| Step | Hyp | Ref
| Expression |
| 1 | | jensen.7 |
. . . . . 6
⊢ (𝜑 → 0 <
(ℂfld Σg 𝑇)) |
| 2 | | jensen.5 |
. . . . . . . . 9
⊢ (𝜑 → 𝑇:𝐴⟶(0[,)+∞)) |
| 3 | 2 | ffnd 6737 |
. . . . . . . 8
⊢ (𝜑 → 𝑇 Fn 𝐴) |
| 4 | | fnresdm 6687 |
. . . . . . . 8
⊢ (𝑇 Fn 𝐴 → (𝑇 ↾ 𝐴) = 𝑇) |
| 5 | 3, 4 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (𝑇 ↾ 𝐴) = 𝑇) |
| 6 | 5 | oveq2d 7447 |
. . . . . 6
⊢ (𝜑 → (ℂfld
Σg (𝑇 ↾ 𝐴)) = (ℂfld
Σg 𝑇)) |
| 7 | 1, 6 | breqtrrd 5171 |
. . . . 5
⊢ (𝜑 → 0 <
(ℂfld Σg (𝑇 ↾ 𝐴))) |
| 8 | | ssid 4006 |
. . . . 5
⊢ 𝐴 ⊆ 𝐴 |
| 9 | 7, 8 | jctil 519 |
. . . 4
⊢ (𝜑 → (𝐴 ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ 𝐴)))) |
| 10 | | jensen.4 |
. . . . 5
⊢ (𝜑 → 𝐴 ∈ Fin) |
| 11 | | sseq1 4009 |
. . . . . . . . 9
⊢ (𝑎 = ∅ → (𝑎 ⊆ 𝐴 ↔ ∅ ⊆ 𝐴)) |
| 12 | | reseq2 5992 |
. . . . . . . . . . . . 13
⊢ (𝑎 = ∅ → (𝑇 ↾ 𝑎) = (𝑇 ↾ ∅)) |
| 13 | | res0 6001 |
. . . . . . . . . . . . 13
⊢ (𝑇 ↾ ∅) =
∅ |
| 14 | 12, 13 | eqtrdi 2793 |
. . . . . . . . . . . 12
⊢ (𝑎 = ∅ → (𝑇 ↾ 𝑎) = ∅) |
| 15 | 14 | oveq2d 7447 |
. . . . . . . . . . 11
⊢ (𝑎 = ∅ →
(ℂfld Σg (𝑇 ↾ 𝑎)) = (ℂfld
Σg ∅)) |
| 16 | | cnfld0 21405 |
. . . . . . . . . . . 12
⊢ 0 =
(0g‘ℂfld) |
| 17 | 16 | gsum0 18697 |
. . . . . . . . . . 11
⊢
(ℂfld Σg ∅) =
0 |
| 18 | 15, 17 | eqtrdi 2793 |
. . . . . . . . . 10
⊢ (𝑎 = ∅ →
(ℂfld Σg (𝑇 ↾ 𝑎)) = 0) |
| 19 | 18 | breq2d 5155 |
. . . . . . . . 9
⊢ (𝑎 = ∅ → (0 <
(ℂfld Σg (𝑇 ↾ 𝑎)) ↔ 0 < 0)) |
| 20 | 11, 19 | anbi12d 632 |
. . . . . . . 8
⊢ (𝑎 = ∅ → ((𝑎 ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ 𝑎))) ↔ (∅ ⊆ 𝐴 ∧ 0 < 0))) |
| 21 | | reseq2 5992 |
. . . . . . . . . . 11
⊢ (𝑎 = ∅ → ((𝑇 ∘f ·
𝑋) ↾ 𝑎) = ((𝑇 ∘f · 𝑋) ↾
∅)) |
| 22 | 21 | oveq2d 7447 |
. . . . . . . . . 10
⊢ (𝑎 = ∅ →
(ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) = (ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾
∅))) |
| 23 | 22, 18 | oveq12d 7449 |
. . . . . . . . 9
⊢ (𝑎 = ∅ →
((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) / (ℂfld
Σg (𝑇 ↾ 𝑎))) = ((ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ ∅)) /
0)) |
| 24 | | reseq2 5992 |
. . . . . . . . . . . . 13
⊢ (𝑎 = ∅ → ((𝑇 ∘f ·
(𝐹 ∘ 𝑋)) ↾ 𝑎) = ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ ∅)) |
| 25 | 24 | oveq2d 7447 |
. . . . . . . . . . . 12
⊢ (𝑎 = ∅ →
(ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) = (ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ ∅))) |
| 26 | 25, 18 | oveq12d 7449 |
. . . . . . . . . . 11
⊢ (𝑎 = ∅ →
((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) / (ℂfld
Σg (𝑇 ↾ 𝑎))) = ((ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ ∅)) / 0)) |
| 27 | 26 | breq2d 5155 |
. . . . . . . . . 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 3444 |
. . . . . . . . 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 2835 |
. . . . . . . 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 344 |
. . . . . . 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 340 |
. . . . . 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 4009 |
. . . . . . . . 9
⊢ (𝑎 = 𝑘 → (𝑎 ⊆ 𝐴 ↔ 𝑘 ⊆ 𝐴)) |
| 33 | | reseq2 5992 |
. . . . . . . . . . 11
⊢ (𝑎 = 𝑘 → (𝑇 ↾ 𝑎) = (𝑇 ↾ 𝑘)) |
| 34 | 33 | oveq2d 7447 |
. . . . . . . . . 10
⊢ (𝑎 = 𝑘 → (ℂfld
Σg (𝑇 ↾ 𝑎)) = (ℂfld
Σg (𝑇 ↾ 𝑘))) |
| 35 | 34 | breq2d 5155 |
. . . . . . . . 9
⊢ (𝑎 = 𝑘 → (0 < (ℂfld
Σg (𝑇 ↾ 𝑎)) ↔ 0 < (ℂfld
Σg (𝑇 ↾ 𝑘)))) |
| 36 | 32, 35 | anbi12d 632 |
. . . . . . . 8
⊢ (𝑎 = 𝑘 → ((𝑎 ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ 𝑎))) ↔ (𝑘 ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ 𝑘))))) |
| 37 | | reseq2 5992 |
. . . . . . . . . . 11
⊢ (𝑎 = 𝑘 → ((𝑇 ∘f · 𝑋) ↾ 𝑎) = ((𝑇 ∘f · 𝑋) ↾ 𝑘)) |
| 38 | 37 | oveq2d 7447 |
. . . . . . . . . 10
⊢ (𝑎 = 𝑘 → (ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) = (ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘))) |
| 39 | 38, 34 | oveq12d 7449 |
. . . . . . . . 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 5992 |
. . . . . . . . . . . . 13
⊢ (𝑎 = 𝑘 → ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎) = ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) |
| 41 | 40 | oveq2d 7447 |
. . . . . . . . . . . 12
⊢ (𝑎 = 𝑘 → (ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) = (ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘))) |
| 42 | 41, 34 | oveq12d 7449 |
. . . . . . . . . . 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 5155 |
. . . . . . . . . 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 3444 |
. . . . . . . . 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 2835 |
. . . . . . . 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 344 |
. . . . . . 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 340 |
. . . . . 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 4009 |
. . . . . . . . 9
⊢ (𝑎 = (𝑘 ∪ {𝑐}) → (𝑎 ⊆ 𝐴 ↔ (𝑘 ∪ {𝑐}) ⊆ 𝐴)) |
| 49 | | reseq2 5992 |
. . . . . . . . . . 11
⊢ (𝑎 = (𝑘 ∪ {𝑐}) → (𝑇 ↾ 𝑎) = (𝑇 ↾ (𝑘 ∪ {𝑐}))) |
| 50 | 49 | oveq2d 7447 |
. . . . . . . . . 10
⊢ (𝑎 = (𝑘 ∪ {𝑐}) → (ℂfld
Σg (𝑇 ↾ 𝑎)) = (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) |
| 51 | 50 | breq2d 5155 |
. . . . . . . . 9
⊢ (𝑎 = (𝑘 ∪ {𝑐}) → (0 < (ℂfld
Σg (𝑇 ↾ 𝑎)) ↔ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) |
| 52 | 48, 51 | anbi12d 632 |
. . . . . . . 8
⊢ (𝑎 = (𝑘 ∪ {𝑐}) → ((𝑎 ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ 𝑎))) ↔ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))))) |
| 53 | | reseq2 5992 |
. . . . . . . . . . 11
⊢ (𝑎 = (𝑘 ∪ {𝑐}) → ((𝑇 ∘f · 𝑋) ↾ 𝑎) = ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) |
| 54 | 53 | oveq2d 7447 |
. . . . . . . . . 10
⊢ (𝑎 = (𝑘 ∪ {𝑐}) → (ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) = (ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐})))) |
| 55 | 54, 50 | oveq12d 7449 |
. . . . . . . . 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 5992 |
. . . . . . . . . . . . 13
⊢ (𝑎 = (𝑘 ∪ {𝑐}) → ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎) = ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) |
| 57 | 56 | oveq2d 7447 |
. . . . . . . . . . . 12
⊢ (𝑎 = (𝑘 ∪ {𝑐}) → (ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) = (ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐})))) |
| 58 | 57, 50 | oveq12d 7449 |
. . . . . . . . . . 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 5155 |
. . . . . . . . . 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 3444 |
. . . . . . . . 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 2835 |
. . . . . . . 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 344 |
. . . . . . 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 340 |
. . . . . 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 4009 |
. . . . . . . . 9
⊢ (𝑎 = 𝐴 → (𝑎 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴)) |
| 65 | | reseq2 5992 |
. . . . . . . . . . 11
⊢ (𝑎 = 𝐴 → (𝑇 ↾ 𝑎) = (𝑇 ↾ 𝐴)) |
| 66 | 65 | oveq2d 7447 |
. . . . . . . . . 10
⊢ (𝑎 = 𝐴 → (ℂfld
Σg (𝑇 ↾ 𝑎)) = (ℂfld
Σg (𝑇 ↾ 𝐴))) |
| 67 | 66 | breq2d 5155 |
. . . . . . . . 9
⊢ (𝑎 = 𝐴 → (0 < (ℂfld
Σg (𝑇 ↾ 𝑎)) ↔ 0 < (ℂfld
Σg (𝑇 ↾ 𝐴)))) |
| 68 | 64, 67 | anbi12d 632 |
. . . . . . . 8
⊢ (𝑎 = 𝐴 → ((𝑎 ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ 𝑎))) ↔ (𝐴 ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ 𝐴))))) |
| 69 | | reseq2 5992 |
. . . . . . . . . . 11
⊢ (𝑎 = 𝐴 → ((𝑇 ∘f · 𝑋) ↾ 𝑎) = ((𝑇 ∘f · 𝑋) ↾ 𝐴)) |
| 70 | 69 | oveq2d 7447 |
. . . . . . . . . 10
⊢ (𝑎 = 𝐴 → (ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) = (ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐴))) |
| 71 | 70, 66 | oveq12d 7449 |
. . . . . . . . 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 5992 |
. . . . . . . . . . . . 13
⊢ (𝑎 = 𝐴 → ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎) = ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐴)) |
| 73 | 72 | oveq2d 7447 |
. . . . . . . . . . . 12
⊢ (𝑎 = 𝐴 → (ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) = (ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐴))) |
| 74 | 73, 66 | oveq12d 7449 |
. . . . . . . . . . 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 5155 |
. . . . . . . . . 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 3444 |
. . . . . . . . 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 2835 |
. . . . . . . 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 344 |
. . . . . . 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 340 |
. . . . . 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 11263 |
. . . . . . . . . 10
⊢ 0 ∈
ℝ |
| 81 | 80 | ltnri 11370 |
. . . . . . . . 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 481 |
. . . . . . 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 450 |
. . . . . . . . . . . 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 4193 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → 𝑘 ⊆ 𝐴) |
| 88 | | simpr 484 |
. . . . . . . . . . . . . . 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 580 |
. . . . . . . . . . . . . . . . . 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 580 |
. . . . . . . . . . . . . . . . . 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 480 |
. . . . . . . . . . . . . . . . . 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 2737 |
. . . . . . . . . . . . . . . . . 18
⊢
(ℂfld Σg (𝑇 ↾ 𝑘)) = (ℂfld
Σg (𝑇 ↾ 𝑘)) |
| 106 | | eqid 2737 |
. . . . . . . . . . . . . . . . . 18
⊢
(ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))) = (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))) |
| 107 | | cnring 21403 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
ℂfld ∈ Ring |
| 108 | | ringcmn 20279 |
. . . . . . . . . . . . . . . . . . . . . . 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 9301 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → 𝑘 ∈ Fin) |
| 112 | | rege0subm 21441 |
. . . . . . . . . . . . . . . . . . . . . . 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 6775 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (𝑇 ↾ 𝑘):𝑘⟶(0[,)+∞)) |
| 116 | | c0ex 11255 |
. . . . . . . . . . . . . . . . . . . . . . . 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 9415 |
. . . . . . . . . . . . . . . . . . . . . 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 19937 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (ℂfld
Σg (𝑇 ↾ 𝑘)) ∈ (0[,)+∞)) |
| 120 | | elrege0 13494 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((ℂfld Σg (𝑇 ↾ 𝑘)) ∈ (0[,)+∞) ↔
((ℂfld Σg (𝑇 ↾ 𝑘)) ∈ ℝ ∧ 0 ≤
(ℂfld Σg (𝑇 ↾ 𝑘)))) |
| 121 | 120 | simplbi 497 |
. . . . . . . . . . . . . . . . . . . . 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 480 |
. . . . . . . . . . . . . . . . . . 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 13074 |
. . . . . . . . . . . . . . . . . 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 6906 |
. . . . . . . . . . . . . . . . . . . . . 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 5153 |
. . . . . . . . . . . . . . . . . . . . 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 3692 |
. . . . . . . . . . . . . . . . . . . 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 218 |
. . . . . . . . . . . . . . . . . . 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 494 |
. . . . . . . . . . . . . . . . . 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 495 |
. . . . . . . . . . . . . . . . . 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 27031 |
. . . . . . . . . . . . . . . . 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 6906 |
. . . . . . . . . . . . . . . . . . 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 5153 |
. . . . . . . . . . . . . . . . . 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 3692 |
. . . . . . . . . . . . . . . . 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 234 |
. . . . . . . . . . . . . . . 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 456 |
. . . . . . . . . . . . . . 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 21368 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ℂ =
(Base‘ℂfld) |
| 141 | | ringmnd 20240 |
. . . . . . . . . . . . . . . . . . . . . 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 9301 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (𝑘 ∪ {𝑐}) ∈ Fin) |
| 144 | 143 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → (𝑘 ∪ {𝑐}) ∈ Fin) |
| 145 | | ssun2 4179 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ {𝑐} ⊆ (𝑘 ∪ {𝑐}) |
| 146 | | vsnid 4663 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ 𝑐 ∈ {𝑐} |
| 147 | 145, 146 | sselii 3980 |
. . . . . . . . . . . . . . . . . . . . . 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 11240 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 · 𝑦) ∈ ℝ) |
| 150 | 149 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 · 𝑦) ∈ ℝ) |
| 151 | | rge0ssre 13496 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
(0[,)+∞) ⊆ ℝ |
| 152 | | fss 6752 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑇:𝐴⟶(0[,)+∞) ∧ (0[,)+∞)
⊆ ℝ) → 𝑇:𝐴⟶ℝ) |
| 153 | 2, 151, 152 | sylancl 586 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → 𝑇:𝐴⟶ℝ) |
| 154 | 98, 89 | fssd 6753 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → 𝑋:𝐴⟶ℝ) |
| 155 | | inidm 4227 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝐴 ∩ 𝐴) = 𝐴 |
| 156 | 150, 153,
154, 10, 10, 155 | off 7715 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → (𝑇 ∘f · 𝑋):𝐴⟶ℝ) |
| 157 | | ax-resscn 11212 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ℝ
⊆ ℂ |
| 158 | | fss 6752 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝑇 ∘f ·
𝑋):𝐴⟶ℝ ∧ ℝ ⊆
ℂ) → (𝑇
∘f · 𝑋):𝐴⟶ℂ) |
| 159 | 156, 157,
158 | sylancl 586 |
. . . . . . . . . . . . . . . . . . . . . . 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 480 |
. . . . . . . . . . . . . . . . . . . . . 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 6775 |
. . . . . . . . . . . . . . . . . . . . 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 480 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → 𝐴 ∈ Fin) |
| 165 | 163, 164 | fexd 7247 |
. . . . . . . . . . . . . . . . . . . . . . . 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 7247 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → 𝑋 ∈ V) |
| 168 | | offres 8008 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑇 ∈ V ∧ 𝑋 ∈ V) → ((𝑇 ∘f ·
𝑋) ↾ (𝑘 ∪ {𝑐})) = ((𝑇 ↾ (𝑘 ∪ {𝑐})) ∘f · (𝑋 ↾ (𝑘 ∪ {𝑐})))) |
| 169 | 165, 167,
168 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . . . . . 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 7446 |
. . . . . . . . . . . . . . . . . . . . . 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 3993 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
(0[,)+∞) ⊆ ℂ |
| 172 | | fss 6752 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑇:𝐴⟶(0[,)+∞) ∧ (0[,)+∞)
⊆ ℂ) → 𝑇:𝐴⟶ℂ) |
| 173 | 163, 171,
172 | sylancl 586 |
. . . . . . . . . . . . . . . . . . . . . . . . 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 6775 |
. . . . . . . . . . . . . . . . . . . . . . . 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 4131 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑥 ∈ ((𝑘 ∪ {𝑐}) ∖ {𝑐}) → 𝑥 ∈ (𝑘 ∪ {𝑐})) |
| 176 | 175 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . 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 6926 |
. . . . . . . . . . . . . . . . . . . . . . . . 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 4481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝑘 ∪ {𝑐}) ∖ {𝑐}) = (𝑘 ∖ {𝑐}) |
| 179 | | difss 4136 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑘 ∖ {𝑐}) ⊆ 𝑘 |
| 180 | 178, 179 | eqsstri 4030 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑘 ∪ {𝑐}) ∖ {𝑐}) ⊆ 𝑘 |
| 181 | 180 | sseli 3979 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑥 ∈ ((𝑘 ∪ {𝑐}) ∖ {𝑐}) → 𝑥 ∈ 𝑘) |
| 182 | | simpr 484 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → 𝑘 ⊆ 𝐴) |
| 184 | 163, 183 | feqresmpt 6978 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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 7447 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → 𝑘 ∈ Fin) |
| 187 | 183 | sselda 3983 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢
(((((𝜑 ∧ ¬
𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) ∧ 𝑥 ∈ 𝑘) → 𝑥 ∈ 𝐴) |
| 188 | 163 | ffvelcdmda 7104 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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 591 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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 | sselid 3981 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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 21452 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2782 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 13494 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((𝑇‘𝑥) ∈ (0[,)+∞) ↔ ((𝑇‘𝑥) ∈ ℝ ∧ 0 ≤ (𝑇‘𝑥))) |
| 194 | 189, 193 | sylib 218 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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 494 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 495 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 15834 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 232 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 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 3251 |
. . . . . . . . . . . . . . . . . . . . . . . . . 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 593 |
. . . . . . . . . . . . . . . . . . . . . . . . 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 2777 |
. . . . . . . . . . . . . . . . . . . . . . . 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 8219 |
. . . . . . . . . . . . . . . . . . . . . . 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 11439 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑥 ∈ ℂ → (0
· 𝑥) =
0) |
| 204 | 203 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . 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 3996 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → 𝐷 ⊆ ℂ) |
| 207 | 166, 206 | fssd 6753 |
. . . . . . . . . . . . . . . . . . . . . . . 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 6775 |
. . . . . . . . . . . . . . . . . . . . . . 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 8224 |
. . . . . . . . . . . . . . . . . . . . . 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 4018 |
. . . . . . . . . . . . . . . . . . . . 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 19980 |
. . . . . . . . . . . . . . . . . . . 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 6926 |
. . . . . . . . . . . . . . . . . . . 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 6737 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → 𝑇 Fn 𝐴) |
| 215 | 166 | ffnd 6737 |
. . . . . . . . . . . . . . . . . . . . 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 3984 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → 𝑐 ∈ 𝐴) |
| 217 | | fnfvof 7714 |
. . . . . . . . . . . . . . . . . . . . 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 839 |
. . . . . . . . . . . . . . . . . . . 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 2781 |
. . . . . . . . . . . . . . . . . . 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 19980 |
. . . . . . . . . . . . . . . . . . . 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 6926 |
. . . . . . . . . . . . . . . . . . . 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 2777 |
. . . . . . . . . . . . . . . . . . 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 7449 |
. . . . . . . . . . . . . . . . . 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 | ffvelcdmd 7105 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → (𝑋‘𝑐) ∈ ℂ) |
| 225 | 173, 216 | ffvelcdmd 7105 |
. . . . . . . . . . . . . . . . . . 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 5169 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → 0 < (𝑇‘𝑐)) |
| 228 | 227 | gt0ne0d 11827 |
. . . . . . . . . . . . . . . . . . 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 12048 |
. . . . . . . . . . . . . . . . . 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 2777 |
. . . . . . . . . . . . . . . . 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 | ffvelcdmd 7105 |
. . . . . . . . . . . . . . . . 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 2841 |
. . . . . . . . . . . . . . . 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 | ffvelcdmd 7105 |
. . . . . . . . . . . . . . . . . 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 11829 |
. . . . . . . . . . . . . . . . 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 6910 |
. . . . . . . . . . . . . . . . 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 6760 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝐹:𝐷⟶ℝ ∧ 𝑋:𝐴⟶𝐷) → (𝐹 ∘ 𝑋):𝐴⟶ℝ) |
| 238 | 91, 98, 237 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → (𝐹 ∘ 𝑋):𝐴⟶ℝ) |
| 239 | 150, 153,
238, 10, 10, 155 | off 7715 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → (𝑇 ∘f · (𝐹 ∘ 𝑋)):𝐴⟶ℝ) |
| 240 | | fss 6752 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝑇 ∘f ·
(𝐹 ∘ 𝑋)):𝐴⟶ℝ ∧ ℝ ⊆
ℂ) → (𝑇
∘f · (𝐹 ∘ 𝑋)):𝐴⟶ℂ) |
| 241 | 239, 157,
240 | sylancl 586 |
. . . . . . . . . . . . . . . . . . . . . . 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 6775 |
. . . . . . . . . . . . . . . . . . . . 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 7247 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld
Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld
Σg (𝑇 ↾ 𝑘))) → (𝐹 ∘ 𝑋) ∈ V) |
| 246 | | offres 8008 |
. . . . . . . . . . . . . . . . . . . . . . . 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 584 |
. . . . . . . . . . . . . . . . . . . . . . 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 7446 |
. . . . . . . . . . . . . . . . . . . . . 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 6752 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝐹 ∘ 𝑋):𝐴⟶ℝ ∧ ℝ ⊆
ℂ) → (𝐹 ∘
𝑋):𝐴⟶ℂ) |
| 250 | 244, 157,
249 | sylancl 586 |
. . . . . . . . . . . . . . . . . . . . . . . 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 6775 |
. . . . . . . . . . . . . . . . . . . . . . 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 8224 |
. . . . . . . . . . . . . . . . . . . . . 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 4018 |
. . . . . . . . . . . . . . . . . . . . 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 19980 |
. . . . . . . . . . . . . . . . . . . 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 6926 |
. . . . . . . . . . . . . . . . . . . 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 6737 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → 𝐹 Fn 𝐷) |
| 257 | | fnfco 6773 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝐹 Fn 𝐷 ∧ 𝑋:𝐴⟶𝐷) → (𝐹 ∘ 𝑋) Fn 𝐴) |
| 258 | 256, 98, 257 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . . . . . 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 7714 |
. . . . . . . . . . . . . . . . . . . . . 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 839 |
. . . . . . . . . . . . . . . . . . . . 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 7008 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑋:𝐴⟶𝐷 ∧ 𝑐 ∈ 𝐴) → ((𝐹 ∘ 𝑋)‘𝑐) = (𝐹‘(𝑋‘𝑐))) |
| 263 | 166, 216,
262 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . . . . 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 7447 |
. . . . . . . . . . . . . . . . . . . . 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 2777 |
. . . . . . . . . . . . . . . . . . . 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 2781 |
. . . . . . . . . . . . . . . . . . 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 7449 |
. . . . . . . . . . . . . . . . . 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 11289 |
. . . . . . . . . . . . . . . . . . 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 12048 |
. . . . . . . . . . . . . . . . . 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 2777 |
. . . . . . . . . . . . . . . . 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 5175 |
. . . . . . . . . . . . . . . 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 3694 |
. . . . . . . . . . . . . . 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 496 |
. . . . . . . . . . . . . . . 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 11347 |
. . . . . . . . . . . . . . . 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 587 |
. . . . . . . . . . . . . . 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 232 |
. . . . . . . . . . . . . 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 961 |
. . . . . . . . . . . . 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 | biimtrid 242 |
. . . . . . . . . . 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 412 |
. . . . . . . . . 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 413 |
. . . . . . . 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 481 |
. . . . . . 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 9205 |
. . . . 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 6737 |
. . . . . 6
⊢ (𝜑 → (𝑇 ∘f · 𝑋) Fn 𝐴) |
| 291 | | fnresdm 6687 |
. . . . . 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 7447 |
. . . 4
⊢ (𝜑 → (ℂfld
Σg ((𝑇 ∘f · 𝑋) ↾ 𝐴)) = (ℂfld
Σg (𝑇 ∘f · 𝑋))) |
| 294 | 293, 6 | oveq12d 7449 |
. . 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 7710 |
. . . . . . . 8
⊢ (𝜑 → (𝑇 ∘f · (𝐹 ∘ 𝑋)) Fn 𝐴) |
| 296 | | fnresdm 6687 |
. . . . . . . 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 7447 |
. . . . . 6
⊢ (𝜑 → (ℂfld
Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐴)) = (ℂfld
Σg (𝑇 ∘f · (𝐹 ∘ 𝑋)))) |
| 299 | 298, 6 | oveq12d 7449 |
. . . . 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 5155 |
. . . 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 3444 |
. . 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 2855 |
. 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 6906 |
. . . 4
⊢ (𝑤 = ((ℂfld
Σg (𝑇 ∘f · 𝑋)) / (ℂfld
Σg 𝑇)) → (𝐹‘𝑤) = (𝐹‘((ℂfld
Σg (𝑇 ∘f · 𝑋)) / (ℂfld
Σg 𝑇)))) |
| 304 | 303 | breq1d 5153 |
. . 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 3692 |
. 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 218 |
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)))) |