Step | Hyp | Ref
| Expression |
1 | | fveq2 6103 |
. . . . . . . . 9
⊢ (𝑛 = 𝑚 → (𝑃‘𝑛) = (𝑃‘𝑚)) |
2 | 1 | fveq1d 6105 |
. . . . . . . 8
⊢ (𝑛 = 𝑚 → ((𝑃‘𝑛)‘𝑥) = ((𝑃‘𝑚)‘𝑥)) |
3 | 2 | cbvmptv 4678 |
. . . . . . 7
⊢ (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) = (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑥)) |
4 | | fveq2 6103 |
. . . . . . . 8
⊢ (𝑥 = 𝑦 → ((𝑃‘𝑚)‘𝑥) = ((𝑃‘𝑚)‘𝑦)) |
5 | 4 | mpteq2dv 4673 |
. . . . . . 7
⊢ (𝑥 = 𝑦 → (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑥)) = (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑦))) |
6 | 3, 5 | syl5eq 2656 |
. . . . . 6
⊢ (𝑥 = 𝑦 → (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) = (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑦))) |
7 | 6 | rneqd 5274 |
. . . . 5
⊢ (𝑥 = 𝑦 → ran (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) = ran (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑦))) |
8 | 7 | supeq1d 8235 |
. . . 4
⊢ (𝑥 = 𝑦 → sup(ran (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)), ℝ, < ) = sup(ran (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑦)), ℝ, < )) |
9 | 8 | cbvmptv 4678 |
. . 3
⊢ (𝑥 ∈ ℝ ↦ sup(ran
(𝑛 ∈ ℕ ↦
((𝑃‘𝑛)‘𝑥)), ℝ, < )) = (𝑦 ∈ ℝ ↦ sup(ran (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑦)), ℝ, < )) |
10 | | itg2i1fseq.3 |
. . . . 5
⊢ (𝜑 → 𝑃:ℕ⟶dom
∫1) |
11 | 10 | ffvelrnda 6267 |
. . . 4
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑃‘𝑚) ∈ dom
∫1) |
12 | | i1fmbf 23248 |
. . . 4
⊢ ((𝑃‘𝑚) ∈ dom ∫1 → (𝑃‘𝑚) ∈ MblFn) |
13 | 11, 12 | syl 17 |
. . 3
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑃‘𝑚) ∈ MblFn) |
14 | | i1ff 23249 |
. . . . 5
⊢ ((𝑃‘𝑚) ∈ dom ∫1 → (𝑃‘𝑚):ℝ⟶ℝ) |
15 | 11, 14 | syl 17 |
. . . 4
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑃‘𝑚):ℝ⟶ℝ) |
16 | | itg2i1fseq.4 |
. . . . . 6
⊢ (𝜑 → ∀𝑛 ∈ ℕ (0𝑝
∘𝑟 ≤ (𝑃‘𝑛) ∧ (𝑃‘𝑛) ∘𝑟 ≤ (𝑃‘(𝑛 + 1)))) |
17 | 1 | breq2d 4595 |
. . . . . . . 8
⊢ (𝑛 = 𝑚 → (0𝑝
∘𝑟 ≤ (𝑃‘𝑛) ↔ 0𝑝
∘𝑟 ≤ (𝑃‘𝑚))) |
18 | | oveq1 6556 |
. . . . . . . . . 10
⊢ (𝑛 = 𝑚 → (𝑛 + 1) = (𝑚 + 1)) |
19 | 18 | fveq2d 6107 |
. . . . . . . . 9
⊢ (𝑛 = 𝑚 → (𝑃‘(𝑛 + 1)) = (𝑃‘(𝑚 + 1))) |
20 | 1, 19 | breq12d 4596 |
. . . . . . . 8
⊢ (𝑛 = 𝑚 → ((𝑃‘𝑛) ∘𝑟 ≤ (𝑃‘(𝑛 + 1)) ↔ (𝑃‘𝑚) ∘𝑟 ≤ (𝑃‘(𝑚 + 1)))) |
21 | 17, 20 | anbi12d 743 |
. . . . . . 7
⊢ (𝑛 = 𝑚 → ((0𝑝
∘𝑟 ≤ (𝑃‘𝑛) ∧ (𝑃‘𝑛) ∘𝑟 ≤ (𝑃‘(𝑛 + 1))) ↔ (0𝑝
∘𝑟 ≤ (𝑃‘𝑚) ∧ (𝑃‘𝑚) ∘𝑟 ≤ (𝑃‘(𝑚 + 1))))) |
22 | 21 | rspccva 3281 |
. . . . . 6
⊢
((∀𝑛 ∈
ℕ (0𝑝 ∘𝑟 ≤ (𝑃‘𝑛) ∧ (𝑃‘𝑛) ∘𝑟 ≤ (𝑃‘(𝑛 + 1))) ∧ 𝑚 ∈ ℕ) →
(0𝑝 ∘𝑟 ≤ (𝑃‘𝑚) ∧ (𝑃‘𝑚) ∘𝑟 ≤ (𝑃‘(𝑚 + 1)))) |
23 | 16, 22 | sylan 487 |
. . . . 5
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) →
(0𝑝 ∘𝑟 ≤ (𝑃‘𝑚) ∧ (𝑃‘𝑚) ∘𝑟 ≤ (𝑃‘(𝑚 + 1)))) |
24 | 23 | simpld 474 |
. . . 4
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 0𝑝
∘𝑟 ≤ (𝑃‘𝑚)) |
25 | | 0plef 23245 |
. . . 4
⊢ ((𝑃‘𝑚):ℝ⟶(0[,)+∞) ↔
((𝑃‘𝑚):ℝ⟶ℝ ∧
0𝑝 ∘𝑟 ≤ (𝑃‘𝑚))) |
26 | 15, 24, 25 | sylanbrc 695 |
. . 3
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑃‘𝑚):ℝ⟶(0[,)+∞)) |
27 | 23 | simprd 478 |
. . 3
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑃‘𝑚) ∘𝑟 ≤ (𝑃‘(𝑚 + 1))) |
28 | | rge0ssre 12151 |
. . . . 5
⊢
(0[,)+∞) ⊆ ℝ |
29 | | itg2i1fseq.2 |
. . . . . 6
⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) |
30 | 29 | ffvelrnda 6267 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝐹‘𝑦) ∈ (0[,)+∞)) |
31 | 28, 30 | sseldi 3566 |
. . . 4
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝐹‘𝑦) ∈ ℝ) |
32 | | itg2i1fseq.1 |
. . . . . . . . 9
⊢ (𝜑 → 𝐹 ∈ MblFn) |
33 | | itg2i1fseq.5 |
. . . . . . . . 9
⊢ (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)) |
34 | 32, 29, 10, 16, 33 | itg2i1fseqle 23327 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑃‘𝑚) ∘𝑟 ≤ 𝐹) |
35 | | ffn 5958 |
. . . . . . . . . 10
⊢ ((𝑃‘𝑚):ℝ⟶ℝ → (𝑃‘𝑚) Fn ℝ) |
36 | 15, 35 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑃‘𝑚) Fn ℝ) |
37 | | ffn 5958 |
. . . . . . . . . . 11
⊢ (𝐹:ℝ⟶(0[,)+∞)
→ 𝐹 Fn
ℝ) |
38 | 29, 37 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐹 Fn ℝ) |
39 | 38 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐹 Fn ℝ) |
40 | | reex 9906 |
. . . . . . . . . 10
⊢ ℝ
∈ V |
41 | 40 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ℝ ∈
V) |
42 | | inidm 3784 |
. . . . . . . . 9
⊢ (ℝ
∩ ℝ) = ℝ |
43 | | eqidd 2611 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃‘𝑚)‘𝑦) = ((𝑃‘𝑚)‘𝑦)) |
44 | | eqidd 2611 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝐹‘𝑦) = (𝐹‘𝑦)) |
45 | 36, 39, 41, 41, 42, 43, 44 | ofrfval 6803 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑃‘𝑚) ∘𝑟 ≤ 𝐹 ↔ ∀𝑦 ∈ ℝ ((𝑃‘𝑚)‘𝑦) ≤ (𝐹‘𝑦))) |
46 | 34, 45 | mpbid 221 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ∀𝑦 ∈ ℝ ((𝑃‘𝑚)‘𝑦) ≤ (𝐹‘𝑦)) |
47 | 46 | r19.21bi 2916 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃‘𝑚)‘𝑦) ≤ (𝐹‘𝑦)) |
48 | 47 | an32s 842 |
. . . . 5
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑃‘𝑚)‘𝑦) ≤ (𝐹‘𝑦)) |
49 | 48 | ralrimiva 2949 |
. . . 4
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ∀𝑚 ∈ ℕ ((𝑃‘𝑚)‘𝑦) ≤ (𝐹‘𝑦)) |
50 | | breq2 4587 |
. . . . . 6
⊢ (𝑧 = (𝐹‘𝑦) → (((𝑃‘𝑚)‘𝑦) ≤ 𝑧 ↔ ((𝑃‘𝑚)‘𝑦) ≤ (𝐹‘𝑦))) |
51 | 50 | ralbidv 2969 |
. . . . 5
⊢ (𝑧 = (𝐹‘𝑦) → (∀𝑚 ∈ ℕ ((𝑃‘𝑚)‘𝑦) ≤ 𝑧 ↔ ∀𝑚 ∈ ℕ ((𝑃‘𝑚)‘𝑦) ≤ (𝐹‘𝑦))) |
52 | 51 | rspcev 3282 |
. . . 4
⊢ (((𝐹‘𝑦) ∈ ℝ ∧ ∀𝑚 ∈ ℕ ((𝑃‘𝑚)‘𝑦) ≤ (𝐹‘𝑦)) → ∃𝑧 ∈ ℝ ∀𝑚 ∈ ℕ ((𝑃‘𝑚)‘𝑦) ≤ 𝑧) |
53 | 31, 49, 52 | syl2anc 691 |
. . 3
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ∃𝑧 ∈ ℝ ∀𝑚 ∈ ℕ ((𝑃‘𝑚)‘𝑦) ≤ 𝑧) |
54 | 1 | fveq2d 6107 |
. . . . . 6
⊢ (𝑛 = 𝑚 → (∫2‘(𝑃‘𝑛)) = (∫2‘(𝑃‘𝑚))) |
55 | 54 | cbvmptv 4678 |
. . . . 5
⊢ (𝑛 ∈ ℕ ↦
(∫2‘(𝑃‘𝑛))) = (𝑚 ∈ ℕ ↦
(∫2‘(𝑃‘𝑚))) |
56 | 55 | rneqi 5273 |
. . . 4
⊢ ran
(𝑛 ∈ ℕ ↦
(∫2‘(𝑃‘𝑛))) = ran (𝑚 ∈ ℕ ↦
(∫2‘(𝑃‘𝑚))) |
57 | 56 | supeq1i 8236 |
. . 3
⊢ sup(ran
(𝑛 ∈ ℕ ↦
(∫2‘(𝑃‘𝑛))), ℝ*, < ) = sup(ran
(𝑚 ∈ ℕ ↦
(∫2‘(𝑃‘𝑚))), ℝ*, <
) |
58 | 9, 13, 26, 27, 53, 57 | itg2mono 23326 |
. 2
⊢ (𝜑 →
(∫2‘(𝑥
∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)), ℝ, < ))) = sup(ran (𝑛 ∈ ℕ ↦
(∫2‘(𝑃‘𝑛))), ℝ*, <
)) |
59 | 29 | feqmptd 6159 |
. . . . 5
⊢ (𝜑 → 𝐹 = (𝑦 ∈ ℝ ↦ (𝐹‘𝑦))) |
60 | 1 | fveq1d 6105 |
. . . . . . . . . 10
⊢ (𝑛 = 𝑚 → ((𝑃‘𝑛)‘𝑦) = ((𝑃‘𝑚)‘𝑦)) |
61 | 60 | cbvmptv 4678 |
. . . . . . . . 9
⊢ (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)) = (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑦)) |
62 | 61 | rneqi 5273 |
. . . . . . . 8
⊢ ran
(𝑛 ∈ ℕ ↦
((𝑃‘𝑛)‘𝑦)) = ran (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑦)) |
63 | 62 | supeq1i 8236 |
. . . . . . 7
⊢ sup(ran
(𝑛 ∈ ℕ ↦
((𝑃‘𝑛)‘𝑦)), ℝ, < ) = sup(ran (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑦)), ℝ, < ) |
64 | | nnuz 11599 |
. . . . . . . . 9
⊢ ℕ =
(ℤ≥‘1) |
65 | | 1zzd 11285 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 1 ∈
ℤ) |
66 | 15 | ffvelrnda 6267 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃‘𝑚)‘𝑦) ∈ ℝ) |
67 | 66 | an32s 842 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑃‘𝑚)‘𝑦) ∈ ℝ) |
68 | 67, 61 | fmptd 6292 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)):ℕ⟶ℝ) |
69 | | peano2nn 10909 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑚 ∈ ℕ → (𝑚 + 1) ∈
ℕ) |
70 | | ffvelrn 6265 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑃:ℕ⟶dom
∫1 ∧ (𝑚
+ 1) ∈ ℕ) → (𝑃‘(𝑚 + 1)) ∈ dom
∫1) |
71 | 10, 69, 70 | syl2an 493 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑃‘(𝑚 + 1)) ∈ dom
∫1) |
72 | | i1ff 23249 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑃‘(𝑚 + 1)) ∈ dom ∫1 →
(𝑃‘(𝑚 +
1)):ℝ⟶ℝ) |
73 | 71, 72 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑃‘(𝑚 +
1)):ℝ⟶ℝ) |
74 | | ffn 5958 |
. . . . . . . . . . . . . . 15
⊢ ((𝑃‘(𝑚 + 1)):ℝ⟶ℝ → (𝑃‘(𝑚 + 1)) Fn ℝ) |
75 | 73, 74 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑃‘(𝑚 + 1)) Fn ℝ) |
76 | | eqidd 2611 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃‘(𝑚 + 1))‘𝑦) = ((𝑃‘(𝑚 + 1))‘𝑦)) |
77 | 36, 75, 41, 41, 42, 43, 76 | ofrfval 6803 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑃‘𝑚) ∘𝑟 ≤ (𝑃‘(𝑚 + 1)) ↔ ∀𝑦 ∈ ℝ ((𝑃‘𝑚)‘𝑦) ≤ ((𝑃‘(𝑚 + 1))‘𝑦))) |
78 | 27, 77 | mpbid 221 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ∀𝑦 ∈ ℝ ((𝑃‘𝑚)‘𝑦) ≤ ((𝑃‘(𝑚 + 1))‘𝑦)) |
79 | 78 | r19.21bi 2916 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃‘𝑚)‘𝑦) ≤ ((𝑃‘(𝑚 + 1))‘𝑦)) |
80 | 79 | an32s 842 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑃‘𝑚)‘𝑦) ≤ ((𝑃‘(𝑚 + 1))‘𝑦)) |
81 | | eqid 2610 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)) = (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)) |
82 | | fvex 6113 |
. . . . . . . . . . . 12
⊢ ((𝑃‘𝑚)‘𝑦) ∈ V |
83 | 60, 81, 82 | fvmpt 6191 |
. . . . . . . . . . 11
⊢ (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘𝑚) = ((𝑃‘𝑚)‘𝑦)) |
84 | 83 | adantl 481 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘𝑚) = ((𝑃‘𝑚)‘𝑦)) |
85 | | fveq2 6103 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = (𝑚 + 1) → (𝑃‘𝑛) = (𝑃‘(𝑚 + 1))) |
86 | 85 | fveq1d 6105 |
. . . . . . . . . . . . 13
⊢ (𝑛 = (𝑚 + 1) → ((𝑃‘𝑛)‘𝑦) = ((𝑃‘(𝑚 + 1))‘𝑦)) |
87 | | fvex 6113 |
. . . . . . . . . . . . 13
⊢ ((𝑃‘(𝑚 + 1))‘𝑦) ∈ V |
88 | 86, 81, 87 | fvmpt 6191 |
. . . . . . . . . . . 12
⊢ ((𝑚 + 1) ∈ ℕ →
((𝑛 ∈ ℕ ↦
((𝑃‘𝑛)‘𝑦))‘(𝑚 + 1)) = ((𝑃‘(𝑚 + 1))‘𝑦)) |
89 | 69, 88 | syl 17 |
. . . . . . . . . . 11
⊢ (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘(𝑚 + 1)) = ((𝑃‘(𝑚 + 1))‘𝑦)) |
90 | 89 | adantl 481 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘(𝑚 + 1)) = ((𝑃‘(𝑚 + 1))‘𝑦)) |
91 | 80, 84, 90 | 3brtr4d 4615 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘𝑚) ≤ ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘(𝑚 + 1))) |
92 | 83 | breq1d 4593 |
. . . . . . . . . . . 12
⊢ (𝑚 ∈ ℕ → (((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘𝑚) ≤ 𝑧 ↔ ((𝑃‘𝑚)‘𝑦) ≤ 𝑧)) |
93 | 92 | ralbiia 2962 |
. . . . . . . . . . 11
⊢
(∀𝑚 ∈
ℕ ((𝑛 ∈ ℕ
↦ ((𝑃‘𝑛)‘𝑦))‘𝑚) ≤ 𝑧 ↔ ∀𝑚 ∈ ℕ ((𝑃‘𝑚)‘𝑦) ≤ 𝑧) |
94 | 93 | rexbii 3023 |
. . . . . . . . . 10
⊢
(∃𝑧 ∈
ℝ ∀𝑚 ∈
ℕ ((𝑛 ∈ ℕ
↦ ((𝑃‘𝑛)‘𝑦))‘𝑚) ≤ 𝑧 ↔ ∃𝑧 ∈ ℝ ∀𝑚 ∈ ℕ ((𝑃‘𝑚)‘𝑦) ≤ 𝑧) |
95 | 53, 94 | sylibr 223 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ∃𝑧 ∈ ℝ ∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘𝑚) ≤ 𝑧) |
96 | 64, 65, 68, 91, 95 | climsup 14248 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)) ⇝ sup(ran (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)), ℝ, < )) |
97 | | fveq2 6103 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑦 → ((𝑃‘𝑛)‘𝑥) = ((𝑃‘𝑛)‘𝑦)) |
98 | 97 | mpteq2dv 4673 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑦 → (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))) |
99 | | fveq2 6103 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦)) |
100 | 98, 99 | breq12d 4596 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑦 → ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥) ↔ (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)) ⇝ (𝐹‘𝑦))) |
101 | 100 | rspccva 3281 |
. . . . . . . . 9
⊢
((∀𝑥 ∈
ℝ (𝑛 ∈ ℕ
↦ ((𝑃‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥) ∧ 𝑦 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)) ⇝ (𝐹‘𝑦)) |
102 | 33, 101 | sylan 487 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)) ⇝ (𝐹‘𝑦)) |
103 | | climuni 14131 |
. . . . . . . 8
⊢ (((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)) ⇝ sup(ran (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)), ℝ, < ) ∧ (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)) ⇝ (𝐹‘𝑦)) → sup(ran (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)), ℝ, < ) = (𝐹‘𝑦)) |
104 | 96, 102, 103 | syl2anc 691 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)), ℝ, < ) = (𝐹‘𝑦)) |
105 | 63, 104 | syl5eqr 2658 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → sup(ran (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑦)), ℝ, < ) = (𝐹‘𝑦)) |
106 | 105 | mpteq2dva 4672 |
. . . . 5
⊢ (𝜑 → (𝑦 ∈ ℝ ↦ sup(ran (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑦)), ℝ, < )) = (𝑦 ∈ ℝ ↦ (𝐹‘𝑦))) |
107 | 59, 106 | eqtr4d 2647 |
. . . 4
⊢ (𝜑 → 𝐹 = (𝑦 ∈ ℝ ↦ sup(ran (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑦)), ℝ, < ))) |
108 | 107, 9 | syl6eqr 2662 |
. . 3
⊢ (𝜑 → 𝐹 = (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)), ℝ, < ))) |
109 | 108 | fveq2d 6107 |
. 2
⊢ (𝜑 →
(∫2‘𝐹)
= (∫2‘(𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)), ℝ, < )))) |
110 | | itg2itg1 23309 |
. . . . . . . 8
⊢ (((𝑃‘𝑚) ∈ dom ∫1 ∧
0𝑝 ∘𝑟 ≤ (𝑃‘𝑚)) → (∫2‘(𝑃‘𝑚)) = (∫1‘(𝑃‘𝑚))) |
111 | 11, 24, 110 | syl2anc 691 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) →
(∫2‘(𝑃‘𝑚)) = (∫1‘(𝑃‘𝑚))) |
112 | 111 | mpteq2dva 4672 |
. . . . . 6
⊢ (𝜑 → (𝑚 ∈ ℕ ↦
(∫2‘(𝑃‘𝑚))) = (𝑚 ∈ ℕ ↦
(∫1‘(𝑃‘𝑚)))) |
113 | | itg2i1fseq.6 |
. . . . . 6
⊢ 𝑆 = (𝑚 ∈ ℕ ↦
(∫1‘(𝑃‘𝑚))) |
114 | 112, 113 | syl6reqr 2663 |
. . . . 5
⊢ (𝜑 → 𝑆 = (𝑚 ∈ ℕ ↦
(∫2‘(𝑃‘𝑚)))) |
115 | 114, 55 | syl6eqr 2662 |
. . . 4
⊢ (𝜑 → 𝑆 = (𝑛 ∈ ℕ ↦
(∫2‘(𝑃‘𝑛)))) |
116 | 115 | rneqd 5274 |
. . 3
⊢ (𝜑 → ran 𝑆 = ran (𝑛 ∈ ℕ ↦
(∫2‘(𝑃‘𝑛)))) |
117 | 116 | supeq1d 8235 |
. 2
⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) = sup(ran
(𝑛 ∈ ℕ ↦
(∫2‘(𝑃‘𝑛))), ℝ*, <
)) |
118 | 58, 109, 117 | 3eqtr4d 2654 |
1
⊢ (𝜑 →
(∫2‘𝐹)
= sup(ran 𝑆,
ℝ*, < )) |