Step | Hyp | Ref
| Expression |
1 | | lmres.2 |
. . . . . . 7
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
2 | | toponmax 21680 |
. . . . . . 7
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽) |
3 | 1, 2 | syl 17 |
. . . . . 6
⊢ (𝜑 → 𝑋 ∈ 𝐽) |
4 | | cnex 10699 |
. . . . . 6
⊢ ℂ
∈ V |
5 | | ssid 3900 |
. . . . . . 7
⊢ 𝑋 ⊆ 𝑋 |
6 | | uzssz 12348 |
. . . . . . . 8
⊢
(ℤ≥‘𝑀) ⊆ ℤ |
7 | | zsscn 12073 |
. . . . . . . 8
⊢ ℤ
⊆ ℂ |
8 | 6, 7 | sstri 3887 |
. . . . . . 7
⊢
(ℤ≥‘𝑀) ⊆ ℂ |
9 | | pmss12g 8482 |
. . . . . . 7
⊢ (((𝑋 ⊆ 𝑋 ∧ (ℤ≥‘𝑀) ⊆ ℂ) ∧ (𝑋 ∈ 𝐽 ∧ ℂ ∈ V)) → (𝑋 ↑pm
(ℤ≥‘𝑀)) ⊆ (𝑋 ↑pm
ℂ)) |
10 | 5, 8, 9 | mpanl12 702 |
. . . . . 6
⊢ ((𝑋 ∈ 𝐽 ∧ ℂ ∈ V) → (𝑋 ↑pm
(ℤ≥‘𝑀)) ⊆ (𝑋 ↑pm
ℂ)) |
11 | 3, 4, 10 | sylancl 589 |
. . . . 5
⊢ (𝜑 → (𝑋 ↑pm
(ℤ≥‘𝑀)) ⊆ (𝑋 ↑pm
ℂ)) |
12 | | fvex 6690 |
. . . . . 6
⊢
(ℤ≥‘𝑀) ∈ V |
13 | | lmres.4 |
. . . . . 6
⊢ (𝜑 → 𝐹 ∈ (𝑋 ↑pm
ℂ)) |
14 | | pmresg 8483 |
. . . . . 6
⊢
(((ℤ≥‘𝑀) ∈ V ∧ 𝐹 ∈ (𝑋 ↑pm ℂ)) → (𝐹 ↾
(ℤ≥‘𝑀)) ∈ (𝑋 ↑pm
(ℤ≥‘𝑀))) |
15 | 12, 13, 14 | sylancr 590 |
. . . . 5
⊢ (𝜑 → (𝐹 ↾ (ℤ≥‘𝑀)) ∈ (𝑋 ↑pm
(ℤ≥‘𝑀))) |
16 | 11, 15 | sseldd 3879 |
. . . 4
⊢ (𝜑 → (𝐹 ↾ (ℤ≥‘𝑀)) ∈ (𝑋 ↑pm
ℂ)) |
17 | 16, 13 | 2thd 268 |
. . 3
⊢ (𝜑 → ((𝐹 ↾ (ℤ≥‘𝑀)) ∈ (𝑋 ↑pm ℂ) ↔ 𝐹 ∈ (𝑋 ↑pm
ℂ))) |
18 | | eqid 2739 |
. . . . . . . . . 10
⊢
(ℤ≥‘𝑀) = (ℤ≥‘𝑀) |
19 | 18 | uztrn2 12346 |
. . . . . . . . 9
⊢ ((𝑗 ∈
(ℤ≥‘𝑀) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝑘 ∈ (ℤ≥‘𝑀)) |
20 | | dmres 5848 |
. . . . . . . . . . . 12
⊢ dom
(𝐹 ↾
(ℤ≥‘𝑀)) = ((ℤ≥‘𝑀) ∩ dom 𝐹) |
21 | 20 | elin2 4088 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ dom (𝐹 ↾ (ℤ≥‘𝑀)) ↔ (𝑘 ∈ (ℤ≥‘𝑀) ∧ 𝑘 ∈ dom 𝐹)) |
22 | 21 | baib 539 |
. . . . . . . . . 10
⊢ (𝑘 ∈
(ℤ≥‘𝑀) → (𝑘 ∈ dom (𝐹 ↾ (ℤ≥‘𝑀)) ↔ 𝑘 ∈ dom 𝐹)) |
23 | | fvres 6696 |
. . . . . . . . . . 11
⊢ (𝑘 ∈
(ℤ≥‘𝑀) → ((𝐹 ↾ (ℤ≥‘𝑀))‘𝑘) = (𝐹‘𝑘)) |
24 | 23 | eleq1d 2818 |
. . . . . . . . . 10
⊢ (𝑘 ∈
(ℤ≥‘𝑀) → (((𝐹 ↾ (ℤ≥‘𝑀))‘𝑘) ∈ 𝑢 ↔ (𝐹‘𝑘) ∈ 𝑢)) |
25 | 22, 24 | anbi12d 634 |
. . . . . . . . 9
⊢ (𝑘 ∈
(ℤ≥‘𝑀) → ((𝑘 ∈ dom (𝐹 ↾ (ℤ≥‘𝑀)) ∧ ((𝐹 ↾ (ℤ≥‘𝑀))‘𝑘) ∈ 𝑢) ↔ (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))) |
26 | 19, 25 | syl 17 |
. . . . . . . 8
⊢ ((𝑗 ∈
(ℤ≥‘𝑀) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → ((𝑘 ∈ dom (𝐹 ↾ (ℤ≥‘𝑀)) ∧ ((𝐹 ↾ (ℤ≥‘𝑀))‘𝑘) ∈ 𝑢) ↔ (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))) |
27 | 26 | ralbidva 3109 |
. . . . . . 7
⊢ (𝑗 ∈
(ℤ≥‘𝑀) → (∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom (𝐹 ↾ (ℤ≥‘𝑀)) ∧ ((𝐹 ↾ (ℤ≥‘𝑀))‘𝑘) ∈ 𝑢) ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))) |
28 | 27 | rexbiia 3161 |
. . . . . 6
⊢
(∃𝑗 ∈
(ℤ≥‘𝑀)∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom (𝐹 ↾ (ℤ≥‘𝑀)) ∧ ((𝐹 ↾ (ℤ≥‘𝑀))‘𝑘) ∈ 𝑢) ↔ ∃𝑗 ∈ (ℤ≥‘𝑀)∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)) |
29 | 28 | imbi2i 339 |
. . . . 5
⊢ ((𝑃 ∈ 𝑢 → ∃𝑗 ∈ (ℤ≥‘𝑀)∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom (𝐹 ↾ (ℤ≥‘𝑀)) ∧ ((𝐹 ↾ (ℤ≥‘𝑀))‘𝑘) ∈ 𝑢)) ↔ (𝑃 ∈ 𝑢 → ∃𝑗 ∈ (ℤ≥‘𝑀)∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))) |
30 | 29 | ralbii 3081 |
. . . 4
⊢
(∀𝑢 ∈
𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ (ℤ≥‘𝑀)∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom (𝐹 ↾ (ℤ≥‘𝑀)) ∧ ((𝐹 ↾ (ℤ≥‘𝑀))‘𝑘) ∈ 𝑢)) ↔ ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ (ℤ≥‘𝑀)∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))) |
31 | 30 | a1i 11 |
. . 3
⊢ (𝜑 → (∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ (ℤ≥‘𝑀)∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom (𝐹 ↾ (ℤ≥‘𝑀)) ∧ ((𝐹 ↾ (ℤ≥‘𝑀))‘𝑘) ∈ 𝑢)) ↔ ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ (ℤ≥‘𝑀)∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)))) |
32 | 17, 31 | 3anbi13d 1439 |
. 2
⊢ (𝜑 → (((𝐹 ↾ (ℤ≥‘𝑀)) ∈ (𝑋 ↑pm ℂ) ∧ 𝑃 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ (ℤ≥‘𝑀)∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom (𝐹 ↾ (ℤ≥‘𝑀)) ∧ ((𝐹 ↾ (ℤ≥‘𝑀))‘𝑘) ∈ 𝑢))) ↔ (𝐹 ∈ (𝑋 ↑pm ℂ) ∧ 𝑃 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ (ℤ≥‘𝑀)∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))))) |
33 | | lmres.5 |
. . 3
⊢ (𝜑 → 𝑀 ∈ ℤ) |
34 | 1, 18, 33 | lmbr2 22013 |
. 2
⊢ (𝜑 → ((𝐹 ↾ (ℤ≥‘𝑀))(⇝𝑡‘𝐽)𝑃 ↔ ((𝐹 ↾ (ℤ≥‘𝑀)) ∈ (𝑋 ↑pm ℂ) ∧ 𝑃 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ (ℤ≥‘𝑀)∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom (𝐹 ↾ (ℤ≥‘𝑀)) ∧ ((𝐹 ↾ (ℤ≥‘𝑀))‘𝑘) ∈ 𝑢))))) |
35 | 1, 18, 33 | lmbr2 22013 |
. 2
⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ (𝐹 ∈ (𝑋 ↑pm ℂ) ∧ 𝑃 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ (ℤ≥‘𝑀)∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))))) |
36 | 32, 34, 35 | 3bitr4rd 315 |
1
⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ (𝐹 ↾ (ℤ≥‘𝑀))(⇝𝑡‘𝐽)𝑃)) |