| Step | Hyp | Ref
| Expression |
| 1 | | smfsupxr.g |
. . . 4
⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ*, <
)) |
| 2 | 1 | a1i 11 |
. . 3
⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐷 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ*, <
))) |
| 3 | | smfsupxr.d |
. . . . . 6
⊢ 𝐷 = {𝑥 ∈ ∩
𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ*, < ) ∈
ℝ} |
| 4 | 3 | a1i 11 |
. . . . 5
⊢ (𝜑 → 𝐷 = {𝑥 ∈ ∩
𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ*, < ) ∈
ℝ}) |
| 5 | | nfv 1914 |
. . . . . . . 8
⊢
Ⅎ𝑛𝜑 |
| 6 | | nfcv 2905 |
. . . . . . . . 9
⊢
Ⅎ𝑛𝑥 |
| 7 | | nfii1 5029 |
. . . . . . . . 9
⊢
Ⅎ𝑛∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) |
| 8 | 6, 7 | nfel 2920 |
. . . . . . . 8
⊢
Ⅎ𝑛 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) |
| 9 | 5, 8 | nfan 1899 |
. . . . . . 7
⊢
Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ ∩
𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) |
| 10 | | smfsupxr.m |
. . . . . . . . 9
⊢ (𝜑 → 𝑀 ∈ ℤ) |
| 11 | | smfsupxr.z |
. . . . . . . . 9
⊢ 𝑍 =
(ℤ≥‘𝑀) |
| 12 | 10, 11 | uzn0d 45436 |
. . . . . . . 8
⊢ (𝜑 → 𝑍 ≠ ∅) |
| 13 | 12 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ ∩
𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) → 𝑍 ≠ ∅) |
| 14 | | smfsupxr.s |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑆 ∈ SAlg) |
| 15 | 14 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑆 ∈ SAlg) |
| 16 | | smfsupxr.f |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) |
| 17 | 16 | ffvelcdmda 7104 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ (SMblFn‘𝑆)) |
| 18 | | eqid 2737 |
. . . . . . . . . 10
⊢ dom
(𝐹‘𝑛) = dom (𝐹‘𝑛) |
| 19 | 15, 17, 18 | smff 46747 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛):dom (𝐹‘𝑛)⟶ℝ) |
| 20 | 19 | adantlr 715 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ∩
𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛):dom (𝐹‘𝑛)⟶ℝ) |
| 21 | | eliinid 45116 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ dom (𝐹‘𝑛)) |
| 22 | 21 | adantll 714 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ∩
𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ dom (𝐹‘𝑛)) |
| 23 | 20, 22 | ffvelcdmd 7105 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ∩
𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑛 ∈ 𝑍) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ) |
| 24 | 9, 13, 23 | supxrre3rnmpt 45440 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ ∩
𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) → (sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ*, < ) ∈
ℝ ↔ ∃𝑦
∈ ℝ ∀𝑛
∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦)) |
| 25 | 24 | rabbidva 3443 |
. . . . 5
⊢ (𝜑 → {𝑥 ∈ ∩
𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ*, < ) ∈
ℝ} = {𝑥 ∈
∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦}) |
| 26 | 4, 25 | eqtrd 2777 |
. . . 4
⊢ (𝜑 → 𝐷 = {𝑥 ∈ ∩
𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦}) |
| 27 | | nfmpt1 5250 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑛(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)) |
| 28 | 27 | nfrn 5963 |
. . . . . . . . . . 11
⊢
Ⅎ𝑛ran
(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)) |
| 29 | | nfcv 2905 |
. . . . . . . . . . 11
⊢
Ⅎ𝑛ℝ* |
| 30 | | nfcv 2905 |
. . . . . . . . . . 11
⊢
Ⅎ𝑛
< |
| 31 | 28, 29, 30 | nfsup 9491 |
. . . . . . . . . 10
⊢
Ⅎ𝑛sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ*, <
) |
| 32 | | nfcv 2905 |
. . . . . . . . . 10
⊢
Ⅎ𝑛ℝ |
| 33 | 31, 32 | nfel 2920 |
. . . . . . . . 9
⊢
Ⅎ𝑛sup(ran
(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ*, < ) ∈
ℝ |
| 34 | 33, 7 | nfrabw 3475 |
. . . . . . . 8
⊢
Ⅎ𝑛{𝑥 ∈ ∩
𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ*, < ) ∈
ℝ} |
| 35 | 3, 34 | nfcxfr 2903 |
. . . . . . 7
⊢
Ⅎ𝑛𝐷 |
| 36 | 6, 35 | nfel 2920 |
. . . . . 6
⊢
Ⅎ𝑛 𝑥 ∈ 𝐷 |
| 37 | 5, 36 | nfan 1899 |
. . . . 5
⊢
Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ 𝐷) |
| 38 | 12 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑍 ≠ ∅) |
| 39 | 19 | adantlr 715 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛):dom (𝐹‘𝑛)⟶ℝ) |
| 40 | | nfcv 2905 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑥𝑍 |
| 41 | | smfsupxr.x |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑥𝐹 |
| 42 | | nfcv 2905 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑥𝑛 |
| 43 | 41, 42 | nffv 6916 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑥(𝐹‘𝑛) |
| 44 | 43 | nfdm 5962 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑥dom
(𝐹‘𝑛) |
| 45 | 40, 44 | nfiin 5024 |
. . . . . . . . . . 11
⊢
Ⅎ𝑥∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) |
| 46 | 45 | ssrab2f 45122 |
. . . . . . . . . 10
⊢ {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ*, < ) ∈
ℝ} ⊆ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) |
| 47 | 3, 46 | eqsstri 4030 |
. . . . . . . . 9
⊢ 𝐷 ⊆ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) |
| 48 | | id 22 |
. . . . . . . . 9
⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ 𝐷) |
| 49 | 47, 48 | sselid 3981 |
. . . . . . . 8
⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ∩
𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) |
| 50 | 49, 21 | sylan 580 |
. . . . . . 7
⊢ ((𝑥 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ dom (𝐹‘𝑛)) |
| 51 | 50 | adantll 714 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ dom (𝐹‘𝑛)) |
| 52 | 39, 51 | ffvelcdmd 7105 |
. . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ) |
| 53 | 48, 3 | eleqtrdi 2851 |
. . . . . . . 8
⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ {𝑥 ∈ ∩
𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ*, < ) ∈
ℝ}) |
| 54 | | rabidim2 45107 |
. . . . . . . 8
⊢ (𝑥 ∈ {𝑥 ∈ ∩
𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ*, < ) ∈
ℝ} → sup(ran (𝑛
∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ*, < ) ∈
ℝ) |
| 55 | 53, 54 | syl 17 |
. . . . . . 7
⊢ (𝑥 ∈ 𝐷 → sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ*, < ) ∈
ℝ) |
| 56 | 55 | adantl 481 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ*, < ) ∈
ℝ) |
| 57 | 49 | adantl 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ ∩
𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) |
| 58 | 57, 24 | syldan 591 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ*, < ) ∈
ℝ ↔ ∃𝑦
∈ ℝ ∀𝑛
∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦)) |
| 59 | 56, 58 | mpbid 232 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦) |
| 60 | 37, 38, 52, 59 | supxrrernmpt 45432 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ*, < ) = sup(ran
(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) |
| 61 | 26, 60 | mpteq12dva 5231 |
. . 3
⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ*, < )) = (𝑥 ∈ {𝑥 ∈ ∩
𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} ↦ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))) |
| 62 | 2, 61 | eqtrd 2777 |
. 2
⊢ (𝜑 → 𝐺 = (𝑥 ∈ {𝑥 ∈ ∩
𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} ↦ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))) |
| 63 | | smfsupxr.n |
. . 3
⊢
Ⅎ𝑛𝐹 |
| 64 | | eqid 2737 |
. . 3
⊢ {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} = {𝑥 ∈ ∩
𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} |
| 65 | | eqid 2737 |
. . 3
⊢ (𝑥 ∈ {𝑥 ∈ ∩
𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} ↦ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) = (𝑥 ∈ {𝑥 ∈ ∩
𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} ↦ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) |
| 66 | 63, 41, 10, 11, 14, 16, 64, 65 | smfsup 46829 |
. 2
⊢ (𝜑 → (𝑥 ∈ {𝑥 ∈ ∩
𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} ↦ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) ∈
(SMblFn‘𝑆)) |
| 67 | 62, 66 | eqeltrd 2841 |
1
⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) |