Step | Hyp | Ref
| Expression |
1 | | nfcv 2900 |
. . 3
⊢
Ⅎ𝑛𝐹 |
2 | | smfsuplem2.z |
. . 3
⊢ 𝑍 =
(ℤ≥‘𝑀) |
3 | | smfsuplem2.s |
. . 3
⊢ (𝜑 → 𝑆 ∈ SAlg) |
4 | | smfsuplem2.f |
. . 3
⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) |
5 | | eqid 2739 |
. . 3
⊢
(topGen‘ran (,)) = (topGen‘ran (,)) |
6 | | eqid 2739 |
. . 3
⊢
(SalGen‘(topGen‘ran (,))) = (SalGen‘(topGen‘ran
(,))) |
7 | | mnfxr 10789 |
. . . . 5
⊢ -∞
∈ ℝ* |
8 | 7 | a1i 11 |
. . . 4
⊢ (𝜑 → -∞ ∈
ℝ*) |
9 | | smfsuplem2.8 |
. . . 4
⊢ (𝜑 → 𝐴 ∈ ℝ) |
10 | 8, 9, 5, 6 | iocborel 43478 |
. . 3
⊢ (𝜑 → (-∞(,]𝐴) ∈
(SalGen‘(topGen‘ran (,)))) |
11 | 1, 2, 3, 4, 5, 6, 10 | smfpimcc 43921 |
. 2
⊢ (𝜑 → ∃ℎ(ℎ:𝑍⟶𝑆 ∧ ∀𝑛 ∈ 𝑍 (◡(𝐹‘𝑛) “ (-∞(,]𝐴)) = ((ℎ‘𝑛) ∩ dom (𝐹‘𝑛)))) |
12 | | smfsuplem2.m |
. . . . . 6
⊢ (𝜑 → 𝑀 ∈ ℤ) |
13 | 12 | adantr 484 |
. . . . 5
⊢ ((𝜑 ∧ (ℎ:𝑍⟶𝑆 ∧ ∀𝑛 ∈ 𝑍 (◡(𝐹‘𝑛) “ (-∞(,]𝐴)) = ((ℎ‘𝑛) ∩ dom (𝐹‘𝑛)))) → 𝑀 ∈ ℤ) |
14 | 3 | adantr 484 |
. . . . 5
⊢ ((𝜑 ∧ (ℎ:𝑍⟶𝑆 ∧ ∀𝑛 ∈ 𝑍 (◡(𝐹‘𝑛) “ (-∞(,]𝐴)) = ((ℎ‘𝑛) ∩ dom (𝐹‘𝑛)))) → 𝑆 ∈ SAlg) |
15 | 4 | adantr 484 |
. . . . 5
⊢ ((𝜑 ∧ (ℎ:𝑍⟶𝑆 ∧ ∀𝑛 ∈ 𝑍 (◡(𝐹‘𝑛) “ (-∞(,]𝐴)) = ((ℎ‘𝑛) ∩ dom (𝐹‘𝑛)))) → 𝐹:𝑍⟶(SMblFn‘𝑆)) |
16 | | smfsuplem2.d |
. . . . . 6
⊢ 𝐷 = {𝑥 ∈ ∩
𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} |
17 | | fveq2 6687 |
. . . . . . . . . 10
⊢ (𝑛 = 𝑚 → (𝐹‘𝑛) = (𝐹‘𝑚)) |
18 | 17 | dmeqd 5758 |
. . . . . . . . 9
⊢ (𝑛 = 𝑚 → dom (𝐹‘𝑛) = dom (𝐹‘𝑚)) |
19 | 18 | cbviinv 4937 |
. . . . . . . 8
⊢ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) = ∩ 𝑚 ∈ 𝑍 dom (𝐹‘𝑚) |
20 | 19 | a1i 11 |
. . . . . . 7
⊢ (𝑥 = 𝑤 → ∩
𝑛 ∈ 𝑍 dom (𝐹‘𝑛) = ∩ 𝑚 ∈ 𝑍 dom (𝐹‘𝑚)) |
21 | | fveq2 6687 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑤 → ((𝐹‘𝑛)‘𝑥) = ((𝐹‘𝑛)‘𝑤)) |
22 | 21 | breq1d 5050 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑤 → (((𝐹‘𝑛)‘𝑥) ≤ 𝑦 ↔ ((𝐹‘𝑛)‘𝑤) ≤ 𝑦)) |
23 | 22 | ralbidv 3110 |
. . . . . . . . 9
⊢ (𝑥 = 𝑤 → (∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦 ↔ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑤) ≤ 𝑦)) |
24 | 17 | fveq1d 6689 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝑚 → ((𝐹‘𝑛)‘𝑤) = ((𝐹‘𝑚)‘𝑤)) |
25 | 24 | breq1d 5050 |
. . . . . . . . . . 11
⊢ (𝑛 = 𝑚 → (((𝐹‘𝑛)‘𝑤) ≤ 𝑦 ↔ ((𝐹‘𝑚)‘𝑤) ≤ 𝑦)) |
26 | 25 | cbvralvw 3350 |
. . . . . . . . . 10
⊢
(∀𝑛 ∈
𝑍 ((𝐹‘𝑛)‘𝑤) ≤ 𝑦 ↔ ∀𝑚 ∈ 𝑍 ((𝐹‘𝑚)‘𝑤) ≤ 𝑦) |
27 | 26 | a1i 11 |
. . . . . . . . 9
⊢ (𝑥 = 𝑤 → (∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑤) ≤ 𝑦 ↔ ∀𝑚 ∈ 𝑍 ((𝐹‘𝑚)‘𝑤) ≤ 𝑦)) |
28 | 23, 27 | bitrd 282 |
. . . . . . . 8
⊢ (𝑥 = 𝑤 → (∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦 ↔ ∀𝑚 ∈ 𝑍 ((𝐹‘𝑚)‘𝑤) ≤ 𝑦)) |
29 | 28 | rexbidv 3208 |
. . . . . . 7
⊢ (𝑥 = 𝑤 → (∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦 ↔ ∃𝑦 ∈ ℝ ∀𝑚 ∈ 𝑍 ((𝐹‘𝑚)‘𝑤) ≤ 𝑦)) |
30 | 20, 29 | cbvrabv2w 42256 |
. . . . . 6
⊢ {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} = {𝑤 ∈ ∩
𝑚 ∈ 𝑍 dom (𝐹‘𝑚) ∣ ∃𝑦 ∈ ℝ ∀𝑚 ∈ 𝑍 ((𝐹‘𝑚)‘𝑤) ≤ 𝑦} |
31 | 16, 30 | eqtri 2762 |
. . . . 5
⊢ 𝐷 = {𝑤 ∈ ∩
𝑚 ∈ 𝑍 dom (𝐹‘𝑚) ∣ ∃𝑦 ∈ ℝ ∀𝑚 ∈ 𝑍 ((𝐹‘𝑚)‘𝑤) ≤ 𝑦} |
32 | | smfsuplem2.g |
. . . . . 6
⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) |
33 | 21 | mpteq2dv 5136 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑤 → (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)) = (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑤))) |
34 | 24 | cbvmptv 5143 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑤)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑤)) |
35 | 34 | a1i 11 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑤 → (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑤)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑤))) |
36 | 33, 35 | eqtrd 2774 |
. . . . . . . . 9
⊢ (𝑥 = 𝑤 → (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑤))) |
37 | 36 | rneqd 5791 |
. . . . . . . 8
⊢ (𝑥 = 𝑤 → ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)) = ran (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑤))) |
38 | 37 | supeq1d 8996 |
. . . . . . 7
⊢ (𝑥 = 𝑤 → sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ) = sup(ran (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑤)), ℝ, < )) |
39 | 38 | cbvmptv 5143 |
. . . . . 6
⊢ (𝑥 ∈ 𝐷 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) = (𝑤 ∈ 𝐷 ↦ sup(ran (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑤)), ℝ, < )) |
40 | 32, 39 | eqtri 2762 |
. . . . 5
⊢ 𝐺 = (𝑤 ∈ 𝐷 ↦ sup(ran (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑤)), ℝ, < )) |
41 | 9 | adantr 484 |
. . . . 5
⊢ ((𝜑 ∧ (ℎ:𝑍⟶𝑆 ∧ ∀𝑛 ∈ 𝑍 (◡(𝐹‘𝑛) “ (-∞(,]𝐴)) = ((ℎ‘𝑛) ∩ dom (𝐹‘𝑛)))) → 𝐴 ∈ ℝ) |
42 | | simprl 771 |
. . . . 5
⊢ ((𝜑 ∧ (ℎ:𝑍⟶𝑆 ∧ ∀𝑛 ∈ 𝑍 (◡(𝐹‘𝑛) “ (-∞(,]𝐴)) = ((ℎ‘𝑛) ∩ dom (𝐹‘𝑛)))) → ℎ:𝑍⟶𝑆) |
43 | | simplrr 778 |
. . . . . 6
⊢ (((𝜑 ∧ (ℎ:𝑍⟶𝑆 ∧ ∀𝑛 ∈ 𝑍 (◡(𝐹‘𝑛) “ (-∞(,]𝐴)) = ((ℎ‘𝑛) ∩ dom (𝐹‘𝑛)))) ∧ 𝑚 ∈ 𝑍) → ∀𝑛 ∈ 𝑍 (◡(𝐹‘𝑛) “ (-∞(,]𝐴)) = ((ℎ‘𝑛) ∩ dom (𝐹‘𝑛))) |
44 | 17 | cnveqd 5728 |
. . . . . . . . 9
⊢ (𝑛 = 𝑚 → ◡(𝐹‘𝑛) = ◡(𝐹‘𝑚)) |
45 | 44 | imaeq1d 5912 |
. . . . . . . 8
⊢ (𝑛 = 𝑚 → (◡(𝐹‘𝑛) “ (-∞(,]𝐴)) = (◡(𝐹‘𝑚) “ (-∞(,]𝐴))) |
46 | | fveq2 6687 |
. . . . . . . . 9
⊢ (𝑛 = 𝑚 → (ℎ‘𝑛) = (ℎ‘𝑚)) |
47 | 46, 18 | ineq12d 4114 |
. . . . . . . 8
⊢ (𝑛 = 𝑚 → ((ℎ‘𝑛) ∩ dom (𝐹‘𝑛)) = ((ℎ‘𝑚) ∩ dom (𝐹‘𝑚))) |
48 | 45, 47 | eqeq12d 2755 |
. . . . . . 7
⊢ (𝑛 = 𝑚 → ((◡(𝐹‘𝑛) “ (-∞(,]𝐴)) = ((ℎ‘𝑛) ∩ dom (𝐹‘𝑛)) ↔ (◡(𝐹‘𝑚) “ (-∞(,]𝐴)) = ((ℎ‘𝑚) ∩ dom (𝐹‘𝑚)))) |
49 | 48 | rspccva 3528 |
. . . . . 6
⊢
((∀𝑛 ∈
𝑍 (◡(𝐹‘𝑛) “ (-∞(,]𝐴)) = ((ℎ‘𝑛) ∩ dom (𝐹‘𝑛)) ∧ 𝑚 ∈ 𝑍) → (◡(𝐹‘𝑚) “ (-∞(,]𝐴)) = ((ℎ‘𝑚) ∩ dom (𝐹‘𝑚))) |
50 | 43, 49 | sylancom 591 |
. . . . 5
⊢ (((𝜑 ∧ (ℎ:𝑍⟶𝑆 ∧ ∀𝑛 ∈ 𝑍 (◡(𝐹‘𝑛) “ (-∞(,]𝐴)) = ((ℎ‘𝑛) ∩ dom (𝐹‘𝑛)))) ∧ 𝑚 ∈ 𝑍) → (◡(𝐹‘𝑚) “ (-∞(,]𝐴)) = ((ℎ‘𝑚) ∩ dom (𝐹‘𝑚))) |
51 | 13, 2, 14, 15, 31, 40, 41, 42, 50 | smfsuplem1 43924 |
. . . 4
⊢ ((𝜑 ∧ (ℎ:𝑍⟶𝑆 ∧ ∀𝑛 ∈ 𝑍 (◡(𝐹‘𝑛) “ (-∞(,]𝐴)) = ((ℎ‘𝑛) ∩ dom (𝐹‘𝑛)))) → (◡𝐺 “ (-∞(,]𝐴)) ∈ (𝑆 ↾t 𝐷)) |
52 | 51 | ex 416 |
. . 3
⊢ (𝜑 → ((ℎ:𝑍⟶𝑆 ∧ ∀𝑛 ∈ 𝑍 (◡(𝐹‘𝑛) “ (-∞(,]𝐴)) = ((ℎ‘𝑛) ∩ dom (𝐹‘𝑛))) → (◡𝐺 “ (-∞(,]𝐴)) ∈ (𝑆 ↾t 𝐷))) |
53 | 52 | exlimdv 1940 |
. 2
⊢ (𝜑 → (∃ℎ(ℎ:𝑍⟶𝑆 ∧ ∀𝑛 ∈ 𝑍 (◡(𝐹‘𝑛) “ (-∞(,]𝐴)) = ((ℎ‘𝑛) ∩ dom (𝐹‘𝑛))) → (◡𝐺 “ (-∞(,]𝐴)) ∈ (𝑆 ↾t 𝐷))) |
54 | 11, 53 | mpd 15 |
1
⊢ (𝜑 → (◡𝐺 “ (-∞(,]𝐴)) ∈ (𝑆 ↾t 𝐷)) |