| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | itgulm.z | . . . . 5
⊢ 𝑍 =
(ℤ≥‘𝑀) | 
| 2 |  | itgulm.m | . . . . . 6
⊢ (𝜑 → 𝑀 ∈ ℤ) | 
| 3 | 2 | adantr 480 | . . . . 5
⊢ ((𝜑 ∧ 𝑟 ∈ ℝ+) → 𝑀 ∈
ℤ) | 
| 4 |  | itgulm.f | . . . . . . . 8
⊢ (𝜑 → 𝐹:𝑍⟶𝐿1) | 
| 5 | 4 | ffnd 6736 | . . . . . . 7
⊢ (𝜑 → 𝐹 Fn 𝑍) | 
| 6 |  | itgulm.u | . . . . . . 7
⊢ (𝜑 → 𝐹(⇝𝑢‘𝑆)𝐺) | 
| 7 |  | ulmf2 26428 | . . . . . . 7
⊢ ((𝐹 Fn 𝑍 ∧ 𝐹(⇝𝑢‘𝑆)𝐺) → 𝐹:𝑍⟶(ℂ ↑m 𝑆)) | 
| 8 | 5, 6, 7 | syl2anc 584 | . . . . . 6
⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑m 𝑆)) | 
| 9 | 8 | adantr 480 | . . . . 5
⊢ ((𝜑 ∧ 𝑟 ∈ ℝ+) → 𝐹:𝑍⟶(ℂ ↑m 𝑆)) | 
| 10 |  | eqidd 2737 | . . . . 5
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ 𝑧 ∈ 𝑆)) → ((𝐹‘𝑛)‘𝑧) = ((𝐹‘𝑛)‘𝑧)) | 
| 11 |  | eqidd 2737 | . . . . 5
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑧 ∈ 𝑆) → (𝐺‘𝑧) = (𝐺‘𝑧)) | 
| 12 | 6 | adantr 480 | . . . . 5
⊢ ((𝜑 ∧ 𝑟 ∈ ℝ+) → 𝐹(⇝𝑢‘𝑆)𝐺) | 
| 13 |  | simpr 484 | . . . . . 6
⊢ ((𝜑 ∧ 𝑟 ∈ ℝ+) → 𝑟 ∈
ℝ+) | 
| 14 |  | itgulm.s | . . . . . . . 8
⊢ (𝜑 → (vol‘𝑆) ∈
ℝ) | 
| 15 | 14 | adantr 480 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑟 ∈ ℝ+) →
(vol‘𝑆) ∈
ℝ) | 
| 16 |  | ulmcl 26425 | . . . . . . . . . . . 12
⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → 𝐺:𝑆⟶ℂ) | 
| 17 |  | fdm 6744 | . . . . . . . . . . . 12
⊢ (𝐺:𝑆⟶ℂ → dom 𝐺 = 𝑆) | 
| 18 | 6, 16, 17 | 3syl 18 | . . . . . . . . . . 11
⊢ (𝜑 → dom 𝐺 = 𝑆) | 
| 19 | 1, 2, 4, 6, 14 | iblulm 26451 | . . . . . . . . . . . 12
⊢ (𝜑 → 𝐺 ∈
𝐿1) | 
| 20 |  | iblmbf 25803 | . . . . . . . . . . . 12
⊢ (𝐺 ∈ 𝐿1
→ 𝐺 ∈
MblFn) | 
| 21 |  | mbfdm 25662 | . . . . . . . . . . . 12
⊢ (𝐺 ∈ MblFn → dom 𝐺 ∈ dom
vol) | 
| 22 | 19, 20, 21 | 3syl 18 | . . . . . . . . . . 11
⊢ (𝜑 → dom 𝐺 ∈ dom vol) | 
| 23 | 18, 22 | eqeltrrd 2841 | . . . . . . . . . 10
⊢ (𝜑 → 𝑆 ∈ dom vol) | 
| 24 |  | mblss 25567 | . . . . . . . . . 10
⊢ (𝑆 ∈ dom vol → 𝑆 ⊆
ℝ) | 
| 25 |  | ovolge0 25517 | . . . . . . . . . 10
⊢ (𝑆 ⊆ ℝ → 0 ≤
(vol*‘𝑆)) | 
| 26 | 23, 24, 25 | 3syl 18 | . . . . . . . . 9
⊢ (𝜑 → 0 ≤ (vol*‘𝑆)) | 
| 27 |  | mblvol 25566 | . . . . . . . . . 10
⊢ (𝑆 ∈ dom vol →
(vol‘𝑆) =
(vol*‘𝑆)) | 
| 28 | 23, 27 | syl 17 | . . . . . . . . 9
⊢ (𝜑 → (vol‘𝑆) = (vol*‘𝑆)) | 
| 29 | 26, 28 | breqtrrd 5170 | . . . . . . . 8
⊢ (𝜑 → 0 ≤ (vol‘𝑆)) | 
| 30 | 29 | adantr 480 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑟 ∈ ℝ+) → 0 ≤
(vol‘𝑆)) | 
| 31 | 15, 30 | ge0p1rpd 13108 | . . . . . 6
⊢ ((𝜑 ∧ 𝑟 ∈ ℝ+) →
((vol‘𝑆) + 1) ∈
ℝ+) | 
| 32 | 13, 31 | rpdivcld 13095 | . . . . 5
⊢ ((𝜑 ∧ 𝑟 ∈ ℝ+) → (𝑟 / ((vol‘𝑆) + 1)) ∈
ℝ+) | 
| 33 | 1, 3, 9, 10, 11, 12, 32 | ulmi 26430 | . . . 4
⊢ ((𝜑 ∧ 𝑟 ∈ ℝ+) →
∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1))) | 
| 34 | 1 | uztrn2 12898 | . . . . . . . 8
⊢ ((𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗)) → 𝑛 ∈ 𝑍) | 
| 35 | 8 | ffvelcdmda 7103 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ (ℂ ↑m 𝑆)) | 
| 36 |  | elmapi 8890 | . . . . . . . . . . . . . . . 16
⊢ ((𝐹‘𝑛) ∈ (ℂ ↑m 𝑆) → (𝐹‘𝑛):𝑆⟶ℂ) | 
| 37 | 35, 36 | syl 17 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛):𝑆⟶ℂ) | 
| 38 | 37 | ffvelcdmda 7103 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ 𝑆) → ((𝐹‘𝑛)‘𝑥) ∈ ℂ) | 
| 39 | 38 | adantllr 719 | . . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ 𝑆) → ((𝐹‘𝑛)‘𝑥) ∈ ℂ) | 
| 40 | 39 | adantlrr 721 | . . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)))) ∧ 𝑥 ∈ 𝑆) → ((𝐹‘𝑛)‘𝑥) ∈ ℂ) | 
| 41 | 37 | feqmptd 6976 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) = (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑛)‘𝑥))) | 
| 42 | 4 | ffvelcdmda 7103 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈
𝐿1) | 
| 43 | 41, 42 | eqeltrrd 2841 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑛)‘𝑥)) ∈
𝐿1) | 
| 44 | 43 | ad2ant2r 747 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)))) → (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑛)‘𝑥)) ∈
𝐿1) | 
| 45 | 6, 16 | syl 17 | . . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐺:𝑆⟶ℂ) | 
| 46 | 45 | ffvelcdmda 7103 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝐺‘𝑥) ∈ ℂ) | 
| 47 | 46 | ad4ant14 752 | . . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)))) ∧ 𝑥 ∈ 𝑆) → (𝐺‘𝑥) ∈ ℂ) | 
| 48 | 45 | feqmptd 6976 | . . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝑆 ↦ (𝐺‘𝑥))) | 
| 49 | 48, 19 | eqeltrrd 2841 | . . . . . . . . . . . . 13
⊢ (𝜑 → (𝑥 ∈ 𝑆 ↦ (𝐺‘𝑥)) ∈
𝐿1) | 
| 50 | 49 | ad2antrr 726 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)))) → (𝑥 ∈ 𝑆 ↦ (𝐺‘𝑥)) ∈
𝐿1) | 
| 51 | 40, 44, 47, 50 | itgsub 25862 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)))) → ∫𝑆(((𝐹‘𝑛)‘𝑥) − (𝐺‘𝑥)) d𝑥 = (∫𝑆((𝐹‘𝑛)‘𝑥) d𝑥 − ∫𝑆(𝐺‘𝑥) d𝑥)) | 
| 52 | 51 | fveq2d 6909 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)))) → (abs‘∫𝑆(((𝐹‘𝑛)‘𝑥) − (𝐺‘𝑥)) d𝑥) = (abs‘(∫𝑆((𝐹‘𝑛)‘𝑥) d𝑥 − ∫𝑆(𝐺‘𝑥) d𝑥))) | 
| 53 | 40, 47 | subcld 11621 | . . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)))) ∧ 𝑥 ∈ 𝑆) → (((𝐹‘𝑛)‘𝑥) − (𝐺‘𝑥)) ∈ ℂ) | 
| 54 | 40, 44, 47, 50 | iblsub 25858 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)))) → (𝑥 ∈ 𝑆 ↦ (((𝐹‘𝑛)‘𝑥) − (𝐺‘𝑥))) ∈
𝐿1) | 
| 55 | 53, 54 | itgcl 25820 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)))) → ∫𝑆(((𝐹‘𝑛)‘𝑥) − (𝐺‘𝑥)) d𝑥 ∈ ℂ) | 
| 56 | 55 | abscld 15476 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)))) → (abs‘∫𝑆(((𝐹‘𝑛)‘𝑥) − (𝐺‘𝑥)) d𝑥) ∈ ℝ) | 
| 57 | 53 | abscld 15476 | . . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)))) ∧ 𝑥 ∈ 𝑆) → (abs‘(((𝐹‘𝑛)‘𝑥) − (𝐺‘𝑥))) ∈ ℝ) | 
| 58 | 53, 54 | iblabs 25865 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)))) → (𝑥 ∈ 𝑆 ↦ (abs‘(((𝐹‘𝑛)‘𝑥) − (𝐺‘𝑥)))) ∈
𝐿1) | 
| 59 | 57, 58 | itgrecl 25834 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)))) → ∫𝑆(abs‘(((𝐹‘𝑛)‘𝑥) − (𝐺‘𝑥))) d𝑥 ∈ ℝ) | 
| 60 |  | rpre 13044 | . . . . . . . . . . . 12
⊢ (𝑟 ∈ ℝ+
→ 𝑟 ∈
ℝ) | 
| 61 | 60 | ad2antlr 727 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)))) → 𝑟 ∈ ℝ) | 
| 62 | 53, 54 | itgabs 25871 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)))) → (abs‘∫𝑆(((𝐹‘𝑛)‘𝑥) − (𝐺‘𝑥)) d𝑥) ≤ ∫𝑆(abs‘(((𝐹‘𝑛)‘𝑥) − (𝐺‘𝑥))) d𝑥) | 
| 63 | 32 | adantr 480 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)))) → (𝑟 / ((vol‘𝑆) + 1)) ∈
ℝ+) | 
| 64 | 63 | rpred 13078 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)))) → (𝑟 / ((vol‘𝑆) + 1)) ∈ ℝ) | 
| 65 | 14 | ad2antrr 726 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)))) → (vol‘𝑆) ∈
ℝ) | 
| 66 | 64, 65 | remulcld 11292 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)))) → ((𝑟 / ((vol‘𝑆) + 1)) · (vol‘𝑆)) ∈
ℝ) | 
| 67 |  | fconstmpt 5746 | . . . . . . . . . . . . . . 15
⊢ (𝑆 × {(𝑟 / ((vol‘𝑆) + 1))}) = (𝑥 ∈ 𝑆 ↦ (𝑟 / ((vol‘𝑆) + 1))) | 
| 68 | 23 | ad2antrr 726 | . . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)))) → 𝑆 ∈ dom vol) | 
| 69 | 63 | rpcnd 13080 | . . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)))) → (𝑟 / ((vol‘𝑆) + 1)) ∈ ℂ) | 
| 70 |  | iblconst 25854 | . . . . . . . . . . . . . . . 16
⊢ ((𝑆 ∈ dom vol ∧
(vol‘𝑆) ∈
ℝ ∧ (𝑟 /
((vol‘𝑆) + 1)) ∈
ℂ) → (𝑆 ×
{(𝑟 / ((vol‘𝑆) + 1))}) ∈
𝐿1) | 
| 71 | 68, 65, 69, 70 | syl3anc 1372 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)))) → (𝑆 × {(𝑟 / ((vol‘𝑆) + 1))}) ∈
𝐿1) | 
| 72 | 67, 71 | eqeltrrid 2845 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)))) → (𝑥 ∈ 𝑆 ↦ (𝑟 / ((vol‘𝑆) + 1))) ∈
𝐿1) | 
| 73 | 64 | adantr 480 | . . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)))) ∧ 𝑥 ∈ 𝑆) → (𝑟 / ((vol‘𝑆) + 1)) ∈ ℝ) | 
| 74 |  | simprr 772 | . . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)))) → ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1))) | 
| 75 |  | fveq2 6905 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝑧 = 𝑥 → ((𝐹‘𝑛)‘𝑧) = ((𝐹‘𝑛)‘𝑥)) | 
| 76 |  | fveq2 6905 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝑧 = 𝑥 → (𝐺‘𝑧) = (𝐺‘𝑥)) | 
| 77 | 75, 76 | oveq12d 7450 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑧 = 𝑥 → (((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧)) = (((𝐹‘𝑛)‘𝑥) − (𝐺‘𝑥))) | 
| 78 | 77 | fveq2d 6909 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑧 = 𝑥 → (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) = (abs‘(((𝐹‘𝑛)‘𝑥) − (𝐺‘𝑥)))) | 
| 79 | 78 | breq1d 5152 | . . . . . . . . . . . . . . . . 17
⊢ (𝑧 = 𝑥 → ((abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)) ↔ (abs‘(((𝐹‘𝑛)‘𝑥) − (𝐺‘𝑥))) < (𝑟 / ((vol‘𝑆) + 1)))) | 
| 80 | 79 | rspccva 3620 | . . . . . . . . . . . . . . . 16
⊢
((∀𝑧 ∈
𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)) ∧ 𝑥 ∈ 𝑆) → (abs‘(((𝐹‘𝑛)‘𝑥) − (𝐺‘𝑥))) < (𝑟 / ((vol‘𝑆) + 1))) | 
| 81 | 74, 80 | sylan 580 | . . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)))) ∧ 𝑥 ∈ 𝑆) → (abs‘(((𝐹‘𝑛)‘𝑥) − (𝐺‘𝑥))) < (𝑟 / ((vol‘𝑆) + 1))) | 
| 82 | 57, 73, 81 | ltled 11410 | . . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)))) ∧ 𝑥 ∈ 𝑆) → (abs‘(((𝐹‘𝑛)‘𝑥) − (𝐺‘𝑥))) ≤ (𝑟 / ((vol‘𝑆) + 1))) | 
| 83 | 58, 72, 57, 73, 82 | itgle 25846 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)))) → ∫𝑆(abs‘(((𝐹‘𝑛)‘𝑥) − (𝐺‘𝑥))) d𝑥 ≤ ∫𝑆(𝑟 / ((vol‘𝑆) + 1)) d𝑥) | 
| 84 |  | itgconst 25855 | . . . . . . . . . . . . . 14
⊢ ((𝑆 ∈ dom vol ∧
(vol‘𝑆) ∈
ℝ ∧ (𝑟 /
((vol‘𝑆) + 1)) ∈
ℂ) → ∫𝑆(𝑟 / ((vol‘𝑆) + 1)) d𝑥 = ((𝑟 / ((vol‘𝑆) + 1)) · (vol‘𝑆))) | 
| 85 | 68, 65, 69, 84 | syl3anc 1372 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)))) → ∫𝑆(𝑟 / ((vol‘𝑆) + 1)) d𝑥 = ((𝑟 / ((vol‘𝑆) + 1)) · (vol‘𝑆))) | 
| 86 | 83, 85 | breqtrd 5168 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)))) → ∫𝑆(abs‘(((𝐹‘𝑛)‘𝑥) − (𝐺‘𝑥))) d𝑥 ≤ ((𝑟 / ((vol‘𝑆) + 1)) · (vol‘𝑆))) | 
| 87 | 61 | recnd 11290 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)))) → 𝑟 ∈ ℂ) | 
| 88 | 65 | recnd 11290 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)))) → (vol‘𝑆) ∈
ℂ) | 
| 89 | 31 | adantr 480 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)))) → ((vol‘𝑆) + 1) ∈
ℝ+) | 
| 90 | 89 | rpcnd 13080 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)))) → ((vol‘𝑆) + 1) ∈
ℂ) | 
| 91 | 89 | rpne0d 13083 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)))) → ((vol‘𝑆) + 1) ≠ 0) | 
| 92 | 87, 88, 90, 91 | div23d 12081 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)))) → ((𝑟 · (vol‘𝑆)) / ((vol‘𝑆) + 1)) = ((𝑟 / ((vol‘𝑆) + 1)) · (vol‘𝑆))) | 
| 93 | 65 | ltp1d 12199 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)))) → (vol‘𝑆) < ((vol‘𝑆) + 1)) | 
| 94 |  | peano2re 11435 | . . . . . . . . . . . . . . . . 17
⊢
((vol‘𝑆)
∈ ℝ → ((vol‘𝑆) + 1) ∈ ℝ) | 
| 95 | 65, 94 | syl 17 | . . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)))) → ((vol‘𝑆) + 1) ∈
ℝ) | 
| 96 |  | rpgt0 13048 | . . . . . . . . . . . . . . . . 17
⊢ (𝑟 ∈ ℝ+
→ 0 < 𝑟) | 
| 97 | 96 | ad2antlr 727 | . . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)))) → 0 < 𝑟) | 
| 98 |  | ltmul2 12119 | . . . . . . . . . . . . . . . 16
⊢
(((vol‘𝑆)
∈ ℝ ∧ ((vol‘𝑆) + 1) ∈ ℝ ∧ (𝑟 ∈ ℝ ∧ 0 <
𝑟)) →
((vol‘𝑆) <
((vol‘𝑆) + 1) ↔
(𝑟 ·
(vol‘𝑆)) < (𝑟 · ((vol‘𝑆) + 1)))) | 
| 99 | 65, 95, 61, 97, 98 | syl112anc 1375 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)))) → ((vol‘𝑆) < ((vol‘𝑆) + 1) ↔ (𝑟 · (vol‘𝑆)) < (𝑟 · ((vol‘𝑆) + 1)))) | 
| 100 | 93, 99 | mpbid 232 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)))) → (𝑟 · (vol‘𝑆)) < (𝑟 · ((vol‘𝑆) + 1))) | 
| 101 | 61, 65 | remulcld 11292 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)))) → (𝑟 · (vol‘𝑆)) ∈ ℝ) | 
| 102 | 101, 61, 89 | ltdivmul2d 13130 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)))) → (((𝑟 · (vol‘𝑆)) / ((vol‘𝑆) + 1)) < 𝑟 ↔ (𝑟 · (vol‘𝑆)) < (𝑟 · ((vol‘𝑆) + 1)))) | 
| 103 | 100, 102 | mpbird 257 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)))) → ((𝑟 · (vol‘𝑆)) / ((vol‘𝑆) + 1)) < 𝑟) | 
| 104 | 92, 103 | eqbrtrrd 5166 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)))) → ((𝑟 / ((vol‘𝑆) + 1)) · (vol‘𝑆)) < 𝑟) | 
| 105 | 59, 66, 61, 86, 104 | lelttrd 11420 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)))) → ∫𝑆(abs‘(((𝐹‘𝑛)‘𝑥) − (𝐺‘𝑥))) d𝑥 < 𝑟) | 
| 106 | 56, 59, 61, 62, 105 | lelttrd 11420 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)))) → (abs‘∫𝑆(((𝐹‘𝑛)‘𝑥) − (𝐺‘𝑥)) d𝑥) < 𝑟) | 
| 107 | 52, 106 | eqbrtrrd 5166 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)))) → (abs‘(∫𝑆((𝐹‘𝑛)‘𝑥) d𝑥 − ∫𝑆(𝐺‘𝑥) d𝑥)) < 𝑟) | 
| 108 | 107 | expr 456 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑛 ∈ 𝑍) → (∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)) → (abs‘(∫𝑆((𝐹‘𝑛)‘𝑥) d𝑥 − ∫𝑆(𝐺‘𝑥) d𝑥)) < 𝑟)) | 
| 109 | 34, 108 | sylan2 593 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)) → (abs‘(∫𝑆((𝐹‘𝑛)‘𝑥) d𝑥 − ∫𝑆(𝐺‘𝑥) d𝑥)) < 𝑟)) | 
| 110 | 109 | anassrs 467 | . . . . . 6
⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑗 ∈ 𝑍) ∧ 𝑛 ∈ (ℤ≥‘𝑗)) → (∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)) → (abs‘(∫𝑆((𝐹‘𝑛)‘𝑥) d𝑥 − ∫𝑆(𝐺‘𝑥) d𝑥)) < 𝑟)) | 
| 111 | 110 | ralimdva 3166 | . . . . 5
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑗 ∈ 𝑍) → (∀𝑛 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)) → ∀𝑛 ∈ (ℤ≥‘𝑗)(abs‘(∫𝑆((𝐹‘𝑛)‘𝑥) d𝑥 − ∫𝑆(𝐺‘𝑥) d𝑥)) < 𝑟)) | 
| 112 | 111 | reximdva 3167 | . . . 4
⊢ ((𝜑 ∧ 𝑟 ∈ ℝ+) →
(∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑛)‘𝑧) − (𝐺‘𝑧))) < (𝑟 / ((vol‘𝑆) + 1)) → ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑗)(abs‘(∫𝑆((𝐹‘𝑛)‘𝑥) d𝑥 − ∫𝑆(𝐺‘𝑥) d𝑥)) < 𝑟)) | 
| 113 | 33, 112 | mpd 15 | . . 3
⊢ ((𝜑 ∧ 𝑟 ∈ ℝ+) →
∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑗)(abs‘(∫𝑆((𝐹‘𝑛)‘𝑥) d𝑥 − ∫𝑆(𝐺‘𝑥) d𝑥)) < 𝑟) | 
| 114 | 113 | ralrimiva 3145 | . 2
⊢ (𝜑 → ∀𝑟 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑗)(abs‘(∫𝑆((𝐹‘𝑛)‘𝑥) d𝑥 − ∫𝑆(𝐺‘𝑥) d𝑥)) < 𝑟) | 
| 115 | 1 | fvexi 6919 | . . . . 5
⊢ 𝑍 ∈ V | 
| 116 | 115 | mptex 7244 | . . . 4
⊢ (𝑘 ∈ 𝑍 ↦ ∫𝑆((𝐹‘𝑘)‘𝑥) d𝑥) ∈ V | 
| 117 | 116 | a1i 11 | . . 3
⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ ∫𝑆((𝐹‘𝑘)‘𝑥) d𝑥) ∈ V) | 
| 118 |  | fveq2 6905 | . . . . . . . 8
⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛)) | 
| 119 | 118 | fveq1d 6907 | . . . . . . 7
⊢ (𝑘 = 𝑛 → ((𝐹‘𝑘)‘𝑥) = ((𝐹‘𝑛)‘𝑥)) | 
| 120 | 119 | adantr 480 | . . . . . 6
⊢ ((𝑘 = 𝑛 ∧ 𝑥 ∈ 𝑆) → ((𝐹‘𝑘)‘𝑥) = ((𝐹‘𝑛)‘𝑥)) | 
| 121 | 120 | itgeq2dv 25818 | . . . . 5
⊢ (𝑘 = 𝑛 → ∫𝑆((𝐹‘𝑘)‘𝑥) d𝑥 = ∫𝑆((𝐹‘𝑛)‘𝑥) d𝑥) | 
| 122 |  | eqid 2736 | . . . . 5
⊢ (𝑘 ∈ 𝑍 ↦ ∫𝑆((𝐹‘𝑘)‘𝑥) d𝑥) = (𝑘 ∈ 𝑍 ↦ ∫𝑆((𝐹‘𝑘)‘𝑥) d𝑥) | 
| 123 |  | itgex 25806 | . . . . 5
⊢
∫𝑆((𝐹‘𝑛)‘𝑥) d𝑥 ∈ V | 
| 124 | 121, 122,
123 | fvmpt 7015 | . . . 4
⊢ (𝑛 ∈ 𝑍 → ((𝑘 ∈ 𝑍 ↦ ∫𝑆((𝐹‘𝑘)‘𝑥) d𝑥)‘𝑛) = ∫𝑆((𝐹‘𝑛)‘𝑥) d𝑥) | 
| 125 | 124 | adantl 481 | . . 3
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ ∫𝑆((𝐹‘𝑘)‘𝑥) d𝑥)‘𝑛) = ∫𝑆((𝐹‘𝑛)‘𝑥) d𝑥) | 
| 126 | 46, 49 | itgcl 25820 | . . 3
⊢ (𝜑 → ∫𝑆(𝐺‘𝑥) d𝑥 ∈ ℂ) | 
| 127 | 38, 43 | itgcl 25820 | . . 3
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∫𝑆((𝐹‘𝑛)‘𝑥) d𝑥 ∈ ℂ) | 
| 128 | 1, 2, 117, 125, 126, 127 | clim2c 15542 | . 2
⊢ (𝜑 → ((𝑘 ∈ 𝑍 ↦ ∫𝑆((𝐹‘𝑘)‘𝑥) d𝑥) ⇝ ∫𝑆(𝐺‘𝑥) d𝑥 ↔ ∀𝑟 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑗)(abs‘(∫𝑆((𝐹‘𝑛)‘𝑥) d𝑥 − ∫𝑆(𝐺‘𝑥) d𝑥)) < 𝑟)) | 
| 129 | 114, 128 | mpbird 257 | 1
⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ ∫𝑆((𝐹‘𝑘)‘𝑥) d𝑥) ⇝ ∫𝑆(𝐺‘𝑥) d𝑥) |