| Step | Hyp | Ref
| Expression |
| 1 | | itg2mono.3 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛):ℝ⟶(0[,)+∞)) |
| 2 | | rge0ssre 13496 |
. . . . . . . . . . . 12
⊢
(0[,)+∞) ⊆ ℝ |
| 3 | | fss 6752 |
. . . . . . . . . . . 12
⊢ (((𝐹‘𝑛):ℝ⟶(0[,)+∞) ∧
(0[,)+∞) ⊆ ℝ) → (𝐹‘𝑛):ℝ⟶ℝ) |
| 4 | 1, 2, 3 | sylancl 586 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛):ℝ⟶ℝ) |
| 5 | 4 | ffvelcdmda 7104 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ) |
| 6 | 5 | an32s 652 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ) |
| 7 | 6 | fmpttd 7135 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)):ℕ⟶ℝ) |
| 8 | 7 | frnd 6744 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) ⊆ ℝ) |
| 9 | | 1nn 12277 |
. . . . . . . . . 10
⊢ 1 ∈
ℕ |
| 10 | | eqid 2737 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) |
| 11 | 10, 6 | dmmptd 6713 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → dom (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) = ℕ) |
| 12 | 9, 11 | eleqtrrid 2848 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 1 ∈ dom (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))) |
| 13 | 12 | ne0d 4342 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → dom (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) ≠ ∅) |
| 14 | | dm0rn0 5935 |
. . . . . . . . 9
⊢ (dom
(𝑛 ∈ ℕ ↦
((𝐹‘𝑛)‘𝑥)) = ∅ ↔ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) = ∅) |
| 15 | 14 | necon3bii 2993 |
. . . . . . . 8
⊢ (dom
(𝑛 ∈ ℕ ↦
((𝐹‘𝑛)‘𝑥)) ≠ ∅ ↔ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) ≠ ∅) |
| 16 | 13, 15 | sylib 218 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) ≠ ∅) |
| 17 | | itg2mono.5 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ ((𝐹‘𝑛)‘𝑥) ≤ 𝑦) |
| 18 | 7 | ffnd 6737 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) Fn ℕ) |
| 19 | | breq1 5146 |
. . . . . . . . . . . 12
⊢ (𝑧 = ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) → (𝑧 ≤ 𝑦 ↔ ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) ≤ 𝑦)) |
| 20 | 19 | ralrn 7108 |
. . . . . . . . . . 11
⊢ ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) Fn ℕ → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))𝑧 ≤ 𝑦 ↔ ∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) ≤ 𝑦)) |
| 21 | 18, 20 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))𝑧 ≤ 𝑦 ↔ ∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) ≤ 𝑦)) |
| 22 | | fveq2 6906 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = 𝑚 → (𝐹‘𝑛) = (𝐹‘𝑚)) |
| 23 | 22 | fveq1d 6908 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = 𝑚 → ((𝐹‘𝑛)‘𝑥) = ((𝐹‘𝑚)‘𝑥)) |
| 24 | | fvex 6919 |
. . . . . . . . . . . . . 14
⊢ ((𝐹‘𝑚)‘𝑥) ∈ V |
| 25 | 23, 10, 24 | fvmpt 7016 |
. . . . . . . . . . . . 13
⊢ (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) = ((𝐹‘𝑚)‘𝑥)) |
| 26 | 25 | breq1d 5153 |
. . . . . . . . . . . 12
⊢ (𝑚 ∈ ℕ → (((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) ≤ 𝑦 ↔ ((𝐹‘𝑚)‘𝑥) ≤ 𝑦)) |
| 27 | 26 | ralbiia 3091 |
. . . . . . . . . . 11
⊢
(∀𝑚 ∈
ℕ ((𝑛 ∈ ℕ
↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) ≤ 𝑦 ↔ ∀𝑚 ∈ ℕ ((𝐹‘𝑚)‘𝑥) ≤ 𝑦) |
| 28 | 23 | breq1d 5153 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝑚 → (((𝐹‘𝑛)‘𝑥) ≤ 𝑦 ↔ ((𝐹‘𝑚)‘𝑥) ≤ 𝑦)) |
| 29 | 28 | cbvralvw 3237 |
. . . . . . . . . . 11
⊢
(∀𝑛 ∈
ℕ ((𝐹‘𝑛)‘𝑥) ≤ 𝑦 ↔ ∀𝑚 ∈ ℕ ((𝐹‘𝑚)‘𝑥) ≤ 𝑦) |
| 30 | 27, 29 | bitr4i 278 |
. . . . . . . . . 10
⊢
(∀𝑚 ∈
ℕ ((𝑛 ∈ ℕ
↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) ≤ 𝑦 ↔ ∀𝑛 ∈ ℕ ((𝐹‘𝑛)‘𝑥) ≤ 𝑦) |
| 31 | 21, 30 | bitrdi 287 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))𝑧 ≤ 𝑦 ↔ ∀𝑛 ∈ ℕ ((𝐹‘𝑛)‘𝑥) ≤ 𝑦)) |
| 32 | 31 | rexbidv 3179 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))𝑧 ≤ 𝑦 ↔ ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ ((𝐹‘𝑛)‘𝑥) ≤ 𝑦)) |
| 33 | 17, 32 | mpbird 257 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))𝑧 ≤ 𝑦) |
| 34 | 8, 16, 33 | suprcld 12231 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ) ∈
ℝ) |
| 35 | 34 | rexrd 11311 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ) ∈
ℝ*) |
| 36 | | 0red 11264 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 0 ∈
ℝ) |
| 37 | | fveq2 6906 |
. . . . . . . . . . 11
⊢ (𝑛 = 1 → (𝐹‘𝑛) = (𝐹‘1)) |
| 38 | 37 | feq1d 6720 |
. . . . . . . . . 10
⊢ (𝑛 = 1 → ((𝐹‘𝑛):ℝ⟶(0[,)+∞) ↔ (𝐹‘1):ℝ⟶(0[,)+∞))) |
| 39 | 1 | ralrimiva 3146 |
. . . . . . . . . 10
⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐹‘𝑛):ℝ⟶(0[,)+∞)) |
| 40 | 9 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → 1 ∈
ℕ) |
| 41 | 38, 39, 40 | rspcdva 3623 |
. . . . . . . . 9
⊢ (𝜑 → (𝐹‘1):ℝ⟶(0[,)+∞)) |
| 42 | 41 | ffvelcdmda 7104 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝐹‘1)‘𝑥) ∈ (0[,)+∞)) |
| 43 | | elrege0 13494 |
. . . . . . . 8
⊢ (((𝐹‘1)‘𝑥) ∈ (0[,)+∞) ↔
(((𝐹‘1)‘𝑥) ∈ ℝ ∧ 0 ≤
((𝐹‘1)‘𝑥))) |
| 44 | 42, 43 | sylib 218 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (((𝐹‘1)‘𝑥) ∈ ℝ ∧ 0 ≤ ((𝐹‘1)‘𝑥))) |
| 45 | 44 | simpld 494 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝐹‘1)‘𝑥) ∈ ℝ) |
| 46 | 44 | simprd 495 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 0 ≤ ((𝐹‘1)‘𝑥)) |
| 47 | 37 | fveq1d 6908 |
. . . . . . . . . 10
⊢ (𝑛 = 1 → ((𝐹‘𝑛)‘𝑥) = ((𝐹‘1)‘𝑥)) |
| 48 | | fvex 6919 |
. . . . . . . . . 10
⊢ ((𝐹‘1)‘𝑥) ∈ V |
| 49 | 47, 10, 48 | fvmpt 7016 |
. . . . . . . . 9
⊢ (1 ∈
ℕ → ((𝑛 ∈
ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘1) = ((𝐹‘1)‘𝑥)) |
| 50 | 9, 49 | ax-mp 5 |
. . . . . . . 8
⊢ ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘1) = ((𝐹‘1)‘𝑥) |
| 51 | | fnfvelrn 7100 |
. . . . . . . . 9
⊢ (((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) Fn ℕ ∧ 1 ∈ ℕ) →
((𝑛 ∈ ℕ ↦
((𝐹‘𝑛)‘𝑥))‘1) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))) |
| 52 | 18, 9, 51 | sylancl 586 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘1) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))) |
| 53 | 50, 52 | eqeltrrid 2846 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝐹‘1)‘𝑥) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))) |
| 54 | 8, 16, 33, 53 | suprubd 12230 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝐹‘1)‘𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) |
| 55 | 36, 45, 34, 46, 54 | letrd 11418 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 0 ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) |
| 56 | | elxrge0 13497 |
. . . . 5
⊢ (sup(ran
(𝑛 ∈ ℕ ↦
((𝐹‘𝑛)‘𝑥)), ℝ, < ) ∈ (0[,]+∞)
↔ (sup(ran (𝑛 ∈
ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ) ∈
ℝ* ∧ 0 ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))) |
| 57 | 35, 55, 56 | sylanbrc 583 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ) ∈
(0[,]+∞)) |
| 58 | | itg2mono.1 |
. . . 4
⊢ 𝐺 = (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) |
| 59 | 57, 58 | fmptd 7134 |
. . 3
⊢ (𝜑 → 𝐺:ℝ⟶(0[,]+∞)) |
| 60 | | itg2cl 25767 |
. . 3
⊢ (𝐺:ℝ⟶(0[,]+∞)
→ (∫2‘𝐺) ∈
ℝ*) |
| 61 | 59, 60 | syl 17 |
. 2
⊢ (𝜑 →
(∫2‘𝐺)
∈ ℝ*) |
| 62 | | itg2mono.6 |
. . 3
⊢ 𝑆 = sup(ran (𝑛 ∈ ℕ ↦
(∫2‘(𝐹‘𝑛))), ℝ*, <
) |
| 63 | | icossicc 13476 |
. . . . . . . 8
⊢
(0[,)+∞) ⊆ (0[,]+∞) |
| 64 | | fss 6752 |
. . . . . . . 8
⊢ (((𝐹‘𝑛):ℝ⟶(0[,)+∞) ∧
(0[,)+∞) ⊆ (0[,]+∞)) → (𝐹‘𝑛):ℝ⟶(0[,]+∞)) |
| 65 | 1, 63, 64 | sylancl 586 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛):ℝ⟶(0[,]+∞)) |
| 66 | | itg2cl 25767 |
. . . . . . 7
⊢ ((𝐹‘𝑛):ℝ⟶(0[,]+∞) →
(∫2‘(𝐹‘𝑛)) ∈
ℝ*) |
| 67 | 65, 66 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) →
(∫2‘(𝐹‘𝑛)) ∈
ℝ*) |
| 68 | 67 | fmpttd 7135 |
. . . . 5
⊢ (𝜑 → (𝑛 ∈ ℕ ↦
(∫2‘(𝐹‘𝑛))):ℕ⟶ℝ*) |
| 69 | 68 | frnd 6744 |
. . . 4
⊢ (𝜑 → ran (𝑛 ∈ ℕ ↦
(∫2‘(𝐹‘𝑛))) ⊆
ℝ*) |
| 70 | | supxrcl 13357 |
. . . 4
⊢ (ran
(𝑛 ∈ ℕ ↦
(∫2‘(𝐹‘𝑛))) ⊆ ℝ* →
sup(ran (𝑛 ∈ ℕ
↦ (∫2‘(𝐹‘𝑛))), ℝ*, < ) ∈
ℝ*) |
| 71 | 69, 70 | syl 17 |
. . 3
⊢ (𝜑 → sup(ran (𝑛 ∈ ℕ ↦
(∫2‘(𝐹‘𝑛))), ℝ*, < ) ∈
ℝ*) |
| 72 | 62, 71 | eqeltrid 2845 |
. 2
⊢ (𝜑 → 𝑆 ∈
ℝ*) |
| 73 | | itg2mono.2 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ MblFn) |
| 74 | 73 | adantlr 715 |
. . . . . . . 8
⊢ (((𝜑 ∧ ((𝑓 ∈ dom ∫1 ∧ 𝑓 ∘r ≤ 𝐺) ∧ ¬
(∫1‘𝑓)
≤ 𝑆)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ MblFn) |
| 75 | 1 | adantlr 715 |
. . . . . . . 8
⊢ (((𝜑 ∧ ((𝑓 ∈ dom ∫1 ∧ 𝑓 ∘r ≤ 𝐺) ∧ ¬
(∫1‘𝑓)
≤ 𝑆)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛):ℝ⟶(0[,)+∞)) |
| 76 | | itg2mono.4 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∘r ≤ (𝐹‘(𝑛 + 1))) |
| 77 | 76 | adantlr 715 |
. . . . . . . 8
⊢ (((𝜑 ∧ ((𝑓 ∈ dom ∫1 ∧ 𝑓 ∘r ≤ 𝐺) ∧ ¬
(∫1‘𝑓)
≤ 𝑆)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∘r ≤ (𝐹‘(𝑛 + 1))) |
| 78 | 17 | adantlr 715 |
. . . . . . . 8
⊢ (((𝜑 ∧ ((𝑓 ∈ dom ∫1 ∧ 𝑓 ∘r ≤ 𝐺) ∧ ¬
(∫1‘𝑓)
≤ 𝑆)) ∧ 𝑥 ∈ ℝ) →
∃𝑦 ∈ ℝ
∀𝑛 ∈ ℕ
((𝐹‘𝑛)‘𝑥) ≤ 𝑦) |
| 79 | | simprll 779 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑓 ∈ dom ∫1 ∧ 𝑓 ∘r ≤ 𝐺) ∧ ¬
(∫1‘𝑓)
≤ 𝑆)) → 𝑓 ∈ dom
∫1) |
| 80 | | simprlr 780 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑓 ∈ dom ∫1 ∧ 𝑓 ∘r ≤ 𝐺) ∧ ¬
(∫1‘𝑓)
≤ 𝑆)) → 𝑓 ∘r ≤ 𝐺) |
| 81 | | simprr 773 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑓 ∈ dom ∫1 ∧ 𝑓 ∘r ≤ 𝐺) ∧ ¬
(∫1‘𝑓)
≤ 𝑆)) → ¬
(∫1‘𝑓)
≤ 𝑆) |
| 82 | 58, 74, 75, 77, 78, 62, 79, 80, 81 | itg2monolem3 25787 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑓 ∈ dom ∫1 ∧ 𝑓 ∘r ≤ 𝐺) ∧ ¬
(∫1‘𝑓)
≤ 𝑆)) →
(∫1‘𝑓)
≤ 𝑆) |
| 83 | 82 | expr 456 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑓 ∘r ≤ 𝐺)) → (¬
(∫1‘𝑓)
≤ 𝑆 →
(∫1‘𝑓)
≤ 𝑆)) |
| 84 | 83 | pm2.18d 127 |
. . . . 5
⊢ ((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑓 ∘r ≤ 𝐺)) →
(∫1‘𝑓)
≤ 𝑆) |
| 85 | 84 | expr 456 |
. . . 4
⊢ ((𝜑 ∧ 𝑓 ∈ dom ∫1) → (𝑓 ∘r ≤ 𝐺 →
(∫1‘𝑓)
≤ 𝑆)) |
| 86 | 85 | ralrimiva 3146 |
. . 3
⊢ (𝜑 → ∀𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐺 →
(∫1‘𝑓)
≤ 𝑆)) |
| 87 | | itg2leub 25769 |
. . . 4
⊢ ((𝐺:ℝ⟶(0[,]+∞)
∧ 𝑆 ∈
ℝ*) → ((∫2‘𝐺) ≤ 𝑆 ↔ ∀𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐺 →
(∫1‘𝑓)
≤ 𝑆))) |
| 88 | 59, 72, 87 | syl2anc 584 |
. . 3
⊢ (𝜑 →
((∫2‘𝐺) ≤ 𝑆 ↔ ∀𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐺 →
(∫1‘𝑓)
≤ 𝑆))) |
| 89 | 86, 88 | mpbird 257 |
. 2
⊢ (𝜑 →
(∫2‘𝐺)
≤ 𝑆) |
| 90 | 22 | feq1d 6720 |
. . . . . . . . . . 11
⊢ (𝑛 = 𝑚 → ((𝐹‘𝑛):ℝ⟶(0[,)+∞) ↔ (𝐹‘𝑚):ℝ⟶(0[,)+∞))) |
| 91 | 90 | cbvralvw 3237 |
. . . . . . . . . 10
⊢
(∀𝑛 ∈
ℕ (𝐹‘𝑛):ℝ⟶(0[,)+∞)
↔ ∀𝑚 ∈
ℕ (𝐹‘𝑚):ℝ⟶(0[,)+∞)) |
| 92 | 39, 91 | sylib 218 |
. . . . . . . . 9
⊢ (𝜑 → ∀𝑚 ∈ ℕ (𝐹‘𝑚):ℝ⟶(0[,)+∞)) |
| 93 | 92 | r19.21bi 3251 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚):ℝ⟶(0[,)+∞)) |
| 94 | | fss 6752 |
. . . . . . . 8
⊢ (((𝐹‘𝑚):ℝ⟶(0[,)+∞) ∧
(0[,)+∞) ⊆ (0[,]+∞)) → (𝐹‘𝑚):ℝ⟶(0[,]+∞)) |
| 95 | 93, 63, 94 | sylancl 586 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚):ℝ⟶(0[,]+∞)) |
| 96 | 59 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐺:ℝ⟶(0[,]+∞)) |
| 97 | 8, 16, 33 | 3jca 1129 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) ⊆ ℝ ∧ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))𝑧 ≤ 𝑦)) |
| 98 | 97 | adantlr 715 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) ⊆ ℝ ∧ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))𝑧 ≤ 𝑦)) |
| 99 | 25 | ad2antlr 727 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) = ((𝐹‘𝑚)‘𝑥)) |
| 100 | 18 | adantlr 715 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) Fn ℕ) |
| 101 | | simplr 769 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝑚 ∈ ℕ) |
| 102 | | fnfvelrn 7100 |
. . . . . . . . . . . . . 14
⊢ (((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) Fn ℕ ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))) |
| 103 | 100, 101,
102 | syl2anc 584 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))) |
| 104 | 99, 103 | eqeltrrd 2842 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑚)‘𝑥) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))) |
| 105 | | suprub 12229 |
. . . . . . . . . . . 12
⊢ (((ran
(𝑛 ∈ ℕ ↦
((𝐹‘𝑛)‘𝑥)) ⊆ ℝ ∧ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))𝑧 ≤ 𝑦) ∧ ((𝐹‘𝑚)‘𝑥) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))) → ((𝐹‘𝑚)‘𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) |
| 106 | 98, 104, 105 | syl2anc 584 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑚)‘𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) |
| 107 | | simpr 484 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ) |
| 108 | | ltso 11341 |
. . . . . . . . . . . . 13
⊢ < Or
ℝ |
| 109 | 108 | supex 9503 |
. . . . . . . . . . . 12
⊢ sup(ran
(𝑛 ∈ ℕ ↦
((𝐹‘𝑛)‘𝑥)), ℝ, < ) ∈ V |
| 110 | 58 | fvmpt2 7027 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ ℝ ∧ sup(ran
(𝑛 ∈ ℕ ↦
((𝐹‘𝑛)‘𝑥)), ℝ, < ) ∈ V) → (𝐺‘𝑥) = sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) |
| 111 | 107, 109,
110 | sylancl 586 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐺‘𝑥) = sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) |
| 112 | 106, 111 | breqtrrd 5171 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑚)‘𝑥) ≤ (𝐺‘𝑥)) |
| 113 | 112 | ralrimiva 3146 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ∀𝑥 ∈ ℝ ((𝐹‘𝑚)‘𝑥) ≤ (𝐺‘𝑥)) |
| 114 | | fveq2 6906 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑧 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑚)‘𝑧)) |
| 115 | | fveq2 6906 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑧 → (𝐺‘𝑥) = (𝐺‘𝑧)) |
| 116 | 114, 115 | breq12d 5156 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑧 → (((𝐹‘𝑚)‘𝑥) ≤ (𝐺‘𝑥) ↔ ((𝐹‘𝑚)‘𝑧) ≤ (𝐺‘𝑧))) |
| 117 | 116 | cbvralvw 3237 |
. . . . . . . . 9
⊢
(∀𝑥 ∈
ℝ ((𝐹‘𝑚)‘𝑥) ≤ (𝐺‘𝑥) ↔ ∀𝑧 ∈ ℝ ((𝐹‘𝑚)‘𝑧) ≤ (𝐺‘𝑧)) |
| 118 | 113, 117 | sylib 218 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ∀𝑧 ∈ ℝ ((𝐹‘𝑚)‘𝑧) ≤ (𝐺‘𝑧)) |
| 119 | 93 | ffnd 6737 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚) Fn ℝ) |
| 120 | 34, 58 | fmptd 7134 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐺:ℝ⟶ℝ) |
| 121 | 120 | ffnd 6737 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐺 Fn ℝ) |
| 122 | 121 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐺 Fn ℝ) |
| 123 | | reex 11246 |
. . . . . . . . . 10
⊢ ℝ
∈ V |
| 124 | 123 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ℝ ∈
V) |
| 125 | | inidm 4227 |
. . . . . . . . 9
⊢ (ℝ
∩ ℝ) = ℝ |
| 126 | | eqidd 2738 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑧 ∈ ℝ) → ((𝐹‘𝑚)‘𝑧) = ((𝐹‘𝑚)‘𝑧)) |
| 127 | | eqidd 2738 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑧 ∈ ℝ) → (𝐺‘𝑧) = (𝐺‘𝑧)) |
| 128 | 119, 122,
124, 124, 125, 126, 127 | ofrfval 7707 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝐹‘𝑚) ∘r ≤ 𝐺 ↔ ∀𝑧 ∈ ℝ ((𝐹‘𝑚)‘𝑧) ≤ (𝐺‘𝑧))) |
| 129 | 118, 128 | mpbird 257 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚) ∘r ≤ 𝐺) |
| 130 | | itg2le 25774 |
. . . . . . 7
⊢ (((𝐹‘𝑚):ℝ⟶(0[,]+∞) ∧ 𝐺:ℝ⟶(0[,]+∞)
∧ (𝐹‘𝑚) ∘r ≤ 𝐺) →
(∫2‘(𝐹‘𝑚)) ≤ (∫2‘𝐺)) |
| 131 | 95, 96, 129, 130 | syl3anc 1373 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) →
(∫2‘(𝐹‘𝑚)) ≤ (∫2‘𝐺)) |
| 132 | 131 | ralrimiva 3146 |
. . . . 5
⊢ (𝜑 → ∀𝑚 ∈ ℕ
(∫2‘(𝐹‘𝑚)) ≤ (∫2‘𝐺)) |
| 133 | 68 | ffnd 6737 |
. . . . . . 7
⊢ (𝜑 → (𝑛 ∈ ℕ ↦
(∫2‘(𝐹‘𝑛))) Fn ℕ) |
| 134 | | breq1 5146 |
. . . . . . . 8
⊢ (𝑧 = ((𝑛 ∈ ℕ ↦
(∫2‘(𝐹‘𝑛)))‘𝑚) → (𝑧 ≤ (∫2‘𝐺) ↔ ((𝑛 ∈ ℕ ↦
(∫2‘(𝐹‘𝑛)))‘𝑚) ≤ (∫2‘𝐺))) |
| 135 | 134 | ralrn 7108 |
. . . . . . 7
⊢ ((𝑛 ∈ ℕ ↦
(∫2‘(𝐹‘𝑛))) Fn ℕ → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦
(∫2‘(𝐹‘𝑛)))𝑧 ≤ (∫2‘𝐺) ↔ ∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦
(∫2‘(𝐹‘𝑛)))‘𝑚) ≤ (∫2‘𝐺))) |
| 136 | 133, 135 | syl 17 |
. . . . . 6
⊢ (𝜑 → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦
(∫2‘(𝐹‘𝑛)))𝑧 ≤ (∫2‘𝐺) ↔ ∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦
(∫2‘(𝐹‘𝑛)))‘𝑚) ≤ (∫2‘𝐺))) |
| 137 | | 2fveq3 6911 |
. . . . . . . . 9
⊢ (𝑛 = 𝑚 → (∫2‘(𝐹‘𝑛)) = (∫2‘(𝐹‘𝑚))) |
| 138 | | eqid 2737 |
. . . . . . . . 9
⊢ (𝑛 ∈ ℕ ↦
(∫2‘(𝐹‘𝑛))) = (𝑛 ∈ ℕ ↦
(∫2‘(𝐹‘𝑛))) |
| 139 | | fvex 6919 |
. . . . . . . . 9
⊢
(∫2‘(𝐹‘𝑚)) ∈ V |
| 140 | 137, 138,
139 | fvmpt 7016 |
. . . . . . . 8
⊢ (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦
(∫2‘(𝐹‘𝑛)))‘𝑚) = (∫2‘(𝐹‘𝑚))) |
| 141 | 140 | breq1d 5153 |
. . . . . . 7
⊢ (𝑚 ∈ ℕ → (((𝑛 ∈ ℕ ↦
(∫2‘(𝐹‘𝑛)))‘𝑚) ≤ (∫2‘𝐺) ↔
(∫2‘(𝐹‘𝑚)) ≤ (∫2‘𝐺))) |
| 142 | 141 | ralbiia 3091 |
. . . . . 6
⊢
(∀𝑚 ∈
ℕ ((𝑛 ∈ ℕ
↦ (∫2‘(𝐹‘𝑛)))‘𝑚) ≤ (∫2‘𝐺) ↔ ∀𝑚 ∈ ℕ
(∫2‘(𝐹‘𝑚)) ≤ (∫2‘𝐺)) |
| 143 | 136, 142 | bitrdi 287 |
. . . . 5
⊢ (𝜑 → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦
(∫2‘(𝐹‘𝑛)))𝑧 ≤ (∫2‘𝐺) ↔ ∀𝑚 ∈ ℕ
(∫2‘(𝐹‘𝑚)) ≤ (∫2‘𝐺))) |
| 144 | 132, 143 | mpbird 257 |
. . . 4
⊢ (𝜑 → ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦
(∫2‘(𝐹‘𝑛)))𝑧 ≤ (∫2‘𝐺)) |
| 145 | | supxrleub 13368 |
. . . . 5
⊢ ((ran
(𝑛 ∈ ℕ ↦
(∫2‘(𝐹‘𝑛))) ⊆ ℝ* ∧
(∫2‘𝐺)
∈ ℝ*) → (sup(ran (𝑛 ∈ ℕ ↦
(∫2‘(𝐹‘𝑛))), ℝ*, < ) ≤
(∫2‘𝐺)
↔ ∀𝑧 ∈ ran
(𝑛 ∈ ℕ ↦
(∫2‘(𝐹‘𝑛)))𝑧 ≤ (∫2‘𝐺))) |
| 146 | 69, 61, 145 | syl2anc 584 |
. . . 4
⊢ (𝜑 → (sup(ran (𝑛 ∈ ℕ ↦
(∫2‘(𝐹‘𝑛))), ℝ*, < ) ≤
(∫2‘𝐺)
↔ ∀𝑧 ∈ ran
(𝑛 ∈ ℕ ↦
(∫2‘(𝐹‘𝑛)))𝑧 ≤ (∫2‘𝐺))) |
| 147 | 144, 146 | mpbird 257 |
. . 3
⊢ (𝜑 → sup(ran (𝑛 ∈ ℕ ↦
(∫2‘(𝐹‘𝑛))), ℝ*, < ) ≤
(∫2‘𝐺)) |
| 148 | 62, 147 | eqbrtrid 5178 |
. 2
⊢ (𝜑 → 𝑆 ≤ (∫2‘𝐺)) |
| 149 | 61, 72, 89, 148 | xrletrid 13197 |
1
⊢ (𝜑 →
(∫2‘𝐺)
= 𝑆) |