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Theorem itg2i1fseq 25790
Description: Subject to the conditions coming from mbfi1fseq 25756, the integral of the sequence of simple functions converges to the integral of the target function. (Contributed by Mario Carneiro, 17-Aug-2014.)
Hypotheses
Ref Expression
itg2i1fseq.1 (𝜑𝐹 ∈ MblFn)
itg2i1fseq.2 (𝜑𝐹:ℝ⟶(0[,)+∞))
itg2i1fseq.3 (𝜑𝑃:ℕ⟶dom ∫1)
itg2i1fseq.4 (𝜑 → ∀𝑛 ∈ ℕ (0𝑝r ≤ (𝑃𝑛) ∧ (𝑃𝑛) ∘r ≤ (𝑃‘(𝑛 + 1))))
itg2i1fseq.5 (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) ⇝ (𝐹𝑥))
itg2i1fseq.6 𝑆 = (𝑚 ∈ ℕ ↦ (∫1‘(𝑃𝑚)))
Assertion
Ref Expression
itg2i1fseq (𝜑 → (∫2𝐹) = sup(ran 𝑆, ℝ*, < ))
Distinct variable groups:   𝑚,𝑛,𝑥,𝐹   𝑃,𝑚,𝑛,𝑥   𝜑,𝑚
Allowed substitution hints:   𝜑(𝑥,𝑛)   𝑆(𝑥,𝑚,𝑛)

Proof of Theorem itg2i1fseq
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6906 . . . . . . . . 9 (𝑛 = 𝑚 → (𝑃𝑛) = (𝑃𝑚))
21fveq1d 6908 . . . . . . . 8 (𝑛 = 𝑚 → ((𝑃𝑛)‘𝑥) = ((𝑃𝑚)‘𝑥))
32cbvmptv 5255 . . . . . . 7 (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) = (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑥))
4 fveq2 6906 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑃𝑚)‘𝑥) = ((𝑃𝑚)‘𝑦))
54mpteq2dv 5244 . . . . . . 7 (𝑥 = 𝑦 → (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑥)) = (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑦)))
63, 5eqtrid 2789 . . . . . 6 (𝑥 = 𝑦 → (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) = (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑦)))
76rneqd 5949 . . . . 5 (𝑥 = 𝑦 → ran (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) = ran (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑦)))
87supeq1d 9486 . . . 4 (𝑥 = 𝑦 → sup(ran (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)), ℝ, < ) = sup(ran (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑦)), ℝ, < ))
98cbvmptv 5255 . . 3 (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)), ℝ, < )) = (𝑦 ∈ ℝ ↦ sup(ran (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑦)), ℝ, < ))
10 itg2i1fseq.3 . . . . 5 (𝜑𝑃:ℕ⟶dom ∫1)
1110ffvelcdmda 7104 . . . 4 ((𝜑𝑚 ∈ ℕ) → (𝑃𝑚) ∈ dom ∫1)
12 i1fmbf 25710 . . . 4 ((𝑃𝑚) ∈ dom ∫1 → (𝑃𝑚) ∈ MblFn)
1311, 12syl 17 . . 3 ((𝜑𝑚 ∈ ℕ) → (𝑃𝑚) ∈ MblFn)
14 i1ff 25711 . . . . 5 ((𝑃𝑚) ∈ dom ∫1 → (𝑃𝑚):ℝ⟶ℝ)
1511, 14syl 17 . . . 4 ((𝜑𝑚 ∈ ℕ) → (𝑃𝑚):ℝ⟶ℝ)
16 itg2i1fseq.4 . . . . . 6 (𝜑 → ∀𝑛 ∈ ℕ (0𝑝r ≤ (𝑃𝑛) ∧ (𝑃𝑛) ∘r ≤ (𝑃‘(𝑛 + 1))))
171breq2d 5155 . . . . . . . 8 (𝑛 = 𝑚 → (0𝑝r ≤ (𝑃𝑛) ↔ 0𝑝r ≤ (𝑃𝑚)))
18 fvoveq1 7454 . . . . . . . . 9 (𝑛 = 𝑚 → (𝑃‘(𝑛 + 1)) = (𝑃‘(𝑚 + 1)))
191, 18breq12d 5156 . . . . . . . 8 (𝑛 = 𝑚 → ((𝑃𝑛) ∘r ≤ (𝑃‘(𝑛 + 1)) ↔ (𝑃𝑚) ∘r ≤ (𝑃‘(𝑚 + 1))))
2017, 19anbi12d 632 . . . . . . 7 (𝑛 = 𝑚 → ((0𝑝r ≤ (𝑃𝑛) ∧ (𝑃𝑛) ∘r ≤ (𝑃‘(𝑛 + 1))) ↔ (0𝑝r ≤ (𝑃𝑚) ∧ (𝑃𝑚) ∘r ≤ (𝑃‘(𝑚 + 1)))))
2120rspccva 3621 . . . . . 6 ((∀𝑛 ∈ ℕ (0𝑝r ≤ (𝑃𝑛) ∧ (𝑃𝑛) ∘r ≤ (𝑃‘(𝑛 + 1))) ∧ 𝑚 ∈ ℕ) → (0𝑝r ≤ (𝑃𝑚) ∧ (𝑃𝑚) ∘r ≤ (𝑃‘(𝑚 + 1))))
2216, 21sylan 580 . . . . 5 ((𝜑𝑚 ∈ ℕ) → (0𝑝r ≤ (𝑃𝑚) ∧ (𝑃𝑚) ∘r ≤ (𝑃‘(𝑚 + 1))))
2322simpld 494 . . . 4 ((𝜑𝑚 ∈ ℕ) → 0𝑝r ≤ (𝑃𝑚))
24 0plef 25707 . . . 4 ((𝑃𝑚):ℝ⟶(0[,)+∞) ↔ ((𝑃𝑚):ℝ⟶ℝ ∧ 0𝑝r ≤ (𝑃𝑚)))
2515, 23, 24sylanbrc 583 . . 3 ((𝜑𝑚 ∈ ℕ) → (𝑃𝑚):ℝ⟶(0[,)+∞))
2622simprd 495 . . 3 ((𝜑𝑚 ∈ ℕ) → (𝑃𝑚) ∘r ≤ (𝑃‘(𝑚 + 1)))
27 rge0ssre 13496 . . . . 5 (0[,)+∞) ⊆ ℝ
28 itg2i1fseq.2 . . . . . 6 (𝜑𝐹:ℝ⟶(0[,)+∞))
2928ffvelcdmda 7104 . . . . 5 ((𝜑𝑦 ∈ ℝ) → (𝐹𝑦) ∈ (0[,)+∞))
3027, 29sselid 3981 . . . 4 ((𝜑𝑦 ∈ ℝ) → (𝐹𝑦) ∈ ℝ)
31 itg2i1fseq.1 . . . . . . . . 9 (𝜑𝐹 ∈ MblFn)
32 itg2i1fseq.5 . . . . . . . . 9 (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) ⇝ (𝐹𝑥))
3331, 28, 10, 16, 32itg2i1fseqle 25789 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → (𝑃𝑚) ∘r𝐹)
3415ffnd 6737 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → (𝑃𝑚) Fn ℝ)
3528ffnd 6737 . . . . . . . . . 10 (𝜑𝐹 Fn ℝ)
3635adantr 480 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → 𝐹 Fn ℝ)
37 reex 11246 . . . . . . . . . 10 ℝ ∈ V
3837a1i 11 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → ℝ ∈ V)
39 inidm 4227 . . . . . . . . 9 (ℝ ∩ ℝ) = ℝ
40 eqidd 2738 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃𝑚)‘𝑦) = ((𝑃𝑚)‘𝑦))
41 eqidd 2738 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝐹𝑦) = (𝐹𝑦))
4234, 36, 38, 38, 39, 40, 41ofrfval 7707 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → ((𝑃𝑚) ∘r𝐹 ↔ ∀𝑦 ∈ ℝ ((𝑃𝑚)‘𝑦) ≤ (𝐹𝑦)))
4333, 42mpbid 232 . . . . . . 7 ((𝜑𝑚 ∈ ℕ) → ∀𝑦 ∈ ℝ ((𝑃𝑚)‘𝑦) ≤ (𝐹𝑦))
4443r19.21bi 3251 . . . . . 6 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃𝑚)‘𝑦) ≤ (𝐹𝑦))
4544an32s 652 . . . . 5 (((𝜑𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑃𝑚)‘𝑦) ≤ (𝐹𝑦))
4645ralrimiva 3146 . . . 4 ((𝜑𝑦 ∈ ℝ) → ∀𝑚 ∈ ℕ ((𝑃𝑚)‘𝑦) ≤ (𝐹𝑦))
47 brralrspcev 5203 . . . 4 (((𝐹𝑦) ∈ ℝ ∧ ∀𝑚 ∈ ℕ ((𝑃𝑚)‘𝑦) ≤ (𝐹𝑦)) → ∃𝑧 ∈ ℝ ∀𝑚 ∈ ℕ ((𝑃𝑚)‘𝑦) ≤ 𝑧)
4830, 46, 47syl2anc 584 . . 3 ((𝜑𝑦 ∈ ℝ) → ∃𝑧 ∈ ℝ ∀𝑚 ∈ ℕ ((𝑃𝑚)‘𝑦) ≤ 𝑧)
491fveq2d 6910 . . . . . 6 (𝑛 = 𝑚 → (∫2‘(𝑃𝑛)) = (∫2‘(𝑃𝑚)))
5049cbvmptv 5255 . . . . 5 (𝑛 ∈ ℕ ↦ (∫2‘(𝑃𝑛))) = (𝑚 ∈ ℕ ↦ (∫2‘(𝑃𝑚)))
5150rneqi 5948 . . . 4 ran (𝑛 ∈ ℕ ↦ (∫2‘(𝑃𝑛))) = ran (𝑚 ∈ ℕ ↦ (∫2‘(𝑃𝑚)))
5251supeq1i 9487 . . 3 sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝑃𝑛))), ℝ*, < ) = sup(ran (𝑚 ∈ ℕ ↦ (∫2‘(𝑃𝑚))), ℝ*, < )
539, 13, 25, 26, 48, 52itg2mono 25788 . 2 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)), ℝ, < ))) = sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝑃𝑛))), ℝ*, < ))
5428feqmptd 6977 . . . . 5 (𝜑𝐹 = (𝑦 ∈ ℝ ↦ (𝐹𝑦)))
551fveq1d 6908 . . . . . . . . . 10 (𝑛 = 𝑚 → ((𝑃𝑛)‘𝑦) = ((𝑃𝑚)‘𝑦))
5655cbvmptv 5255 . . . . . . . . 9 (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)) = (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑦))
5756rneqi 5948 . . . . . . . 8 ran (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)) = ran (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑦))
5857supeq1i 9487 . . . . . . 7 sup(ran (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)), ℝ, < ) = sup(ran (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑦)), ℝ, < )
59 nnuz 12921 . . . . . . . . 9 ℕ = (ℤ‘1)
60 1zzd 12648 . . . . . . . . 9 ((𝜑𝑦 ∈ ℝ) → 1 ∈ ℤ)
6115ffvelcdmda 7104 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃𝑚)‘𝑦) ∈ ℝ)
6261an32s 652 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑃𝑚)‘𝑦) ∈ ℝ)
6362, 56fmptd 7134 . . . . . . . . 9 ((𝜑𝑦 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘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)
6610, 64, 65syl2an 596 . . . . . . . . . . . . . . . 16 ((𝜑𝑚 ∈ ℕ) → (𝑃‘(𝑚 + 1)) ∈ dom ∫1)
67 i1ff 25711 . . . . . . . . . . . . . . . 16 ((𝑃‘(𝑚 + 1)) ∈ dom ∫1 → (𝑃‘(𝑚 + 1)):ℝ⟶ℝ)
6866, 67syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ ℕ) → (𝑃‘(𝑚 + 1)):ℝ⟶ℝ)
6968ffnd 6737 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → (𝑃‘(𝑚 + 1)) Fn ℝ)
70 eqidd 2738 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃‘(𝑚 + 1))‘𝑦) = ((𝑃‘(𝑚 + 1))‘𝑦))
7134, 69, 38, 38, 39, 40, 70ofrfval 7707 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → ((𝑃𝑚) ∘r ≤ (𝑃‘(𝑚 + 1)) ↔ ∀𝑦 ∈ ℝ ((𝑃𝑚)‘𝑦) ≤ ((𝑃‘(𝑚 + 1))‘𝑦)))
7226, 71mpbid 232 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → ∀𝑦 ∈ ℝ ((𝑃𝑚)‘𝑦) ≤ ((𝑃‘(𝑚 + 1))‘𝑦))
7372r19.21bi 3251 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃𝑚)‘𝑦) ≤ ((𝑃‘(𝑚 + 1))‘𝑦))
7473an32s 652 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑃𝑚)‘𝑦) ≤ ((𝑃‘(𝑚 + 1))‘𝑦))
75 eqid 2737 . . . . . . . . . . . 12 (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)) = (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))
76 fvex 6919 . . . . . . . . . . . 12 ((𝑃𝑚)‘𝑦) ∈ V
7755, 75, 76fvmpt 7016 . . . . . . . . . . 11 (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘𝑚) = ((𝑃𝑚)‘𝑦))
7877adantl 481 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘𝑚) = ((𝑃𝑚)‘𝑦))
79 fveq2 6906 . . . . . . . . . . . . . 14 (𝑛 = (𝑚 + 1) → (𝑃𝑛) = (𝑃‘(𝑚 + 1)))
8079fveq1d 6908 . . . . . . . . . . . . 13 (𝑛 = (𝑚 + 1) → ((𝑃𝑛)‘𝑦) = ((𝑃‘(𝑚 + 1))‘𝑦))
81 fvex 6919 . . . . . . . . . . . . 13 ((𝑃‘(𝑚 + 1))‘𝑦) ∈ V
8280, 75, 81fvmpt 7016 . . . . . . . . . . . 12 ((𝑚 + 1) ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘(𝑚 + 1)) = ((𝑃‘(𝑚 + 1))‘𝑦))
8364, 82syl 17 . . . . . . . . . . 11 (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘(𝑚 + 1)) = ((𝑃‘(𝑚 + 1))‘𝑦))
8483adantl 481 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘(𝑚 + 1)) = ((𝑃‘(𝑚 + 1))‘𝑦))
8574, 78, 843brtr4d 5175 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘𝑚) ≤ ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘(𝑚 + 1)))
8677breq1d 5153 . . . . . . . . . . . 12 (𝑚 ∈ ℕ → (((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘𝑚) ≤ 𝑧 ↔ ((𝑃𝑚)‘𝑦) ≤ 𝑧))
8786ralbiia 3091 . . . . . . . . . . 11 (∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘𝑚) ≤ 𝑧 ↔ ∀𝑚 ∈ ℕ ((𝑃𝑚)‘𝑦) ≤ 𝑧)
8887rexbii 3094 . . . . . . . . . 10 (∃𝑧 ∈ ℝ ∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘𝑚) ≤ 𝑧 ↔ ∃𝑧 ∈ ℝ ∀𝑚 ∈ ℕ ((𝑃𝑚)‘𝑦) ≤ 𝑧)
8948, 88sylibr 234 . . . . . . . . 9 ((𝜑𝑦 ∈ ℝ) → ∃𝑧 ∈ ℝ ∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘𝑚) ≤ 𝑧)
9059, 60, 63, 85, 89climsup 15706 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)) ⇝ sup(ran (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)), ℝ, < ))
91 fveq2 6906 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((𝑃𝑛)‘𝑥) = ((𝑃𝑛)‘𝑦))
9291mpteq2dv 5244 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)))
93 fveq2 6906 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
9492, 93breq12d 5156 . . . . . . . . . 10 (𝑥 = 𝑦 → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) ⇝ (𝐹𝑥) ↔ (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)) ⇝ (𝐹𝑦)))
9594rspccva 3621 . . . . . . . . 9 ((∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) ⇝ (𝐹𝑥) ∧ 𝑦 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)) ⇝ (𝐹𝑦))
9632, 95sylan 580 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)) ⇝ (𝐹𝑦))
97 climuni 15588 . . . . . . . 8 (((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)) ⇝ sup(ran (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)), ℝ, < ) ∧ (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)) ⇝ (𝐹𝑦)) → sup(ran (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)), ℝ, < ) = (𝐹𝑦))
9890, 96, 97syl2anc 584 . . . . . . 7 ((𝜑𝑦 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)), ℝ, < ) = (𝐹𝑦))
9958, 98eqtr3id 2791 . . . . . 6 ((𝜑𝑦 ∈ ℝ) → sup(ran (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑦)), ℝ, < ) = (𝐹𝑦))
10099mpteq2dva 5242 . . . . 5 (𝜑 → (𝑦 ∈ ℝ ↦ sup(ran (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑦)), ℝ, < )) = (𝑦 ∈ ℝ ↦ (𝐹𝑦)))
10154, 100eqtr4d 2780 . . . 4 (𝜑𝐹 = (𝑦 ∈ ℝ ↦ sup(ran (𝑚 ∈ ℕ ↦ ((𝑃𝑚)‘𝑦)), ℝ, < )))
102101, 9eqtr4di 2795 . . 3 (𝜑𝐹 = (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)), ℝ, < )))
103102fveq2d 6910 . 2 (𝜑 → (∫2𝐹) = (∫2‘(𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)), ℝ, < ))))
104 itg2i1fseq.6 . . . . . 6 𝑆 = (𝑚 ∈ ℕ ↦ (∫1‘(𝑃𝑚)))
105 itg2itg1 25771 . . . . . . . 8 (((𝑃𝑚) ∈ dom ∫1 ∧ 0𝑝r ≤ (𝑃𝑚)) → (∫2‘(𝑃𝑚)) = (∫1‘(𝑃𝑚)))
10611, 23, 105syl2anc 584 . . . . . . 7 ((𝜑𝑚 ∈ ℕ) → (∫2‘(𝑃𝑚)) = (∫1‘(𝑃𝑚)))
107106mpteq2dva 5242 . . . . . 6 (𝜑 → (𝑚 ∈ ℕ ↦ (∫2‘(𝑃𝑚))) = (𝑚 ∈ ℕ ↦ (∫1‘(𝑃𝑚))))
108104, 107eqtr4id 2796 . . . . 5 (𝜑𝑆 = (𝑚 ∈ ℕ ↦ (∫2‘(𝑃𝑚))))
109108, 50eqtr4di 2795 . . . 4 (𝜑𝑆 = (𝑛 ∈ ℕ ↦ (∫2‘(𝑃𝑛))))
110109rneqd 5949 . . 3 (𝜑 → ran 𝑆 = ran (𝑛 ∈ ℕ ↦ (∫2‘(𝑃𝑛))))
111110supeq1d 9486 . 2 (𝜑 → sup(ran 𝑆, ℝ*, < ) = sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝑃𝑛))), ℝ*, < ))
11253, 103, 1113eqtr4d 2787 1 (𝜑 → (∫2𝐹) = sup(ran 𝑆, ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wral 3061  wrex 3070  Vcvv 3480   class class class wbr 5143  cmpt 5225  dom cdm 5685  ran crn 5686   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  r cofr 7696  supcsup 9480  cr 11154  0cc0 11155  1c1 11156   + caddc 11158  +∞cpnf 11292  *cxr 11294   < clt 11295  cle 11296  cn 12266  [,)cico 13389  cli 15520  MblFncmbf 25649  1citg1 25650  2citg2 25651  0𝑝c0p 25704
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cc 10475  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233  ax-addf 11234
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-disj 5111  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-ofr 7698  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-omul 8511  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fi 9451  df-sup 9482  df-inf 9483  df-oi 9550  df-dju 9941  df-card 9979  df-acn 9982  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-q 12991  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-ioo 13391  df-ioc 13392  df-ico 13393  df-icc 13394  df-fz 13548  df-fzo 13695  df-fl 13832  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-rlim 15525  df-sum 15723  df-rest 17467  df-topgen 17488  df-psmet 21356  df-xmet 21357  df-met 21358  df-bl 21359  df-mopn 21360  df-top 22900  df-topon 22917  df-bases 22953  df-cmp 23395  df-ovol 25499  df-vol 25500  df-mbf 25654  df-itg1 25655  df-itg2 25656  df-0p 25705
This theorem is referenced by:  itg2i1fseq2  25791
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