Step | Hyp | Ref
| Expression |
1 | | itgulm.z |
. . . 4
⊢ 𝑍 =
(ℤ≥‘𝑀) |
2 | | itgulm.m |
. . . 4
⊢ (𝜑 → 𝑀 ∈ ℤ) |
3 | | itgulm.f |
. . . . . 6
⊢ (𝜑 → 𝐹:𝑍⟶𝐿1) |
4 | | ffn 6045 |
. . . . . 6
⊢ (𝐹:𝑍⟶𝐿1 → 𝐹 Fn 𝑍) |
5 | 3, 4 | syl 17 |
. . . . 5
⊢ (𝜑 → 𝐹 Fn 𝑍) |
6 | | itgulm.u |
. . . . 5
⊢ (𝜑 → 𝐹(⇝𝑢‘𝑆)𝐺) |
7 | | ulmf2 24138 |
. . . . 5
⊢ ((𝐹 Fn 𝑍 ∧ 𝐹(⇝𝑢‘𝑆)𝐺) → 𝐹:𝑍⟶(ℂ ↑𝑚
𝑆)) |
8 | 5, 6, 7 | syl2anc 693 |
. . . 4
⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑𝑚
𝑆)) |
9 | | eqidd 2623 |
. . . 4
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ 𝑥 ∈ 𝑆)) → ((𝐹‘𝑘)‘𝑥) = ((𝐹‘𝑘)‘𝑥)) |
10 | | eqidd 2623 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝐺‘𝑥) = (𝐺‘𝑥)) |
11 | | 1rp 11836 |
. . . . 5
⊢ 1 ∈
ℝ+ |
12 | 11 | a1i 11 |
. . . 4
⊢ (𝜑 → 1 ∈
ℝ+) |
13 | 1, 2, 8, 9, 10, 6,
12 | ulmi 24140 |
. . 3
⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑥 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑥) − (𝐺‘𝑥))) < 1) |
14 | 1 | r19.2uz 14091 |
. . 3
⊢
(∃𝑗 ∈
𝑍 ∀𝑘 ∈
(ℤ≥‘𝑗)∀𝑥 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑥) − (𝐺‘𝑥))) < 1 → ∃𝑘 ∈ 𝑍 ∀𝑥 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑥) − (𝐺‘𝑥))) < 1) |
15 | 13, 14 | syl 17 |
. 2
⊢ (𝜑 → ∃𝑘 ∈ 𝑍 ∀𝑥 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑥) − (𝐺‘𝑥))) < 1) |
16 | | ulmcl 24135 |
. . . . . . 7
⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → 𝐺:𝑆⟶ℂ) |
17 | 6, 16 | syl 17 |
. . . . . 6
⊢ (𝜑 → 𝐺:𝑆⟶ℂ) |
18 | 17 | adantr 481 |
. . . . 5
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ ∀𝑥 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑥) − (𝐺‘𝑥))) < 1)) → 𝐺:𝑆⟶ℂ) |
19 | 18 | feqmptd 6249 |
. . . 4
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ ∀𝑥 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑥) − (𝐺‘𝑥))) < 1)) → 𝐺 = (𝑧 ∈ 𝑆 ↦ (𝐺‘𝑧))) |
20 | 8 | ffvelrnda 6359 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ (ℂ ↑𝑚
𝑆)) |
21 | | elmapi 7879 |
. . . . . . . . 9
⊢ ((𝐹‘𝑘) ∈ (ℂ ↑𝑚
𝑆) → (𝐹‘𝑘):𝑆⟶ℂ) |
22 | 20, 21 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘):𝑆⟶ℂ) |
23 | 22 | adantrr 753 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ ∀𝑥 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑥) − (𝐺‘𝑥))) < 1)) → (𝐹‘𝑘):𝑆⟶ℂ) |
24 | 23 | ffvelrnda 6359 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ ∀𝑥 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑥) − (𝐺‘𝑥))) < 1)) ∧ 𝑧 ∈ 𝑆) → ((𝐹‘𝑘)‘𝑧) ∈ ℂ) |
25 | 18 | ffvelrnda 6359 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ ∀𝑥 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑥) − (𝐺‘𝑥))) < 1)) ∧ 𝑧 ∈ 𝑆) → (𝐺‘𝑧) ∈ ℂ) |
26 | 24, 25 | nncand 10397 |
. . . . 5
⊢ (((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ ∀𝑥 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑥) − (𝐺‘𝑥))) < 1)) ∧ 𝑧 ∈ 𝑆) → (((𝐹‘𝑘)‘𝑧) − (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) = (𝐺‘𝑧)) |
27 | 26 | mpteq2dva 4744 |
. . . 4
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ ∀𝑥 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑥) − (𝐺‘𝑥))) < 1)) → (𝑧 ∈ 𝑆 ↦ (((𝐹‘𝑘)‘𝑧) − (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧)))) = (𝑧 ∈ 𝑆 ↦ (𝐺‘𝑧))) |
28 | 19, 27 | eqtr4d 2659 |
. . 3
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ ∀𝑥 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑥) − (𝐺‘𝑥))) < 1)) → 𝐺 = (𝑧 ∈ 𝑆 ↦ (((𝐹‘𝑘)‘𝑧) − (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))))) |
29 | 23 | feqmptd 6249 |
. . . . 5
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ ∀𝑥 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑥) − (𝐺‘𝑥))) < 1)) → (𝐹‘𝑘) = (𝑧 ∈ 𝑆 ↦ ((𝐹‘𝑘)‘𝑧))) |
30 | 3 | ffvelrnda 6359 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈
𝐿1) |
31 | 30 | adantrr 753 |
. . . . 5
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ ∀𝑥 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑥) − (𝐺‘𝑥))) < 1)) → (𝐹‘𝑘) ∈
𝐿1) |
32 | 29, 31 | eqeltrrd 2702 |
. . . 4
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ ∀𝑥 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑥) − (𝐺‘𝑥))) < 1)) → (𝑧 ∈ 𝑆 ↦ ((𝐹‘𝑘)‘𝑧)) ∈
𝐿1) |
33 | 24, 25 | subcld 10392 |
. . . 4
⊢ (((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ ∀𝑥 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑥) − (𝐺‘𝑥))) < 1)) ∧ 𝑧 ∈ 𝑆) → (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧)) ∈ ℂ) |
34 | | ulmscl 24133 |
. . . . . . . . 9
⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → 𝑆 ∈ V) |
35 | 6, 34 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑆 ∈ V) |
36 | 35 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ ∀𝑥 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑥) − (𝐺‘𝑥))) < 1)) → 𝑆 ∈ V) |
37 | 36, 24, 25, 29, 19 | offval2 6914 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ ∀𝑥 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑥) − (𝐺‘𝑥))) < 1)) → ((𝐹‘𝑘) ∘𝑓 − 𝐺) = (𝑧 ∈ 𝑆 ↦ (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧)))) |
38 | | iblmbf 23534 |
. . . . . . . 8
⊢ ((𝐹‘𝑘) ∈ 𝐿1 → (𝐹‘𝑘) ∈ MblFn) |
39 | 31, 38 | syl 17 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ ∀𝑥 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑥) − (𝐺‘𝑥))) < 1)) → (𝐹‘𝑘) ∈ MblFn) |
40 | | iblmbf 23534 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ 𝐿1
→ 𝑥 ∈
MblFn) |
41 | 40 | ssriv 3607 |
. . . . . . . . . 10
⊢
𝐿1 ⊆ MblFn |
42 | | fss 6056 |
. . . . . . . . . 10
⊢ ((𝐹:𝑍⟶𝐿1 ∧
𝐿1 ⊆ MblFn) → 𝐹:𝑍⟶MblFn) |
43 | 3, 41, 42 | sylancl 694 |
. . . . . . . . 9
⊢ (𝜑 → 𝐹:𝑍⟶MblFn) |
44 | 1, 2, 43, 6 | mbfulm 24160 |
. . . . . . . 8
⊢ (𝜑 → 𝐺 ∈ MblFn) |
45 | 44 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ ∀𝑥 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑥) − (𝐺‘𝑥))) < 1)) → 𝐺 ∈ MblFn) |
46 | 39, 45 | mbfsub 23429 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ ∀𝑥 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑥) − (𝐺‘𝑥))) < 1)) → ((𝐹‘𝑘) ∘𝑓 − 𝐺) ∈ MblFn) |
47 | 37, 46 | eqeltrrd 2702 |
. . . . 5
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ ∀𝑥 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑥) − (𝐺‘𝑥))) < 1)) → (𝑧 ∈ 𝑆 ↦ (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) ∈ MblFn) |
48 | | eqid 2622 |
. . . . . . . 8
⊢ (𝑧 ∈ 𝑆 ↦ (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) = (𝑧 ∈ 𝑆 ↦ (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) |
49 | 48, 33 | dmmptd 6024 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ ∀𝑥 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑥) − (𝐺‘𝑥))) < 1)) → dom (𝑧 ∈ 𝑆 ↦ (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) = 𝑆) |
50 | 49 | fveq2d 6195 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ ∀𝑥 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑥) − (𝐺‘𝑥))) < 1)) → (vol‘dom (𝑧 ∈ 𝑆 ↦ (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧)))) = (vol‘𝑆)) |
51 | | itgulm.s |
. . . . . . 7
⊢ (𝜑 → (vol‘𝑆) ∈
ℝ) |
52 | 51 | adantr 481 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ ∀𝑥 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑥) − (𝐺‘𝑥))) < 1)) → (vol‘𝑆) ∈
ℝ) |
53 | 50, 52 | eqeltrd 2701 |
. . . . 5
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ ∀𝑥 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑥) − (𝐺‘𝑥))) < 1)) → (vol‘dom (𝑧 ∈ 𝑆 ↦ (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧)))) ∈ ℝ) |
54 | | 1re 10039 |
. . . . . 6
⊢ 1 ∈
ℝ |
55 | 22 | ffvelrnda 6359 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑥 ∈ 𝑆) → ((𝐹‘𝑘)‘𝑥) ∈ ℂ) |
56 | 17 | adantr 481 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐺:𝑆⟶ℂ) |
57 | 56 | ffvelrnda 6359 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑥 ∈ 𝑆) → (𝐺‘𝑥) ∈ ℂ) |
58 | 55, 57 | subcld 10392 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑥 ∈ 𝑆) → (((𝐹‘𝑘)‘𝑥) − (𝐺‘𝑥)) ∈ ℂ) |
59 | 58 | abscld 14175 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑥 ∈ 𝑆) → (abs‘(((𝐹‘𝑘)‘𝑥) − (𝐺‘𝑥))) ∈ ℝ) |
60 | | ltle 10126 |
. . . . . . . . . . 11
⊢
(((abs‘(((𝐹‘𝑘)‘𝑥) − (𝐺‘𝑥))) ∈ ℝ ∧ 1 ∈ ℝ)
→ ((abs‘(((𝐹‘𝑘)‘𝑥) − (𝐺‘𝑥))) < 1 → (abs‘(((𝐹‘𝑘)‘𝑥) − (𝐺‘𝑥))) ≤ 1)) |
61 | 59, 54, 60 | sylancl 694 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑥 ∈ 𝑆) → ((abs‘(((𝐹‘𝑘)‘𝑥) − (𝐺‘𝑥))) < 1 → (abs‘(((𝐹‘𝑘)‘𝑥) − (𝐺‘𝑥))) ≤ 1)) |
62 | | fveq2 6191 |
. . . . . . . . . . . . . . 15
⊢ (𝑧 = 𝑥 → ((𝐹‘𝑘)‘𝑧) = ((𝐹‘𝑘)‘𝑥)) |
63 | | fveq2 6191 |
. . . . . . . . . . . . . . 15
⊢ (𝑧 = 𝑥 → (𝐺‘𝑧) = (𝐺‘𝑥)) |
64 | 62, 63 | oveq12d 6668 |
. . . . . . . . . . . . . 14
⊢ (𝑧 = 𝑥 → (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧)) = (((𝐹‘𝑘)‘𝑥) − (𝐺‘𝑥))) |
65 | | ovex 6678 |
. . . . . . . . . . . . . 14
⊢ (((𝐹‘𝑘)‘𝑥) − (𝐺‘𝑥)) ∈ V |
66 | 64, 48, 65 | fvmpt 6282 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ 𝑆 → ((𝑧 ∈ 𝑆 ↦ (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧)))‘𝑥) = (((𝐹‘𝑘)‘𝑥) − (𝐺‘𝑥))) |
67 | 66 | adantl 482 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑥 ∈ 𝑆) → ((𝑧 ∈ 𝑆 ↦ (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧)))‘𝑥) = (((𝐹‘𝑘)‘𝑥) − (𝐺‘𝑥))) |
68 | 67 | fveq2d 6195 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑥 ∈ 𝑆) → (abs‘((𝑧 ∈ 𝑆 ↦ (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧)))‘𝑥)) = (abs‘(((𝐹‘𝑘)‘𝑥) − (𝐺‘𝑥)))) |
69 | 68 | breq1d 4663 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑥 ∈ 𝑆) → ((abs‘((𝑧 ∈ 𝑆 ↦ (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧)))‘𝑥)) ≤ 1 ↔ (abs‘(((𝐹‘𝑘)‘𝑥) − (𝐺‘𝑥))) ≤ 1)) |
70 | 61, 69 | sylibrd 249 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑥 ∈ 𝑆) → ((abs‘(((𝐹‘𝑘)‘𝑥) − (𝐺‘𝑥))) < 1 → (abs‘((𝑧 ∈ 𝑆 ↦ (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧)))‘𝑥)) ≤ 1)) |
71 | 70 | ralimdva 2962 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (∀𝑥 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑥) − (𝐺‘𝑥))) < 1 → ∀𝑥 ∈ 𝑆 (abs‘((𝑧 ∈ 𝑆 ↦ (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧)))‘𝑥)) ≤ 1)) |
72 | 71 | impr 649 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ ∀𝑥 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑥) − (𝐺‘𝑥))) < 1)) → ∀𝑥 ∈ 𝑆 (abs‘((𝑧 ∈ 𝑆 ↦ (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧)))‘𝑥)) ≤ 1) |
73 | 49 | raleqdv 3144 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ ∀𝑥 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑥) − (𝐺‘𝑥))) < 1)) → (∀𝑥 ∈ dom (𝑧 ∈ 𝑆 ↦ (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧)))(abs‘((𝑧 ∈ 𝑆 ↦ (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧)))‘𝑥)) ≤ 1 ↔ ∀𝑥 ∈ 𝑆 (abs‘((𝑧 ∈ 𝑆 ↦ (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧)))‘𝑥)) ≤ 1)) |
74 | 72, 73 | mpbird 247 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ ∀𝑥 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑥) − (𝐺‘𝑥))) < 1)) → ∀𝑥 ∈ dom (𝑧 ∈ 𝑆 ↦ (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧)))(abs‘((𝑧 ∈ 𝑆 ↦ (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧)))‘𝑥)) ≤ 1) |
75 | | breq2 4657 |
. . . . . . . 8
⊢ (𝑟 = 1 → ((abs‘((𝑧 ∈ 𝑆 ↦ (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧)))‘𝑥)) ≤ 𝑟 ↔ (abs‘((𝑧 ∈ 𝑆 ↦ (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧)))‘𝑥)) ≤ 1)) |
76 | 75 | ralbidv 2986 |
. . . . . . 7
⊢ (𝑟 = 1 → (∀𝑥 ∈ dom (𝑧 ∈ 𝑆 ↦ (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧)))(abs‘((𝑧 ∈ 𝑆 ↦ (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧)))‘𝑥)) ≤ 𝑟 ↔ ∀𝑥 ∈ dom (𝑧 ∈ 𝑆 ↦ (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧)))(abs‘((𝑧 ∈ 𝑆 ↦ (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧)))‘𝑥)) ≤ 1)) |
77 | 76 | rspcev 3309 |
. . . . . 6
⊢ ((1
∈ ℝ ∧ ∀𝑥 ∈ dom (𝑧 ∈ 𝑆 ↦ (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧)))(abs‘((𝑧 ∈ 𝑆 ↦ (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧)))‘𝑥)) ≤ 1) → ∃𝑟 ∈ ℝ ∀𝑥 ∈ dom (𝑧 ∈ 𝑆 ↦ (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧)))(abs‘((𝑧 ∈ 𝑆 ↦ (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧)))‘𝑥)) ≤ 𝑟) |
78 | 54, 74, 77 | sylancr 695 |
. . . . 5
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ ∀𝑥 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑥) − (𝐺‘𝑥))) < 1)) → ∃𝑟 ∈ ℝ ∀𝑥 ∈ dom (𝑧 ∈ 𝑆 ↦ (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧)))(abs‘((𝑧 ∈ 𝑆 ↦ (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧)))‘𝑥)) ≤ 𝑟) |
79 | | bddibl 23606 |
. . . . 5
⊢ (((𝑧 ∈ 𝑆 ↦ (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) ∈ MblFn ∧ (vol‘dom (𝑧 ∈ 𝑆 ↦ (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧)))) ∈ ℝ ∧ ∃𝑟 ∈ ℝ ∀𝑥 ∈ dom (𝑧 ∈ 𝑆 ↦ (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧)))(abs‘((𝑧 ∈ 𝑆 ↦ (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧)))‘𝑥)) ≤ 𝑟) → (𝑧 ∈ 𝑆 ↦ (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) ∈
𝐿1) |
80 | 47, 53, 78, 79 | syl3anc 1326 |
. . . 4
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ ∀𝑥 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑥) − (𝐺‘𝑥))) < 1)) → (𝑧 ∈ 𝑆 ↦ (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) ∈
𝐿1) |
81 | 24, 32, 33, 80 | iblsub 23588 |
. . 3
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ ∀𝑥 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑥) − (𝐺‘𝑥))) < 1)) → (𝑧 ∈ 𝑆 ↦ (((𝐹‘𝑘)‘𝑧) − (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧)))) ∈
𝐿1) |
82 | 28, 81 | eqeltrd 2701 |
. 2
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ ∀𝑥 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑥) − (𝐺‘𝑥))) < 1)) → 𝐺 ∈
𝐿1) |
83 | 15, 82 | rexlimddv 3035 |
1
⊢ (𝜑 → 𝐺 ∈
𝐿1) |