| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | fveq2 6906 | . . . . . . . . 9
⊢ (𝑛 = 𝑚 → (𝑃‘𝑛) = (𝑃‘𝑚)) | 
| 2 | 1 | fveq1d 6908 | . . . . . . . 8
⊢ (𝑛 = 𝑚 → ((𝑃‘𝑛)‘𝑥) = ((𝑃‘𝑚)‘𝑥)) | 
| 3 | 2 | cbvmptv 5255 | . . . . . . 7
⊢ (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) = (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑥)) | 
| 4 |  | fveq2 6906 | . . . . . . . 8
⊢ (𝑥 = 𝑦 → ((𝑃‘𝑚)‘𝑥) = ((𝑃‘𝑚)‘𝑦)) | 
| 5 | 4 | mpteq2dv 5244 | . . . . . . 7
⊢ (𝑥 = 𝑦 → (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑥)) = (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑦))) | 
| 6 | 3, 5 | eqtrid 2789 | . . . . . 6
⊢ (𝑥 = 𝑦 → (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) = (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑦))) | 
| 7 | 6 | rneqd 5949 | . . . . 5
⊢ (𝑥 = 𝑦 → ran (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) = ran (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑦))) | 
| 8 | 7 | supeq1d 9486 | . . . 4
⊢ (𝑥 = 𝑦 → sup(ran (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)), ℝ, < ) = sup(ran (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑦)), ℝ, < )) | 
| 9 | 8 | cbvmptv 5255 | . . 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 | ffvelcdmda 7104 | . . . 4
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑃‘𝑚) ∈ dom
∫1) | 
| 12 |  | i1fmbf 25710 | . . . 4
⊢ ((𝑃‘𝑚) ∈ dom ∫1 → (𝑃‘𝑚) ∈ MblFn) | 
| 13 | 11, 12 | syl 17 | . . 3
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑃‘𝑚) ∈ MblFn) | 
| 14 |  | i1ff 25711 | . . . . 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 5155 | . . . . . . . 8
⊢ (𝑛 = 𝑚 → (0𝑝
∘r ≤ (𝑃‘𝑛) ↔ 0𝑝
∘r ≤ (𝑃‘𝑚))) | 
| 18 |  | fvoveq1 7454 | . . . . . . . . 9
⊢ (𝑛 = 𝑚 → (𝑃‘(𝑛 + 1)) = (𝑃‘(𝑚 + 1))) | 
| 19 | 1, 18 | breq12d 5156 | . . . . . . . 8
⊢ (𝑛 = 𝑚 → ((𝑃‘𝑛) ∘r ≤ (𝑃‘(𝑛 + 1)) ↔ (𝑃‘𝑚) ∘r ≤ (𝑃‘(𝑚 + 1)))) | 
| 20 | 17, 19 | anbi12d 632 | . . . . . . 7
⊢ (𝑛 = 𝑚 → ((0𝑝
∘r ≤ (𝑃‘𝑛) ∧ (𝑃‘𝑛) ∘r ≤ (𝑃‘(𝑛 + 1))) ↔ (0𝑝
∘r ≤ (𝑃‘𝑚) ∧ (𝑃‘𝑚) ∘r ≤ (𝑃‘(𝑚 + 1))))) | 
| 21 | 20 | rspccva 3621 | . . . . . 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)))) | 
| 22 | 16, 21 | sylan 580 | . . . . 5
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) →
(0𝑝 ∘r ≤ (𝑃‘𝑚) ∧ (𝑃‘𝑚) ∘r ≤ (𝑃‘(𝑚 + 1)))) | 
| 23 | 22 | simpld 494 | . . . 4
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 0𝑝
∘r ≤ (𝑃‘𝑚)) | 
| 24 |  | 0plef 25707 | . . . 4
⊢ ((𝑃‘𝑚):ℝ⟶(0[,)+∞) ↔
((𝑃‘𝑚):ℝ⟶ℝ ∧
0𝑝 ∘r ≤ (𝑃‘𝑚))) | 
| 25 | 15, 23, 24 | sylanbrc 583 | . . 3
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑃‘𝑚):ℝ⟶(0[,)+∞)) | 
| 26 | 22 | simprd 495 | . . 3
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑃‘𝑚) ∘r ≤ (𝑃‘(𝑚 + 1))) | 
| 27 |  | rge0ssre 13496 | . . . . 5
⊢
(0[,)+∞) ⊆ ℝ | 
| 28 |  | itg2i1fseq.2 | . . . . . 6
⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) | 
| 29 | 28 | ffvelcdmda 7104 | . . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝐹‘𝑦) ∈ (0[,)+∞)) | 
| 30 | 27, 29 | sselid 3981 | . . . 4
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝐹‘𝑦) ∈ ℝ) | 
| 31 |  | itg2i1fseq.1 | . . . . . . . . 9
⊢ (𝜑 → 𝐹 ∈ MblFn) | 
| 32 |  | itg2i1fseq.5 | . . . . . . . . 9
⊢ (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)) | 
| 33 | 31, 28, 10, 16, 32 | itg2i1fseqle 25789 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑃‘𝑚) ∘r ≤ 𝐹) | 
| 34 | 15 | ffnd 6737 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑃‘𝑚) Fn ℝ) | 
| 35 | 28 | ffnd 6737 | . . . . . . . . . 10
⊢ (𝜑 → 𝐹 Fn ℝ) | 
| 36 | 35 | adantr 480 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐹 Fn ℝ) | 
| 37 |  | reex 11246 | . . . . . . . . . 10
⊢ ℝ
∈ V | 
| 38 | 37 | a1i 11 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ℝ ∈
V) | 
| 39 |  | inidm 4227 | . . . . . . . . 9
⊢ (ℝ
∩ ℝ) = ℝ | 
| 40 |  | eqidd 2738 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃‘𝑚)‘𝑦) = ((𝑃‘𝑚)‘𝑦)) | 
| 41 |  | eqidd 2738 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝐹‘𝑦) = (𝐹‘𝑦)) | 
| 42 | 34, 36, 38, 38, 39, 40, 41 | ofrfval 7707 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑃‘𝑚) ∘r ≤ 𝐹 ↔ ∀𝑦 ∈ ℝ ((𝑃‘𝑚)‘𝑦) ≤ (𝐹‘𝑦))) | 
| 43 | 33, 42 | mpbid 232 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ∀𝑦 ∈ ℝ ((𝑃‘𝑚)‘𝑦) ≤ (𝐹‘𝑦)) | 
| 44 | 43 | r19.21bi 3251 | . . . . . 6
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃‘𝑚)‘𝑦) ≤ (𝐹‘𝑦)) | 
| 45 | 44 | an32s 652 | . . . . 5
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑃‘𝑚)‘𝑦) ≤ (𝐹‘𝑦)) | 
| 46 | 45 | ralrimiva 3146 | . . . 4
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ∀𝑚 ∈ ℕ ((𝑃‘𝑚)‘𝑦) ≤ (𝐹‘𝑦)) | 
| 47 |  | brralrspcev 5203 | . . . 4
⊢ (((𝐹‘𝑦) ∈ ℝ ∧ ∀𝑚 ∈ ℕ ((𝑃‘𝑚)‘𝑦) ≤ (𝐹‘𝑦)) → ∃𝑧 ∈ ℝ ∀𝑚 ∈ ℕ ((𝑃‘𝑚)‘𝑦) ≤ 𝑧) | 
| 48 | 30, 46, 47 | syl2anc 584 | . . 3
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ∃𝑧 ∈ ℝ ∀𝑚 ∈ ℕ ((𝑃‘𝑚)‘𝑦) ≤ 𝑧) | 
| 49 | 1 | fveq2d 6910 | . . . . . 6
⊢ (𝑛 = 𝑚 → (∫2‘(𝑃‘𝑛)) = (∫2‘(𝑃‘𝑚))) | 
| 50 | 49 | cbvmptv 5255 | . . . . 5
⊢ (𝑛 ∈ ℕ ↦
(∫2‘(𝑃‘𝑛))) = (𝑚 ∈ ℕ ↦
(∫2‘(𝑃‘𝑚))) | 
| 51 | 50 | rneqi 5948 | . . . 4
⊢ ran
(𝑛 ∈ ℕ ↦
(∫2‘(𝑃‘𝑛))) = ran (𝑚 ∈ ℕ ↦
(∫2‘(𝑃‘𝑚))) | 
| 52 | 51 | supeq1i 9487 | . . 3
⊢ sup(ran
(𝑛 ∈ ℕ ↦
(∫2‘(𝑃‘𝑛))), ℝ*, < ) = sup(ran
(𝑚 ∈ ℕ ↦
(∫2‘(𝑃‘𝑚))), ℝ*, <
) | 
| 53 | 9, 13, 25, 26, 48, 52 | itg2mono 25788 | . 2
⊢ (𝜑 →
(∫2‘(𝑥
∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)), ℝ, < ))) = sup(ran (𝑛 ∈ ℕ ↦
(∫2‘(𝑃‘𝑛))), ℝ*, <
)) | 
| 54 | 28 | feqmptd 6977 | . . . . 5
⊢ (𝜑 → 𝐹 = (𝑦 ∈ ℝ ↦ (𝐹‘𝑦))) | 
| 55 | 1 | fveq1d 6908 | . . . . . . . . . 10
⊢ (𝑛 = 𝑚 → ((𝑃‘𝑛)‘𝑦) = ((𝑃‘𝑚)‘𝑦)) | 
| 56 | 55 | cbvmptv 5255 | . . . . . . . . 9
⊢ (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)) = (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑦)) | 
| 57 | 56 | rneqi 5948 | . . . . . . . 8
⊢ ran
(𝑛 ∈ ℕ ↦
((𝑃‘𝑛)‘𝑦)) = ran (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑦)) | 
| 58 | 57 | supeq1i 9487 | . . . . . . 7
⊢ sup(ran
(𝑛 ∈ ℕ ↦
((𝑃‘𝑛)‘𝑦)), ℝ, < ) = sup(ran (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑦)), ℝ, < ) | 
| 59 |  | nnuz 12921 | . . . . . . . . 9
⊢ ℕ =
(ℤ≥‘1) | 
| 60 |  | 1zzd 12648 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 1 ∈
ℤ) | 
| 61 | 15 | ffvelcdmda 7104 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃‘𝑚)‘𝑦) ∈ ℝ) | 
| 62 | 61 | an32s 652 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑃‘𝑚)‘𝑦) ∈ ℝ) | 
| 63 | 62, 56 | fmptd 7134 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)):ℕ⟶ℝ) | 
| 64 |  | peano2nn 12278 | . . . . . . . . . . . . . . . . 17
⊢ (𝑚 ∈ ℕ → (𝑚 + 1) ∈
ℕ) | 
| 65 |  | ffvelcdm 7101 | . . . . . . . . . . . . . . . . 17
⊢ ((𝑃:ℕ⟶dom
∫1 ∧ (𝑚
+ 1) ∈ ℕ) → (𝑃‘(𝑚 + 1)) ∈ dom
∫1) | 
| 66 | 10, 64, 65 | syl2an 596 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑃‘(𝑚 + 1)) ∈ dom
∫1) | 
| 67 |  | i1ff 25711 | . . . . . . . . . . . . . . . 16
⊢ ((𝑃‘(𝑚 + 1)) ∈ dom ∫1 →
(𝑃‘(𝑚 +
1)):ℝ⟶ℝ) | 
| 68 | 66, 67 | syl 17 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑃‘(𝑚 +
1)):ℝ⟶ℝ) | 
| 69 | 68 | ffnd 6737 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑃‘(𝑚 + 1)) Fn ℝ) | 
| 70 |  | eqidd 2738 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃‘(𝑚 + 1))‘𝑦) = ((𝑃‘(𝑚 + 1))‘𝑦)) | 
| 71 | 34, 69, 38, 38, 39, 40, 70 | ofrfval 7707 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑃‘𝑚) ∘r ≤ (𝑃‘(𝑚 + 1)) ↔ ∀𝑦 ∈ ℝ ((𝑃‘𝑚)‘𝑦) ≤ ((𝑃‘(𝑚 + 1))‘𝑦))) | 
| 72 | 26, 71 | mpbid 232 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ∀𝑦 ∈ ℝ ((𝑃‘𝑚)‘𝑦) ≤ ((𝑃‘(𝑚 + 1))‘𝑦)) | 
| 73 | 72 | r19.21bi 3251 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃‘𝑚)‘𝑦) ≤ ((𝑃‘(𝑚 + 1))‘𝑦)) | 
| 74 | 73 | an32s 652 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑃‘𝑚)‘𝑦) ≤ ((𝑃‘(𝑚 + 1))‘𝑦)) | 
| 75 |  | eqid 2737 | . . . . . . . . . . . 12
⊢ (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)) = (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)) | 
| 76 |  | fvex 6919 | . . . . . . . . . . . 12
⊢ ((𝑃‘𝑚)‘𝑦) ∈ V | 
| 77 | 55, 75, 76 | fvmpt 7016 | . . . . . . . . . . 11
⊢ (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘𝑚) = ((𝑃‘𝑚)‘𝑦)) | 
| 78 | 77 | adantl 481 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘𝑚) = ((𝑃‘𝑚)‘𝑦)) | 
| 79 |  | fveq2 6906 | . . . . . . . . . . . . . 14
⊢ (𝑛 = (𝑚 + 1) → (𝑃‘𝑛) = (𝑃‘(𝑚 + 1))) | 
| 80 | 79 | fveq1d 6908 | . . . . . . . . . . . . 13
⊢ (𝑛 = (𝑚 + 1) → ((𝑃‘𝑛)‘𝑦) = ((𝑃‘(𝑚 + 1))‘𝑦)) | 
| 81 |  | fvex 6919 | . . . . . . . . . . . . 13
⊢ ((𝑃‘(𝑚 + 1))‘𝑦) ∈ V | 
| 82 | 80, 75, 81 | fvmpt 7016 | . . . . . . . . . . . 12
⊢ ((𝑚 + 1) ∈ ℕ →
((𝑛 ∈ ℕ ↦
((𝑃‘𝑛)‘𝑦))‘(𝑚 + 1)) = ((𝑃‘(𝑚 + 1))‘𝑦)) | 
| 83 | 64, 82 | syl 17 | . . . . . . . . . . 11
⊢ (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘(𝑚 + 1)) = ((𝑃‘(𝑚 + 1))‘𝑦)) | 
| 84 | 83 | adantl 481 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘(𝑚 + 1)) = ((𝑃‘(𝑚 + 1))‘𝑦)) | 
| 85 | 74, 78, 84 | 3brtr4d 5175 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘𝑚) ≤ ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘(𝑚 + 1))) | 
| 86 | 77 | breq1d 5153 | . . . . . . . . . . . 12
⊢ (𝑚 ∈ ℕ → (((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘𝑚) ≤ 𝑧 ↔ ((𝑃‘𝑚)‘𝑦) ≤ 𝑧)) | 
| 87 | 86 | ralbiia 3091 | . . . . . . . . . . 11
⊢
(∀𝑚 ∈
ℕ ((𝑛 ∈ ℕ
↦ ((𝑃‘𝑛)‘𝑦))‘𝑚) ≤ 𝑧 ↔ ∀𝑚 ∈ ℕ ((𝑃‘𝑚)‘𝑦) ≤ 𝑧) | 
| 88 | 87 | rexbii 3094 | . . . . . . . . . 10
⊢
(∃𝑧 ∈
ℝ ∀𝑚 ∈
ℕ ((𝑛 ∈ ℕ
↦ ((𝑃‘𝑛)‘𝑦))‘𝑚) ≤ 𝑧 ↔ ∃𝑧 ∈ ℝ ∀𝑚 ∈ ℕ ((𝑃‘𝑚)‘𝑦) ≤ 𝑧) | 
| 89 | 48, 88 | sylibr 234 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ∃𝑧 ∈ ℝ ∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘𝑚) ≤ 𝑧) | 
| 90 | 59, 60, 63, 85, 89 | climsup 15706 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)) ⇝ sup(ran (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)), ℝ, < )) | 
| 91 |  | fveq2 6906 | . . . . . . . . . . . 12
⊢ (𝑥 = 𝑦 → ((𝑃‘𝑛)‘𝑥) = ((𝑃‘𝑛)‘𝑦)) | 
| 92 | 91 | mpteq2dv 5244 | . . . . . . . . . . 11
⊢ (𝑥 = 𝑦 → (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))) | 
| 93 |  | fveq2 6906 | . . . . . . . . . . 11
⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦)) | 
| 94 | 92, 93 | breq12d 5156 | . . . . . . . . . 10
⊢ (𝑥 = 𝑦 → ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥) ↔ (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)) ⇝ (𝐹‘𝑦))) | 
| 95 | 94 | rspccva 3621 | . . . . . . . . 9
⊢
((∀𝑥 ∈
ℝ (𝑛 ∈ ℕ
↦ ((𝑃‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥) ∧ 𝑦 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)) ⇝ (𝐹‘𝑦)) | 
| 96 | 32, 95 | sylan 580 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)) ⇝ (𝐹‘𝑦)) | 
| 97 |  | climuni 15588 | . . . . . . . 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)) | 
| 98 | 90, 96, 97 | syl2anc 584 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)), ℝ, < ) = (𝐹‘𝑦)) | 
| 99 | 58, 98 | eqtr3id 2791 | . . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → sup(ran (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑦)), ℝ, < ) = (𝐹‘𝑦)) | 
| 100 | 99 | mpteq2dva 5242 | . . . . 5
⊢ (𝜑 → (𝑦 ∈ ℝ ↦ sup(ran (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑦)), ℝ, < )) = (𝑦 ∈ ℝ ↦ (𝐹‘𝑦))) | 
| 101 | 54, 100 | eqtr4d 2780 | . . . 4
⊢ (𝜑 → 𝐹 = (𝑦 ∈ ℝ ↦ sup(ran (𝑚 ∈ ℕ ↦ ((𝑃‘𝑚)‘𝑦)), ℝ, < ))) | 
| 102 | 101, 9 | eqtr4di 2795 | . . 3
⊢ (𝜑 → 𝐹 = (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)), ℝ, < ))) | 
| 103 | 102 | fveq2d 6910 | . 2
⊢ (𝜑 →
(∫2‘𝐹)
= (∫2‘(𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)), ℝ, < )))) | 
| 104 |  | itg2i1fseq.6 | . . . . . 6
⊢ 𝑆 = (𝑚 ∈ ℕ ↦
(∫1‘(𝑃‘𝑚))) | 
| 105 |  | itg2itg1 25771 | . . . . . . . 8
⊢ (((𝑃‘𝑚) ∈ dom ∫1 ∧
0𝑝 ∘r ≤ (𝑃‘𝑚)) → (∫2‘(𝑃‘𝑚)) = (∫1‘(𝑃‘𝑚))) | 
| 106 | 11, 23, 105 | syl2anc 584 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) →
(∫2‘(𝑃‘𝑚)) = (∫1‘(𝑃‘𝑚))) | 
| 107 | 106 | mpteq2dva 5242 | . . . . . 6
⊢ (𝜑 → (𝑚 ∈ ℕ ↦
(∫2‘(𝑃‘𝑚))) = (𝑚 ∈ ℕ ↦
(∫1‘(𝑃‘𝑚)))) | 
| 108 | 104, 107 | eqtr4id 2796 | . . . . 5
⊢ (𝜑 → 𝑆 = (𝑚 ∈ ℕ ↦
(∫2‘(𝑃‘𝑚)))) | 
| 109 | 108, 50 | eqtr4di 2795 | . . . 4
⊢ (𝜑 → 𝑆 = (𝑛 ∈ ℕ ↦
(∫2‘(𝑃‘𝑛)))) | 
| 110 | 109 | rneqd 5949 | . . 3
⊢ (𝜑 → ran 𝑆 = ran (𝑛 ∈ ℕ ↦
(∫2‘(𝑃‘𝑛)))) | 
| 111 | 110 | supeq1d 9486 | . 2
⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) = sup(ran
(𝑛 ∈ ℕ ↦
(∫2‘(𝑃‘𝑛))), ℝ*, <
)) | 
| 112 | 53, 103, 111 | 3eqtr4d 2787 | 1
⊢ (𝜑 →
(∫2‘𝐹)
= sup(ran 𝑆,
ℝ*, < )) |