ↾ 𝑘)) ∧ ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘)))})) → (((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ 𝐷 ∧ (𝐹‘((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))))) |
134 | | fveq2 6674 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑤 = ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) → (𝐹‘𝑤) = (𝐹‘((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))))) |
135 | 134 | breq1d 5040 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑤 = ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) → ((𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ↔ (𝐹‘((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))))) |
136 | 135 | elrab 3588 |
. . . . . . . . . . . . . . . . 17
⊢
(((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))} ↔ (((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ 𝐷 ∧ (𝐹‘((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))))) |
137 | 133, 136 | sylibr 237 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld
Σg (𝑇 ↾ 𝑘)) ∧ ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘)))})) → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))}) |
138 | 137 | expr 460 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 < (ℂfld
Σg (𝑇 ↾ 𝑘))) → (((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘)))} → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))})) |
139 | 88, 138 | embantd 59 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 < (ℂfld
Σg (𝑇 ↾ 𝑘))) → ((0 < (ℂfld
Σg (𝑇 ↾ 𝑘)) → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘)))}) → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))})) |
140 | | cnfldbas 20221 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ℂ =
(Base‘ℂfld) |
141 | | ringmnd 19426 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(ℂfld ∈ Ring → ℂfld ∈
Mnd) |
142 | 107, 141 | mp1i 13 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → ℂfld ∈
Mnd) |
143 | 110, 86 | ssfid 8819 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (𝑘 ∪ {𝑐}) ∈ Fin) |
144 | 143 | adantr 484 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → (𝑘 ∪ {𝑐}) ∈ Fin) |
145 | | ssun2 4063 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ {𝑐} ⊆ (𝑘 ∪ {𝑐}) |
146 | | vsnid 4553 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ 𝑐 ∈ {𝑐} |
147 | 145, 146 | sselii 3874 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 𝑐 ∈ (𝑘 ∪ {𝑐}) |
148 | 147 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → 𝑐 ∈ (𝑘 ∪ {𝑐})) |
149 | | remulcl 10700 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 · 𝑦) ∈ ℝ) |
150 | 149 | adantl 485 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 · 𝑦) ∈ ℝ) |
151 | | rge0ssre 12930 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
(0[,)+∞) ⊆ ℝ |
152 | | fss 6521 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑇:𝐴⟶(0[,)+∞) ∧ (0[,)+∞)
⊆ ℝ) → 𝑇:𝐴⟶ℝ) |
153 | 2, 151, 152 | sylancl 589 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → 𝑇:𝐴⟶ℝ) |
154 | 98, 89 | fssd 6522 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → 𝑋:𝐴⟶ℝ) |
155 | | inidm 4109 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝐴 ∩ 𝐴) = 𝐴 |
156 | 150, 153,
154, 10, 10, 155 | off 7442 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → (𝑇 ∘f · 𝑋):𝐴⟶ℝ) |
157 | | ax-resscn 10672 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ℝ
⊆ ℂ |
158 | | fss 6521 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝑇 ∘f ·
𝑋):𝐴⟶ℝ ∧ ℝ ⊆
ℂ) → (𝑇
∘f · 𝑋):𝐴⟶ℂ) |
159 | 156, 157,
158 | sylancl 589 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → (𝑇 ∘f · 𝑋):𝐴⟶ℂ) |
160 | 159 | ad3antrrr 730 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → (𝑇 ∘f · 𝑋):𝐴⟶ℂ) |
161 | 86 | adantr 484 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → (𝑘 ∪ {𝑐}) ⊆ 𝐴) |
162 | 160, 161 | fssresd 6545 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐})):(𝑘 ∪ {𝑐})⟶ℂ) |
163 | 2 | ad3antrrr 730 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → 𝑇:𝐴⟶(0[,)+∞)) |
164 | 110 | adantr 484 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → 𝐴 ∈ Fin) |
165 | 163, 164 | fexd 7000 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → 𝑇 ∈ V) |
166 | 98 | ad3antrrr 730 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → 𝑋:𝐴⟶𝐷) |
167 | 166, 164 | fexd 7000 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → 𝑋 ∈ V) |
168 | | offres 7709 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑇 ∈ V ∧ 𝑋 ∈ V) → ((𝑇 ∘f ·
𝑋) ↾ (𝑘 ∪ {𝑐})) = ((𝑇 ↾ (𝑘 ∪ {𝑐})) ∘f · (𝑋 ↾ (𝑘 ∪ {𝑐})))) |
169 | 165, 167,
168 | syl2anc 587 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐})) = ((𝑇 ↾ (𝑘 ∪ {𝑐})) ∘f · (𝑋 ↾ (𝑘 ∪ {𝑐})))) |
170 | 169 | oveq1d 7185 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → (((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐})) supp 0) = (((𝑇 ↾ (𝑘 ∪ {𝑐})) ∘f · (𝑋 ↾ (𝑘 ∪ {𝑐}))) supp 0)) |
171 | 151, 157 | sstri 3886 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
(0[,)+∞) ⊆ ℂ |
172 | | fss 6521 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑇:𝐴⟶(0[,)+∞) ∧ (0[,)+∞)
⊆ ℂ) → 𝑇:𝐴⟶ℂ) |
173 | 163, 171,
172 | sylancl 589 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → 𝑇:𝐴⟶ℂ) |
174 | 173, 161 | fssresd 6545 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → (𝑇 ↾ (𝑘 ∪ {𝑐})):(𝑘 ∪ {𝑐})⟶ℂ) |
175 | | eldifi 4017 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑥 ∈ ((𝑘 ∪ {𝑐}) ∖ {𝑐}) → 𝑥 ∈ (𝑘 ∪ {𝑐})) |
176 | 175 | adantl 485 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
(((((𝜑 ∧ ¬
𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) ∧ 𝑥 ∈ ((𝑘 ∪ {𝑐}) ∖ {𝑐})) → 𝑥 ∈ (𝑘 ∪ {𝑐})) |
177 | 176 | fvresd 6694 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
(((((𝜑 ∧ ¬
𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) ∧ 𝑥 ∈ ((𝑘 ∪ {𝑐}) ∖ {𝑐})) → ((𝑇 ↾ (𝑘 ∪ {𝑐}))‘𝑥) = (𝑇‘𝑥)) |
178 | | difun2 4370 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝑘 ∪ {𝑐}) ∖ {𝑐}) = (𝑘 ∖ {𝑐}) |
179 | | difss 4022 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑘 ∖ {𝑐}) ⊆ 𝑘 |
180 | 178, 179 | eqsstri 3911 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑘 ∪ {𝑐}) ∖ {𝑐}) ⊆ 𝑘 |
181 | 180 | sseli 3873 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑥 ∈ ((𝑘 ∪ {𝑐}) ∖ {𝑐}) → 𝑥 ∈ 𝑘) |
182 | | simpr 488 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) |
183 | 87 | adantr 484 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → 𝑘 ⊆ 𝐴) |
184 | 163, 183 | feqresmpt 6738 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → (𝑇 ↾ 𝑘) = (𝑥 ∈ 𝑘 ↦ (𝑇‘𝑥))) |
185 | 184 | oveq2d 7186 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → (ℂfld
Σg (𝑇 ↾ 𝑘)) = (ℂfld
Σg (𝑥 ∈ 𝑘 ↦ (𝑇‘𝑥)))) |
186 | 111 | adantr 484 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → 𝑘 ∈ Fin) |
187 | 183 | sselda 3877 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢
(((((𝜑 ∧ ¬
𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) ∧ 𝑥 ∈ 𝑘) → 𝑥 ∈ 𝐴) |
188 | 163 | ffvelrnda 6861 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢
(((((𝜑 ∧ ¬
𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) ∧ 𝑥 ∈ 𝐴) → (𝑇‘𝑥) ∈ (0[,)+∞)) |
189 | 187, 188 | syldan 594 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢
(((((𝜑 ∧ ¬
𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) ∧ 𝑥 ∈ 𝑘) → (𝑇‘𝑥) ∈ (0[,)+∞)) |
190 | 171, 189 | sseldi 3875 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢
(((((𝜑 ∧ ¬
𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) ∧ 𝑥 ∈ 𝑘) → (𝑇‘𝑥) ∈ ℂ) |
191 | 186, 190 | gsumfsum 20284 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → (ℂfld
Σg (𝑥 ∈ 𝑘 ↦ (𝑇‘𝑥))) = Σ𝑥 ∈ 𝑘 (𝑇‘𝑥)) |
192 | 182, 185,
191 | 3eqtrrd 2778 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → Σ𝑥 ∈ 𝑘 (𝑇‘𝑥) = 0) |
193 | | elrege0 12928 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((𝑇‘𝑥) ∈ (0[,)+∞) ↔ ((𝑇‘𝑥) ∈ ℝ ∧ 0 ≤ (𝑇‘𝑥))) |
194 | 189, 193 | sylib 221 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢
(((((𝜑 ∧ ¬
𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) ∧ 𝑥 ∈ 𝑘) → ((𝑇‘𝑥) ∈ ℝ ∧ 0 ≤ (𝑇‘𝑥))) |
195 | 194 | simpld 498 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢
(((((𝜑 ∧ ¬
𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) ∧ 𝑥 ∈ 𝑘) → (𝑇‘𝑥) ∈ ℝ) |
196 | 194 | simprd 499 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢
(((((𝜑 ∧ ¬
𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) ∧ 𝑥 ∈ 𝑘) → 0 ≤ (𝑇‘𝑥)) |
197 | 186, 195,
196 | fsum00 15246 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → (Σ𝑥 ∈ 𝑘 (𝑇‘𝑥) = 0 ↔ ∀𝑥 ∈ 𝑘 (𝑇‘𝑥) = 0)) |
198 | 192, 197 | mpbid 235 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → ∀𝑥 ∈ 𝑘 (𝑇‘𝑥) = 0) |
199 | 198 | r19.21bi 3121 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
(((((𝜑 ∧ ¬
𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) ∧ 𝑥 ∈ 𝑘) → (𝑇‘𝑥) = 0) |
200 | 181, 199 | sylan2 596 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
(((((𝜑 ∧ ¬
𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) ∧ 𝑥 ∈ ((𝑘 ∪ {𝑐}) ∖ {𝑐})) → (𝑇‘𝑥) = 0) |
201 | 177, 200 | eqtrd 2773 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(((((𝜑 ∧ ¬
𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) ∧ 𝑥 ∈ ((𝑘 ∪ {𝑐}) ∖ {𝑐})) → ((𝑇 ↾ (𝑘 ∪ {𝑐}))‘𝑥) = 0) |
202 | 174, 201 | suppss 7889 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → ((𝑇 ↾ (𝑘 ∪ {𝑐})) supp 0) ⊆ {𝑐}) |
203 | | mul02 10896 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑥 ∈ ℂ → (0
· 𝑥) =
0) |
204 | 203 | adantl 485 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((((𝜑 ∧ ¬
𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) ∧ 𝑥 ∈ ℂ) → (0 · 𝑥) = 0) |
205 | 89 | ad3antrrr 730 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → 𝐷 ⊆ ℝ) |
206 | 205, 157 | sstrdi 3889 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → 𝐷 ⊆ ℂ) |
207 | 166, 206 | fssd 6522 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → 𝑋:𝐴⟶ℂ) |
208 | 207, 161 | fssresd 6545 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → (𝑋 ↾ (𝑘 ∪ {𝑐})):(𝑘 ∪ {𝑐})⟶ℂ) |
209 | 116 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → 0 ∈ V) |
210 | 202, 204,
174, 208, 144, 209 | suppssof1 7894 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → (((𝑇 ↾ (𝑘 ∪ {𝑐})) ∘f · (𝑋 ↾ (𝑘 ∪ {𝑐}))) supp 0) ⊆ {𝑐}) |
211 | 170, 210 | eqsstrd 3915 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → (((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐})) supp 0) ⊆ {𝑐}) |
212 | 140, 16, 142, 144, 148, 162, 211 | gsumpt 19201 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → (ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) = (((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))‘𝑐)) |
213 | 148 | fvresd 6694 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → (((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))‘𝑐) = ((𝑇 ∘f · 𝑋)‘𝑐)) |
214 | 163 | ffnd 6505 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → 𝑇 Fn 𝐴) |
215 | 166 | ffnd 6505 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → 𝑋 Fn 𝐴) |
216 | 161, 148 | sseldd 3878 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → 𝑐 ∈ 𝐴) |
217 | | fnfvof 7441 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑇 Fn 𝐴 ∧ 𝑋 Fn 𝐴) ∧ (𝐴 ∈ Fin ∧ 𝑐 ∈ 𝐴)) → ((𝑇 ∘f · 𝑋)‘𝑐) = ((𝑇‘𝑐) · (𝑋‘𝑐))) |
218 | 214, 215,
164, 216, 217 | syl22anc 838 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → ((𝑇 ∘f · 𝑋)‘𝑐) = ((𝑇‘𝑐) · (𝑋‘𝑐))) |
219 | 212, 213,
218 | 3eqtrd 2777 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → (ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) = ((𝑇‘𝑐) · (𝑋‘𝑐))) |
220 | 140, 16, 142, 144, 148, 174, 202 | gsumpt 19201 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))) = ((𝑇 ↾ (𝑘 ∪ {𝑐}))‘𝑐)) |
221 | 148 | fvresd 6694 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → ((𝑇 ↾ (𝑘 ∪ {𝑐}))‘𝑐) = (𝑇‘𝑐)) |
222 | 220, 221 | eqtrd 2773 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))) = (𝑇‘𝑐)) |
223 | 219, 222 | oveq12d 7188 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) = (((𝑇‘𝑐) · (𝑋‘𝑐)) / (𝑇‘𝑐))) |
224 | 207, 216 | ffvelrnd 6862 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → (𝑋‘𝑐) ∈ ℂ) |
225 | 173, 216 | ffvelrnd 6862 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → (𝑇‘𝑐) ∈ ℂ) |
226 | | simplrr 778 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) |
227 | 226, 222 | breqtrd 5056 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → 0 < (𝑇‘𝑐)) |
228 | 227 | gt0ne0d 11282 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → (𝑇‘𝑐) ≠ 0) |
229 | 224, 225,
228 | divcan3d 11499 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → (((𝑇‘𝑐) · (𝑋‘𝑐)) / (𝑇‘𝑐)) = (𝑋‘𝑐)) |
230 | 223, 229 | eqtrd 2773 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) = (𝑋‘𝑐)) |
231 | 166, 216 | ffvelrnd 6862 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → (𝑋‘𝑐) ∈ 𝐷) |
232 | 230, 231 | eqeltrd 2833 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ 𝐷) |
233 | 91 | ad3antrrr 730 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → 𝐹:𝐷⟶ℝ) |
234 | 233, 231 | ffvelrnd 6862 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → (𝐹‘(𝑋‘𝑐)) ∈ ℝ) |
235 | 234 | leidd 11284 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → (𝐹‘(𝑋‘𝑐)) ≤ (𝐹‘(𝑋‘𝑐))) |
236 | 230 | fveq2d 6678 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → (𝐹‘((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) = (𝐹‘(𝑋‘𝑐))) |
237 | | fco 6528 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝐹:𝐷⟶ℝ ∧ 𝑋:𝐴⟶𝐷) → (𝐹 ∘ 𝑋):𝐴⟶ℝ) |
238 | 91, 98, 237 | syl2anc 587 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → (𝐹 ∘ 𝑋):𝐴⟶ℝ) |
239 | 150, 153,
238, 10, 10, 155 | off 7442 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → (𝑇 ∘f · (𝐹 ∘ 𝑋)):𝐴⟶ℝ) |
240 | | fss 6521 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝑇 ∘f ·
(𝐹 ∘ 𝑋)):𝐴⟶ℝ ∧ ℝ ⊆
ℂ) → (𝑇
∘f · (𝐹 ∘ 𝑋)):𝐴⟶ℂ) |
241 | 239, 157,
240 | sylancl 589 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → (𝑇 ∘f · (𝐹 ∘ 𝑋)):𝐴⟶ℂ) |
242 | 241 | ad3antrrr 730 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → (𝑇 ∘f · (𝐹 ∘ 𝑋)):𝐴⟶ℂ) |
243 | 242, 161 | fssresd 6545 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐})):(𝑘 ∪ {𝑐})⟶ℂ) |
244 | 238 | ad3antrrr 730 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → (𝐹 ∘ 𝑋):𝐴⟶ℝ) |
245 | 244, 164 | fexd 7000 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → (𝐹 ∘ 𝑋) ∈ V) |
246 | | offres 7709 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑇 ∈ V ∧ (𝐹 ∘ 𝑋) ∈ V) → ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐})) = ((𝑇 ↾ (𝑘 ∪ {𝑐})) ∘f · ((𝐹 ∘ 𝑋) ↾ (𝑘 ∪ {𝑐})))) |
247 | 165, 245,
246 | syl2anc 587 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐})) = ((𝑇 ↾ (𝑘 ∪ {𝑐})) ∘f · ((𝐹 ∘ 𝑋) ↾ (𝑘 ∪ {𝑐})))) |
248 | 247 | oveq1d 7185 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → (((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐})) supp 0) = (((𝑇 ↾ (𝑘 ∪ {𝑐})) ∘f · ((𝐹 ∘ 𝑋) ↾ (𝑘 ∪ {𝑐}))) supp 0)) |
249 | | fss 6521 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝐹 ∘ 𝑋):𝐴⟶ℝ ∧ ℝ ⊆
ℂ) → (𝐹 ∘
𝑋):𝐴⟶ℂ) |
250 | 244, 157,
249 | sylancl 589 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → (𝐹 ∘ 𝑋):𝐴⟶ℂ) |
251 | 250, 161 | fssresd 6545 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → ((𝐹 ∘ 𝑋) ↾ (𝑘 ∪ {𝑐})):(𝑘 ∪ {𝑐})⟶ℂ) |
252 | 202, 204,
174, 251, 144, 209 | suppssof1 7894 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → (((𝑇 ↾ (𝑘 ∪ {𝑐})) ∘f · ((𝐹 ∘ 𝑋) ↾ (𝑘 ∪ {𝑐}))) supp 0) ⊆ {𝑐}) |
253 | 248, 252 | eqsstrd 3915 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → (((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐})) supp 0) ⊆ {𝑐}) |
254 | 140, 16, 142, 144, 148, 243, 253 | gsumpt 19201 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → (ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) = (((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))‘𝑐)) |
255 | 148 | fvresd 6694 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → (((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))‘𝑐) = ((𝑇 ∘f · (𝐹 ∘ 𝑋))‘𝑐)) |
256 | 91 | ffnd 6505 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → 𝐹 Fn 𝐷) |
257 | | fnfco 6543 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝐹 Fn 𝐷 ∧ 𝑋:𝐴⟶𝐷) → (𝐹 ∘ 𝑋) Fn 𝐴) |
258 | 256, 98, 257 | syl2anc 587 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → (𝐹 ∘ 𝑋) Fn 𝐴) |
259 | 258 | ad3antrrr 730 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → (𝐹 ∘ 𝑋) Fn 𝐴) |
260 | | fnfvof 7441 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑇 Fn 𝐴 ∧ (𝐹 ∘ 𝑋) Fn 𝐴) ∧ (𝐴 ∈ Fin ∧ 𝑐 ∈ 𝐴)) → ((𝑇 ∘f · (𝐹 ∘ 𝑋))‘𝑐) = ((𝑇‘𝑐) · ((𝐹 ∘ 𝑋)‘𝑐))) |
261 | 214, 259,
164, 216, 260 | syl22anc 838 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → ((𝑇 ∘f · (𝐹 ∘ 𝑋))‘𝑐) = ((𝑇‘𝑐) · ((𝐹 ∘ 𝑋)‘𝑐))) |
262 | | fvco3 6767 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑋:𝐴⟶𝐷 ∧ 𝑐 ∈ 𝐴) → ((𝐹 ∘ 𝑋)‘𝑐) = (𝐹‘(𝑋‘𝑐))) |
263 | 166, 216,
262 | syl2anc 587 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → ((𝐹 ∘ 𝑋)‘𝑐) = (𝐹‘(𝑋‘𝑐))) |
264 | 263 | oveq2d 7186 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → ((𝑇‘𝑐) · ((𝐹 ∘ 𝑋)‘𝑐)) = ((𝑇‘𝑐) · (𝐹‘(𝑋‘𝑐)))) |
265 | 261, 264 | eqtrd 2773 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → ((𝑇 ∘f · (𝐹 ∘ 𝑋))‘𝑐) = ((𝑇‘𝑐) · (𝐹‘(𝑋‘𝑐)))) |
266 | 254, 255,
265 | 3eqtrd 2777 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → (ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) = ((𝑇‘𝑐) · (𝐹‘(𝑋‘𝑐)))) |
267 | 266, 222 | oveq12d 7188 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) = (((𝑇‘𝑐) · (𝐹‘(𝑋‘𝑐))) / (𝑇‘𝑐))) |
268 | 234 | recnd 10747 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → (𝐹‘(𝑋‘𝑐)) ∈ ℂ) |
269 | 268, 225,
228 | divcan3d 11499 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → (((𝑇‘𝑐) · (𝐹‘(𝑋‘𝑐))) / (𝑇‘𝑐)) = (𝐹‘(𝑋‘𝑐))) |
270 | 267, 269 | eqtrd 2773 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) = (𝐹‘(𝑋‘𝑐))) |
271 | 235, 236,
270 | 3brtr4d 5062 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → (𝐹‘((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) |
272 | 135, 232,
271 | elrabd 3590 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))}) |
273 | 272 | a1d 25 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → ((0 < (ℂfld
Σg (𝑇 ↾ 𝑘)) → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘)))}) → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))})) |
274 | 120 | simprbi 500 |
. . . . . . . . . . . . . . . 16
⊢
((ℂfld Σg (𝑇 ↾ 𝑘)) ∈ (0[,)+∞) → 0 ≤
(ℂfld Σg (𝑇 ↾ 𝑘))) |
275 | 119, 274 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → 0 ≤ (ℂfld
Σg (𝑇 ↾ 𝑘))) |
276 | | leloe 10805 |
. . . . . . . . . . . . . . . 16
⊢ ((0
∈ ℝ ∧ (ℂfld Σg (𝑇 ↾ 𝑘)) ∈ ℝ) → (0 ≤
(ℂfld Σg (𝑇 ↾ 𝑘)) ↔ (0 < (ℂfld
Σg (𝑇 ↾ 𝑘)) ∨ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))))) |
277 | 80, 122, 276 | sylancr 590 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (0 ≤ (ℂfld
Σg (𝑇 ↾ 𝑘)) ↔ (0 < (ℂfld
Σg (𝑇 ↾ 𝑘)) ∨ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))))) |
278 | 275, 277 | mpbid 235 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (0 < (ℂfld
Σg (𝑇 ↾ 𝑘)) ∨ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘)))) |
279 | 139, 273,
278 | mpjaodan 958 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → ((0 < (ℂfld
Σg (𝑇 ↾ 𝑘)) → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘)))}) → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))})) |
280 | 87, 279 | embantd 59 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → ((𝑘 ⊆ 𝐴 → (0 < (ℂfld
Σg (𝑇 ↾ 𝑘)) → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘)))})) → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))})) |
281 | 85, 280 | syl5bi 245 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (((𝑘 ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ 𝑘))) → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘)))}) → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))})) |
282 | 281 | ex 416 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) → (((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) → (((𝑘 ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ 𝑘))) → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘)))}) → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))}))) |
283 | 282 | com23 86 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) → (((𝑘 ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ 𝑘))) → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘)))}) → (((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))}))) |
284 | 283 | expcom 417 |
. . . . . . . 8
⊢ (¬
𝑐 ∈ 𝑘 → (𝜑 → (((𝑘 ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ 𝑘))) → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘)))}) → (((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))})))) |
285 | 284 | adantl 485 |
. . . . . . 7
⊢ ((𝑘 ∈ Fin ∧ ¬ 𝑐 ∈ 𝑘) → (𝜑 → (((𝑘 ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ 𝑘))) → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘)))}) → (((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))})))) |
286 | 285 | a2d 29 |
. . . . . 6
⊢ ((𝑘 ∈ Fin ∧ ¬ 𝑐 ∈ 𝑘) → ((𝜑 → ((𝑘 ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ 𝑘))) → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld
Σg (𝑇 ↾ 𝑘)))})) → (𝜑 → (((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))})))) |
287 | 31, 47, 63, 79, 84, 286 | findcard2s 8764 |
. . . . 5
⊢ (𝐴 ∈ Fin → (𝜑 → ((𝐴 ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ 𝐴))) → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐴)) / (ℂfld
Σg (𝑇 ↾ 𝐴))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐴)) / (ℂfld
Σg (𝑇 ↾ 𝐴)))}))) |
288 | 10, 287 | mpcom 38 |
. . . 4
⊢ (𝜑 → ((𝐴 ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ 𝐴))) → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐴)) / (ℂfld
Σg (𝑇 ↾ 𝐴))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐴)) / (ℂfld
Σg (𝑇 ↾ 𝐴)))})) |
289 | 9, 288 | mpd 15 |
. . 3
⊢ (𝜑 → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐴)) / (ℂfld
Σg (𝑇 ↾ 𝐴))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐴)) / (ℂfld
Σg (𝑇 ↾ 𝐴)))}) |
290 | 156 | ffnd 6505 |
. . . . . 6
⊢ (𝜑 → (𝑇 ∘f · 𝑋) Fn 𝐴) |
291 | | fnresdm 6455 |
. . . . . 6
⊢ ((𝑇 ∘f ·
𝑋) Fn 𝐴 → ((𝑇 ∘f · 𝑋) ↾ 𝐴) = (𝑇 ∘f · 𝑋)) |
292 | 290, 291 | syl 17 |
. . . . 5
⊢ (𝜑 → ((𝑇 ∘f · 𝑋) ↾ 𝐴) = (𝑇 ∘f · 𝑋)) |
293 | 292 | oveq2d 7186 |
. . . 4
⊢ (𝜑 → (ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐴)) = (ℂfld
Σg (𝑇 ∘f · 𝑋))) |
294 | 293, 6 | oveq12d 7188 |
. . 3
⊢ (𝜑 → ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐴)) / (ℂfld
Σg (𝑇 ↾ 𝐴))) = ((ℂfld
Σg (𝑇 ∘f · 𝑋)) / (ℂfld
Σg 𝑇))) |
295 | 3, 258, 10, 10, 155 | offn 7437 |
. . . . . . . 8
⊢ (𝜑 → (𝑇 ∘f · (𝐹 ∘ 𝑋)) Fn 𝐴) |
296 | | fnresdm 6455 |
. . . . . . . 8
⊢ ((𝑇 ∘f ·
(𝐹 ∘ 𝑋)) Fn 𝐴 → ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐴) = (𝑇 ∘f · (𝐹 ∘ 𝑋))) |
297 | 295, 296 | syl 17 |
. . . . . . 7
⊢ (𝜑 → ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐴) = (𝑇 ∘f · (𝐹 ∘ 𝑋))) |
298 | 297 | oveq2d 7186 |
. . . . . 6
⊢ (𝜑 → (ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐴)) = (ℂfld
Σg (𝑇 ∘f · (𝐹 ∘ 𝑋)))) |
299 | 298, 6 | oveq12d 7188 |
. . . . 5
⊢ (𝜑 → ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐴)) / (ℂfld
Σg (𝑇 ↾ 𝐴))) = ((ℂfld
Σg (𝑇 ∘f · (𝐹 ∘ 𝑋))) / (ℂfld
Σg 𝑇))) |
300 | 299 | breq2d 5042 |
. . . 4
⊢ (𝜑 → ((𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐴)) / (ℂfld
Σg (𝑇 ↾ 𝐴))) ↔ (𝐹‘𝑤) ≤ ((ℂfld
Σg (𝑇 ∘f · (𝐹 ∘ 𝑋))) / (ℂfld
Σg 𝑇)))) |
301 | 300 | rabbidv 3381 |
. . 3
⊢ (𝜑 → {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐴)) / (ℂfld
Σg (𝑇 ↾ 𝐴)))} = {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg (𝑇 ∘f · (𝐹 ∘ 𝑋))) / (ℂfld
Σg 𝑇))}) |
302 | 289, 294,
301 | 3eltr3d 2847 |
. 2
⊢ (𝜑 → ((ℂfld
Σg (𝑇 ∘f · 𝑋)) / (ℂfld
Σg 𝑇)) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg (𝑇 ∘f · (𝐹 ∘ 𝑋))) / (ℂfld
Σg 𝑇))}) |
303 | | fveq2 6674 |
. . . 4
⊢ (𝑤 = ((ℂfld
Σg (𝑇 ∘f · 𝑋)) / (ℂfld
Σg 𝑇)) → (𝐹‘𝑤) = (𝐹‘((ℂfld
Σg (𝑇 ∘f · 𝑋)) / (ℂfld
Σg 𝑇)))) |
304 | 303 | breq1d 5040 |
. . 3
⊢ (𝑤 = ((ℂfld
Σg (𝑇 ∘f · 𝑋)) / (ℂfld
Σg 𝑇)) → ((𝐹‘𝑤) ≤ ((ℂfld
Σg (𝑇 ∘f · (𝐹 ∘ 𝑋))) / (ℂfld
Σg 𝑇)) ↔ (𝐹‘((ℂfld
Σg (𝑇 ∘f · 𝑋)) / (ℂfld
Σg 𝑇))) ≤ ((ℂfld
Σg (𝑇 ∘f · (𝐹 ∘ 𝑋))) / (ℂfld
Σg 𝑇)))) |
305 | 304 | elrab 3588 |
. 2
⊢
(((ℂfld Σg (𝑇 ∘f · 𝑋)) / (ℂfld
Σg 𝑇)) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld
Σg (𝑇 ∘f · (𝐹 ∘ 𝑋))) / (ℂfld
Σg 𝑇))} ↔ (((ℂfld
Σg (𝑇 ∘f · 𝑋)) / (ℂfld
Σg 𝑇)) ∈ 𝐷 ∧ (𝐹‘((ℂfld
Σg (𝑇 ∘f · 𝑋)) / (ℂfld
Σg 𝑇))) ≤ ((ℂfld
Σg (𝑇 ∘f · (𝐹 ∘ 𝑋))) / (ℂfld
Σg 𝑇)))) |
306 | 302, 305 | sylib 221 |
1
⊢ (𝜑 → (((ℂfld
Σg (𝑇 ∘f · 𝑋)) / (ℂfld
Σg 𝑇)) ∈ 𝐷 ∧ (𝐹‘((ℂfld
Σg (𝑇 ∘f · 𝑋)) / (ℂfld
Σg 𝑇))) ≤ ((ℂfld
Σg (𝑇 ∘f · (𝐹 ∘ 𝑋))) / (ℂfld
Σg 𝑇)))) |