Proof of Theorem smflimsuplem3
Step | Hyp | Ref
| Expression |
1 | | nfv 1915 |
. 2
⊢
Ⅎ𝑛𝜑 |
2 | | nfv 1915 |
. 2
⊢
Ⅎ𝑥𝜑 |
3 | | nfv 1915 |
. 2
⊢
Ⅎ𝑘𝜑 |
4 | | smflimsuplem3.m |
. 2
⊢ (𝜑 → 𝑀 ∈ ℤ) |
5 | | smflimsuplem3.z |
. 2
⊢ 𝑍 =
(ℤ≥‘𝑀) |
6 | | fvex 6685 |
. . . 4
⊢ (𝐻‘𝑛) ∈ V |
7 | 6 | dmex 7618 |
. . 3
⊢ dom
(𝐻‘𝑛) ∈ V |
8 | 7 | a1i 11 |
. 2
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → dom (𝐻‘𝑛) ∈ V) |
9 | | fvexd 6687 |
. 2
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ dom (𝐻‘𝑛)) → ((𝐻‘𝑛)‘𝑥) ∈ V) |
10 | | smflimsuplem3.s |
. 2
⊢ (𝜑 → 𝑆 ∈ SAlg) |
11 | 10 | adantr 483 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑆 ∈ SAlg) |
12 | | smflimsuplem3.e |
. . . . . . . . . . . . 13
⊢ 𝐸 = (𝑛 ∈ 𝑍 ↦ {𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈
ℝ}) |
13 | 12 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐸 = (𝑛 ∈ 𝑍 ↦ {𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈
ℝ})) |
14 | | eqid 2823 |
. . . . . . . . . . . . 13
⊢ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈
ℝ} = {𝑥 ∈
∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈
ℝ} |
15 | 5 | eluzelz2 41683 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℤ) |
16 | | eqid 2823 |
. . . . . . . . . . . . . . . 16
⊢
(ℤ≥‘𝑛) = (ℤ≥‘𝑛) |
17 | 15, 16 | uzn0d 41706 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ 𝑍 → (ℤ≥‘𝑛) ≠ ∅) |
18 | | fvex 6685 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐹‘𝑚) ∈ V |
19 | 18 | dmex 7618 |
. . . . . . . . . . . . . . . . 17
⊢ dom
(𝐹‘𝑚) ∈ V |
20 | 19 | rgenw 3152 |
. . . . . . . . . . . . . . . 16
⊢
∀𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∈ V |
21 | 20 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ 𝑍 → ∀𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∈ V) |
22 | 17, 21 | iinexd 41407 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ 𝑍 → ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∈ V) |
23 | 22 | adantl 484 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∈ V) |
24 | 14, 23 | rabexd 5238 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → {𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈
ℝ} ∈ V) |
25 | 13, 24 | fvmpt2d 6783 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) = {𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈
ℝ}) |
26 | | fvres 6691 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑚 ∈
(ℤ≥‘𝑛) → ((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚) = (𝐹‘𝑚)) |
27 | 26 | eqcomd 2829 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑚 ∈
(ℤ≥‘𝑛) → (𝐹‘𝑚) = ((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚)) |
28 | 27 | adantl 484 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → (𝐹‘𝑚) = ((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚)) |
29 | 28 | dmeqd 5776 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → dom (𝐹‘𝑚) = dom ((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚)) |
30 | 29 | iineq2dv 4946 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) = ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom ((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚)) |
31 | 30 | eleq2d 2900 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ↔ 𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom ((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚))) |
32 | 27 | fveq1d 6674 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑚 ∈
(ℤ≥‘𝑛) → ((𝐹‘𝑚)‘𝑥) = (((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚)‘𝑥)) |
33 | 32 | mpteq2ia 5159 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑚 ∈
(ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑚 ∈ (ℤ≥‘𝑛) ↦ (((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚)‘𝑥)) |
34 | 33 | rneqi 5809 |
. . . . . . . . . . . . . . . 16
⊢ ran
(𝑚 ∈
(ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) = ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ (((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚)‘𝑥)) |
35 | 34 | supeq1i 8913 |
. . . . . . . . . . . . . . 15
⊢ sup(ran
(𝑚 ∈
(ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) = sup(ran
(𝑚 ∈
(ℤ≥‘𝑛) ↦ (((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚)‘𝑥)), ℝ*, <
) |
36 | 35 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) = sup(ran
(𝑚 ∈
(ℤ≥‘𝑛) ↦ (((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚)‘𝑥)), ℝ*, <
)) |
37 | 36 | eleq1d 2899 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈
ℝ ↔ sup(ran (𝑚
∈ (ℤ≥‘𝑛) ↦ (((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚)‘𝑥)), ℝ*, < ) ∈
ℝ)) |
38 | 31, 37 | anbi12d 632 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∧ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈
ℝ) ↔ (𝑥 ∈
∩ 𝑚 ∈ (ℤ≥‘𝑛)dom ((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚) ∧ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ (((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚)‘𝑥)), ℝ*, < ) ∈
ℝ))) |
39 | 38 | rabbidva2 3478 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → {𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈
ℝ} = {𝑥 ∈
∩ 𝑚 ∈ (ℤ≥‘𝑛)dom ((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ (((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚)‘𝑥)), ℝ*, < ) ∈
ℝ}) |
40 | 25, 39 | eqtrd 2858 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) = {𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom ((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ (((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚)‘𝑥)), ℝ*, < ) ∈
ℝ}) |
41 | 40, 36 | mpteq12dv 5153 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < )) = (𝑥 ∈ {𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom ((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ (((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚)‘𝑥)), ℝ*, < ) ∈
ℝ} ↦ sup(ran (𝑚
∈ (ℤ≥‘𝑛) ↦ (((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚)‘𝑥)), ℝ*, <
))) |
42 | | nfcv 2979 |
. . . . . . . . . 10
⊢
Ⅎ𝑚(𝐹 ↾ (ℤ≥‘𝑛)) |
43 | | nfcv 2979 |
. . . . . . . . . 10
⊢
Ⅎ𝑥(𝐹 ↾ (ℤ≥‘𝑛)) |
44 | 15 | adantl 484 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ ℤ) |
45 | | smflimsuplem3.f |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) |
46 | 45 | adantr 483 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝐹:𝑍⟶(SMblFn‘𝑆)) |
47 | 5 | eleq2i 2906 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ 𝑍 ↔ 𝑛 ∈ (ℤ≥‘𝑀)) |
48 | 47 | biimpi 218 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ (ℤ≥‘𝑀)) |
49 | | uzss 12268 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈
(ℤ≥‘𝑀) → (ℤ≥‘𝑛) ⊆
(ℤ≥‘𝑀)) |
50 | 48, 49 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ 𝑍 → (ℤ≥‘𝑛) ⊆
(ℤ≥‘𝑀)) |
51 | 50, 5 | sseqtrrdi 4020 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ 𝑍 → (ℤ≥‘𝑛) ⊆ 𝑍) |
52 | 51 | adantl 484 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (ℤ≥‘𝑛) ⊆ 𝑍) |
53 | 46, 52 | fssresd 6547 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹 ↾ (ℤ≥‘𝑛)):(ℤ≥‘𝑛)⟶(SMblFn‘𝑆)) |
54 | | eqid 2823 |
. . . . . . . . . 10
⊢ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom ((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ (((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚)‘𝑥)), ℝ*, < ) ∈
ℝ} = {𝑥 ∈
∩ 𝑚 ∈ (ℤ≥‘𝑛)dom ((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ (((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚)‘𝑥)), ℝ*, < ) ∈
ℝ} |
55 | | eqid 2823 |
. . . . . . . . . 10
⊢ (𝑥 ∈ {𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom ((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ (((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚)‘𝑥)), ℝ*, < ) ∈
ℝ} ↦ sup(ran (𝑚
∈ (ℤ≥‘𝑛) ↦ (((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚)‘𝑥)), ℝ*, < )) = (𝑥 ∈ {𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom ((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ (((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚)‘𝑥)), ℝ*, < ) ∈
ℝ} ↦ sup(ran (𝑚
∈ (ℤ≥‘𝑛) ↦ (((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚)‘𝑥)), ℝ*, <
)) |
56 | 42, 43, 44, 16, 11, 53, 54, 55 | smfsupxr 43097 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ {𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom ((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ (((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚)‘𝑥)), ℝ*, < ) ∈
ℝ} ↦ sup(ran (𝑚
∈ (ℤ≥‘𝑛) ↦ (((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚)‘𝑥)), ℝ*, < )) ∈
(SMblFn‘𝑆)) |
57 | 41, 56 | eqeltrd 2915 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < )) ∈
(SMblFn‘𝑆)) |
58 | | smflimsuplem3.h |
. . . . . . . 8
⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, <
))) |
59 | 57, 58 | fmptd 6880 |
. . . . . . 7
⊢ (𝜑 → 𝐻:𝑍⟶(SMblFn‘𝑆)) |
60 | 59 | ffvelrnda 6853 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐻‘𝑛) ∈ (SMblFn‘𝑆)) |
61 | | eqid 2823 |
. . . . . 6
⊢ dom
(𝐻‘𝑛) = dom (𝐻‘𝑛) |
62 | 11, 60, 61 | smff 43016 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐻‘𝑛):dom (𝐻‘𝑛)⟶ℝ) |
63 | 62 | feqmptd 6735 |
. . . 4
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐻‘𝑛) = (𝑥 ∈ dom (𝐻‘𝑛) ↦ ((𝐻‘𝑛)‘𝑥))) |
64 | 63 | eqcomd 2829 |
. . 3
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ dom (𝐻‘𝑛) ↦ ((𝐻‘𝑛)‘𝑥)) = (𝐻‘𝑛)) |
65 | 64, 60 | eqeltrd 2915 |
. 2
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ dom (𝐻‘𝑛) ↦ ((𝐻‘𝑛)‘𝑥)) ∈ (SMblFn‘𝑆)) |
66 | | eqid 2823 |
. 2
⊢ {𝑥 ∈ ∪ 𝑘 ∈ 𝑍 ∩ 𝑛 ∈
(ℤ≥‘𝑘)dom (𝐻‘𝑛) ∣ (𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑥)) ∈ dom ⇝ } = {𝑥 ∈ ∪
𝑘 ∈ 𝑍 ∩ 𝑛 ∈
(ℤ≥‘𝑘)dom (𝐻‘𝑛) ∣ (𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑥)) ∈ dom ⇝ } |
67 | | eqid 2823 |
. 2
⊢ (𝑥 ∈ {𝑥 ∈ ∪
𝑘 ∈ 𝑍 ∩ 𝑛 ∈
(ℤ≥‘𝑘)dom (𝐻‘𝑛) ∣ (𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑥)) ∈ dom ⇝ } ↦ ( ⇝
‘(𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑥)))) = (𝑥 ∈ {𝑥 ∈ ∪
𝑘 ∈ 𝑍 ∩ 𝑛 ∈
(ℤ≥‘𝑘)dom (𝐻‘𝑛) ∣ (𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑥)) ∈ dom ⇝ } ↦ ( ⇝
‘(𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑥)))) |
68 | 1, 2, 3, 4, 5, 8, 9, 10, 65, 66, 67 | smflimmpt 43091 |
1
⊢ (𝜑 → (𝑥 ∈ {𝑥 ∈ ∪
𝑘 ∈ 𝑍 ∩ 𝑛 ∈
(ℤ≥‘𝑘)dom (𝐻‘𝑛) ∣ (𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑥)) ∈ dom ⇝ } ↦ ( ⇝
‘(𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑥)))) ∈ (SMblFn‘𝑆)) |